Aspect Math
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HiI am M.SASPECT MATH fwZ© cixÿvq MwY‡Zi ¸iæZ¡
© Medistry 1
.. .. . .. .. . .. ..... .. .. .. .. .. .. . .. . .. .. fwZ© cix¶vq
WHY
.. . . . . . MATH . . . . .. . . .
MwY‡Zi ¸iæZ¡
wek^we`¨vjq, BwÄwbqvwis Ges GBPGmwm cix¶vq MwYZ GKwU Avek¨Kxq welq| cÖvq mKj cvewjK wek¦we`¨vjq fwZ© cix¶v‡ZB MwYZ
Ask †_‡K D‡jøL‡hvM¨ msL¨K cÖkœ _v‡K| ZvQvov BwÄwbqvwis Gi g‡Zv DbœZgv‡bi K¨vwiqvi Mo‡Z MwY‡Zi weKí †bB| ZvB mevi Av‡M
Rvb‡Z nq, †Kvb RvqMvq KZ b¤^i _v‡K...
cÖm½ 01 mvaviY wek¦we`¨vjq
µwgK wek¦we`¨vj‡qi bvg BDwbU cix¶v c×wZ †gvU b¤^i MwYZ As‡ki b¤^i DËi Kivi aiY
01 XvKv wek^we`¨vjq A MCQ + SAQ 100 15 + 10 Avek¨K
02 Rvnv½xibMi wek^we`¨vjq A MCQ 80 22 Avek¨K
03 ivRkvnx wek^we`¨vjq C MCQ 80 25/12 Avek¨K
A MCQ 100 25 Avek¨K
04 PÆMÖvg wek^we`¨vjq D MCQ 100 25 Avek¨K
05 evsjv‡`k BDwbfvwm©wU Ae cÖ‡dkbvjÕm FST MCQ 100 25 Avek¨K
06 XvKv Awaf~³ 7-K‡jR A MCQ 100 25 Avek¨K
FEOS
07 e½eÜz †kL gywReyi ingvb †gwiUvBg wek¦we`¨vjq FET
MCQ + SAQ 100 12 + 08 Avek¨K
cÖm½ 02 ¸”Q (GST) wek¦we`¨vjq
MwYZ As‡ki
µwgK ¸”Qfy³ mvaviY wek¦we`¨vj‡qi bvg µwgK ¸”Qfy³ weÁvb I cÖhyw³ wek¦we`¨vj‡qi bvg
b¤^i
01 RMbœv_ wek^we`¨vjq 01 kvnRvjvj weÁvb I cÖhyw³ wek^we`¨vjq
02 Lyjbv wek^we`¨vjq 02 h‡kvi weÁvb I cÖhyw³ wek¦we`¨vjq
03 Bmjvgx wek^we`¨vjq 03 nvRx ‡gvnv¤§` `v‡bk weÁvb I cÖhyw³ wek¦we`¨vjq
04 Kzwgjøv wek^we`¨vjq 04 †bvqvLvjx weÁvb I cÖhyw³ wek¦we`¨vjq
05 †eMg †iv‡Kqv wek^we`¨vjq 05 gvIjvbv fvmvbx weÁvb I cÖhyw³ wek¦we`¨vjq
06 ewikvj wek^we`¨vjq 06 cvebv weÁvb I cÖhyw³ wek¦we`¨vjq 25
07 RvZxq Kwe KvRx bRiæj Bmjvg wek¦we`¨vjq 07 cUzqvLvjx weÁvb I cÖhyw³ wek¦we`¨vjq
08 †kL nvwmbv wek¦we`¨vjq 08 e½eÜz †kL gywReyi ingvb weÁvb I cÖhyw³ wek¦we`¨vjq
09 iex›`ª wek¦we`¨vjq 09 iv½vgvwU weÁvb I cÖhyw³ wek^we`¨vjq
10 e½eÜz †kL gywReyi ingvb wWwRUvj wek¦we`¨vjq 10 e½gvZv †kL dwRjvZz‡bœQv gywRe weÁvb I cÖhyw³ wek¦we`¨vjq
11 e½eÜz †kL gywReyi ingvb wek¦we`¨vjq 11 Puv`cyi weÁvb I cÖhyw³ wek¦we`¨vjq
cÖm½ 03 K…wl wek¦we`¨vjq
µwgK wek¦we`¨vj‡qi bvg cix¶v c×wZ †gvU b¤^i MwYZ As‡ki b¤^i DËi Kivi aiY
01 mgwš^Z K…wl wek^we`¨vjq fwZ© cixÿv (08wU) MCQ 100 20 Avek¨K
cÖm½ 04 BwÄwbqvwis
µwgK wek¦we`¨vj‡qi bvg cix¶v c×wZ †gvU b¤^i MwYZ As‡ki b¤^i DËi Kivi aiY
MCQ 100 34 Avek¨K
01 BUET
Written 400 140 Avek¨K
02 CKRUET (Engineering Cluster) MCQ 500 150 Avek¨K
03 BUTex Written 200 60 Avek¨K
04 IUT MCQ 100 35 Avek¨K
05 MIST Written 100 40 Avek¨K
06 DU-Technology MCQ 120 35 Avek¨K
07 Textile Engineering MCQ 200 60 Avek¨K
08 Sylhet Engineering College Written 100 30 Avek¨K
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
2 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
Mn
„ wkÿ‡Ki weKí (Mí bvwK mZ¨!!!!)
GKRb M„nwkÿK †hgb K‡i Zvi wkÿv_©xi cwiPh©v K‡i, nv‡Z-Kj‡g MwYZ †kLvq ASPECT MATH
kZfvM †Póv K‡i‡Q wVK †Zgb K‡iB GWwgkb MwYZ‡K wkÿv_©x‡`i mvg‡b Dc¯’vcb Ki‡Z| Zvi wKQz
bgybv wb‡P Dc¯’vcb Kiv n‡jv|
nv‡Z Kj‡g UwcK we‡kølY
01 A = 4 7 Ges B = 0 1 n‡j AB = ?
1 3 1 0
EXAMPLE
1 0 = (1 1) + (3 0) (1 0) + (3 1) ¸‡Yi gZ RwUj welq‡K wP‡Î wP‡Î Dc¯’vcb Kiv
AB = 4 7
1 3
0 1 (4 1) + (7 0) (4 0) + (7 1)
n‡q‡Q| wVK †hgbfv‡e †Zvgvi M„nwkÿK †Zvgv‡K
1 0 = (1 1) + (3 0) (1 0) + (3 1)
LvZvq wPÎ Gu‡K wkLv‡Zv| Zvn‡j e‡jv n‡jv bv M„n
AB = 4 7
1 3
0 1 (4 1) + (7 0) (4 0) + (7 1)
wkÿ‡Ki weKí!!!!!!!
1 2 3
EXAMPLE 02 4 5 6 wbY©vq‡Ki gvb KZ?
0 8 0
Procedure With Steps and Figure
†h fv‡e AsKwU Ki‡Z n‡e: cv‡ki wPGwU fv‡jv K‡i jÿ Ki| c×wZwUi myweav :
†Kvb mvwi Kjvg GK Kivi †Kvb Sv‡gjv/Tension _v‡K bv|
0
48 †hvMdj
cix¶vi n‡j wM‡q ‡PvL eyu‡S AsK Kiv ïiy Ki‡Z cvi‡e| 1 2 3 0
4 5 6
gy‡L gy‡L Kiv m¤¢e| 40 sec Gi †ewk mgq jvM‡e bv| cix¶vq G ai‡bi AsKB †ewk we‡qvM
0 8 0
Av‡m |
1 2 3
Step-1: 1st Ges 2nd Row `ywU cv‡ki wP‡Îi gZ wb‡P wb‡P wjL| 4 5 6 0
96
Step-2: Zvici Ggb fv‡e Zxi KvU‡Z n‡e †hb cÖwZwU Zx‡i wZbwU K‡i msL¨v _v‡K | 0 †hvMdj
Step-3: cÖwZwU Zx‡ii msL¨v¸‡jv Avjv`v Avjv`v fv‡e ¸b K‡i †hvM Ki |
Step-4: AZtci wb‡Pi Zx‡ii †hvMdj n‡Z Dc‡ii Zx‡ii †hvMdj we‡qvM Ki‡e| hv
wbY©vq‡Ki gvb = (0 + 96 + 0) – (0 + 48 + 0) = 48
cv‡e ZvB Answer|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT MATH g¨vwRK¨vj †cÖ‡R‡›Ukb
© Medistry 3
؇›Ø ؇›Ø Math
GWwgk‡b Ggb A‡bK g¨v_ Av‡m, hvi DËi AvcvZ`„wó‡Z GKwU g‡b n‡jI Avm‡j DËi nq Ab¨wU| wkÿv_©x‡`i CONCEPT wK¬qvi bv _vKvq
fzj DËiwU †`q Ges cieZ©x‡Z †iRvë †`‡L AvkvnZ n‡q hvq| P‡jv †`‡L †bB GB mKj Ø›Ø ASPECT MATH wKfv‡e mgvavb K‡i‡Q!!!!!
EXAMPLE 02 lim |x| = ?
x0 x
A. 1 B. –1 C. 1 D. wjwg‡Ui Aw¯ÍZ¡ †bB
†Zvgv‡`i A‡b‡Ki Kv‡Q g‡b n‡Z cv‡i, Wvbw`KeZ©x wjwgU w`‡q mgvavb Ki‡j Ans: 1
evgw`KeZ©x w`‡q mgvavb Ki‡j Ans: –1
Avevi, A‡b‡KB fve‡Z cvi Ans: 1 n‡e wKš‘ Avmj welqwU wfbœ|
Explanation : y = f(x) dvskbwUi x = a we›`y‡Z wjwg‡Ui Aw¯ÍZ¡ _vK‡e hw` Wvb w`KeZ©x I evg w`K eZ©x wjwgU mgvb nq|
A_©vr, lim + f(a) = lim – f(a) nq
xa xa
Zvn‡j Gevi †`‡L †bqv hvK mwVK DËi wK n‡e?
–x
L.S.L = lim – = lim – (–1) = – 1
xa x xa
x
R.S.L = lim + = lim + (1) = 1
xa x xa
GLv‡b, lim – f(x) lim + f(x) ∵ Wvbw`KeZ©x I evgw`KeZ©x wjwgU mgvb bq ZvB Ans: wjwg‡Ui Aw¯ÍZ¡ †bB|
xa xa
Gevi †`L‡j Concept clear bv _vK‡j DËwU ev` n‡q †hZ|
x–2
EXAMPLE 01 f(x) = 2x – 4 dvskbwUi †iÄ KZ? ax + b
†Zvgiv hviv GB f(x) =
cx + d
dvsk‡bi
1 1 1 1 Technique
A. B. R – – C. R – D. j‡e x Gi mnM
2 2 2 2 e¨envi Ki‡e| †iÄ = R –
n‡i x Gi mnM
1
Zv‡`i g‡Z Ans: R – 2 hv fzj
Explanation : Gevi Zvn‡j †`‡L †bqv hvK mwVK DËi wK n‡e........
Avgiv hLb †`L‡ev f(x) = 2 ZLb Avgiv ewj dvskbwUi †iÄ = {2} KviY aªyeK dvsk‡bi †ÿ‡Î aªæeKwU n‡e †iÄ|
x–2 (x – 2) 1
†`‡L †bqv hvK, f(x) = 2x – 4 = 2(x – 2) = 2
1
Zvn‡j GwUI GKwU aªæeK dvskb Ges †iÄ n‡e = 2
1
KviY x Gi mKj gv‡bi Rb¨ f(x) Gi gvb 2 Qvov Ab¨ wKQz Avm‡e bv|
02 y = (x 1) (x 10) eµ‡iLvwU x Aÿ‡K KZevi †Q` Ki‡e?
2 4
EXAMPLE
A. 8 B. 6 C. 4 D. 5
AvgivRvwb, mgxKi‡Yi mgvavb hZwU _vK‡e †mwU ZZevi †Q` Ki‡e| cÖ`Ë mgxKi‡Y x Gi NvZ 6 ZvB g‡b n‡Z cv‡i †h, GwU 6 evi †Q` Ki‡e| A_©vr
DËi n‡e Option: B|
Explanation : y = (x2 1) (x4 10) (x2 – 1) (x2 + 10) (x2 – 10) = 0
wKš‘ (x2 + 10) Gi †Kv‡bv ev¯Íe mgvavb †bB|
ev¯Íe mgvavb¸‡jv n‡e x = ± 1, ± 10 †h‡n‡Zz ev¯Íe mgvavb 4wU| ZvB †Q`we›`yI n‡e 4wU|
wK ev”Pviv †Kgb w`jvg!!!!!!!!
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
4 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
M‡í
M‡í
wØc`x we¯Í…wZ
eo fvB‡qi Av‡M †QvU fvB‡qi we‡q: (mv‡_ wØc`xi µwgK c`)
wkÿv_©x eÜziv, ejZ †Zvgvi Av‡k cv‡k †Kvb family †Z †QvU fvB Zvi eo fvB‡K †i‡L we‡q K‡i †dj‡j MÖv‡gi †jvK‡`i K_vi Kvi‡Y eo fvB evmv‡Z
Problem •Zwi K‡i bv? wK K‡i? nv K‡i|
ej‡Zv H mgq cwiw¯’wZ ¯^vfvweK K‡i †K? †n †Zvgiv wVKB e‡jQ, `v`v †QvU bvZxi cÿ wb‡q welqUv wVK K‡i, ZvB bv?
Zvn‡j welqUv †Kgb `vovq †`L‡Zv Math-G †M‡j?
2q c` (†QvU) eo fvB
1g c` (eo) = `v`v-†QvU fvB
`v`v †QvU fvB eo fvB
EXAMPLE 03
3 + x -Gi we¯Í…wZ‡Z x7 I x8 Zg c‡`i mnM mgvb n‡j n = ? [mvaviYZ (axp + bxq)n AvKv‡i _vK‡e]
n
2
1
2
(2q c‡`i x Gi mnM) 8 (eo fvB)
=
3 (1g c‡`i x Gi mnM) n – 7
GiKg gRvi gRvi wUªKm
`v`v †QvU fvB wk‡L AsK Ki GK wbwg‡l
1 8
= n – 7 = 48 n = 55
6 n–7
`v`v
Ex: (1 + x)44 Gi we¯Í…wZ‡Z 21 Zg I 22 Zg c` mgvb n‡j, x = ?
(20 + 1) (21 + 1)
x 21 21 7
=
†QvU eo = =
1 44 – 20 24 8
nv‡Z
nv‡Z
K¨vjKz‡jkb wkLvB
c„w_exi †Kvb msL¨v‡K 9 Øviv fvM gv‡b `kwg‡Ki c‡i H msL¨v evi evi Av‡m|
1 cÖ‡qvM :
†hgb: 9 = 0.111....
wØc`x we¯Í…wZi avivq
2 Ex : 0.3 + 0.03 + 0.003 + .....
= 0.222....
9
= 0.333 .....
3 1
= = 0.333..... 3 1
9 3 = = nv... nv... Kx gRv GZ mnR?
9 3
5
= 0.55....
9
6 2
= : 0.666....
9 3
nv... nv... Avwg `yó 9, hv‡K Dc‡i ivwL, `kwg‡Ki c‡i Zv‡KB evi evi wdwi‡q †`B|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT MATH g¨vwRK¨vj †cÖ‡R‡›Ukb
© Medistry 5
sin/cos Gi ARvbv gvb m¤úwK©Z
sin = 0 15 30 37.5 45 52.5 60 75 90
sin = 0 0.25 0.50.6 0.707 0.79 0.87 0.96 1
0 + 30 0 + 0.5
GLv‡b GKwU welq †f‡e †`L, 2 = 15, ZvB gvbwUI n‡e Mo, †hgb 2 = 0.25, GKBfv‡e Ab¨ gvb ¸‡jvI n‡q‡Q|
GLb, hw` ejv nq, sin = 0.125, GLv‡b 0.125 hv, 0.25 Gi A‡a©K, ZvB 15 Gi A‡a©K n‡e, A_©vr 7.5
GLb, sin 37.5 = ?
sin
30 + 45 0.5 + 0.707
2 = 2
= 0.6
Gfv‡eB mg¯Í gvb †ei Ki‡Z nq|
fveyb, sin = ln2 n‡j a = ? †`Lyb, sin 45 = 0.707
sin = 0.693 GLv‡b, 0.693 gvb 0.707
= sin–1 (0.693) = 43 †_‡K Kg, gv‡bB 45 A‡cÿv Kg|
A‡b‡KiB sin I cos Gi gvb wb‡q mgm¨v| P‡jv Ø›ØwU clear Kwi|
sin 0 †Lqvj K‡iQ? sin I cos Gi †Kv‡Yi gv‡bi †hvMdj hLbB 90 ZLbB Ans. same nq|
0
cos 90
sin 30 1 A_©vr sin A = cos B if A + B = 90
cos 60 2 Ex: sin 15 = cos 75
sin 60 3 wK Ø›Ø Clear?
cos 30 2
Magical Calculation:
(–4, 3) I (12, –1) we›`y؇qi ms‡hvM †iLv‡K e¨vm a‡i AswKZ e„‡Ëi mgxKiY?
–48 –3
GUv †Zvgiv mevB Rvb, but Gi c‡iB calculation eo K‡i
(x + 4) (x – 12) + (y – 3) (y + 1) = 0
w`j ZvB g‡b †iL- †hvM Ki‡j mnM Avi ¸Y Ki‡j aªæeK
–8x –2y cvIqv hvq|
x2 + y2 – 8x – 2y – 51 = 0
eo Concept wPÎ w`‡q
†QvU Kwi....... Rq Kwi
🙆 wP‡Î wP‡Î mvaviY ¯úk©K msL¨v wbY©qÑ
Ae¯’v wPÎ mvaviY ¯úk©K msL¨v kZ©
1. AšÍt¯úk©x e„Ë r1
c1 c2
r2 1wU c1c2 = r1 – r2
2. ci¯úi †Q`x e„Ë r1
c1 c2
r2 2wU c 1c 2 r 1 – r 2
r1 r2
3. ci¯úi¯úk©xe„Ë c1 c2 3wU c1c2 = r1 + r2
r1 r2
4. KLbI ¯úk© Ki‡ebv Ggb 2wU e„Ë c1 c2
4wU c1c2 > r1 + r2
r1
5. GKB †K›`ª wewkó wfbœ e¨vmv‡a©i e„Ë c1 mvaviY ¯úk©K †bB c1c2 r1 r2
c2
r2
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
6 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
GiKg Av‡ivI KwZcq Tricks hv †Zvgvi Problems Solving Gi MwZ evwo‡q w`‡e ...
GENERAL RULES [WRITTEN] 3 in 1 ASPECT SUPER TRICKS [MCQ]
lim sin ax = a
1. x0 lim sin 4x Gi gvb-4 (Ans)
bx b x0 7x 7
lim tan ax a lim tan 3x 3
2. x0
bx
=
b x0 5x Gi gvb 5 (Ans.)
–1
lim tan ax = a lim tan 3x 3
3. x0
bx b x0 5x Gi gvb 5 (Ans).
lim tan ax = a
4. x0 lim tan 5x 5
sin bx b x0 sin 2x = 2 (Ans)
–1 –1
lim sin ax a lim sin (2x) = 2 (Ans)
5. x0 =
bx b x0 x
lim sin ax a lim sin 3x 3
6. x0 = x0 sin 5x = 5 (Ans)
sin bx b
GK Q‡K ch©vqµwgK AšÍixKiY:
y = x4 ; y1 = 4x3 = 4p1 x4-1 y = xn n‡j y = (ax + b)n 1 (–1)n.n!
y= Gi nZg AšÍiK = xn+1
y2 = 4.3x2 = 4p2 x4-2 yn = n! yn = n! an x
y3 = 4.3.2.x = 4p3 x4-3 ym = 0 [m > n] ym = 0 [m > n] 1
y4 = 4.3.2.1 = 4p4 x4-4 = 4!
y= Gi nZg AšÍiK =
ym = npm xn-m [m < n] ym = npm (ax + b)n–m am x+a
y5 = 0 [5 > 4] [m < n] (–1)n.n!
(x+a)n+1
y = sinx y = cosx y = emx
y1 = cosx y1 = –sinx y1 = m.emx y = Sin(ax + b) Gi nZg AšÍiK,
y2 = –sinx y2 = –cosx y2 = m2.emx n
y3 = –cosx y3 = sinx y3 = m3.emx yn = an sin
ax b
2
y4 = sinx = y y4 = cosx = y yn = mn.emx
A_©vr cÖwZ 4n Zg AšÍi‡Ki ci A_©vr cÖwZ 4n Zg AšÍi‡Ki ci y = ax
cybive„wË NU‡e cybive„wË NU‡e y1 = axlna
y2 = ax(lna)2 y = cos(ax + b) Gi nZg AšÍiK,
y = sinx n‡j, yn = sin + x
n y = cosx n‡j, yn = cos
2
y3 = ax(lna)3 yn = a cos n ax b
n
n + x yn = ax(lna)n 2
2
x2 + y2 + 2gx + 2fy + c = 0 e„ËwUi †¶‡Î
(i) x A¶‡K ¯ck© Ki‡j, c = g2 Y Y Y
e¨vmva© = |e„‡Ëi †K‡›`ªi †KvwU|
(ii) y A¶‡K ¯ck© Ki‡j, c = f2 Ges C(x, y) C(x,y) C(x, y)
e¨vmva© = |e„‡Ëi †K‡›`ªi fyR| y r=x
x y
(iii) Dfq A¶‡K ¯ck© Ki‡j, c = g2 = f2 Ges X
X X
e¨vmva© = |e„‡Ëi †K‡›`ªi fyR| = |e„‡Ëi †K‡›`ªi †KvwU| O O O
x A¶‡K ¯ck© Ki‡j y A¶‡K ¯ck© Ki‡j Dfq A¶‡K ¯ck© Ki‡j
GK Q‡K mKj KwYK:
mvaviY wØNvZ mgxKiY ax2 + bxy + cy2 + dx + ey + f = 0 mgxKiYwU‡Z-
b2 4ac
<0 = 0 cive„Ë (Prabola) > 0 Awae„Ë (Hyperbola)
b = 0; a = c; e„Ë (Circle)
b = 0; a c; Dce„Ë (Ellipse)
Note: GB c×wZ e¨envi K‡iI mgxKi‡Yi R¨vwgwZK cÖK…wZ wbY©q|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT MATH g¨vwRK¨vj †cÖ‡R‡›Ukb
© Medistry 7
🙆 K_vq K_vq n‡e mÂvic_ Rq................
RwUj term (?) I aªæeK term (a, b, c ----) mgvb n‡j mij‡iLv, aªæeK hw` †ewk nq e„Ë n‡e g‡b Ki, GKB aªæe‡Ki †hvMwe‡qvM Dce„Ë Zvnvi iƒc| _vwK‡j
Extra PjK Zvnv n‡e cive„Ë mv‡ne|
(i) |z + a| = |z + b| mij‡iLv (aªæeK I RwUj term mgvb) (ii) |z + a| = b|z + c| e„Ë (aªæeK > RwUj)
(iii) |z a| = b|z + a| Dce„Ë (aªæe‡Ki †hvM we‡qvM) (iv) |z + a| = x or y cive„Ë
sin 15 = ?
wP‡Î w·KvYwgwZ cos 75 = ?
wP‡Î
w·KvYwgwZi wKQz g¨vwRK¨vj †cÖ‡R‡›Ukb
Nwoi KvUvi w`‡K Aewkó Av½yj 30
sin =
4
1wU Av½yj
45 1 1
60 30 Example: sin 30 = =
4 2
90
Nwoi KvUvi wecixZ w`‡K Aewkó Av½yj 45
cos =
0 4 2wU Av½yj
2 1
Example: cos 45 = =
4 2
Nwoi KvUvi w`‡K Aewkó Av½yj 60
tan = 1wU Av½yj 3wU Av½yj
Nwoi KvUvi wecixZ w`‡K Aewkó Av½yj
3
Example: tan 60 = = 3
1
weKí c×wZ: †KvY m¤úwK©Z As‡Ki mgm¨v AvR‡KB f¯§xf~Z
sin/cos m¤úwK©Z
0 30 45 60 90
Step- 01 0 1 2 3 4
0 1 2 3 4
Step- 02
4 4 4 4 4
0 1 2 3 4
Step- 03
4 4 4 4 4
1 1 3
Step- 04 0 1
2 2 2
Step- 05: sin = 0 0.5 0.707 0.87 1
cos = 1 0.87 0.707 0.5 0
tan/cot m¤úwK©Z
0 30 45 60 90
Step-01 0 1 3 9
0 1 3 9
Step-02
3 3 3 3 3
Step-03
0 1 3 9
3 3 3 3 3
1
Step-04 0 1 3
3
Step-05: tan = 0 0.57 1 1.73
cot = 1.73 1 0.57 0
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
8 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
Super Hexagon :
sin cos
tan cot
1
sec cosec
b
Type-1 cvkvcvwk 3 we›`y wb‡j a = [†h‡Kv‡bv w`‡K cÖ‡hvR¨]
c
sin cos tan cosec
Ex : tan = sin = sin = sec =
cos cot sec cot
tan cos cot sin
sec = cot = cos = cos =
sin sin cosec tan
Type-2
evgcv‡ki wP‡Î jÿ¨ Kwi sin cos
`vM KvUv wÎfzR ¸‡jv‡Z Wvb †_‡K ev‡g A_©vr Nwoi KvUvi w`‡K
†M‡j a2 + b2 = c2 m~ÎwU †g‡b P‡j| tan cot
1
Example: sin2 + cos2 = 1
1 + cot2 = cosec2 sin2 = 1 – cos2 sec cosec
tan2 + 1 = sec2 cos2 = 1 – sin2
cosec2 – cot2 = 1 sec2 – tan2 = 1
cot2 = cosec2 – 1 tan2 = sec2 – 1
Type-3 †h hvi eivei †m Zvi wecixZ| sin cos
1 1 1
sin = cos = tan =
cosec sec cot
1 1 1 tan cot
cosec = sec = cot = 1
sin cos tan
sec cosec
sin/cos Gi avivi gvb wbY©q
c` msL¨v
Avgiv wk‡LwQ, eM©vKv‡ii w·KvYwgwZK †hvMd‡ji gvb =
2
2 2 8 2
EXAMPLE 01 sin 10 + sin 20 + -------- + sin 80 = = 4
2
Last – First 8010 n 8
Magical Solve: n = +1= +1=8 Now, gvb = = = 4
Difference 10 2 2
2 2 2 2
EXAMPLE 02 sin 3 + sin 9 + sin 15 + ............ + sin 177 = ?
177 – 3 n
Magical Solve : n = + 1 = 30 Ans. = 15
6 2
EXAMPLE 03 2sin 128 = 2sin 27 [2 Gi power hZ n‡e Zvi †_‡K 1 Kg n‡e memgq Z‡e, sin0 Gi mgq GKUv plus, GKUv minus Avi cos
Gi mgq meB plus] = 2 – 2 + 2 – 2 + 2 – 2
†h‡nZz sin ZvB 1wU plus Ges 1wU Minus Gi 2 Gi power 7 wQj ZvB 6Uv n‡q‡Q|
1
cos Gi µgea©gvb ¸Yd‡ji ivwki gvb = n
2
1
EXAMPLE 01 cos.cos2.cos4 =
23 (3wU cos Gi term)
1
=
8
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT MATH wUªKm
© Medistry 9
cv‡ë †M‡Q ASPECT MATH e`jv‡Z n‡e
cixÿv c×wZ Awfbe Dc¯’vcbv cÖ¯‘wZ ce©
†Zvgv‡`i cÖ‡qvR‡bB Avgv‡`i Av‡qvRb †Rbv‡ij g¨v_W (WRITTEN) I kU©KvU© wUªK‡mi •ØZ Dc¯’vcb|
mv‡f© †Uwej (MAGNETIC DECISION) Gi gva¨‡g UwcK wm‡jKkb| cÖwZwU
cÖ_gZ
Dc¯’vcbvi wØZxqZ
wek¦we`¨vj‡qi Rb¨ Avjv`v Avjv`v UwcK †ivWg¨vc
GKwU Uwc‡K m¤¢ve¨ mKj ai‡bi cÖkœ wb‡q MODEL EXAMPLE Dc¯’vcb
Z…ZxqZ †Rbv‡ij g¨v_W (WRITTEN) I kU©KvU© wUªK‡mi •ØZ Dc¯’vcb
Awfbe c×wZ PZz_©Z
cÂgZ
INSTANT PRACTICE & SOLVE ms‡hvRb
m‡e©v”P msL¨K weMZ cÖ‡kœi kU©KvU© e¨vL¨vmn we‡kølY
lôZ CONCEPT TEST [MCQ I wjwLZ Gi mgwš^Z Abykxjb]
bgybv
Dc¯’vcb
TACTIC 01 5 6 g¨vwUª·wUi wecixZ g¨vwUª· Gi †Uªm †KvbwU?
2 4 [JU-A, SetB. 20-21]
S General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
Gi wecixZ g¨vwU©· =
5 6 1 4 –6 cÖ`Ë g¨vwUª‡·i †Uªm
2 4 20 –12 – 2 5 wecixZ g¨vwUª‡·i †Uªm = wbY©vq‡Ki gvb
1 –3 =
5+4
=
9
1
= – 2
4 6 2 4 1 5 9
†Uªm = + =
20 12 8
5 – 1 5
=
8 2 8 8
4 8
02 hw` A =
1 2 2
TACTIC 3 4 Z‡e A + 3A – 10I n‡e GKwUÑ [RU. Moderna, Set-2. 20-21]
S g¨vwUª·
A. A‡f`K B. cÖwZmg g¨vwUª· C. k~b¨ g¨vwUª· D. †KvbwUB bq Ans C
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
6
A2 = 3 . = =
1 2 1 2 1 + 6 2 8 7 †Kv‡bv g¨vwUª· A Gi Rb¨ A2 †Uªm A + |A| I = 0
4 3 4 3 12 6 + 16 9 22
GLv‡b A = 3 –4 A2 – (1 – 4) A + (–4 – 6) I = 0
1 2
7 6 1 2 1 0
A + 3A10I=
2
100 1
9 22
+ 33
4 A2 + 3A – 10 I = 0
=
7 + 3 10 6 + 6 0 0 0
9 + 9 0 22 12 10 = 0 0 = k~b¨ g¨vwUª·
1 2
TACTIC 03 2 1 Gi wgvb-; GK‡Ki GKwU RwUj Nbg~j| [DU.2009-10]
S 2 1
A. 1 B. 3 C. D. 0 Ans D
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
2 wbY©vq‡Ki As‡K hw` †Kv_vI _v‡K Zvn‡j = 1 ewm‡q wbY©vq‡Ki gvb
1 1 2 2
wbY©q Ki‡j Answer cvIqv hv‡e|
2
1 = 2 1 2 /
1 [c1 = c1+c2+c3]
1 2
2 1 2 1 1 1 1 1
2 1 = 1 1 1 = 0
0 2 1 1 1
2 1
= 0 2 1 [ 1 + + 2 = 0] = 0 Ans.
0 1
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10 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
3 –4
TACTIC 04 A = n‡j det (2A–1) Gi gvb n‡jv-
2 –3
[DU. 19-20]
S
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
3 –4 [GLv‡b DwjøwLZ g¨vwUª· Gi order = 2 2
A = 2 –3
3 –4
A = 2 –3 n‡j,
A_©vr 2 Ges ¸wYZK n‡”Q A–1 Gi mv‡_
1 –3 4 3 –4 |A| = – 9 + 8 = –1 KZ¸Y Av‡Q|
A–1 =
– 9 + 8 –2 3 2 –3
=
det (2A–1)=
2 GLv‡b 2A–1 A_©vr ¸wYZK = 2
6 –8
2A–1 = 4 –6 det (2A–1) = – 36 + 32 = – 4
A
Note : ïay eM©g¨vwUª‡·i †ÿ‡Î cÖ‡hvR¨|]
(2)2
= (¸wYZK)order gvb = =–4
(1)
TACTIC 05 ABC G A(3, 3) B(–1, 5) C(4, –2) n‡j wÎfz‡Ri †¶Îdj wbY©q Ki? [JU. 2018-19]
S 8
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
3 3 Zero Method: A(3, 3), B(–1, 5), C(4, – 2)
1 †h‡Kvb GKwUi ¯’vbvsK (0, 0) Ki‡Z n‡e-
ABC = 1 5
2 4 2 A(3–3, 3–3), B(–1 –3, 5–3), C(4 –3, – 2–3)
3 3 cÖwZ¯’vwcZ we›`y¸‡jv A(0, 0), B(–4, 2), C(1, –5)
1 Kv‡Q = 2 1 = 2
= [(15 + 2 + 12) – (–3 + 20 – 6)]
2
1 B(–4, 2) C(1, –5)
= (29 – 11) = 9 eM© GKK Ans.
2
`~‡i = (–4) (–5) = 20
1 1
†ÿÎdj = | `~‡i – Kv‡Q | = |20 – 2| = 9 eM©GKK
2 2
TACTIC 06 (1, 2) we›`yi mv‡c‡¶ (4, 3) we›`yi cÖwZwe¤^ KZ? [IU. 2017-18]
S General Rules [Written] ASPECT Tricks & Tips [MCQ]
3 in 1
(1, 2) we›`yi mv‡c‡¶ (4, 3) we›`yi cÖwZwe¤^ (x, y) 1 K‡g‡Q 1 Kgv‡ev
1= 4 x 2=
3 y (4,3) (1, 2 (x, y)
(4, 3) (1, 2) (–2, 1) Ans.
2 2
x = 2 y=1 Ans.(2, 1) 3 K‡g‡Q 3 Kgv‡ev
1
TACTIC 07 y 1 eµ‡iLv x A¶‡K A we›`y‡Z Ges y A¶‡K B we›`y‡Z †Q` Ki‡j AB mij‡iLvi mgxKiY n‡e- [DU.12-13. BUTex.19-20]
2x
S
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
cÖ`Ë mij‡iLvwU x A¶‡K †h we›`y‡Z †Q` K‡i H we›`y‡Z y ¯’vbv¼ k~b¨ eµ‡iLv †_‡K mij‡iLvi Eqution †ei Ki‡Z PvB‡j eµ‡iLvwU‡K
1 Simplify Ki‡Z n‡e| Zvici hv mij‡iLvi •ewkó¨ bq †m c` ¸‡jv‡K
0 1 x + 2= 1 x = 3 A (3, 0)
2x ev` w`‡q w`‡jB mij‡iLvi mgxKiY cvIqv hvq|
Avevi, cÖ`Ë mij‡iLvwU y A¶‡K †h we›`y‡Z †Q` K‡i H we›`y‡Z x ¯’vbv¼ k~b¨ 1
y 1
y 1
1
y
3
0, 3 2x
20 2 2 2 + x +1
y= 2y + xy = x + 3
2+x
AB mij‡iLvi mgxKiY, x 3 y 0 x 2 y 3 0 x – 2y + 3 = 0 [mij‡iLvi mgxKi‡Y †h‡nZz xy AvKv‡i †Kvb c`
30 3
0 _v‡K bv| ZvB xy ev` w`‡q]
2
TACTIC 08 x + y = 81 e„ËwUi R¨v (2, 3) we›`y‡Z mgwØLwÛZ nq, R¨v Gi mgxKiY KZ? [JU.03-04; MBSTU.15-16;KU.01-02;SUST.09-10]
2 2
S General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
†K›`ª (0, 0) Ges mgwØLÛb we›`y (2, 3) Gi ms‡hvM †iLvi
x2 + y2 = a2 e„ËwUi R¨v (x1, y1) we›`y‡Z mgwØLwÛZ n‡j, R¨v Gi mgxKiY:
Xvj = 3 0 3 wb‡Y©q R¨v Gi Xvj = 2 ; x.x1 + y.y1 = (x1)2 + (y1)2
20 2 3
x.(–2) + y.3 = (–2)2 + (3)2
myZivs R¨v Gi mgxKiY, y3 = 2 (x + 2) 2x –3y + 13 = 0 Ans. 2x – 3y + 13 = 0 Ans.
3
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ASPECT MATH wUªKm
© Medistry 11
TACTIC 09 cÖ‡Z¨K msL¨vq cÖ‡Z¨KwU AsK †Kej GKevi e¨envi K‡i 2,3,5,7,8,9 Øviv wZb AsK wewkó KZ¸‡jv msL¨v MVb Kiv hvq|
S General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
†gvU AsK i‡q‡Q 6 wU (2, 3, 5, 7, 8, 9) hv Øviv 3 AsK wewkó msL¨v MVb
Kiv hv‡e- 6P3 = 120 wU 6 5 4 = 6 5 4 = 120
GLv‡b, AsK msL¨v = 6 Ges †Kej GKevi e¨envi Ki‡Z
ej‡Q , ZvB GK GK Kwg‡q Ni¸‡jv c~ib Kiv nj|
TACTIC 10 4, 5, 6, 7, 8 A¼¸‡jvi cÖ‡Z¨KwU‡K †h‡Kvb msL¨Kevi wb‡q Pvi AsKwewkó KZ¸‡jv msL¨v MVb Kiv hvq|
S General Rules [Written] ASPECT Tricks & Tips [MCQ]
3 in 1
Pvi AsKwewkó msL¨v MV‡bi †ÿ‡Î †gvU cyib‡hvM¨ ¯’vb 4wU| msL¨v¸‡jv‡K 5 5 5 5
†h‡Kv‡bv msL¨K evi e¨envi Ki‡j- 1g ¯’vbwU cyi‡bi Dcvq 5P1 jÿ¨ Ki: Ò†h‡Kvb msL¨Kevi
2q ¯’vbwU cyi‡bi Dcvq 5P1 ; 3q ¯’vbwU cyi‡bi Dcvq 5P1
= 5555 = 625
wb‡qÓ e‡j‡Q ZvB GK GK K‡i
5
4_© ¯’vbwU cyi‡bi Dcvq P1 Kg‡jv bv|
†gvU Dcvq = 5P1 5P1 5P1 5P1 = 625
x2
TACTIC 11 f(x) = Øviv msÁvwqZ, f1 (x) wbY©q Ki| [DU. 2019-20]
x3
S
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
x2 ax + b
n‡j f1(x) = –dx + b
y = f(x) = xy 3y = x 2 2 3y = x xy f(x) =
x3 cx + d cx a
2 3y 3y 2 3x 2 x 2 1 3x2
= x f1(y) = f1(x) = f(x) = ; f (x) =
1y y1 x1 x3 x1
sin75 + sin15
TACTIC 12 =? [DU. 2019-20; RU 15-16, DU: 11-12, RU: 12-13,JNU: 10-11, DU: 04-05]
sin75 – sin15
S
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
sin75 + sin15 sin(90 – 15) + sin15 sinA + sinB
L.H.S = = = tan(45 + B) [hLb A + B = 90˚; B < A]
sin75 – sin15 sin(90 – 15) – sin15 sinA – sinB
cos15 1 +
sin15 sin75 + sin15
= tan(45 + B)= tan(45 + 15) = tan60 = 3
cos15 + sin15 cos15 sin75 – sin15
= =
cos15 1 –
cos15 – sin15 sin15
cos15
1 + tan15 tan45 + tan15
= = = tan(45 + 15) = tan60 = 3
1 – tan15 1 – tan45 tan15
2
lim 2x 2 + 3x + 5 Gi gvb|
13 x
TACTIC 3x + 5x – 6
S
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
x2 2 + + 2
3 5 3 5
2+ + 2
2
lim 2x2+3x+5 lim x x x x
lim
x Ges j‡e x Gi m‡e©v”P NvZ = n‡i x Gi m‡e©v”P NvZ n‡j
x 3 + – 2
x 3x +5x–6 x 2 5 6 x 5 6
3+ – 2
x x x x Ans. m‡e©v”P NvZhy³ x Gi mn‡Mi AbycvZ|
2+0+0 2
3 + 0 –0 3 (Ans)
14 lim cos7x – cos9x = ?
TACTIC x0 cos3x – cos5x
[Ref. †KZve DwÏb; wm. †ev.- 10]
S
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
1 1
2sin (7x + 9x) sin (9x – 7x)
lim cos7x – cos9x lim 2 2
x0 cos3x – cos5x x0
2 2
1 1 lim cos ax – cos bx = b2 – a2
2sin (3x + 5x) sin (5x – 3x) x0 cos cx – cos dx d – c
2 2 2 2
sin8x sinx 2sin4x cos4x lim cos 7x – cos 9x = 92 – 72 = 32 = 2 (Ans.)
x0
lim
x0 lim
x0 cos 3x– cos 5x 5 – 3 16
sin4x sinx sin4x
2.cos0 = 2.1 = 2 (Ans.)
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12 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
TACTIC 15 y = sinx + sinx + sin x + .....................
S
GENERAL RULES [WRITTEN] 3 in 1 ASPECT SUPER TRICKS [MCQ]
y= sinx + sinx + sin x + ..................... y = (x) + (x) + (x) + .....................
y = sinx + y dy (x)
=
y2 = sinx + y y2 – y – sinx = 0 dx 2y – 1 (x) = sinx
Differentiate with respect to x dy cosx
= (x) = cosx
dy dy dy dx 2y – 1
2y – – cosx = 0 (2y – 1) = cosx
dx dx dx
dy cosx
=
dx 2y – 1
TACTIC 16 y = x2 Ges y = 2x Øviv Ave× GjvKvi †¶Îdj- [DU: 06-07, SUST: 12-13]
S
GENERAL RULES [WRITTEN] 3 in 1 TRICKS [MCQ]
y
y = x2
y = 2x
x2 = 4ay Ges y = mx Øviv Ave×
x 8
O(0, 0)
†ÿ‡Îi †ÿÎdj = 3 a2m3
1
GLv‡b a = 4 Ges m = 2
cÖ`Ë y = x2 Ges y = 2x mgxKiY mgvavb K‡i cvB, x = 0 Ges x =2.
†ÿÎdj = . . 23 = sq.
8 1 2 4
3 4
1
x3 x2
2 2
8 4 3
wb‡Y©q †¶Îdj = y1 y 2 dx x 2 x dx = 2. 4 sq. units.
2
units. Ans.
0 0 3 2 0 3 3
4
wb‡Y©q †¶Îdj = sq. units. Ans. [†¶Îdj FYvZ¡K n‡Z cv‡i bv|]
3
1
TACTIC 17 †Kv‡bv wØNvZ mgxKi‡Yi GKwU g~j n‡j mgxKiYwU n‡eÑ
1+i
S
GENERAL RULES 3 in 1 SHORTCUT TRICKS & TIPS
GKwU g~j 1 n‡j, AciwU 1
1+i 1i 1
= x 1 = x + xi 1 x = xi (1 x)2 = (xi)2 1
1+i
mgxKiY, x2 1 + 1 x + 1 . 1 = 0
1 + i 1 i 1 + i 1 i 2x + x2 = x2 2x2 2x + 1 = 0
1+i+1i 1 2 1
x2 x + 2 2 = 0 x2 x + = 0 2x2 2x + 1 = 0
12 i2 l i 2 2
TACTIC 18 †Kv‡bv we›`y‡Z P Ges 2P gv‡bi `yBwU ej wµqvkxj| cÖ_g ejwU‡K wظY K‡i wØZxqwUi gvb 8 GKK e„w× Kiv n‡j Zv‡`i jwäi
S w`K AcwiewZ©Z _v‡K| P Gi gvb- [DU 13-14, RU 14-15, KU 12-13, 09-10, IU 04-05]
GENERAL RULES [WRITTEN] 3 in 1 ASPECT SUPER TRICKS [MCQ]
awi, P I 2P gv‡bi `yBwU e‡ji ga¨eZ©x †KvY Ges jwä, P Gi w`‡Ki P Ges Q gv‡bi `yBwU ej ‡Kv‡Y wµqvkxj Ges P I Q Gi cwie‡Z© P’
mv‡_ †KvY Drcbœ K‡i| I Q’ ejØq wµqv Ki‡j hw` jwäi w`K AcwiewZ©Z _v‡K Zvn‡j
2P sin (2P 8) sin α P P'
cÖkœg‡Z, tan θ Ges tan θ =
Q Q'
P 2P cos 2P (2P 8) cos α
P 2P
2P sin α
(2P 8) sin α
1
4 =
2P 2P + 8
P = 4 GKK
P 2P cosα 2P (2P 8) cosα 1 2 cosα P 8 cosα
P = 4 GKK Ans.
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ASPECT MATH wUªKm
© Medistry 13
TACTIC 19 2 lb wt, 4 lb wt Ges 6 lb wt gv‡bi wZbwU ej GKwU we›`y‡Z ci¯ci 120 †Kv‡Y wµqvkxj, Zv‡`i jwäi gvb KZ?
S
GENERAL RULES [WRITTEN] 3 in 1 ASPECT SUPER TRICKS [MCQ]
OX eivei ej¸‡jvi j¤^vsk wb‡q cvB,
Rcos = 2 cos0 + 4 cos120 + 6 cos240 Y
1 1
= 2 + 4 + 6 = 2 2 3 4 R
2 2 120
Rcos = 3 --- --- -- (i) 120 X
Avevi, OX ‡iLvi Dj¤^ eivei ej¸‡jvi O 2
j¤^vsk wb‡q cvB, 120
6 ejÎq mgvšÍi avivq wµqviZ _vK‡j jwä,
R sin = 2 sin0+ 4 sin120 + 6 sin 240 Z
F = 3 mgvšÍi avivq mvaviY AšÍi
3 3
= 0 + 4. + 6 = 3 2 Ans.
2 2
= 2 3 3 3 3 --- --- (ii)
(i) + (ii)2 n‡Z cvB, R2 (cos2 + sin2) = 9 + 3
2
R2 = 12
R= 12 = 2 3
wb‡Y©q jwä = 2 3 lb-wt Ans.
TACTIC 20 4001, 4002, ...... 4030 msL¨v¸wji †f`vsK wbY©q Ki|
S
GENERAL RULES [WRITTEN] 3 in 1 ASPECT SUPER TRICKS [MCQ]
awi, xi = 4001, 4002, ....... 4030 Ges ui = xi – 4000
ui = 1, 2, 3, 4, ......, 30
Avgiv Rvwb, n2 – 1 302 – 1
n –12
†f`vsK = (e¨eavb)2 12 = 12 12
= 74.92 (cÖvq)
†f`vsK g~j n‡Z ¯^vaxb Ges 1g ¯^vfvweK msL¨vi †f`vsK = 12
302 – 1 900 – 1
wb‡Y©q †f`vsK = = = 74.92 (cÖvq)
12 12
TACTIC 21 hw` x2 5x 3 = 0 mgxKi‡Yi g~jØq , nq, Z‡e 1 1 m¤^wjZ mgxKiYwU wbY©q Ki|
S
GENERAL RULES [WRITTEN] 3 in 1 ASPECT SUPER TRICKS [MCQ]
†`Iqv Av‡Q, x 5x 3 = 0
2
f
1 1 2
= 0 5 3 = 0
1
mgxKi‡Yi g~jØq Ges x x x
5 3 1 5x 3x2
+= = 5, = =3 =0
1 1 x2
3x2 + 5x 1 = 0
I g~j؇qi mgwó = 1 + 1 = + = 5 Ges ¸Ydj 1 . 1 = 1 = 1
1 1
1 1
3 3 weKí c×wZ: ax2 + bx + c = 0 mgxKiY g~jØq, , n‡j, ,
wb‡Y©q mgxKiY, x x + = 0
2 5 1
3 3 g~jwewkó mgxKiY n‡e cx2 + bx + a = 0
3x2 + 5x 1 = 0 (Ans.) †hgb- x2 5x 3 = 0 ZvB 3x2 5x + 1 = 0 3x2 + 5x 1 = 0
TACTIC 22 GKwU mgevû wÎfz‡Ri evû·qi mgvšÍiv‡j GKB µ‡g mgwe›`y‡Z Kvh©iZ 6N, 10N, 14N gv‡bi wZbwU †e‡Mi jwäi gvb KZ?
S Solve : = (gv‡bi cv_©K¨) 3
Magical 3 memgq constant _vK‡eB
= (10 – 6) 3 = 4 3
TACTIC 23 (9, 9) I (5, 5) we›`y؇qi ms‡hvRK †iLv‡K e¨vm a‡i AswKZ e„‡Ëi mgxKiY-
2
S 2
A. x + y + 4x + 14y = 0 B. x2 + y2 + 4x 14y = 0
C. x + y 4x + 14y = 0
2 2
D. x2 + y2 4x 14y = 0
n †h Option wU (9, 9) ev (5, 5) we›`y Øviv wm× n‡e †mwUB DËi| GLv‡b B OptionwU(9, 9) ev (5, 5) we›`y Øviv wm× nq| ZvB GwUB DËi|
olve
S B Sol
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14 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
? `„wk‡ãi DrmY
ó AvKl© cov ïiæ Kivi Av‡M Rvb‡Z n‡e fwZ© cÖ‡kœi c¨vUvb©- eyS‡Z n‡e cÖ‡kœi MwZ-cÖK…wZ A_©vr wK ÷vB‡j cÖkœ nq|
†mRb¨ mv¤úªwZK mv‡ji XvKv wek¦we`¨vjq, BwÄwbqvwis ¸”Q wek¦we`¨vjq LywUbvwU Aa¨vqwfwËK QvovI ïiæ‡ZB Zz‡j aiv n‡jv hv‡Z †Zvgiv
mn‡RB aviYv wb‡Z cv‡iv| †Zvgiv hLb †h wek¦we`¨vjq cixÿv w`‡e ïiæ‡ZB †m wek¦we`¨vj‡qi cÖkœ¸‡jv †`‡L wb‡e|
c~Y©gvb: 100
cÖ_g el© ¯œvZK (m¤§vb) †kÖwYi fwZ© cix¶v 2022-2023 mgq: 1.30 wgwbU
cixÿv_©x‡`i cÖwZ wb‡`©kvewj
01. K-BDwbU fwZ© cixÿv `yB As‡k wef³: MCQ Ask I wjwLZ Ask| MCQ As‡ki Dˇii Rb¨ OMR wkU Ges wjwLZ As‡ki Dˇii Rb¨
Avjv`v DËicÎ mieivn Kiv n‡q‡Q|
02. cixÿv_©x wb‡R cÖkc œ ‡Îi †fZi †_‡K DËicÎ (OMR wkU) †ei Ki‡e| OMR wk‡Ui Dcwifv‡M cÖ‡ekcÎ Abyhvqx Bs‡iwR eo nv‡Zi Aÿ‡i
wb‡Ri bvg, wcZv I gvZvi bvg wjL‡Z n‡e Ges ¯^vÿi Ki‡Z n‡e| cixÿv_©x‡K evsjvq †ivj I wmwiqvj b¤^i wj‡L mswkøó e„Ë c~iY Ki‡Z n‡e|
wjwLZ cixÿvi DËi c‡Îi Dc‡ii As‡k wb‡Ri bvg, †ivj b¤^i I wmwiqvj b¤^i ¯úó K‡i wjL‡Z n‡e|
03. MCQ As‡ki cÖkc œ ‡Î cÖ‡Z¨K cÖ‡kœi PviwU DËi †`Iqv Av‡Q| mwVK DËi †e‡Q wb‡q DËic‡Îi (OMR wkU) mswkøó Ni Kv‡jv Kvwji ej‡cb
w`‡q m¤ú~Y©iƒ‡c fivU Ki‡Z n‡e| G As‡ki Dˇii Rb¨ m‡e©v”P 45 wgwbU wba©vwiZ Av‡Q| 45 wgwbU Gi c~‡e© G As‡ki DËi †kl n‡j OMR
wkU Rgv w`‡q wjwLZ As‡ki DËi ïiæ Ki‡Z cvi‡e|
04. MCQ As‡ki †gvU b¤^i 60| cÖwZ wel‡q 15wU K‡i DËi w`‡Z n‡e| cÖwZ cÖ‡kœi b¤^i 1.00| cÖwZwU fzj Dˇii Rb¨ 0.25 b¤^i KvUv hv‡e
Ges Zv welqwfwËK mgš^q Kiv n‡e|
05. GKB cÖ‡kœi Dˇii Rb¨ GKvwaK e„Ë c~iY MÖnY‡hvM¨ n‡e bv|
06. Calculator e¨envi Kiv hv‡e bv| cÖkc œ ‡Îi duvKv RvqMvq cÖ‡qvRb‡ev‡a Calculation Kiv hv‡e|
07. wjwLZ As‡ki †gvU b¤^i 40| wjwLZ As‡ki Dˇii Rb¨ mieivnK…Z DËic‡Îi wba©vwiZ ¯’vb e¨envi Ki‡e|
08. MCQ As‡k †h mKj welq DËi w`‡e †m mKj wel‡q wjwLZ As‡ki DËi cÖ`vb eva¨Zvg~jK|
09. cÖkc
œ Î †diZ †`Iqvi cÖ‡qvRb †bB|
cixÿv_©x‡`i we‡klfv‡e jÿ ivL‡Z n‡e
K) mvaviYfv‡e cixÿv_©x‡`i Physics, Chemistry, Mathematics Ges Biology GB PviwU wel‡qiB MCQ Ges wjwLZ As‡ki DËi w`‡Z n‡e|
Z‡e GBme wel‡qi g‡a¨ Physics I Chemistry eva¨Zvg~jK|
L) Mathematics Ges Biology D”P gva¨wgK A_ev mggvb ch©v‡q Aa¨qb Kiv m‡Ë¡I †KD B”Qv Ki‡j ïaygvÎ PZz_© wel‡qi cwie‡Z© Bangla
A_ev English wel‡q cixÿv w`‡q PviwU welq c~iY Ki‡e|
M) A-Level ch©v‡q Aa¨qbK…Z cixÿv_©x c`v_©weÁvb I imvqbmn Ab¨ (MwYZ/RxeweÁvb/evsjv/Bs‡iwR wel‡qi g‡a¨) †h †Kvb `ywU wel‡q cixÿv
w`‡q PviwU welq c~Y© Ki‡e|
N) PviwUi AwaK wel‡q DËi Ki‡j DËicÎ g~j¨vqb Kiv n‡e bv|
O) cixÿvq †h †Kv‡bv iKg Am`ycvq Aej¤^b ev Aej¤^‡bi †Póv Ki‡j cixÿv_©x‡K ewn®‹vi Kiv n‡e Ges Zvi cixÿv evwZj e‡j MY¨ n‡e|
P) †gvevBj †dvb A_ev †h †Kv‡bv ai‡bi Electronic device wb‡q cixÿvi n‡j cÖ‡ek m¤ú~Y© wbwl× Ges †KD hw` Z_¨ †Mvcb K‡i Gme device
m‡½ iv‡L Zv cixÿvq Am`ycvq Aej¤^b wn‡m‡e MY¨ Kiv n‡e|
AevK mvdj¨
G eQi XvKv wek¦we`¨vjq fwZ© cixÿvq MCQ ‡Z 14wU Ges WRITTEN-G 4wU cÖkœ Aspect Math eB †_‡K mivmwi ev mv`„k¨c~Y©fv‡e Kgb
c‡o‡Q| [we¯ÍvwiZ A‡±vei 2022 GwWkb †_‡K c„ôv b¤^imn Aci c„ôvq cÖgvY †`Iqv n‡jv|]
MATHEMATICS
Per MCQ 1 MCQ PART MARKS
151 = 15
d2y
01. y = x2 lnx n‡j, Gi gvb KZ?
dx2
A. x4 lnx 2x2 3x4 B. 6x4 lnx 5x4 C. 6x4 lnx 2x2 3x4 D. x4 lnx 2x2 + 3x4
mwVK DËi B. 6x4 lnx 5x4
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! AšÍixKiY c„ôv: 242, IU: 04 bs cÖkœ Concept-10 Abyiƒc
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH weMZ 2022-23 mv‡ji cÖkœ e¨vL¨vmn we‡kølY
© Medistry 15
–2 1
mwVK Dˇii c‡ÿ hyw³: y = x lnx ; y1 = – 2x lnx + x x = – 2x lnx + x
–2 –3 –3 –3
y2 = 6x–4lnx + –2x–3 ×
1
+ (– 3)x–4 = 6x–4 lnx – 2x–4 – 3x–4 = 6x– 4 lnx – 5x–4
x
02. x dx x Gi gvb KZ?
e +e
A. tan1 (ex) B. tan(ex) C. tan1 (ex) D. tan(ex)
mwVK DËi C. tan1 (ex)
cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
†cv÷ g‡U©g
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! †hvMRxKiY c„ôv: 270, DU: 03 bs cÖkœ Concept-02 ûeû
x
mwVK Dˇii c‡ÿ hyw³: = = =
dx dx e dx dz
[Let, ex = z exdx = dz] = tan–1 (z) + c = tan–1(ex) + c
ex + e–x e + 1x
x (ex)2 + 1 z2 + 1
e
03. hw` H m‡e©v”P D”PZv Ges R Avbyf‚wgK cvjøv nq, Z‡e GKwU e¯‘‡K f‚wgi mv‡_ 30 †Kv‡Y wb‡ÿc Kiv n‡j wb‡Pi †KvbwU mwVK?
A. R = 3H B. R = 4H C. R = 4 3H D. R = 3 2H
mwVK DËi C. R = 4 3H
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! mgZ‡j Pjgvb e¯‘i MwZ c„ôv: 580, BUET: 01 bs cÖkœ
Concept-07 mv`„k¨c~Y©
4H 4H 1 4H
mwVK Dˇii c‡ÿ hyw³: Avgiv Rvwb, tan = R tan30 = R = R R = 4 3 H
3
04. tan + cot = 2cosec, 0 < n‡j -Gi gvb KZ?
2
5
A. B. C. D.
4 3 6 3
mwVK DËi D.
3
cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
†cv÷ g‡U©g
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! wecixZ w·KvYwgwZK c„ôv: 531, MEx: 03 bs cÖkœ Concept-03 ûeû
sin cos 2 sin + cos
2 2
2 sin2 + cos2 1
mwVK Dˇii c‡ÿ hy w ³: tan + cot = 2cosec + =
cos sin sin
sincos
=
sin
cos
= 2 cos = =
2 3
weKít tan + cot = 2 cosec 1 + tan = 2 cosec 1 + tan2 = 2 × 1 × sin 1 + tan2 = 2 sec2 = 2sec
2
tan sin cos cos
sec = 2 sec = sec =
3 3
lim 2x3 (2k + 1)x2 + 2x + k
05. = 6 n‡j, k-Gi gvb KZ?
x1 x1
1
A. 1 B. 1 C. 3 D.
2
mwVK DËi C. 3
cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
†cv÷ g‡U©g
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! AšÍixKiY (wjwgU) c„ôv: 200, CKRUET: 02 bs cÖkœ Concept-03 Abyiƒc
lim 3 2
2x3 – (2k + 1)x2 + 2x + k x1 {2x – (2k + 1)x + 2x + k}
mwVK Dˇii c‡ÿ hyw³: lim =–6 = – 6 x1
lim
lim {2x3 – (2k + 1)x2 + 2x + k}
x1 x –1 x1 (x – 1)
= – 6 x1 (x – 1) 2 – 2k – 1 + 2 + k = 0 – k = – 3 k = 3
lim
2x3 – (2k + 1)x2 + 2x + k 6x2 –2(2k + 1)x + 2 + 0
weKí: lim = – 6 lim =–6
x1 x –1 x1 1
2
6(1) – (4k + 2) + 2
= – 6 6 – 4k – 2 + 2 = – 6 4k = 12 k = 3
1
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
16 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
06. tan + sec = x n‡j cosec Gi gvb KZ?
x2 + 1 x2 1 1 x2 1 + x2
A. 2 B. 2 C. D.
x 1 x +1 1 + x2 1 x2
x2 + 1
mwVK DËi A. 2
x 1
cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
†cv÷ g‡U©g
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! mshy³ w·KvYwgwZ c„ôv: 161, RU: 11 bs cÖkœ Concept-03 ûeû
2
1 + sin (1 + sin) 1 + sin 1 + sin + 1 – sin x2 + 1
mwVK Dˇii c‡ÿ hyw³: tan + sec = x cos = x
cos
2 = x2
1 – sin
= x2 = 2
1 + sin – 1 + sin x – 1
2 x2 + 1 x2 + 1
= 2 cosec = 2
2 sin x – 1 x –1
2x + 1 1
07. hw` f(x) = nq, Z‡e f (x) Gi †Kv‡Wv‡gb †KvbwU?
x3
A. ℝ B. (3, ) C. (, 3) D. ℝ {3}
mwVK DËi D. ℝ {3}
cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
†cv÷ g‡U©g
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!!
dvskb I dvsk‡bi †jLwPÎ c„ôv: 411, CU: 12 bs cÖkœ Concept-03 Abyiƒc
mwVK Dˇii c‡ÿ hyw³: f(x) Gi †Wv‡gb Ges f –1(x) Gi †Kv‡Wv‡gb GKB|
2x + 1
f(x) = ; f(x) ev¯Íe n‡Z n‡j, x – 3 0 x 3 f(x) Gi †Wv‡gb = ℝ – {3} = f –1(x) Gi †Kv‡Wv‡gb
x–3
08. P = a i 2j + k Ges Q = 2a i aj 4k ci¯úi j¤^ n‡j, a-Gi gvb KZ?
A. 1, 2 B. 1, 2 C. 1, 2 D. 1, 2
mwVK DËi C. 1, 2
cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
†cv÷ g‡U©g
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! †f±i c„ôv: 331, JU: 05 bs cÖkœ Concept-02 ûeû
mwVK Dˇii c‡ÿ hyw³: P . Q = 0 a.2a + (– 2) (– a) + 1(– 4) = 0 2a2 + 2a – 4 = 0 a2 + a – 2 = 0
a + 2a – a – 2 = 0 a(a + 2) – 1 (a + 2) = 0 (a – 1) (a + 2) = 0 a = 1, – 2
2
09. (0, 2) Ges (2, 0) we›`yMvgx mij‡iLv x-A‡ÿi abvZ¥K w`‡Ki mv‡_ Kx †KvY Drcbœ K‡i?
A. 30 B. 45 C. 60 D. 120
mwVK DËi B. 45
cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
†cv÷ g‡U©g
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! mij‡iLv c„ôv: 91, Concept Test-03 Concept-07 Abyiƒc
0–2
mwVK Dˇii c‡ÿ hyw³: m = tan = –2 – 0 = 1 = 45
10. y-A‡ÿi mgvšÍivj Ges 2x 7y + 11 = 0 I x + 3y 8 = 0 †iLv؇qi †Q`we›`y w`‡q AwZµgKvix mij‡iLvi mgxKiY wb‡Pi †KvbwU?
A. 13x 23 = 0 B. 3x 7 = 0 C. 7x 3 = 0 D. 23x 13 = 0
mwVK DËi A. 13x 23 = 0
cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
†cv÷ g‡U©g
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! mij‡iLv c„ôv: 106, DU: 01 bs cÖkœ Concept-07 ûeû
mwVK Dˇii c‡ÿ hyw³: 2x – 7y = – 11......... (i); x + 3y = 8 ................. (ii)
27
{(i) – (ii) × 2} K‡i cvB, –13y = – 11 – 16 y =
13
–
(ii) n‡Z cvB, x = 8 – 3 × =
27 104 81 23
=
13 13 13
23
†Q`we›`y = ; y A‡ÿi mgvšÍivj †iLv, x = a x = 13x – 23 = 0
23 27
13 13 13
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH weMZ 2022-23 mv‡ji cÖkœ e¨vL¨vmn we‡kølY
© Medistry 17
11. 7 Rb wm‡bUi I 5 Rb Mfb©‡ii GKwU `j †_‡K KZ Dcv‡q 4 Rb wm‡bUi I 3 Rb Mfb©‡ii GKwU KwgwU MVb Kiv hvq?
A. 350 B. 10 C. 35 D. 30
mwVK DËi A. 350
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! web¨vm I mgv‡ek c„ôv: 376, DU: 01 bs cÖkœ Concept-04 Abyiƒc
7.6.5.4 5.4.3
mwVK Dˇii c‡ÿ hyw³: 7C4 × 5C3 = ×
4.3.2.1 3.2.1
= 350
12. hw` 240(72x) = 1 nq, Z‡e x Gi gvb KZ?
A. 4 B. 3 C. 5 D. 2
mwVK DËi D. 2
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! wm‡jevm ewnf‚©Z - - -
mwVK Dˇii c‡ÿ hyw³: 2401(7 ) = 1 7 = 2401 7 = 7 2x = 4 x = 2
–2x 2x 2x 4
1
13. > 4 AmgZvwUi mgvavb †mU n‡e wb‡Pi †KvbwU?
|x + 2|
A. , x 2 B. C. , x 2 D.
9 7 7 1 9 1 7 1
4 4 4 4 4 4 4 4
mwVK DËi A. , x 2
9 7
4 4
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! ev¯Íe msL¨v c„ôv: 607, DU: 04 bs cÖkœ Concept-07 Abyiƒc
mwVK Dˇii c‡ÿ hyw³: [we:`ª: cÖ‡kœ mwVK Dˇii †Kvb Ackb †bB wKš‘ KvQvKvwQ A‡_© Ackb A †bIqv n‡jv]
1 1 1 1 1 1 1
> 4; x + 2 0 x – 2; GLb, > 4 |x + 2| < – < x + 2 < – – 2 < x + 2 – 2 < – 2
|x + 2| |x + 2| 4 4 4 4 4
9 7 9 7 9 7
– < x < – – – , x – 2 A_©vr, – , – , x–2
4 4 4 4 4 4
1
14. y = 1 + eµ‡iLv x-Aÿ‡K A we›`y‡Z Ges y-Aÿ‡K B we›`y‡Z †Q` Ki‡j AB mij‡iLvi mgxKiY wb‡Pi †KvbwU?
2+x
A. x + 2y + 3 = 0 B. x + 2y 3 = 0 C. x 2y + 3 = 0 D. x 2y 3 = 0
mwVK DËi C. x 2y + 3 = 0
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! mij‡iLv c„ôv: 105, MEx: 08 bs cÖkœ Concept-07 ûeû
mwVK Dˇii c‡ÿ hyw³: y = 1 + 1
2+x
1 y A‡ÿ x = 0
x A‡ÿ y = 0 0 = 1 + 3
B (0,2)
B 0
2+x 1 3 3
y=1+ =
1
= – 1 2 + x = – 1 x = – 3 A (– 3, 0) 2+0 2 2 (–3,0) A
2+x (0, 0)
x y
wb‡Y©q mgxKiY: + = 1 – x + 2y = 3 x – 2y + 3 = 0
–3 3
2
15. cosec 10 4sin 70 Gi gvb KZ?
1
A. 1 B. C. 2 D. 2
2
mwVK DËi D. 2
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Concept †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! mshy³ w·KvYwgwZ c„ôv: 152, CU: 10 bs cÖkœ Concept-03 mv`„k¨c~Y©
mwVK Dˇii c‡ÿ hyw³: cosec 10 – 4 sin 70
1 – 2 – cos80 1 – 2 . + 2 cos80
1 1
=
1
– 4 sin70 =
1 – 4 sin70 sin10 1 – 2(2 sin70 sin10) 1 – 2(cos60 – cos80)
= = =
2 = 2
=2
sin10 sin10 sin 10 sin10 sin10 cos80
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
18 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
MATHEMATICS WRITTEN PART MARKS
42.5 = 10
1
01. 3x2 6x + 2 = 0 mgxKi‡Yi g~jØq m Ges n n‡j m + Ges n + 1 g~j wewkó mgxKiYwU wbY©q Ki|
n m
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Topic †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! eûc`x 465 Concept-04 Abyiƒc
–6
m + + n + = m + n +
2 1 1 m+n 2
mgvavb: 3x m+n=–
2
– 6x + 2 = 0; g~jØq m, n = 2 Ges mn = =2+2=2+3=5
3 3 n m mn
3
m + n + = mn + 1 + 1 +
1 1 1 2 1 2 3 4 + 12 + 9 25
= +2+2= +2+ = =
n m mn 3 3 2 6 6
3
25
wb‡Y©q mgxKiY: x – (g~j؇qi mgwó) x + (g~j؇qi ¸Ydj) = 0 x2 – 5x +
2
= 0 6x2 – 30x + 25 = 0
6
02. p Gi †Kvb gv‡bi Rb¨ (4, 4) we›`ywU x2 8x + py + 7 = 0 cive„‡Ëi Dc‡K›`ª n‡e?
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Topic †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! KwYK 480 Concept-02 mv`„k¨c~Y©
mgvavb: x – 8x + py + 7 = 0 x – 8x + 16 = – py – 7 + 16
2 2
–p
(x – 4)2 = – py + 9 = – p y – (x – 4)2 = 4. y – X2 = 4. a. Y
9 9
p 4 p
Dc‡K›`ª: X = 0 Y=a
x–4=0 9 p 9 p 36 – p2
y– =– y= – =
x=4 p 4 p 4 4p
36 – p2
cÖkœg‡Z, = 4 36 – p = 16p p + 16p – 36 = 0 p2 + 18p – 2p – 36 = 0
2 2
4p
p(p + 18) – 2 (p + 18) = 0 (p + 18) (p – 2) = 0 p = 2, – 18 (Ans.)
2
03. y = cosx ln n‡j, d y2 + y Gi gvb wbY©q Ki|
1
sec x + tan x dx
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Topic †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! AšÍixKiY 238 Concept-10 -
y = cosx ln = cosx ln = cosx ln cosx
1 1
mgvavb: secx + tanx 1 + sinx
1 + sinx
cosx cosx
1 + sinx –sinx(1+ sinx)–cosx(0 + cosx)
= – sinx ln
dy cos x
+ cosx. .
dx 1 + sinx cosx (1 + sinx)2
2 2
cos x – sinx – (sin x + cos x) cos x – (1 + sinx)
= – sinx ln = – sinx ln = – sinx ln
cos x
+ + –1
1 + sinx 1 + sinx 1 + sinx 1+ sinx 1 + sinx
d2y 1 + sinx (– sinx) (1 + sinx) – cosx(0 + cosx)
= – cosx ln
cos x
– sinx .
dx2 1 + sinx cosx (1 + sinx)2
sinx –sinx – (sin2x + cos2x) d2y
=–y– . = – y + tanx 2 + y = tanx
cosx 1 + sinx dx
+ + a2 b2
04. hw` acos + bsin = acos + bsin nq, Z‡e †`LvI †h, cos2 sin2 = 2 .
2 2 a + b2
†cv÷ g‡U©g cÖkœwU †h Aa¨vq †_‡K cÖkœwU ASPECT MATH Gi †h cÖkœwU ASPECT MATH Gi †h Topic †_‡K †hfv‡e Kgb
Kiv n‡q‡Q c„ôv †_‡K Kgb c‡o‡Q Kgb c‡o‡Q c‡o‡Q
inm¨!!! w·KvYwgwZ 165 Concept-04 Abyiƒc
+ –
2 sin sin
cos – cos b 2 2 b
mgvavb: acos + b sin = a cos + b sin a cos – acos = bsin – b sin =
sin – sin a +
=
– a
2 cos sin
2 2
+ + + + +
sin sin2 cos2 cos2 + sin2
2 b 2 b2 2 a2 2 2 a2 + b2 + + a2 – b2
= = 2 = 2 = 2 2 cos2 – sin2 = 2 2 (Showed)
+ a + a + b + + a –b 2 2 a +b
cos cos2 sin2 cos2 – sin2
2 2 2 2 2
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ASPECT MATH weMZ 2022-23 mv‡ji cÖkœ e¨vL¨vmn we‡kølY
© Medistry 19
SB Why GKB mij‡iLvq Aew¯’Z n‡j †h †Kvb `ywU we›`y wb‡q MwVZ
¸”Q fwZ© cixÿv Xvj mgvb|
weÁvb ¯œvZK cÖ_g el© mgwš^Z
BDwbU-A
a1 13
fwZ© cixÿv- 2022-23
†mU-04
=
A(–1, 3) B(–2, 1)
C(a, a)
a+2 2+1
01. f‚wg †_‡K k~‡b¨ wbwÿß GKwU ej 100 wgUvi `~‡i f‚wg‡Z wd‡i Av‡m| a1 2
=
75 a + 2 1
†mUvi wePiYc‡_i me©vwaK D”PZv 4 wgUvi n‡j wb‡ÿcY †KvY KZ?
a 1 = 2a + 4
1 4 1 4 a = 5 we›`ywU (5, 5)
A. tan B. cos
3 5
06. 4y 3x + 12 = 0 Ges 4y 3x + 3 = 0 †iLv؇qi ga¨eZ©x `~iZ¡ KZ GKK?
C. sin1 D. sin1
5 3
3 4 A.
9
B.
12
5 5
75
4 3 6
4H 4
S B Why tan = R tan = 100 C.
5
D.
5
c1 c 2
3 3 4 SA Why d =
tan = = tan1 = cos1 a 2 b2
4 4 5
02. †h KwY‡Ki c¨vivwgwZK mgxKiY x = 3 + at2, y = 2at †mUvi kxl©we›`yi ¯’vbv¼- 12 3 9
= =
4 3
A. (0, 0) B. (2, 0) 2 2 5
C. (3, 0) D. (2, 3)
y 2 07. x Gi ‡Kvb gv‡bi Rb¨ y = x ln x Gi jNy gvb wbY©q Kiv hv‡e?
S C Wh x = 3 + at y = 2at
x3 2 y A. e B. e
= t ------------ (i) t = --------- (ii)
a 2a 1 1
C. D.
x 3 y2 e e
(i) n‡Z cvB, =
a 4a SC Why f(x) = xlnx
y2 = 4(x 3) kxl© = (3, 0) 1 1
f(x) = x. + lnx = 1 + lnx ; f(x) =
x x
03. lim x2
2 3 5 6
+ 3 + 2 + 2 Gi gvb KZ? m‡e©v”P ev me©wb¤œ gv‡bi Rb¨ f(x) = 0
x x4
+ 1 x + 7 x + 1 x 6
A. 8 B. 10 1 + lnx = 0
C. 11 D. 16 lnx = 1
lim 1
x2
2 3 5 6 x = e1 x =
SC Why + + +
x x4 +1 x3 + 7 x2 + 1 x2 5 e
x = n‡j f = = e > 0
lim 1 1 1
= 2
+
3
+
5
+
6 = 0+0 + 5 + 6 = 11 e e 1
x 5
x + 12
2 1
x+ 2
1
1+ 2 1 e
x x x x2
1
x2 y2 x= g~j dvsk‡b emv‡j, me©wb¤œ gvb cvIqv hvq|
04. + = 1 Dce„‡Ëi wbqvgK †iLv؇qi ga¨eZ©x `~iZ¡ KZ GKK? e
30 14
A. 7 B. 14 08. 2x2 + y2 8x 2y + 1 = 0 Dce„ËwUi †K‡›`ªi ¯’vbv¼ †KvbwU?
C. 15 D. 30 A. (2, 1) B. ( 2, 1)
x2
y2
C. (1, 2) D. (1, 2)
S C Why 30 + 14 = 1 e = 1 14 = 16 = 4 y 2 2
30 30 30 S Wh 2x + y – 8x – 2y + 1 = 0
A
x2 42 2x2 – 8x = – (y2 – 2y + 1) = – (y – 1)2
+ =1
( 30)2 ( 14)2 2 (x2 – 4x + 4) + (y – 1)2 = 8
2a 2 30 30 (x – 2)2 (y – 1)2
wbqvg‡Ki `~iZ¡ = = = = 15 + =1
e 4 2 4 8
30 †K›`ª = (2, 1)
05. A ( 1, 3) Ges B ( 2, 1) we›`yMvgx mij‡iLvi Dcwiw¯’Z P (a, a) x Gi mnM y Gi mnM
Aspect Special: †K›`ª =
(–2) x Gi mnM (–2) y Gi mnM
2 2
we›`yi ¯’vbv¼ †KvbwU?
B. ( 5, 5) –8 –2
=
A. (5, 5)
(2, 1)
C. (4, 4) D. ( 4, 4) (–2) 2 (–2) 1
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20 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
09. x2 8x + 4y 4 = 0 KwYKwUi w`Kv‡ÿi cv`we›`yi ¯’vbv¼-
14. (x 3)2 + (y 2)2 = 25 e„‡Ëi GKwU R¨v †K‡›`ª ‡KvY •Zwi K‡i|
A. (4, 6) B. (4, 6) 2
C. ( 4, 6) D. (6, 4) R¨vwUi •`N©¨ KZ GKK?
S A Why x2 + 8x + 4y – 4 = 0 Y 2
X = 4aY 5 3
A. 5 3 B.
x – 8x + 16 = – 4y + 4 + 16
2
6
X
(x – 4)2 = – 4 (y – 5) = 4(–1) (y – 5) C. 5 2 D. 7 3
w`Kvÿ (0, – a)
wbqvg‡Ki cv`we›`y: y – 5 = – (– 1) 1 R¨v
SC Why = 2sin
y=6 x–4=0 2
x=4 R¨v
= 2sin1
wbqvg‡Ki cv`we›`yi ¯’vbv¼, (x, y) = (4, 6) 2 25
R¨v 1 R¨v 10
2
sin4 = = = R¨v
10 2 10
10. 3 sin 3x ecos3x dx = ? 2
0 R¨v = 5 2 GKK|
A. 3e B. 1 e
15. sin + sin + Gi gvb KZ?
5
C. e 1 D. 3e 1 6 6
2 A. 1 B. 0
SC Why 3 sin 3x ecos3x dx Let, z = cos3x
0 dz = 3sin3xdx C. cos D. 3 sin
S B Why sin 6 + sin + 6
x /2 0 5
0
e
1
1
z
dz ez 0 e1 e0 e 1 z 0 1
= sin + sin +
11. a > 1 n‡j
d
(ln ax) = ? 6 6
dx
ax = sin sin = 0
A.
ln a
B. ln a 6 6
C. ax D. x ln a 16. A Ges (AT + B)C g¨vwUª· `yBwUi µg h_vµ‡g 4 5 Ges 5 2 n‡j C
d 1 d x g¨vwUª· Gi µg Kx n‡e?
SB Why (ln ax) = x . (a )
dx a dx A. 4 2 B. 4 3
1 C. 4 4 D. 4 5
= x . ax . ln a = ln a
a
S A Wh y (AT
+ B)C Gi µg 5 2
dx
12. ey = tan1 x n‡j =? Avevi, A Gi µg 4 5
dy
(AT + B) Gi µg 5 4
A. 1 + x2 tan1 x B. (1 + x2) tan1 x
C n‡e 4 2
C. 1 x2 tan1 D. (1 x2) tan1x
1 x2) = sin cos1
S B Why ey = tan1 x 1
17. hw` tan (sin1 nq Zvn‡j x = ?
y = ln(tan1 x) 5
dy 1 1 5 5
= A. B.
dx tan1 x 1 + x2 3 3
dx
= (1 + x2) tan1x C.
5
D.
5
dy 3 3
13. y = x x2 + x3 x4 + ........... n‡j x = ? SA Why tan(sin
1
1 x2) = sincos1
1 5
y y 5 2
A. B.
1y 1
tantan1
1+y 1 x2
= sinsin1
2
C.
y
D.
y x 5 1
1x2
1+y y1
1 x2 2
tan tan1
x
S A Why y = x x2 x2 + x3 x4 + --------- x
=
5
y = x + x2 x3 + x4 + ----------
1 y = 1 x + x2 x3 + x4 ---------- 1 x2 2
=
x
1 y = (1 + x)1 5
1 1x 4
2
1=x = 5 5x2 = 4x2
1y x2 5
y 5
x= 9x2 = 5 x=
1y 3
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ASPECT SERIES
ASPECT MATH weMZ 2022-23 mv‡ji cÖkœ e¨vL¨vmn we‡kølY
© Medistry 21
Why 2cos x + 3cosx = 2
2
18. r = 8 cos + 6 sin KwYK Øviv x- A‡ÿi LwÛZ As‡ki •`N©¨ KZ GKK? SA
A. 8 B. 6 2cos2x + 4cox cosx 2 = 0
C. 4 D. 3 2cosx (cosx+2) 1(cosx+2) = 0
(cosx+2) (2cosx 1) = 0
S A Why r = 8cos + 6sin
nq, cosx + 2 = 0 A_ev, 2cosx 1 = 0 cosx = 1
r2 = 8r cos + 6r sin cosx = 2; hv Am¤¢e 2
x2 + y2 = 8x + 8y
cosx = cos x = 2n
x2 + y2 8x 8y = 0 3 3
( g, f) = (4, 4) GLb, n = 0 n‡j x = ,
3 3
x A‡ÿi LwÐZvsk = 2 g2 c = 2 16 = 8 GKK| 7
n = 1 n‡j x = Ges 5
3 3
19. (3 3 3i) ( 3 3 + 9i) Gi gWzjvm = ?
, 5
x= [∵ 0 < < 2]
A. 54 3 B. 27 3 3 3
C. 36 3 D. 45 3 23. k Gi †Kvb gv‡bi Rb¨ nPr = k (n+1Cr nCr1) n‡e?
SC Why (3 3 3i) ( 3 3 + 9i) Gi gWzjvm A. r B. (r 1)!
C. r! D. r 1
= ( (3 3)2 + 32)( (3 3)2 + 92) [∵|z1z2| = |z1| |z2| ] S C Why nPr = k(n+1Cr nCr1)
= 6 6 3 = 36 3 nCrr! = knCr [∵ nCr+nCr–1
= n +1Cr Ges nPr = nCrr!] k = r!
20. 2 cosec sin1
1
1
dx = ?
0 x 24. f(x) = log (x x2 4) Gi †Wv‡gb †KvbwU?
A. [ 2, 2] B. ( , 2]
A. 2.5 B. 21.0
C. [4, ) D. [2, )
C. 1.5 D. 1.0 y
S D Wh y = log a x
S D Why 2 cosecsin1 xdx [a > 0; a 1; x > 0]
1 1
0 GLb, f(x) = log (x – x2 – 4) Avevi, x2 – 4 0
1 GLb, x – x2 – 4 > 0 (x + 2) (x – 2) 0
= 2 cosec cosec1 x dx x > x2 – 4
0
(x + 2) (x – 2) 0
2 1
– –
1
= 2 xdx = 2
x
=10=1
0 2 0 x> x –42
–
21. Gi †Kvb †Wv‡g‡bi Rb¨ x2 + ax + 3 = 0 Gi g~jØq ev¯Íe I Amgvb n‡e? †Wv‡gb = [2,
25. wZbwU mgZjxq ej P, Q Ges R †Kv‡bv we›`y‡Z wµqv K‡i mvg¨ve¯’vq
A. (–2 3, 2 3) B. (–, – 2 3)
Av‡Q| hw` P Ges Q Gi gvb h_vµ‡g 5 3N I 5N Ges Zv‡`i ga¨eZ©x
C. (– , –2 3) (2 3, ) D. (2 3, )
SC Why ev¯Íe I Amgvb g~j n‡j, D > 0 †KvY 2 nq Zvn‡j R, Q Gi m‡½ KZ †KvY •Zwi Ki‡e?
a2 – 4.1.(3) > 0
A. B.
4 3
a – (2 3) > 0
2 2
2 3
(a + 2 3) (a – 2 3) > 0 C.
3
D.
4
y
S C Wh awi, R I Q Gi ga¨eZ©x Q=5N
– 3 3
†KvY Ges GLv‡b P I Q Gi jwä ej R
†Wv‡gb = (– , – 2 3) (2 3, ) wn‡m‡e R ejwU KvR Ki‡Q|
22. 2 cos2x + 3cosx = 2, 0 < < 2 Gi mgvavb †mU-
5 3 sin90
tan = = 3 30
P=5 3N
5 5 + 5 3 cos90 60
A. , B. ,
3 3 3 = R
3
2 5 2
C. , D. , = 90+30=120 =
3 3 2 3 3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
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22 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
4
SA Why e =
KUET, CUET, RUET MwYZ
5
COMBINED ADMISSION a b2
TEST 2022-2023
c~Y©gvb-150 ae = 9 e2 = 1 2
e a
a2 b 2
a e = 9
2 2
1 a e =
01. y2 kx + 8 = 0, (k 0) cive„‡Ëi wbqvg‡Ki mgxKiY x 1 = 0 n‡j
e b = a (1 e2)
2 2
a =9 = (20)2 1 = 144
5 4 16
Ges cive„ËwU x2 + y2 = 4 e„ˇK `ywU Avjv`v ev¯Íe we›`y‡Z †Q` Ki‡j, k 4 5 25
Gi gvb KZ n‡e? a = 20 cm b = 12cm
A. 4 B. 8 C. 4 e„nr Aÿ 20cm I ÿz`ª Aÿ 12cm
D. 8 E. 2
04. sec1 (x) = cos1 + sin1
1 5
2 mgxKiYwUi mgvavb †KvbwU?
S B Why y2 kx + 8 = 0 Y
3 3
y2 = kx 8 18 18 6+3
k 8 A. B. C.
y = 4. x
2 O (2,0) 3 6 6 3 18
4 k
X (1,0) X
3 18
D. E.
kxl© = 8
k 0 x=1
5 15 + 6
S E Why sec (x) = cos1 2 + sin1
Y 1 1 5
8 k 3 3
GLb, k 1 = 4
cos1 = cos1 + cos1
1 1 2
k = 4 n‡j, kxl© = (2, 0) x 2 3 3
8 k
1=0 3 3 5
k 4 k = 8 n‡j, kxl© = ( 1, 0)
cos1
1 2
32 k2 4k = 0 kxl© (1, 0) n‡j cive„ Ë wU e„ Ë wU‡K x 1 5
k2 + 4k 32 = 0 sin
= cos1
`ywU we›`y‡Z †Q` K‡i| Avevi, kxl© 1 2 1 1 1 2
3 3
(k + 8) (k 4) = 0
(2, 0) n‡j cive„ËwU e„ËwU‡K ¯úk© 2 3 3 4 27 2
k = 8, 4 = cos1
K‡i| 1 2 5 1 25 3
= cos = cos 3 3
k=8 6 3 6 6 3
6 15
02. x2 8y2 = 2 Awae„‡Ëi w`Kv‡ÿi mgxKiY 3x = 4 n‡j Gi Dc‡Kw›`ªK cos1 = cos1
1
j‡¤^i •`N©¨ KZ n‡e? x 18
1 15 + 6
1 1 1 =
A. B. C. x 18
2 3 3 3 2
18
1 1 x=
D. E. 15 + 6
2 3 2 2 1
2 2 05. cos (x ) sin1 (x) = sin1 (1 x) mgxKiYwUi mgvavb †KvbwU?
x y
S E Why x2 8y2 = 2 2 1 = 1 1 1 17
A. 0, B. C. 0, 2
4 2 2
D. 1, 2 E. 0,1
x2 y2 1
2 2 =1 S A Wh y cos (x) sin1(x) = sin1 (1 x)
( 2) 1
2 sin1 ( 1 x2) sin1 (x) = sin1 (1 x)
a= 2;b=
1
2
sin1 { }
1 x2 1 x2 x 1 ( 1 x2)2 = sin1 (1 x)
1 x2 x2 = 1 x
12
2 2x2 x = 0
2b2 2
Dc‡Kw›`ªK j‡¤^i •`N©¨ = a = 1
2 x(2x 1) = 0 x = 0,
2
1 06. †Kvb GKwU we›`y ‡ Z F I 3F gv‡bi ej `ywU wµqviZ| cÖ_gwU‡K Pvi¸Y
= GKK
2 2 Ki‡j Ges wØZxqwUi gvb AviI 18 GKK e„w× Ki‡j jwäi w`K
03. †Kvb Dce„‡Ëi GKwU Dc‡K›`ª I Zvi wbKUZg wbqvg‡Ki ga¨eZ©x `~iZ¡ AcwiewZ©Z _v‡K| F Gi gvb KZ?
4 A. 2 B. 4 C. 1
9cm Ges Dr‡Kw›`ªKZv n‡j Zvi e„nr Aÿ I ÿz`ª Aÿ-Gi ‣`N©¨ KZ
5 D. 8 E. 3
n‡e? F 4F
A. 20cm and 12cm B. 30cm and 24cm S A Why 3F = 3F + 18
C. 15cm and 12cm D. 40cm and 24cm 3F + 18 = 12F
E. 400cm and 144cm 9F = 18 F = 2 GKK
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ASPECT MATH weMZ 2022-23 mv‡ji cÖkœ e¨vL¨vmn we‡kølY
© Medistry 23
07. 16m •`‡N©¨i GKwU mylg exg AB Gi IRb 60kg hvi A I B cÖv‡šÍ 10. hw` A = [aij] GKwU eM© g¨vwUª· nq, †hLv‡b aij = 2i j ; i, j = 1, 2,
33
h_vµ‡g 20kg I 45kg IRb Szjv‡bv| A cÖvšÍ †_‡K KZ `~i‡Z¡ ïaygvÎ 3. Zvn‡j A g¨vwUª·wU GKwU-
GKwU Aej¤^b ¯’vcb Ki‡j e¨e¯’vwU myw¯’wZ _vK‡e? A. Involutory matrix B. Idempotent matrix
240 48 48 C. Nilpotent matrix D. Singular matrix
A. m B. m C. m E. Orthogonal matrix
7 7 25
48 36 S D Why A = [aij]3 3
D. m E. m aij = 2i j
5 5 a11 a12 a13
S D Wh y A C B A = a 21 a 22 a 23 a11 = 1 a12 = 0
AB = 16m a 31 a 32 a 33 a 13 = 1 a21 = 3
x (8x) 1 0 1 a = 2 a23 =1
8 R 22
= 3 2 1 a31 = 5 a32 = 4
45kg-wt
5 4 3 a 33 = 3
20kg-wt
1 0 1
60kg-wt
MC = 0 |A| = 3 2 1 = 1 (6 4) (12 10) = 2 2 = 0
(20 x) 60 (8 x) 45 (16 x) = 0 5 4 3
48 A g¨vwUª·wU GKwU e¨wZµgx (Signature) g¨vwUª·|
x= m
5
2 3 x
08. `ywU †b․Kv 5km/hr †e‡M P‡j 3 km/hr †e‡M cÖevwnZ 500m PIov GKwU 11. 0 4 x wbY©vq‡Ki (2, 1)th fzw³i mn¸YK 9 n‡j x Gi gvb †KvbwU?
b`x cvwo w`‡Z Pvq| GKwU †b․Kv b~¨bZg c‡_ I AciwU b~¨bZg mg‡q 1 3 1 x
1 3
b`xwU cvwo w`‡Z B”QzK| Dfq †b․Kv GKB mg‡q hvÎv ïiæ Ki‡j Zv‡`i A. B. C. 2
2 2
Aci cv‡o †cu․Qv‡bvi mg‡qi cv_©K¨ KZ n‡e?
D. 0 E. 2
A. 1 minute B. 1.25 minutes C. 1.5 minutes
3 x
2
D. 1.75 minutes E. 2 minutes S C Wh 0 4 x
y
S C Why 1 3 1 x
(2, 1) th fzw³i mn¸YK =
d 3 x
b~¨bZg c‡_, t1 = 2 2 3 1 x = 9
v u
0.5 0.5 (3 3x 3x) = 9 6x 3 = 9 x = 2
= = hr 12. x = y Ges x + y = 1 †iLv `ywUi AšÍf©y³ †KvY¸wji mgwØLÐK¸wji
5 2 32 4
mgxKiY †KvbwU?
b~b¨Zg mg‡q, A. x + 1 = 0 and y + 1 = 0 B. 2x + 1 = 0 and 2y + 1 = 0
d 0.5 C. 2x 1 = 0 and 2y 1 = 0 D. x = 1 and y = 1
t2 = tmin = = = 0.1 hr
v 5 E. x = 0 and y = 0
mg‡qi cv_©K¨ = t1 t2 = 4 0.1 = 4 60 = 1.5 minutes
0.5 0.1 S C Why x y = 0...... (i)
x + y 1 = 0......... (ii)
09. GKRb UªvwdK AvBb Agvb¨Kvix PvjK 2m/sec2 Z¡i‡Y Mvwo Pvjbv ïiæ xy x+y1
Ki‡j UnjZ UªvwdK cywjk 5 sec ci Zv‡K avIqv ïiæ Kij| cywj‡ki Mvox mgxKiY `ywUi mgwØLÛK¸wji mgxKiY, =
2 2
20 m/sec mg‡e‡M Pj‡j, KZ mgq ci †mwU AvBb Agvb¨Kvix Pvj‡Ki (+) wb‡q, x y = x + y 1 2y 1 = 0
Mvox‡K AwZµg Ki‡Z cvi‡e? () wb‡q, x y = x y + 1 2x 1 = 0
A. 3 sec B. 4 sec C. 5 sec 2
D. 6 sec E. 7 sec 13. OP †iLvsk‡K Nwoi KuvUvi w`‡K †Kv‡Y Nyiv‡bv‡Z Zvi bZzb Ae¯’vb
3
S C Why n‡jv OQ| P Gi ¯’vbv¼ ( 3 3) n‡j P I Q Gi ga¨eZ©x `~iZ¡
u = 0 + 2 5 = 10ms1 KZ n‡e?
cywj‡ki V = 20ms1 a = 2ms2 A. 4 3 B. 12 C. 6
Mvwo UªvK
D. 2 3 E. 6
3
S A Why tan = = 3
1 s1
s=0+ 2 52
2
= 25 3
Y
s2 = ( 333)
3
awi, cywj‡ki Mvwo t mgq ci UªvKwU‡K ai‡Z cvi‡e| P
cÖkœg‡Z, s2 = 25 + s1 P ( 33)
+3
2
PQ = ( 3 3 ) + ( 3 3)2 X
20 t = 25 + (10 t) + 2 t2
1 X
2
O
= 12 + 36 +3
t2 10t + 25 = 0 (t 5)2 = 0 =4 3 Q
( 3 3)
t5=0 t=5s Y
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
24 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
5
14. 0 x e¨ewa‡Z 2sinx + cos2x dvsk‡bi m‡e©v”P I me©wb¤œ gv‡bi S E Why tan =
2 12
Rb¨ bx‡Pi †Kvb DËiwU mwVK? †h‡nZz cos FYvZ¥K wKš‘ tan abvZ¥K
3 myZivs sin I FYvZ¥K n‡e
A. At x = , there is a minimum which is
6 2 5 12
3 sin = Ges cos = 13
13
B. At x = , there is a maximum which is
6 2 5 12
+
1+ 3 sin + cos ( ) sin + cos 13 13 51
C. Al x = , there is a maximum which is cÖ` Ë ivwk = = = =
6 2 sec ( ) + tan sec + tan 13 5 26
+
12 12
D. At x = , there is a minimum which is 3
x x 2 Gi gvb KZ n‡e?
6 2 x
17. lim
1 x2 x 4
E. At x = , there is a minimum which is
6 2 1 + ln2 1 ln2 2 ln2
S B Why f(x) = 2 sinx + cos2x A. B. C.
1 ln2 1 + ln2 2 + ln2
For maximum and minimum value, f(x) = 0 2 + ln2 ln2 1
2cosx 2 sin2x = 0 2 cosx = 2.2 sinx.cosx D.
2 ln2
E.
ln2 + 1
1 5 2x
S B Why lim xx 4 ; 0 form
2
2 sinx = 1 sinx = x = , x 0
2 6 6
x2
x = ∵ 0 x f (x) = 2 sinx 4 cos 2x 2x 2x ln2
6 2 = lim x x ; [Using L Hospital Rule]
x2 x + x lnx
f = 2 sin 4cos 2 2.2 22 ln2 4 (1 ln2) 1 ln2
6 6 6 = 2 = =
2 + 22 ln2 4 (1 + ln2) 1 + ln2
1
= 2 4 =12=3<0
1 1
2 2 18. x3 + siny = x3 mgxKi‡Y x = 1 Gi Rb¨
dy
Gi gvb KZ n‡e?
dx
At, x = , f(x) has a maximum value 8 2
6 A. B. C. 0
3 3
1 1 3
and that is = f
6 = 2 sin 6 + cos 2 6 = 2 2 + 2 = 2 D.
3
E.
3
15. †K›`ª ( 3, 2) I 2 e¨vmva© wewkó e„‡Ëi †h Rb¨ ( 4, 3) we›`y‡Z 2 2
1
mgwØLwÐZ n‡q‡Q Zvi mgvšÍivj ¯úk©‡Ki mgxKiY †KvbwU? SA Why x3 + siny = x
3
A. x + y + 5 2 = 0 B. x + y 5 2 = 0 x = 1 n‡j, 1 + siny = 1 siny = 0 y = 0
C. x + y 5 2 2 = 0 D. x + y + 5 + 2 2 = 0 1 2
1 dy
E. x + y + 7 2 2 = 0 GLb, x3 + siny = x3 3 x 3 + cosy dx = 3x2
. 1 + 1. = 3 [∵ x = 1; y = 0]
C (3, 2) 1 dy dy 1 8
=3 =
S D Why E
D ( 4, 3)
F 3 dx dx 3 3
1
A B 19. hw` y = Acos3x + B sin3x + x sin3x nq, Zvn‡j bx‡Pi †Kvb DËiwU mwVK?
2
2+3
mCD = =1 A. y2 + 9y1 = 3cos3x B. y2 9y1 = 3cos3x
3+4 3
R¨v Gi Xvj, MEF = 1 [∵ ci¯úi `ywU j¤^ †iLvi Xv‡ji ¸Ydj = 1] C. y2 + 9y = cos3x
2
D. y2 + 9y = 3cos3x
¯úk©K Gi Xvj, MAB = 1 [∵ ¯úk©K R¨v Gi mgvšÍivj] E. y2 + 9y = 3sin3x
1
¯úk©K Gi mgxKiY, y = mx + c S D Why y = A cos3x + B sin3x + 2 x sin3x
y=x+c x+yc=0 1
3 2 c c+5 y1 = 3A sin3x + 3B cos3x + [3x cos 3x + sin3x]
GLb, =2 =2 2
1+1 2
= 3A + sin 3x + 3B + cos 3x
1 3x
c+5=2 2 c=52 2 2 2
wb‡Y©q ¯úk©K Gi mgxKiY : x + y ( 5 2 2) = 0
y2 = 3A + 3 cos 3x + 3B + 3 ( sin3x) + cos 3x
1 3x 3
x+y+52 2=0 2 2 2
= cos 3x 9A + + 9 B + sin 3x
5 sin + cos) 3 3 x
16. hw` tan =
12
Ges cos FYvZ¥K nq, Zvn‡j
sec ( ) + tan
Gi gvb 2 2 2
KZ n‡e? 1
= 9A cos 3x + 3 cos 3x 9 B sin 3x 9. .x sin 3x
14 21 2
A. 313 B. C. 1
39 26 = 9 [A cos 3x + B sin 3x + x sin 3x] + 3 cos 3x
14 51 2
D.
39
E.
26 y2 = 9y + 3 cos 3x y2 + 9y = 3 cos 3x
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH weMZ 2022-23 mv‡ji cÖkœ e¨vL¨vmn we‡kølY
© Medistry 25
x2 1 1
20. x = 1 Gi Rb¨ dx Gi gvb n‡j, †hvMRxKiY aªæeK c Gi
Area = 2 (y1 y2) dx = 2
1 + x6 12 ( 1 x2 1 x) dx
0 0
gvb KZ? 3 1
x 1 x 1 1
2
2
=2 + sin x + (1 x)2
A. B. 0 C.
1 2 2 3 0
12 3
1 2 2
= 2 sin1 (1) = 2 . = 2 eM© GKK
1 2
D. E. 1 2 3 2 2 3 4 3
12
2 2 3
x 6 dx = 1 3x dx3 2 = 1 d(x )3 2 23. 2x + i (x2 1) Gi eM©g~j †KvbwU?
SB Why
1+x 3 1 + (x ) 3 1 + (x ) 1 1
A. {(x + 1) + i (x 1)} B. {(x 1) + i (x + 1)}
1 1 3 2 2
= tan (x ) + c 1
3 C. 2 {(x + 1) + i(x 1)} D. {(x + 1)u + i(x 1)}
1 3
GLb, x = 1 n‡j, 3 tan1 (1) + c = 12 1
E. {(x 1) + i (x + 1)}
1 3
. +c=
34 12 S A Why z = 2x + i (x2 1)
1
+c= c=0 z= { |z| + Re(z) + i |z| Re (z)}
12 12 2
0 (2x)2 + (x2 1)2 = (x2 1)2 + 4(x2)
|z| =
21. tan + x dx Gi gvb KZ? = (x2 + 1)2 = x2 + 1
4 |z| + Re (z) = x + 1 + 2x = (x + 1)2
2
4
Ges |z| Re (z) = x2 + 1 2x = (x 1)2
B. ln C. ln
1 1 1 1
A. 2 ln (2) 1
2 2 4 2 z = { (x + 1)2 + i (x 1)2}
2
1
D. ln (2) E. None 1
2 = {(x + 1) + i (x 1)}
2
0
24. 27x2 + 6x (P + 2) = 0 mgxKi‡Yi GKwU g~j AciwUi e‡M©i mgvb
S E Why tan 4 + x dx
n‡j, P Gi gvbmg~n KZ?
4 A. 1 or 6 B. 1 or 6 C. 1 or 6
0 ut, D. 1 or 6 E. 6 or 6
1 + tanx
= dx
(cos x sinx) = z S A Wh y 27x2 + 6x (P + 2) = 0 ; & 2
1 tanx (sinx + cosx) dx = dx 6 2
4 + 2 = = 9 + 92 = 2
27 3
0 x 0
cosx + sinx 4
92 + 9 + 2 = 0 = ,
1 2
= dx 3 3
cosx sinx z 2 1
4 Ges . 2 = 3 = 27
(P + 2)
1
=
1 1 1 1 1 (P + 2)
dz = [lnz] 2 = ln2 GLb, = 3 n‡j, 27 = 27
2 z 2
22. x2 + y2 = 1 Ges y2 = 1 x eµ‡iLv `ywU Øviv Ave× †ÿ‡Îi †ÿÎdj P+2=1 P=1
KZ? 2 8 (P + 2)
Avevi, = 3 n‡j, 27 = 27
2 2 2
A. B. + C. 2 P + 2 = 8 P = 6 P = 1, 6
4 3 4 3 4 3
1 2 1
D. 4 E. 2 + 25. †Kvb wØNvZ mgxKi‡Yi GKwU g~j
2 +i
n‡j Dnvi mgxKiY †KvbwU n‡e?
2 3 4 3
A. 2x2 4x + 5 = 0 B. 5x2 4x + 1 = 0
S C Why y2 = 1 x = (x 1) Y
C. 4x 5x + 1 = 0
2
D. 5x2 + 4x 1 = 0
x2 + 1 x = 1 E. 5x 7x + 2 = 0
2
x (x 1) = 0 X X 1 2i
x = 0, 1
O
x2 + y2 = 1
S B Why x = 2 + i = 4 + 1
y = 1, 0 y2 = 1 x 5x = 2 i (5x 2)2 = ( i)2
Y 25x2 20x + 4 = i2 = 1 5x2 4x + 1 = 0
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
26 cvV¨eB‡K mnR Kivi cÖqvm
© Medistry Avm‡c± wmwiR
MwY‡Zi cÖ‡qvRbxq m~Îvejx 1g cÎ
cÖ_g Aa¨vq: g¨vwUª· I wbY©vqK
AbyeÜx g¨vwU· Ges wecixZ g¨vwUª·
Adjoint g¨vwU·: †Kvb eM© g¨vwUª· A Gi wbY©vqK |A| Gi mn¸YK Øviv MwVZ g¨vwUª‡·i (fzw³¸‡jvi µg Abymv‡i) Transpose g¨vwUª·‡K cÖ`Ë g¨vwUª· A Gi
Adjoint matrix ejv nq| GwU‡K m~wPZ Kiv nq Adj (A)
A = c
a b
d g¨vwUª· Gi AbyeÜx g¨vwUª· ev adjoint of A/Adj (A)
= cÖvBgvwi K‡Y©i Dcv`vb¸‡jvi ¯’vb cwieZ©b Ges †m‡KÛvwi K‡Y©i Dcv`vb¸‡jvi wPý cwieZ©b Ki‡j hv cvIqv hvq ZvB Adjoint
–b
Adjoint of A = –c
d
a
A g¨vwUª‡·i wecixZ ev Inverse g¨vwUª· =
1 1 d –b
ad – bc –c a
Adj (A) =
Det (A)
Note: mKj g¨vwUª‡·i wecixZ g¨vwUª· _v‡K bv| Inverse g¨vwUª· _vKvi kZ© `ywU:
(i) g¨vwUª·wU Aek¨B eM© g¨vwUª· n‡Z n‡e| (ii) g¨vwUª·wUi wbY©vq‡Ki gvb k~b¨ nIqv hv‡e bv|
wecixZ g¨vwUª‡·i •ewkó¨: (i) (A1)1 = A (ii) (AB)1 (iii) (AT)1 = (A1)T (iv) (BA) A1 = B (AA1) = B. (v) I1 = In
(vi) AB = C n‡j A = CB1 Ges B = A1 C.
d b
a 0 0 1/a 0 0
a b 1
ad bc c a Gi wecixZ g¨vwUª· 0 1/b 0
[MCQ Gi Rb¨: c d Gi wecix‡Z g¨vwUª· ; 0 b 0
0 0 c 0 0 1/c
wbY©vq‡Ki gvb wbY©q
wbY©vq‡Ki gvb: †Kvb wbY©vq‡Ki †h †Kvb mvwi ev Kjv‡gi Gi Dcv`vbmg~n I Zv‡`i wbR wbR mnivwki ¸Yd‡ji mgwóB wbY©vq‡Ki gvb|
a1 b1 c1
a2 b2 c2 wbY©vq‡Ki a1, a2, a3 Gi mn¸YK h_vµ‡g A1, A2, A3 n‡j wbY©vq‡Ki gvb n‡e
a3 b 3 c3
= a1A1 + a2A2 + a3A3 = a1
b2 c2 b1 c1 b1 c1
b3 c3+ a2 – b3 c3+ a3b2 c2
Abyivwk I mn¸YK wbY©q msµvšÍ
Abyivwk:
FORMULA Step-01: †h ivwk ev msL¨vi Abyivwk †ei Ki‡Z ej‡e wVK †mB ivwk eivei Row Ges Column ev` `vI|
Step-02: evwK Dcv`vb ¸‡jv w`‡q wbY©vqK MVb Ki| †mwUB Abyivwk|
Step-03: gvb ‡ei Ki‡Z ej‡j mvaviY wbq‡g wbY©vq‡Ki gvb †ei Ki‡Z n‡e|
a1 b1 c1 Magic!!!
mn¸YK: a2 b2 c2 Gi b3 Gi mn¸YK KZ?
a3 b3 c3 mn¸YK = (–1) mvwi + Kjvg Abyivwk
Finix Tecnique: mn¸YK = wPý Abyivwk
Step-1: Abyivwk †ei Kivi c×wZ Aej¤^b K‡i cÖ_‡g Abyivwk †ei Ki|
Step-2: Abyivwk mvg‡b (– 1) R + C m~Î e¨envi K‡i h_vh_ wPý emvI| †mwUB mn¸YK |
Shortcut Soln 1 1
3 2 a c1 a1 c1
a2 c2 = –a2 c2
a1 a2 a3
Shortcut Tricks. b1 b2 b3 mn¸YK wbY©q Gi e· Gi wfZi wPwýZ Dcv`vb ¸‡jvi mvg‡b (+) Ges evwK Dcv`vb ¸‡jvi mvg‡b (–) em‡e|
c1 c2 c3
Z…Zxq Aa¨vq: mij‡iLv
x A‡¶i mgvšÍivj ev y-A‡ÿi Dci j¤^ †iLvi mgxKiY, y = b y A‡¶i mgvšÍivj ev x-A‡ÿi Dci j¤^ †iLvi mgxKiY, x = a
x A‡ÿi mgxKib: y = 0 Y A‡ÿi mgxKib: x = 0
y Aÿ n‡Z †Qw`Z Ask abvZ¥K n‡j †iLvwU DaŸ©Mvgx Ges FYvZ¥K n‡j †iLvwU wb¤œMvgx ‡KvwU؇qi AšÍi
‡Kvb †iLvi Xvj m = f~R؇qi AšÍi = tan
x-Gi mnM y Gi mnM
Kvb mij‡iLv ax + by + c = 0 Gi Xvj, m = ax + by + c = 0 †iLvi j¤^ †iLvi Xvj =
y-Gi mnM x Gi mnM
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH GKb‡Ri cÖ‡qvRbxq m~Îvejx
© Medistry 27
†KvwU؇qi AšÍi a1x + b1y + c1 = 0 I a2x + b2y + c2 = 0 †iLv؇qi †Q`we›`y
`yBwU wbw`©ó we›`y (x1, y1) I (x2, y2) Mvgx mij‡iLvi Xvj = fzR؇qi AšÍi b1c2 – c1b2 c1a2 – c2a1
=
a1b2 – a2b1 a1b2 – a2b1
†h †Kvb cÖKvi `yiZ¡ memgqB abvZ¥K n‡e, Z‡e `yiZ¡ w`‡q †Kvb ARvbv gvb †ei Ki‡Z ej‡j †m‡¶‡Î `yiZ¡ `y‡UvB n‡e
MvwYwZK mgm¨vmg~n mgvav‡bi Rb¨ cÖ‡qvRbxq m~Î I cÖwµqvmg~n
(i) r = x2 + y2 (iii) x = r cos
P(r, ) Gi Rb¨ †cvjvi n‡Z Kv‡Z©mxq
P(x, y) Gi Rb¨ Kv‡Z© m xq n‡Z †cvjvi (iv) y = r sin
(ii) = tan–1
y
x
P(x,y) we›`ywUi PviwU PZzf©v‡M Ae¯’v‡bi Dci wfwË K‡i AvM©~‡g‡›Ui gvb () wbY©‡qi mvaviY wbqg
P(x, 0) Gi †¶‡Î: = 0 P(x, y) Gi †¶‡Î: = tan
–1 y
P(– x, y) Gi †¶‡Î: = – tan
–1 y
x x
P(0, y) Gi †¶‡Î: = –1y –1y
2 P( x, y) Gi †¶‡Î: = + tan = – + tan
P(–x, 0) Gi †¶‡Î: = x x
–1y –1y
P(x, y) Gi †¶‡Î: = 2 – tan or – tan
P(0, –y) Gi †¶‡Î: = –
2 x x
`~iZ¡ m~Î
Kv‡Z©mxq ¯’vbv‡¼ P(x1, y1) I Q(x2, y2) `ywU we›`y n‡j D³ we›`y؇qi ga¨eZx© `~iZ¡, PQ = x1 x 2 2 y1 y 2 2 GKK
†cvjvi ¯’vbv‡¼ P(r1, 1) I Q(r2, 2) `ywU we›`y n‡j D³ we›`y؇qi ga¨eZ©x `~iZ¡, PQ r12 r2 2 2r1r2 cos(1 2 ) GKK
we›`y P(x1,y1) †_‡K †iLv ax+by+c=0 Gi j¤^ `~iZ¡ D = 1 2 1 2
ax +by +c
j¤^ `~iZ¡
a +b
`yBwU mgvšÍivj †iLvi ga¨eZ©x c –c
GKwU †iLv ax+by+c1=0 †_‡K Aci GKwU †iLv ax+by+c2=0 Gi ga¨eZx© `~iZ¡, D = 12 2 2
`~iZ¡ a +b
j¤^ AvK…wZi mij‡iLvi Av`k© xcos + ysin = p †hLv‡b, p = g~jwe›`y n‡Z mij‡iLvi Dci Aw¼Z j¤^ `~iZ¡ = x A‡ÿi abvZ¥K w`‡Ki mv‡_
mgxKiY Drcbœ †KvY|
wefw³KiY m~Î
A(x1, y1) Ges B(x2, y2) we›`y؇qi ms‡hvRK mij‡iLv‡K C(x, y) we›`ywU m1:m2 Abycv‡Z AšÍwe©f³ Ki‡j,
AšÍwe©fw³ KiY m x m 2 x1 m1 y 2 m 2 y1
x 1 2 Ges y
m1 m 2 m1 m 2
A(x1, y1) Ges B (x2, y2) we›`y؇qi ms‡hvRK mij‡iLv‡K C(x, y) we›`ywU m 1 : m 2 Abycv‡Z ewnwe©f³ Ki‡j,
ewnwe©fw³ KiY m1 x 2 m 2 x 1 m1 y 2 m 2 y1
x Ges y
m1 m 2 m1 m 2
x x 2 y1 y 2
ga¨we›`y wbY©q A(x1, y1) Ges B (x2, y2) we›`y؇qi ms‡hvRK mij‡iLvi ga¨we›`y : 1 ,
2 2
y x
AÿØq‡K wef³ Ki‡j (x1, y1) I (x2, y2) we›`y؇qi ms‡hvM †iLv‡K x Aÿ‡K: 1 , y Aÿ‡K: 1 Abycv‡Z wef³ K‡i
y2 x2
(x1, y1) , (x2, y2) , (x3 , y3) kxl© wewkó wÎfz‡Ri fi‡K‡›`ªi ¯’vbv¼ (x, y) n‡j,
fi‡K›`ª msµvšÍ x x x y y y
x= 1 2 3 ,y= 1 2 3
3 3
†¶Îdj wbY©q msµvšÍ
wÎfy‡Ri wZbwU kxl© we›`yi ¯’vbv¼ A(x1, y1), B (x2, y2), C(x3, y3) †`Iqv _vK‡j, wÎfy‡Ri †¶Îdj
wÎfy‡Ri †¶Îdj 1
= {(x1y2 + x2y3 + x3y1) – (x2y1 + x3y2 + x1y3)} eM© GKK
2
eM©/mvgvšÍwi‡Ki †¶Îdj GKwU eM© ev mvgšÍwiK‡K `yBwU me©mg wÎfy‡R wef³ Kiv hvq †¶Îdj = 2 ABC
a1 b1 c1
mg‡iL/mgwe›`y/GKB a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, a3x + b3y + c3 = 0 ‡iLvÎq mgwe›`y n‡j, a2 b2 c2 = 0
mgZ‡j/GKB mij‡iLv Aew¯’Z a3 b3 c3
n‡j y 2 y3
y1 y 2
wZbwU (x1, y1) (x2, y2), (x3, y3) we›`y mg‡iL nIqvi kZ© n‡”Q we›`y·qi Xvj mgvb n‡e, x -x = x - x
1 2 2 3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
28 cvV¨eB‡K mnR Kivi cÖqvm
© Medistry Avm‡c± wmwiR
mij‡iLvi mgxKiY
GKwU mij‡iLvi Xvj m Ges y A‡ÿi †Q`K Ask c n‡j Zvi mgxKiY n‡e y = mx + c
‡bvU: c = 0 n‡j, y = mx, hv g~jwe›`y w`‡q AwεgKvix †iLvi mvaviY mgxKiY wb‡`©k K‡i|
GKwU ‡iLvi Xvj m Ges †iLvwU (x1 , y1) we›`yMvgx n‡j, †iLvwUi mgxKiY n‡e y y1 = (x x1)
x x1 y y1
(x1, y1) Ges (x2, y2) we›`yMvgx †iLvi mgxKiY x =
1 x2 y1 y2
xy
†Q`K AvKvi: x Aÿ Ges y Aÿ n‡Z h_vµ‡g a Ges b Ask †Q`Kvix †iLvi mgxKiY =1
ab
x y
+ = 1 ‡iLvvwU x Aÿ‡K A (a, 0) we›`y‡Z Ges y Aÿ‡K B (0 , b) we›`y‡Z †Q` K‡i|
a b
1 1
AB = OA + OB = a + b Ges OAB = |OA OB| eM© GKK |ab| eM© GKK|
2 2 2 2
2 2
AÿØq Øviv + = 1 †iLvwUi †Q` As‡ki ga¨we›`yi ¯’vbv¼ .
x y a b
a b 2 2
j¤^ AvKvi mgxKiY: g~jwe›`y n‡Z ‡Kv‡bv mij‡iLvi Dci Aw¼Z j‡¤^i •`N¨© p Ges j¤^wU x A‡ÿi
abvZ¥K w`‡Ki mv‡_ †KvY Drcbœ Ki‡j, †iLvwUi mgxKiY n‡e x cos + y sin = P.
†iLv؇qi ga¨eZ©x †Kv‡Yi gvb wbY©q
m –m a b –a b
`yBwU †iLvi a1x+b1y+c1=0 I a2x+b2y+c2=0 Gi ga¨eZx© †KvY n‡j, tan = 1+m 1 2
=
2 1 1 2
1m2 a1a2+b1b2
Note: mij‡iLvi Xvj tan ¯’yj‡KvY n‡j (–)ve Ges m~²‡KvY n‡j (+)ve
we›`yi mv‡c‡¶ we›`yi cÖwZwe¤^
mv‡c¶ we›`ywU cÖ`Ë we›`y I cÖwZwe¤^ we›`yi ga¨we›`y nq|
x x2 y y2
†hgb- (x1, y1) we›`yi mv‡c‡¶ (x2, y2) we›`yi cÖwZwe¤^ (x,y) n‡j,= x1 Ges = y1
2 2
A‡ÿi mv‡c‡ÿ cÖwZwe¤^ (i) x A‡ÿi mv‡c‡ÿ (x, –y) (ii) y A‡ÿi mv‡c‡ÿ (–x, y)
ci¯úi mgvšÍvivj Ges j¤^ mij‡iLv ؇qi Xvj
`yBwU mij‡iLv a1x + b1y + c1 = 0 Ges a2x + b2y + c2 = 0 ci¯úi mgvšÍivj n‡j Zv‡`i XvjØq mgvb n‡e A_©vr m1= m2 A_ev †iLv؇qi mgvšÍivj
a b
n‡j a1 = b1 Ges ci¯úi j¤^ n‡j Xvj؇qi ¸bdj = –1 A_©vr m1 × m2 = –1 A_ev a1a2 + b1b2 = 0
2 2
a1x + b1y + c1= 0 Ges a2x + b2y + c2 = 0 †iLv؇qi AšÍf©~³ (i) a1a2 + b1b2 > 0 n‡j, (+) wPý wb‡q ¯’~j‡KvY Ges () wPý wb‡q
†KvY mg~‡ni mgwØLÛK †iLvmgy‡ni mgxKiY, m~²‡Kv‡Yi mgwØLÛ‡Ki mgxKiY cvIqv hv‡e|
a 1 x b 1 y c1 a 2 x b2 y c2 (ii) a1a2 + b1b2 < 0 n‡j, (+) wPý wb‡q m~²‡KvY Ges () wPý wb‡q
a 1 b1
2 2
a 2 b2
2 2
¯’~j‡Kv‡Yi mgwØLÛ‡Ki mgxKiY cvIqv hv‡e
MCQ Gi Rb¨ we‡kl m~Î/‡K․kj
A
F(x3, y3) E(x2, y2)
D(x3, y3) C(x, y)
B C
D(x1, y1)
A = (x2 + x3 x1, y2 + y3 y1)
B = (x1 + x3 x2,y1 + y3 y2)
C = (x1 + x2 x3,y1 + y2 y3) A(x1, y1) B(x2, y2)
A(x1, y1)
ABCD: mvgvšÍwi‡Ki PZz_© kx‡l©i ¯’vbv¼|
F E (x, y) = (x2 + x3 x1, y2 + y3 y1)
B(x2, y2) D C(x3, y3)
AD ga¨gvi mgxKiY,
(2y1 y2 y3)x (2x1 x2 x3)y
= (2y1 y2 y3)x1 (2x1 x2 x3)y1
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH GKb‡Ri cÖ‡qvRbxq m~Îvejx
© Medistry 29
c
ax + by + c = 0 Øviv x- A‡ÿi †Q`vsk =
a
y A‡ÿi †Q`vsk = c/b; Aÿ؇qi ga¨eZ©x LwÛZ
2
2 2 c
Ask = (c/a) + (c/b) ; Aÿ؇qi Øviv MwVZ wÎfz‡Ri †ÿÎdj =
2|ab|
x y
GKwU †iLvi Aÿ؇qi ga¨eZ©x Ask (, ) we›`y‡Z mgwØLwÛZ n‡j Zvi mgxKiY, + =1
2 2
x y a
g~jwe›`y n‡Z †Kvb †iLvi Dci Aw¼Z j¤^ x A‡ÿi abvZ¥K w`‡Ki mv‡_ †KvY Drcbœ Ki‡j Zvi mgxKiY: a + b = 1; †hLv‡b tan = b
a1x + b1y + c1 = 0 .........(i)
a2x + b2y + c2 = 0 .........(ii)
a3x + b3y + c3 = 0 .........(i) †iLv wZbwU Øviv MwVZ wÎfz‡Ri †ÿÎdj
{c (a b3 – a3b2) – c2(a1b3 – a3b1) + c3(a1b2 – a2b1)}2
= 1 22|(a
2b3 – a3b2)(a1b3 – a3b1)(a1b2 – a2b1)|
a1x + b1y + c1 a1b3 – a3b1 a1x + b1y + c1 a1a3 – b1b3
(1) I (2) †iLvi †Q`we›`yMvgx Ges (3) Gi mgvßivj I j¤^ Giƒc †iLvi mgxKiY h_vµ‡g = ; =
a2x + b2y + c2 a3b3 – a3b2 a2x + b2y + c2 a2a3 – b2b2
a1y + b1y + c1 = 0 I a2x + b2y + c2 = 0
|c a 2 + b 2 – c a 2 + b 2|
mgvšÍivj †iLv؇qi ga¨eZ©x `~iZ¡ = 1 2 2 2 2 2 2 1 2 1
a1 + b 1 a2 + b 1
f(x) ax + by + c = 0 †iLv g(x) a1x + b1y + c1 = 0 I AB ‡iLv؇qi AšÍf~©³ †KvY¸‡jvi GKwU mgwØLÛK n‡j AB Gi mgxKiY (a2 + b2)
g(x) – 2(aa1 + bb1) f(x) = 0
A(x1,y1), B(x2,y2) we›`y؇qi ms‡hvM †iLvsk‡K ax + by + c = 0 mij‡iLvwU |ax1 + by1 + c| : |ax2 + by2 + c| Abycv‡Z wef³ K‡i|
†h †Kv‡bv wÎfz‡Ri j¤^‡K›`ª, fi‡K›`ª Ges cwi‡K›`ª h_vµ‡g A, B, C n‡j A, B, C mg‡iL Ges AB : BC = 2 : 1
PZz_© Aa¨vq: e„Ë
MvwYwZK mgm¨vmg~n mgvav‡bi Rb¨ cÖ‡qvRbxq m~Î I cÖwµqvmg~n
x2 + y2 + 2gx + 2fy + c = 0 e„‡Ëi Rb¨
(i ) †K›`ª (–g, –f) (vi) x-A¶‡K ¯úk© Ki‡j, c = g2
(ix) y-A¶ †_‡K †Qw`Z R¨vÕi •`N©¨ = 2 f 2 c (ii) e¨vmva© = g 2 f 2 c
(vii) y-A¶‡K ¯úk© Ki‡j, c = f2 (x) x, y Dfq A¶‡K ¯úk© Ki‡j, c = g2 = f2
(iii) †K›`ª y-A‡¶i Dci n‡j g = 0 (viii) x-A¶ †_‡K †Qw`Z R¨vÕi •`N©¨ = 2 g 2 c
(x) Area = r2 (iv) †K›`ª x-A‡¶i Dci n‡j f = 0
†K›`ª (h, k) wewkó GKwU e„Ë x ev y A¶‡K ¯úk© Ki‡j Zvi mgxKiY
i. e¨vmva© = |e„‡Ëi †K‡›`ªi †KvwU| = k
x A¶‡K ¯úk© Ki‡j
ii. mgxKiY, (x – h)2 + (y – k)2 = k2; †hLv‡b, k = e¨vmva
i. e¨vmva© = |e„‡Ëi †K‡›`ªi fyR| = h
y A¶‡K ¯úk© Ki‡j
ii. mgxKiY, (x – h)2 + (y – k)2 = h2; †hLv‡b, h = e¨vmva©
wbw`©ó †K›`ª Ges Aci †Kvb we›`yMvgx I †Kvb e„‡Ëi mv‡_ GK‡Kw›`ªK Ges †Kvb wbw`©ó we›`yMvgx e„‡Ëi mgxKiY wbY©q
wbw`©ó †K›`ª Ges Aci †Kvb Step-1: e„Ë †Kvb wbw`©ó we›`yMvgx n‡j †K›`ª †_‡K D³ we›`yi `yiZ¡ wb‡Y©q Ki‡Z n‡e hv e„‡Ëi e¨vmva© mgvb|
we›`yMvgx e„‡Ëi mgxKiY wbY©q Step-2: †K›`ª I e¨vmva© †_‡K mgxKiY wbY©‡qi m~Î cÖ‡qvM Ki‡Z n‡e|
†Kvb e„‡Ëi mv‡_ GK‡Kw›`ªK Step-1: cÖ`Ë e„‡Ëi †K›`ª wb‡Y©q Ki‡Z n‡e, H †K›`ªB n‡e wb‡Y©q e„‡Ëi †K›`ª|
Ges †Kvb wbw`©ó we›`yMvgx e„‡Ëi Step-2: D³ †K›`ª Ges cÖ`Ë we›`yi `yiZ¡ wbY©q Ki‡Z n‡e, D³ `yiZ¡B n‡e wb‡Y©q e„‡Ëi e¨vmva©|
mgxKiY Step-3: †K›`ª I e¨vmva© †_‡K mgxKiY wbY©‡qi m~Î cÖ‡qvM Ki‡Z n‡e|
GKwU mij‡iLv GKwU e„ˇK ¯ck© Kivi kZ©
i. e„‡Ëi e¨vm¨va© = †K›`ª †_‡K †iLvi Dci AswKZ j¤^`~iZ¡
ax + by + c = 0 mij‡iLv x2 +y2 + 2gx + 2fy + c = 0 e„ˇK ¯ck© Kivi kZ©:
ag bf c
g2 f 2 c
a 2 b2
ii. y = mx + c †iLvwU x2 + y2 = a2 e„‡Ëi ¯úk©K n‡j, c = a 1 m 2
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
30 cvV¨eB‡K mnR Kivi cÖqvm
© Medistry Avm‡c± wmwiR
¯ck©‡Ki •`N¨©
a2
2 2
i. (x1, y1) we›`y n‡Z x2 + y2 = a2 e„‡Ë AswKZ x1 y1
ii. (x1, y1) we›`y n‡Z x2 + y2 + 2gx + 2fy + c = 0 e„‡Ë AswKZ ¯ck©‡Ki •`N¨©,
x1 y1 2gx 1 2fy 1 c
2 2
PT =
iii. g~jwe›`y (0, 0) n‡Z †Kvb e„‡Ëi ¯ck©‡Ki •`N¨© = aªæeK
e„‡Ëi mv‡c‡¶ †Kvb we›`yi Ae¯’vb wbY©q
we›`yi Ae¯’vb (x1, y1) we›`ywU x2 + y2 + 2gx + 2fy + c = 0 e„‡Ëi (x1, y1) we›`ywU x2 + y2 = a2 e„‡Ëi
evB‡i n‡j x12 y12 2gx1 2fy1 c > 0 x12 + y12 – a2 Gi gvb > 0
e„‡Ëi (cwiwai) Dci n‡j x12 y12 2gx1 2fy1 c =0 x12 + y12 – a2 = 0
wfZ‡i n‡j x12 y12 2gx1 2fy1 c < 0 x12 + y12 – a2 < 0
R¨v Gi •`N©¨
e„‡Ëi R¨v KZ…©K †K‡›`ª Drcbœ †KvY, = 2 sin–1 R¨v Gi ‣`N¨© = 2 r 2 d 2
e¨vm
mßg Aa¨vq: mshy³ I †hŠwMK †Kv‡Yi w·KvYwgwZK AbycvZ
MvwYwZK mgm¨vmg~n mgvav‡bi Rb¨ cÖ‡qvRbxq m~Î I cÖwµqvmg~n:
sin/cos Gi †h․wMK †Kv‡Yi m~Î
C D sin C D
sin (A + B) = sinA cosB + cosA sinB sinC sinD = 2cos sin2A = 2sinA cosA
2 2
– sinA
cos(A B) = cosA cosB + CD CD
cosC + cosD = 2cos cos cos2A = cos2A sin2A
sinB 2 2
CD DC cos2A = 1 2 sin2A ; 2 sin2A = 1
2sinA cosB = sin(A+B) + sin(AB) cosCcosD = 2sin sin
2 2 cos2A
2cosA sinB = sin(A+B) sin (AB) sin (A + B) sin(AB) = sin2A sin2B = cos2B cos2Acos2A = 2cos2A 1; 2cos2A = 1 + cos2A
2 tan A
2cosA cosB = cos(A+B) + cos(AB) cos(A + B)cos(A B) = cos2A sin2B =cos2B sin2A sin2A =
1 tan 2 A
1 1 tan 2 A
2sinA sinB = cos (AB) cos (A+B) sin3A = 3sinA4sin3A sin3A= (3sinAsin3A) cos2A =
4 1 tan 2 A
cos3A= 4cos A 3cosA cos A
3 3
sinC + sinD = 2sin C D cos C D 1
2 2 = (cos3A+3cosA)
4
tan/cot Gi †h․wMK †Kv‡Yi m~Î
tanA tanB 2 tan A 3 tan A tan 3 A
tan (A + B) = tan2A =
1+– tanA tanB 1 tan A
2 tan3A =
1 3 tan 2 A
3
cotA cotB +–1 cot A – 3cotA 1 cos 2A
cot3A = tan2A =
cot (A + B) = 3cot2A – 1
cotB cotA 1 cos 2A
wÎfz‡Ri m~Î mg~n
a b c
wÎfz‡Ri sine Gi m~Î = = = 2R [R = cwie¨vmva©]
sinA sinB sinC
wÎfz‡Ri cosine Gi m~Î b2 c2 a 2 c2 a 2 b2 a 2 b2 c2
1. cosA = 2. cosB = 3. cosC =
2bc 2ac 2ab
1
wÎfz‡Ri †¶Îdj evû؇qi ¸Ydj evû؇qi AšÍf~©³ †Kv‡Yi sin
2
wÎfz‡Ri Aa© cwimxgv
abc abc
GKwU wÎfz‡Ri Aa© cwimxgv = †¶Îdj = s(s a) (s b) (s c) wÎfz‡Ri cwie¨vmva R = wÎfz‡Ri AšÍte¨vmva© r
2 4 s
Awf‡¶c- m~Î i) a = b cosC + c cosB ii) b = c cosA + a cosC iii)c = a cosB + b cosC
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH GKb‡Ri cÖ‡qvRbxq m~Îvejx
© Medistry 31
sin2 I cos2 Gi aviv _vK‡j gvb wbY©q
†kl c` 1g c`
avivi mgwói †ÿ‡Î sin2 Ges cos2 Gi Rb¨ Av‡M c` msL¨v †ei Ki‡Z n‡e| c` msL¨v = e¨eavb +1
n n1
(i) c` msL¨v †Rvo n‡j Answer = 2 (ii) c` msL¨v we‡Rvo n‡j Answer = + (sin20 /sin290 /cos20 /cos2 90)
2
Note:
(i) c` msL¨v we‡Rvo n‡j sin20 /sin290 /cos20 /cos2 90 GB PviwU c‡`i †h †Kv‡bv GKwU c` cÖ‡kœ _vK‡e Ges †hwU _vK‡e Zvi gvb †hvM Ki‡Z n‡e|
1
(ii) c` msL¨v we‡Rvo n‡j Ges sin20 /sin290 /cos20 /cos2 90 GB PviwU c‡`i †Kv‡bvwU bv _vK‡j †hvM Ki‡Z n‡e|
2
[A_©vr c`msL¨v we‡Rvo n‡j ga¨c` Avjv`v K‡i wb‡q Gi mv‡_ evwK As‡ki mgwó ‡hvM Ki‡Z n‡e]
beg Aa¨vq: AšÍixKiY
La Hospital’ Rule
x Qvov Ab¨ wKQz †hgb x0, x2, x BZ¨vw` n‡j ïiæ‡ZB limit ewm‡q w`‡Z n‡e †hB gvb cvIqv hv‡e ZvB Ans. Example : xlim
0
(secx)x = ? Ans: 1
0
x Gi cwie‡Z© limit ewm‡q mvaviYZ ev, GB AvK…wZ Av‡m| Gme †¶‡Î La Hospital’s Rule Apply Ki‡Z n‡e|
0
0
La Hospital’s Rule: je I ni‡K x Gi mv‡c‡¶ Differentiate Ki‡Z n‡e| AZtci x Gi gvb ewm‡q ev, AvK…wZ Remove n‡j †h gvb cvIqv hvq ZvB Ans.
0
0
Note: gvb ewm‡q ev AvK…wZ Remove bv nIqv ch©šÍ je I ni‡K Differentiate Ki‡Z n‡e|
0
lim =
x lim(x)
lim(x y) = lim (x) lim (y)
lim (xy) = lim(x) lim (y)
y lim(y)
n n
lim f(x) = limf(x) lim(cx) = c.lim(x) lim(c) = c
lim sin lim = lim sin k = k lim = lim tan = 1
= lim
0
=1
0 sin 0 sin k 0 0 tan 0
–1 –1
lim –1 = lim sin = 1 lim –1 = lim tan = 1 lim = lim tan k = k
0 sin 0 0 tan 0 0 tan k 0
sin
lim =1
2
lim sin 2
lim sinn
k
0
= =kn
0 0 n
x n
lim e – 1 =1 lim ln(1 + x) =1 lim (1 + x) – 1 = n
0 x 0 x 0 x
lim 1 + 1 = e lim 1 + m = emn lim 1 + m = emn
x x nx
0 x 0 x 0 x
n n x
lim x – a = nan – 1 lim a – 1 = ln a lim f(x + h) – f(x) = d f(x) = dy = f(x)
0 x – a 0 x 0 h dx dx
ASPECT EXCLUSIVE: Awfbe †UKwbK (SHORTCUT SOLUTION):
ASPECT SHORTCUT TRICKS & TIPS MODEL EXAMPLE
lim sin ax a lim sin 4x 4
x0 bx = b x0 7x Gi gvb 7
lim tan ax a lim tan 3x Gi gvb 3
x0 bx = b x0 5x 5
–1
lim tan ax = a lim tan 3x 3
x0 bx b x0 5x Gi gvb 5
lim tan ax a lim tan 5x 5
x0 sin bx = b x0 sin 2x = 2
–1 –1
lim sin ax = a lim sin (2x) = 2
x0 bx b x0 x
lim sin ax a lim sin 3x 3
x0 sin bx = b x0 sin 5x = 5
2 2
lim cos(ax) –2cos (bx) = b – a cos2x – cos3x 32–22 5
lim = =
x0 x 2 x2 2 2
lim sinax – sinbx a – b lim sin8x – sin3x 8 – 3 5
x0 sincx – sindx = c – d x0 sin5x – sin2x = 5 – 2 = 3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
32 cvV¨eB‡K mnR Kivi cÖqvm
© Medistry Avm‡c± wmwiR
ASPECT SHORTCUT TRICKS & TIPS MODEL EXAMPLE
lim cos ax – cos bx a2 – b2 lim cos 7x – cos 9x 92 – 72 32 1
x0 = x0 cos 3x– cos 5x 52 – 32 = 13 = 2
=
cos cx – cos dx c2 – d2
2 2
lim 1– cos ax = a 2
x0 lim 1– cos2x = 22 = 4
1 – cos bx b x0 1 – cos3x 3 9
2 2
lim 1– cos2 ax = a
x0 lim 1– cos 7x
=
7
=
49
bx 2b x0 3x2 2.3 6
exRMwYZxq Amxg wjwgU x n‡j
fMœvs‡ki ni Ges j‡ei m‡ev©”P Power mgvb lim a x a x GB AvKv‡i _vK‡j
fMœvs‡ki j‡ei m‡e©v”P Power < n‡ii m‡e©v”P PowerAns:0 x
j‡ei m‡e©v”P Power hy³ x Gi mnM a x a x
fMœvs‡ki j‡ei m‡e©v”P Power > n‡ii m‡e©v”P PowerAns: Ans:
n‡ii m‡e©v”P Power hy³ x Gi mnM Ans: ax Gi mn‡Mi AbycvZ
eM©g~j msµvšÍ dvskb
lim a+x– a–x GB AvKv‡ii cÖ‡kœi Ans: 1 lim a+bx – cdx
GB AvKv‡ii cÖ‡kœi Ans:
b∓d
x0 x a x0 x 2 a
evB‡bvwgqvj AvKvi msµvšÍ
xn – an
) = lim 1 + bx = eab
b b a bx a x
eax.x = eab lim (1+ lim AvKv‡i _vK‡j Ans. = nan–1 nq|
lim (1+ax) =
x0
x
x x x xa xa x – a
Required Formulas for Differentiation
d
dx
( x n ) n x n 1
d n
dx
(cx ) c n x n1
d
dx
x 2 1x d
dx
(c ) 0
d d d 1 d 1 1 d
(e x ) e x (log e x) or, (ln x) (log x) log e
a a (a x ) a x log e a a x ln a
dx dx dx x dx x x ln a dx
d d d d
(sin x) cos x (cos x) sin x (tan x ) sec2 x (secx) secx tan x
dx dx dx dx
1
(sin 1 x ) (cos 1 x )
d d d 1 d
(cot x) cos ec2 x (cos ecx) cos ecxcot x
dx dx 1 x2
dx dx 1 x2
d (tan-1x) = 1 d
d (cot-1x) = 1 (sec1 x)
1 d 1
1 x2 cos ec1x
dx
dx 1 x2 dx x x21 dx x x2 1
d ( U V ) dU dV d ( UV ) U dV V dU d dU dV dW V dU U dV
dx dx dx dx dx dx (UVW) = VW + UW + UV d U dx dx
dx dx dx dx
dx V V2
MvwYwZK mgm¨vmg~n mgvav‡bi Rb¨ cÖ‡qvRbxq m~Î I cÖwµqvmg~n
function d
i. (Constant) GB ai‡bi Problem Solve Kivi Rb¨ dx (ax) = ax lna GB m~Î e¨envi Ki‡Z n‡e|
function d
function ii. (Exponential) GB ai‡bi Problem Solve Kivi Rb¨ dx (ex) = ex GB m~Î e¨envi Ki‡Z n‡e|
(Constant)
GB ai‡bi Problem iii. (Function)Constant GB ai‡bi Problem Solve Kivi Rb¨ d (xn) = nxn1 GB m~Î e¨envi Ki‡Z n‡e|
dx
Note: †h‡Kv‡bv ai‡bi AšÍixKiY Ki‡Z †M‡j cÖ_‡g cÖ‡kœi x Gi RvqMvq hv _vK‡e Zv‡K x a‡i m~Î cÖ‡qvM Ki‡Z n‡e|
Zvici hv‡K x aiv n‡jv Zvi AšÍixKiY Ki‡Z n‡e| me‡k‡l `yB AšÍixKiY ¸Y Ki‡Z n‡e|
g ( x) d d
{f(x)} g ( x ) i.Y= {f(x)} GB AvKv‡ii Problem Solve Ki‡Z n‡e, dx{f(x) g ( x ) } = {f(x)} g ( x ) dx{g(x) ln(f(x)} m~‡Îi mvnv‡h¨
AvKv‡ii Problem g ( x)
ii.{f(x)} GB AvKv‡ii Problem Dfq c‡ÿ ln wb‡qI Solve Kiv hvq|
Ae¨³ dvskb Implicit function †Pbvi Dcvq: (mgxKiY AvKv‡i _vK‡e Ges mgxKi‡Yi GKB cv‡k GKvwaK PjK we`¨gvb _vK‡e)|
[Implicit dy x Gi mv‡c‡ÿ Differentiate (Yconstant)
Technique: =–
Function] dx y Gi mv‡c‡ÿ Differentiate (Xconstant)
Parametric Function †Pbvi Dcvq: x I y Gi gvb Z…Zxq Ab¨ GKwU Pj‡Ki gva¨‡g †`qv _vK‡e Ges Z…Zxq PjKwU‡K e‡j civwgwZ|
civwgwZK dvskb dy civwgwZi mv‡cÿ y Gi diff dy/d
Technique: = = [ civwgwZ]
dx civwgwZi mv‡c‡ÿ x Gi diff dx/d
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH GKb‡Ri cÖ‡qvRbxq m~Îvejx
© Medistry 33
†`Iqv dvskbwU‡K y = f(x) ai‡Z n‡e|
Step-1:
f ' (x) Ges f '' (x) wbY©q Ki‡Z n‡e|
Step-2:
f ' (x) = 0 a‡i x Gi gvb (awi x Gi gvb¸wj a,b,c BZ¨vw` ) wbY©q Ki‡Z n‡e|
Step-3:
x = a Gi Rb¨ f "(a) ‡ei Ki‡Z n‡e| hw` f "(a) > 0 nq Z‡e x = a Gi Rb¨ f (x) Gi me©wb¤œ gvb ev jNygvb _vK‡e Ges
Step-4:
Pig gvb ev mwÜ gvb
hw` f ''(x) < 0 nq Z‡e x = a Gi Rb¨ f(x) Gi m‡e©v”P gvb ev ¸iægvb _vK‡e|
(¸iægvb I jNy gvb)
Step-5: Abyiƒcfv‡e x = b, c BZ¨vw`i Rb¨ Pig gvb wbb©q Ki‡Z n‡e|
Note: (i) f '(x) dvsk‡bi m‡e©v”P ev me©wb¤œ gvb _vK‡e hw` f '(x) = 0 nq|
(ii) f ''(x)= 0 n‡j dvsk‡bi †Kvb Piggvb _vK‡e bv| f ''(a) = 0 n‡j cieZ©x ch©v‡q AšÍiK mnM wbY©q K‡i dvsk‡bi Pig gvb wbY©q Kiv hvq|
(iii) †Kv‡bv we›`y‡Z dvsk‡bi m‡e©v”P gvb Aci GKwU we›`yi me©wb¤œ gvb A‡cÿv ÿz`ªZg n‡Z cv‡i|
ch©vq µwgK AšÍixKiY
dy d
y = f(x) GKwU dvskb n‡j AšÍixKi‡Yi wewfbœ AvKvi: y1 = dx = f(x) = dx {f (x)}
dy d2y d3y dny
y = f(x) Gi 1g AšÍixKiY, y1 = dx = f(x); 2q AšÍixKiY, y2 = 2 = f(x); 3q AšÍixKiY, y3 = 3 = f(x); n Zg AšÍixKiY, yn = n = fn(x)
dx dx dx
Technique:
y = sinx y = cosx
y = x4 y1 = –sinx
y = xn n‡j y1 = cosx
y1 = 4x3 = 4p1 x4-1 y = –sinx y2 = –cosx
yn = n! 2
y2 = 4.3x2 = 4p2 x4-2 y3 = –cosx y3 = sinx
ym = 0 [m > n] 4 4-3
y3 = 4.3.2.x = p3 x
ym = npm xn-m [m < n] 4 4-4 y4 = sinx = y y4 = cosx = y
y4 = 4.3.2.1 = p4 x = 4! A_©vr cÖwZ 4n Zg AšÍi‡Ki A_©vr cÖwZ 4n Zg AšÍi‡Ki ci
y5 = 0 [5 > 4]
ci cybive„wË nq cybive„wË NU‡e
y = (ax + b) n
1 (–1) .n! n y = emx y = ax
y = Gi nZg AšÍiK = y1 = m.emx y1 = axlna
yn = n! an x xn+1
y2 = m2.emx y2 = ax(lna)2
ym = 0 [m > n] 1 (–1)n.n!
y= Gi nZg AšÍiK = y3 = m3.emx y3 = ax(lna)3
ym = npm (ax + b)n–m am [m < n] x+a (x+a)n+1
yn = mn.emx yn = ax(lna)n
y = Sin(ax+b) Gi nZg AšÍiK,
y = cos(ax + b) Gi nZg AšÍiK,
n
yn = an sin
ax b yn = an cos n ax b
2
2
MCQ Gi Rb¨ we‡kl m~Î/†K․kj
(n 1)!
y= xn lnx, nN n‡j yn = y = (ax + b)m, (n N, a 0) n‡j
x
m! (ax + b)m n
yn = an , hLb n < m y = ann! , hLb n = m
(m n)!
`kg Aa¨vq: †hvMRxKiY
Required Formulas for Integration
n 1
2
1 1
x n dx (n 1)x n 1
1 1
x dx n 1 c ,
x dx x c
n
n 1 dx 2 x c
x x
a dx c a 0, a 1
x x (u v)dx udx v dx
e dx e c
x dx ln x c
1 x ax
lna
sinxdx cosx c cosxdx sinx c cosec xdx cotx c
d 2
uvdx u vdx [ (u ) vdx]dx
dx
f ( x )
sec2x dx = tanx + c cosecxcotxdx cosecx c secx tanxdx secx c f (x) dx ln f (x) c
f (x)
f(x)
dx 2 f(x) c tanxdx ln(cosx) c ln(secx) c sec xdx ln tan 4 x2 c = ln secx + tanx+ c
a 2 x 2 a tan
1 x
1 x 2 tan
1
cot xdx ln(sin x ) c = - ln (cosecx) + c
dx 1 dx
c xc
a
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
34 cvV¨eB‡K mnR Kivi cÖqvm
© Medistry Avm‡c± wmwiR
1 x 1
dx dx
cosecxdx ln | tan 2 | c = ln (cosecx + cotx) + c
x
sin c sin xc
2 2 a 2
a x 1 x
ax xa ax
a 2 x 2 2a ln a x c
dx 1
x 2 a 2 2a ln x a c 1 x
dx 1
ax
dx a sin
a
a2 x2 c
ln x x a c ln x x a c ax
dx dx 1 x
2 2 2 2
2 2 dx a sin a2 x2 c
x a 2
x a
2 ax a
dx 1 1 mx dx 1 mx
a 2 (mx) 2 am tan a c a 2 ( mx ) 2
m
sin 1
a
c
dx 1 mx a + c dx 1 a mx
(mx) 2
a2
2am
log
mx a a 2
(mx ) 2
2am
log
a mx
c
mLwÛZ dvskb
e {f(x) + f(x)}dx = e {f(x)}+c e
x x ax
{af(x) + f(x)}dx = eaxf(x)+c
x a2 x 2 2 a2
x + a dx = 2 x + a + 2 ln |x + x + a | + c ; [awi, x = a, tan]
x a dx = 2 x a 2 ln |x + x a | + c ; [awi, x = a, tan]
2 2 2 2 2 2 2 2 2 2
x a2 x2 a2 1 x x 2 a 1 x
2
x a dx =
2 2
2
+ sin
2 a
+c
x a dx = 2 a x + 2 sin a + c
2 2 2
uvdx = u vdx dx (u) vdx dx {d } eax {a. f(x) + f (x)} dx = eax. f(x) + c
eax eax
eax cosbx dx = (asinbx bcosbx) + c eax cosbx dx = (acosbx + bcosbx) + c
a + b2
2
a + b 2
2
2 dx 2
tan1
2ax + b 1 2ax + b D + c, hLb D > 0, a > 0
ln
ax + bx + c
, (a 0, D = b2 4ac) = + c, hLb D < 0, a > 0 =
|D| |D| |D| 2ax + b + D
D c~Y© eM© n‡j (ax b)m dx AvKv‡i cÖKvk K‡i mgvKjb Ki‡Z n‡e|
n
nxn 1 n(n 1) xn 2 n(n 1) (n 2)xn 3
x e dx = e
ax x
n ax
+ + ........ x hy³ c` ch©šÍ
a a2 a3 a4
b
f(a + b x)dx b
ba 8
10 xdx 82
ev
f(x)dx
= = =3
a f(x) + f(a + b x) a f(x) + f(a + b x) 2 2 x + 10 x 2
b
f(x)dx = f(a + b x) dx
b
(f x)dx
3
xdx 2
3
2=
a a 2 2 + 3 (5 x) 2 2 + 3x
b c f(x c)dx = h f(x)dx (x 1)dx 2 = xdx 2
4 3
ac a 2 4 + (x 1) 1 4 + x
b nb 3 12
f(nx)dx = 1 f(x)dx
4xdx 2 = 1 xdx 2
a n nn 2 3 + (4x) 4 8 3 + x
(x a) (b x) dx = (b a)2 , b > a
b b
dx
= , b > a
a 8 a (x a) (b x)
n–1 n–3 n–5 3.1
/2sinn xdx = /2cosn xdx =
. . ....... . (n = †Rvo c~Y©msL¨v n‡e|)
0 0 n n–2 n–4 4.2 2
/2 /2 n – 1 n – 3 n – 5 3
0 sin xdx = 0 cos xdx =
n n
. . ..... (n = we‡Rvo c~Y©msL¨v n‡e|)
n n–2 n–4 2
MvwYwZK mgm¨vmg~n mgvav‡bi Rb¨ cÖ‡qvRbxq m~Î I cÖwµqvmg~n
cÖwZ¯’vcb c×wZ:
AsKwU cix¶vi n‡j wKfv‡e wPb‡e gy‡L gy‡L Kivi Technique
As‡Ki `ywU Ask †`Iqv _vK‡e| d
Step-01: myweavRbK Ask‡K Z ai | g‡b g‡b Ki|
Ask `ywUi †h †Kvb GKwU‡K ev Gi †Kvb dx
d Step-02: †`L AsKwU‡Z Complete result Av‡Q wKbv| Complete result _vK‡j Zv AsK n‡Z ev`
Ask we‡kl‡K dx Ki‡j Ab¨ AskwU ûeû
`vI| bv _vK‡j Operator Gi ‡fZ‡i-evB‡i wPý ev msL¨v Input Output K‡i Complete
wKsevmvgvb¨ cwiewZ©Z iƒ‡c wn‡m‡e H result evwb‡q bvI (Aek¨B cÖkœc‡Î As‡Ki Wv‡Ki g‡a¨B), Zvici Zv ev` `vI|
As‡K ¸b Ae¯’vq _vK‡e| Step-03: AZtci Zzwg †hfv‡e Integration Gi cÖv_wgK m~Î wk‡LQ †mfv‡e m~Î †dj |
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH GKb‡Ri cÖ‡qvRbxq m~Îvejx
© Medistry 35
wbw`©ó †hvMR Gi cÖ‡qvM:
b
y Gi GKwU dvskb _vK‡j †¶Îdj = ydx
a
b
y Gi `ywU dvskb _vK‡j †¶Îdj =
a
( y1 y2 )dx [g‡b ivL‡Z n‡e a I b n‡e x Gi Value]
a2
e„‡Ëi mgxKiY x2 + y2 = a2 Øviv Ave×- (i) †gvU †¶Îdj = a2 (ii) GKwU PZz©fv‡Mi Ave× †¶‡Îi †¶Îdj = 4
x2 y2 ab
Dce„‡Ëi mgxKiY a2 + b2 = 1 Øviv Ave× (‡hLv‡b a > b) (i) †gvU †¶Îdj = ab (ii) GKwU PZz©fv‡Mi Ave× †¶‡Îi †¶Îdj = 4
8a2
y2 = 4ax cive„Ë I y = mx Øviv Ave× †¶‡Îi †¶Îdj =
3m3
16
`ywU cive„‡Ëi a mgvb n‡j †¶Îdj = 3 a2
MwY‡Zi cÖ‡qvRbxq m~Îvejx 2q cÎ
Z…Zxq Aa¨vq: RwUj msL¨v
MvwYwZK mgm¨vmg~n mgvav‡bi Rb¨ cÖ‡qvRbxq m~Î I cÖwµqvmg~n:
y
Z = x + iy Gi gWzjvm |z| = r = x 2 y 2 , tan 1 ( )
x
RwUj msL¨vi Z = – x + iy Gi gWzjvm |z| = r = x 2 y 2 , tan 1 ( )
y
x
gWzjvm I
AvMy©‡g›U Z = – x – iy RwUj msL¨v gWzjvm |z| = r = x 2 y2 , tan 1( y )
x
Z = x – iy Gi gWzjvm |z| = r = x 2 y 2 , 2 tan 1 ( y ) = –tan–1
y
x x
gvb wbY©q i 2 1 i 3 i 2 .i i i 4 i 2 .i 2 1 i 4n 3 i 4n .i 3 1.(i) i
m¤úwK©Z in + in+1 + in+2 + in+3 = 0 [†hLv‡b n †h‡Kvb c~Y© msL¨v]
–1+ –3 2 –1– –3
GK‡Ki Nbg~j wZbwU 1, , 2 = ; =
2 2
GK‡Ki Nbg~j ¸‡jvi GKwU ev¯Íe Ges Aci `ywU KvíwbK
GK‡Ki Nbg~j GK‡Ki KvíwbK g~j¸‡jvi †hvMdj = –1
GK‡Ki KvíwbK g~j¸‡jvi ¸Ydj =1
GK‡Ki Nbg~j ¸‡jvi †hvMdj = 0; n + n+1 + n+2 = 0 [†hLv‡b n †h‡Kv‡bv c~Y© msL¨v]
3 1 => 3n 2 3n .2 2
Awfbe ‡UKwbK (Shortcut solution)
n n
(i) ni = ( 1 + i) (ii) –ni = ( 1 – i)
2 2
eM©g~j wbY©q eM©g~j †ei Kivi wbqg:
Step-1: i hy³ c`wU‡K 2 Øviv fvM K‡i Ggb 2wU Drcv`K •Zwi Ki‡Z n‡e| hv‡`i e‡M©i AšÍidj ev¯Íe Ask|
Step-2: ev¯Íe Ask negative n‡j i n‡e eo Drcv`‡Ki mv‡_| ev¯Íe Ask positive n‡j i n‡e †QvU Drcv`‡Ki mv‡_|
Step-3: eM©g~‡ji gvSLv‡bi wPý n‡e cÖ`Ë RwUj msL¨vi Aev¯Íe As‡ki wP‡ýi g‡Zv|
3 1 3 1 3
–a3 = –a, –a, – a2 a , a ( ),a ( )
2 2
a = a, a, a2 = a, a
Nbg~j; 4-Zg 3 3 1 + 3 1 1 3
,a
g~j I 6-Zg 2 2
g~j a
4
a2 (1 i)
2
6 3 3 3
a2 = ai, a i, n i2
z|=k Gi mÂvi c_ e„Ë wb‡`©k K‡i |z – k1| = k2 n‡j Gi mÂvi c_ e„Ë wb‡`©k K‡i
mÂvi c‡_i | az k1|| bz k 2 | Gi mÂvi c_ e„Ë wb‡`©k K‡i | z k1|| z k 2 | Gi mÂvi c_ mij‡iLv wb‡`©k K‡i
cÖK…wZ wbY©q
| z k1| | z k 2 |k 3 Gi mÂvi c_ Awae„Ë wb‡`©k K‡i | z k1| | z k 2 |k 3 Gi mÂvi c_ Dce„Ë wb‡`©k K‡i
msµvšÍ
| z k1| x ev y Gi mÂvi c_ cive„Ë wb‡`©k K‡i
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
36 cvV¨eB‡K mnR Kivi cÖqvm
© Medistry Avm‡c± wmwiR
PZz_© Aa¨vq: eûc`x I eûc`x mgxKiY
mgxKi‡Yi g~j †_‡K mgxKiY wbY©q msµvšÍ
f(x) = 0 mgxKi‡Yi g~j¸‡jv , , n‡j-
i. – , – , – g~j wewkó mgxKiY f( – x) = 0
ii. g~j wewkó mgxKiY f = 0
n n n n
x
iii. g~j wewkó mgxKiY f- = 0
n n n n
x
iv. n, n, n g~j wewkó mgxKiY f = 0
x
n
v. (+h), (+h), (+h) g~j wewkó mgxKiY f(x–h)=0
vi. ( – h), ( – h), ( – h) g~j wewkó mgxKiY f(x+h) = 0
vii. 2, 2 g~j wewkó mgxKiY f( x) = 0
viii. ( + ) Ges g~jwewkó mgxKiY x2 {( + ) + } x + ( + ) = 0
wØNvZ mgxKi‡Yi m‡ev©”P ev me©wb¤œ gvb msµvšÍ
b2
ax2 + bx + c = 0 ivwki m‡ev©”P ev me©wb¤œ gvb = c
4a
i. a > 0(x2 Gi mnM abvZ¥K) n‡j gvb me©wb¤œ
ii. a < 0(x2 Gi mnM FYvZ¥K) n‡j gvb m‡e©v”P
m‡ev©”P ev me©wb¤œ gv‡bi †¶‡Î, x = b n‡e
2a
fvM‡kl Dccv`¨ msµvšÍ
fvM‡kl Dccv`¨ (Remainder Theorem): †Kvb eûc`x f(x) †K (x a) Øviv fvM Ki‡j fvM‡kl n‡e f(a) A_©vr x Gi cwie‡Z© a ewm‡q †h gvb cvIqv
hvq ZvB fvM‡kl|
Drcv`K Dccv`¨ (Factor theorem): x PjKhy³ †Kvb eûc`x ivwk‡Z x Gi ¯’v‡b a emv‡j hw` gvb k~b¨ nq Z‡e xa H ivwkwUi GKwU Drcv`K n‡e|
kZ©g~jK MvwYwZK cÖ‡qvM
hv mivmwi g‡b ivLvB fv‡jv:
i. x2 – px + q I x2 – qx + p ivwk؇qi GKwU mvaviY Drcv`K _vKvi kZ©, p + q + 1 = 0
a1 b 1 c1
ii. a1x2 + b1x + c1 = 0 Ges a2x2 + b2x + c2 = 0 mgxKiY `yÕwUi `yÕwU g~jB mvaviY nIqvi kZ©- = =
a2 b 2 c2
iii. x2 – px + q = 0 I x2 – qx + p = 0 mgxKi‡Yi GKwU mvaviY g~j _vKvi kZ©, p + q + 1 = 0
iv. x2 – bx + c = 0 I x2 – cx + b = 0 Gi g~j؇qi cv_©K¨ GKwU aªæe ivwk nevi kZ©, b + c + 4 = 0
v. x2 – px + q = 0 mgxKi‡Yi g~j `yBwUi cv_©K¨ 1 n‡j, p2 = 1 + 4q n‡e|
m n b
vi. ax2 + bx + b = 0 mgxKi‡Yi g~j؇qi AbycvZ m : n n‡j + m+ =0
n a
MCQ ‡K․kj
f(x) = 0 mgxKi‡Yi g~j¸wj , , .... n‡j,
(a) , , ... g~j wewkó mgxKiY f( x) = 0
(b) , , ........ g~j wewkó mgxKiY f = 0
1 1 1 1
x
(c) + h, + h, + h ..... g~j wewkó mgxKiY f(x h) = 0
(d) h, h, h ....... g~j wewkó mgxKiY f(x + h) = 0
ax2 + bx + c = 0 mgxKi‡Yi-
(a) g~jØq ¸YvZ¥K wecixZ n‡j, a = c
(b) g~jØq ¸YvZ¥K wecixZ Ges wecixZ wPýhy³ n‡j, a = c
(c) GKwU g~j AciwUi n ¸Y n‡j, nb2 = ac (a + n)2
ax2 + bx + c = 0 mgxKi‡Yi g~jØq , n‡j,
(a) , g~j wewkó mgxKiY, ax2 + bx + c = 0
1 1
(b) , g~j wewkó mgxKiY, cx2 + bx + a = 0
1 1
(c) , g~j wewkó mgxKiY, cx2 bx + a = 0
b2 b
ax2 + bx + c ivwki m‡e©v”P ev me©wb¤œ gvb c n‡e x = Gi Rb¨| a > 0, a < 0 n‡j cÖ`Ë ivwki h_vµ‡g me©wb¤œ I m‡e©v”P gvb cvIqv hv‡e|
4a 2a
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH GKb‡Ri cÖ‡qvRbxq m~Îvejx
© Medistry 37
lô Aa¨vq: KwYK
Kwb‡Ki mvaviY mgxKiY: x2 + y2 + 2gx + 2fy + 2hxy + c = 0
(ab – h2) = 1 n‡j KwbKwU cive„Ë
0 < (ab – h2) < 1 n‡j KwbKwU Dce„Ë
(ab – h2) > 1 n‡j KwbKwU Awae„Ë
a = b, h = 0 n‡j e„Ë
Dr‡Kw›`ªKZv e n‡j, e = 1 n‡j KwbKwU cive„Ë
0 < e < 1 n‡j KwbKwU Dce„Ë
e > 1 n‡j KwbKwU Awae„Ë
e = 0 n‡j e„Ë
e = n‡j mij‡iLv
‡dvKvm (, ) wbqvgK‡iLv ax + by + c = 0, Dr‡Kw›`ªKZv e n‡j Kwb‡Ki mgxKiY,
(ax + by + c)2
(x – )2 + (x – )2 = e2
a2 + b 2
cive„Ë
†Kvb wØNvZ mgxKi‡Yi wØNvZ m¤^wjZ c`¸‡jv c~Y©eM© m„wó Ki‡j GwU GKwU cive„Ë|
b2 – 4ac b
x A‡ÿi mgvšÍivj Aÿ wewkó cive„‡Ëi mgxKiY, x = ay1 + by + c, kxl© –
1
– Ges Gi Dc‡Kw›`ªK j¤^ = |a|
4a 2a
b b2 – 4ac
y A‡ÿi mgvšÍivj Aÿ wewkó cive„‡Ëi mgxKiY, y = ax2 + bx + c, kxl© – –
1
2a 4a
Ges Gi Dc‡Kw›`ªK j¤^ = |a|
y2 = 4ax cive„‡Ëi (x1,y1) we›`y‡Z ¯úk©‡Ki mgxKiY, yy1 = 2a (x + x1)
Ges ¯úk©we›`yi ¯’vbv¼ – m2 – m
a a 2a
y = mx + c ‡iLvwU y2 = 4ax cive„ˇK ¯úk© Ki‡j c =
m
y = mx + c ‡iLvwU x2 = 4ay cive„ˇK ¯úk© Ki‡j c = –am2 Ges ¯úk©we›`yi ¯’vbv¼ (2am, am2)
(x1, y1) we›`ywU y2 = 4ax cive„‡Ëi, evB‡i Ae¯’vb Ki‡j, y12 – 4ax1 > 0
Dc‡i Ae¯’vb Ki‡j, y12 – 4ax1 = 0
†fZ‡i Ae¯’vb Ki‡j, y12 – 4ax1 <0
{P(x,y)we›`y †_‡K Aÿ‡iLvi j¤^ `~iZ¡}2
= 4 |Dc‡Kw›`ªK j‡¤^i •`N©¨|
P(x,y) we›`y †_‡K kxl© we›`y‡Z ¯úk©‡Ki Dci j¤^ `~iZ¡
cive„‡Ëi AvKvi y2 = 4ax x2 = 4ay (y – )2 = 4a(x – ) (x – )2 = 4a(y – )
kxl©we›`y (0, 0) (0, 0) (, ) (, )
Dc‡K›`ª (a, 0) (0, a) (a + , ) (, a + )
wbqvgK †iLvi cv`we›`y (–a, 0) (0, –a) (–a + , ) (, – a + )
Aÿ‡iLvi mgxKiY y=0 x=0 y–=0 x–=0
wbqvgK †iLvi mgxKiY x+a=0 y+a=0 x–+a=0 y–+a=0
Dc‡Kw›`ªK j‡¤^i mgxKiY x=a y=a x – a y – a
kxl© ¯úk©‡Ki mgxKiY x=0 y=0 x–=0 y–=0
Dc‡Kw›`ªK j‡¤^i •`N¨© 4|a| 4|a| 4|a| 4|a|
Dc‡Kw›`ªK j‡¤^i cÖvšÍwe›`y (a 2a) ( 2a, a) (a + a + ) a + a +
(x, y) we›`yi Dc‡Kw›`ªK `~iZ¡ x+a y+a x–+a y–+a
Dce„Ë
(ax + by + c)2
†dvKvm (, ) wbqvgK‡iLv ax + by + c = 0 Ges Dr‡Kw›`ªKZv e n‡j, Dce„‡Ëi mgxKiY, (x – )2 + (y – )2 = e2 a2 + b 2
x2 y2 x12 y12
2 + 2 = 1 Dce„‡Ëi, evB‡i Ae¯’vb Ki‡j, 2 + 2 – 1 > 0
(x1, y1) we›`ywU
a b a b
x12 y12
Dc‡i Ae¯’vb Ki‡j, a2 + b2 – 1 = 0
x2 y2
‡fZ‡i Ae¯’vb Ki‡j, a12 + b12 – 1 < 0
a2 m b2
y = mx a2m2 + b2 †iLvwU m Gi mKj gv‡bi Rb¨ Dce„ˇK ( 2 2 2 2 2 2) we›`y‡Z ¯úk© Ki‡e|
a m +b a m +b
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
38 cvV¨eB‡K mnR Kivi cÖqvm
© Medistry Avm‡c± wmwiR
xx yy
(x1, y1) we›`y‡Z ¯úk©‡Ki mgxKiY, a21 + b21 = 1
x2 y2
y = mx + c ‡iLvwU + = 1 Dce„ˇK ¯úk© Ki‡j c2 = a2m2 + b2
a2 b 2
†h‡Kv‡bv we›`y‡Z Dc‡Kw›`ªK `~iZ¡Ø‡qi †hvMdj e„nr A‡ÿi mgvb|
Dce„‡Ëi mKj †K›`ªMvgx R¨v H we›`y‡Z mgwØLwÛZ nq|
P (x, y) we›`ywU Dce„‡Ëi Dci Aew¯’Z n‡j Gi civwgwZK ¯’vbv¼ (acos, bsin)
civwgwZK mgxKiY, x = acos, y = bsin
x y
GB Dce„‡Ëi Dci 1 we›`y‡Z ¯úk©‡Ki mgxKiY, a cos b cos 1 = 1
Ax2 + By2 + Cx + Dy + E = 0; [A, B, C, D, E Constant] Ges A B n‡j,
(x1, y1) we›`y‡Z ¯úk©‡Ki mgxKiY, Axx1 + Byy1 + C 2 1 + D 2 1 + E = 0
x+x y+y
x2 y2 x2 y2 (x – a)2 (y – b)2 (x – a)2 (y – b)2
+ =1 + =1 + =1 + =1
Dce„‡Ëi AvKvi a2 b 2 a2 b 2 a2 b2 a2 b2
a<b b<a a<b b<a
‡K‡›`ªi ¯’vbv¼ (0, 0) (0, 0) (, ) (, )
b2 a2 b2 a2
Dr‡Kw›`ªKZv e= 1– e= 1– e= 1– 2 e= 1– 2
a2 b2 a b
e„nr A‡ÿi •`N¨© 2a 2b 2a 2b
ÿz`ª A‡ÿi ‣`N¨© 2b 2a 2b 2a
e„nrA‡ÿi mgxKiY y=0 x=0 y–=0 x–=0
ÿz`ªA‡ÿi mgxKiY x=0 y=0 x–=0 y–=0
kxl©Ø‡qi ¯’vbv¼ ( a, 0) (0, b) ( a + , ) (, b + )
( ae, 0) (0, be) ( ae + , ) (, be + )
‡dvKvm؇qi ¯’vbv¼
( a2 – b2, 0) (0, b2 – a2) ( a2 – b2 + , ) (, b2 – a2 + b)
‡dvKvm؇qi `~iZ¡ 2ae = 2 a2 – b2 2be = 2 b2 – a2 2ae = 2 a2 – b2 2be = 2 b2 – a2
a 0 0 b a + a b a b + b
e e e e
wbqvgK‡iLvi cv`we›`y
a 0 0 b a
+ a b a b + b
2 2 2 2
a2 – b 2 b2 – a2 a2 – b 2 b 2 – a2
2a 2a2 2b 2b2 2a 2a2 2b 2b2
wbqvgK‡iLv؇qi `~iZ¡ = 2 = 2 2 = 2 = 2 2
e a – b2 e b –a e a – b2 e b –a
a b a b
wbqvgK‡iLv؇qi mgxKiY x= y= x–= y–=
e e e e
2 2 2 2
2b 2a 2b 2a
Dc‡Kw›`ªK j‡¤^i •`N¨©
a b a b
Dc‡Kw›`ªK j‡¤^i mgxKiY x = ae y = be x – = ae y – = be
‡ÿÎdj , ab , ab , ab , ab
a b a b
Dc‡K›`ª I Abyiƒc wbqvg‡Ki `~iZ¡ – ae – be – ae – be
e e e e
Awae„Ë
(x – h)2 (y – k)2
(h, k) ‡K›`ª wewkó Awae„‡Ëi mgxKiY, – =1
a2 b2
xx1 yy1
(x1, y1) we›`y‡Z ¯úk©‡Ki mgxKiY, 2 – 2 = 1
a b
civwgwZK ¯’vbv¼, (asec, btan)
civwgwZK mgxKiY, x = asec, y = btan
x 1 y 1 1
asec1, btan1 I (asec2, btan2) we›`yMvgx R¨v Gi mgxKiY, cos (1 – 2) – sin (1 – 2) = cos (1 + 2)
a 2 b 2 2
†Kvb kZ© D‡jøL bv _vK‡j a > b aiv nq| Z‡e †dvKv‡mi †KvwU w¯’i _vK‡j a > b Ges f~R w¯’i _vK‡j b > a aiv nq|
Awae„‡Ëi Dci †Kvb we›`y P(x,y) Dc‡K›`ªØq S, S n‡j |PS – PS| = 2a
x2 y2
y = mx + c †iLvwU 2 – 2 = 1 Awae„ˇK ¯úk© Ki‡e hw` c2 = a2m2 – b2 nq|
a b
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH GKb‡Ri cÖ‡qvRbxq m~Îvejx
© Medistry 39
x2 y2 y2 x2 (x – a)2 (y – b)2 (y – b)2 (y – a)2
Awae„‡Ëi AvKvi – =1 – =1 – =1 – =1
a2 b 2 b 2 a2 a2 b2 b2 a2
‡K‡›`ªi ¯’vbv¼ (0, 0) (0, 0) (, ) (, )
Dr‡Kw›`ªKZv b2 a2 b2 a2
e= 1– e= 1+ e= 1+ 2 e= 1+ 2
a2 b2 a b
Avo A‡ÿi •`N¨© 2a 2b 2a 2b
AbyeÜx A‡ÿi •`N¨© 2b 2a 2b 2a
AvoA‡ÿi mgxKiY y=0 x=0 y–=0 x–=0
AbyeÜx A‡ÿi mgxKiY x=0 y=0 x–=0 y–=0
kxl©Ø‡qi ¯’vbv¼ (a, 0) (0, b) (a + , ) (, b + )
(ae, 0) (0, be) (ae + , ) (, b + )
‡dvKvm؇qi ¯’vbv¼
( a2 + b2, 0) (0, a2 + b2) ( a2 – b2 + , ) (, a2 + b2 + )
‡dvKvm؇qi `~iZ¡ 2ae = 2 a2 + b2 2ae = 2 a2 + b2 2ae = 2 a2 + b2 2ae = 2 a2 + b2
a 0 0 b a + ab a b + b
2 2 2 2
wbqvgK‡iLvi cv`we›`y
a2 + b 2 a2 + b2 a2 + b 2 a2 + b 2
2a 2a2 2b 2b2 2a 2a2 2b 2b2
wbqvgK‡iLv؇qi `~iZ¡ = 2 = 2 = 2 = 2
e a + b2 e a + b2 e a + b2 e a + b2
a b a b
wbqvgK‡iLv؇qi mgxKiY x= y= x–= y–=
e e e e
2b2 2a2 2b2 2a2
Dc‡Kw›`ªK j‡¤^i ‣`N¨©
a b a b
x = ae y = be x – = ae y – = be
Dc‡Kw›`ªK j‡¤^i mgxKiY
x = a2 + b 2 y = a2 + b 2 x – = a2 – b 2 y – = a2 + b 2
b b b b
Awae„‡Ëi AmxgZU y= x y= x y – = (x – ) y – = (x – )
a a a a
mßg Aa¨vq: wecixZ w·KvYwgwZK dvsk I w·KvYwgwZK mgxKiY
w·KvYwgwZK mgxKiY mgvav‡bi Rb¨ cÖ‡qvRbxq mvaviY m~Îvejx
sec–1(–x) = –sec–1x cosec–1(–x) = – cosec–1x 2sin–1x = sin–1(2x 1 – x2)
2cos–1x = cos–1(2x2 – 1) 3sin–1x = sin–1(3x – 4x3) 3cos–1x = cos–1(4x3 – 3x)
3x – x3 1 –1 1 – 1 – x2 1 –1 1 + x2 – 1
3 tan–x = tan–1 sin x = tan–1 tan x = tan–1
1 – 3x2 2 x 2 x
ax + by p by –1ax + by –1 a y –1 ax – by –1 a y
tan–1 = + tan–1 tan = tan + tan–1 tan = tan – tan–1
ax – by 4 ax bx – ay b x bx + ay b x
–1 1 – x p
tan = – tan x–1
sin = sin n‡j = n–1n cos = cos n‡j = 2n
1+x 4
p
tan = tan n‡j = n + sin = 0 n‡j = (2n + 1) tan = 0 n‡j n
2
p p
sin = 1 n‡j = (4n + 1) 2 sin = – 1 n‡j n – 1
2
cos = 1 n
p
cos–1 (–x) = – cos–1x cot–1x + tan–1x = sin–1x + sin–1y = sin–1[x 1 – y2 + y 1 – x2]
2
p x+y
cos–1x + cosec–1x = sin–1x – sin–1y = sin–1[x 1 – y2 + y 1 – x2] tan–1x + tan–1y = tan–1
2 1 – xy
x–y
cos–1x + cos–1y = cos–1[xy – 1 – x2 1 – y2] tan–1x – tan–1y = tan–1
1 + xy
1
cos–1x – cos–1y = cos–1[xy + 1 – x2 1 – y2] sin–1 = cosec–1x
x
x + y + z – xyz 1
tan–1x + tan–1y + tan–1z = tan–1 tan–1 = cot–1x
1 – xy – yz – zx x
2
2x –1 1 – x 2x
2tan–1x = tan–1 2 = cos = sin–1 sin–1(–x) = – sin–1x
1–x 1 + x2 1 – x2
1 1+x 1–x 1–x
cos–1x = cos–1 = sin–1 = tan–1 tan–1(–x) = – cot–1x
2 2 2 1+x
2
x –1 1 – x 1 1
sin–1x = cos–1 1 – x2 = tan–1 2 = cot = sec–1 = cosec–1
1–x x 1 – x2 x
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
40 cvV¨eB‡K mnR Kivi cÖqvm
© Medistry Avm‡c± wmwiR
Aóg Aa¨vq: w¯’wZwe`¨v
** `ywU e‡ji ‡ÿ‡Î jwäi gvb I w`K wbY©q:
B C
jwä R = P2+Q2+2PQcos
Q
ej؇qi AšÍ©MZ †KvY R
Q sin
tan O A
P Q cos P
Note:
i) 0 n‡j, R max P Q ii) 180 n‡j, Rmin = P Q α 90 n‡j, R 2 P 2 Q 2
iii)
iv) `yBwU e‡ji jwä (R) e„nËg ej (P) Gi mv‡_ †h †Kvb Drcbœ K‡i, e„nËg ejwU‡K wظb Kivq hw` D³ †KvbwU A‡a©K n‡q hvq Z‡e P I Q Gi ga¨eZx©
†KvY = 120
v) `yBwU ej P I Q Gi jwä R n‡j Ges P I Q Gi ga¨eZx© †KvY n‡j,
GKwU ej AciwUi wظY Ges jwä j¤^ eivei wµqviZ n‡j, = 120
GKwU ej AciwUi wظY n‡j, = 120
`yBwU mgvb e‡ji jwä k~b¨ n‡j, = 180
MvwYwZK mgm¨vmg~n mgvav‡bi Rb¨ cÖ‡qvRbxq m~Î I cÖwµqvmg~n
`ywU e‡ji ‡ÿ‡Î I jwäi gvb I w`K wbY©q
Q sin
jwä R = P2+Q2+2PQcos ; ej؇qi AšÍ©MZ †KvY tan
P Q cos
`ywU mggv‡bi e‡ji †ÿ‡Î jwäi gvb I jwäi w`K wbY©q
α α
P I Q mgvb n‡j, jwä R 2 Pcos Ges jwäi w`K θ
2 2
`ywU ej j¤^ eivei wµqvkxj n‡j jwäi gvb I jwäi w`K wbY©q
Q
jwä, R = P2 + Q2 Ges jwäi w`K, = tan–1 P
jwä †QvU e‡ji mv‡_ j¤^ eivei wµqvkxj n‡j jwäi gvb I ej؇qi ga¨eZ©x †KvY wbY©q
Qsin
jwä, R = P2 – Q2 ; †hLv‡b (P Q); tan90 = P + Qcos = 0
P + Qcos
ej؇qi ga¨eZx© ‡KvY n‡j, cos = – e„nËg ej = cos–1 – Q
¶z`ªZg ej P
Note: i) 0 n‡j, R max P Q ii) 180 n‡j, Rmin = PQ iii) α 90 n‡j, R 2 P 2 Q 2
iv) `yBwU e‡ji jwä (R) e„nËg ej (P) Gi mv‡_ †h †KvY Drcbœ K‡i, e„nËg ejwU‡K wظb Kivq hw` D³ †KvYwU A‡a©K n‡q hvq Z‡e P I Q Gi
ga¨eZx© †KvY = 120
v) `yBwU ej P I Q Gi jwä R n‡j Ges P I Q Gi ga¨eZx© †KvY n‡j,
GKwU ej AciwUi wظY Ges jwä j¤^ eivei wµqviZ n‡j, = 120
GKwU ej AciwUi wظY n‡j, = 120 `yBwU mgvb e‡ji jwä k~b¨ n‡j, = 180
jvwgi Dccv`¨ msµvšÍ
GK we›`y‡Z wfbœ †iLv eivei wµqviZ wZbwU GKZjxq ej mvg¨ve¯’vq _vK‡j, cÖ‡Z¨KwU e‡ji gvb Aci `yÕwU e‡ji AšÍf©y³ †Kv‡Yi mvB‡bi mgvbycvwZK|
cv‡ki wP‡Îi †¶‡Î, jvwgi Dccv`¨ Abymv‡i,
P Q R
sinYOZ sinZOX sinXOY
Y Q
O
X Z R P
2 2
Avevi, †h‡Kv‡bv e‡ji gvb Aci ej `ywUi jwäi gv‡bi mgvb n‡e| †hgb- Q = P + R + 2P Rcos;
P= Q2 + R2 + 2Q Rcos
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH GKb‡Ri cÖ‡qvRbxq m~Îvejx
© Medistry 41
j¤^vsk Dccv`¨ msµvšÍ
j¤^vsk Dccv`¨ (Resolved part’s Theorem): GKwU we›`y‡Z †Kvb wbw`©ó w`‡K `yB ev, Z‡ZvwaK Q
R
e‡ji j¤^vs‡ki exRMvwYwZK †hvMdj GKB w`K eivei Zv‡`i jwäi j¤^vs‡ki mgvb|
P I Q Gi jwä ej R n‡j, j¤^vsk Dccv`¨ Abymv‡i,
R cos P cos Q cos .
P
X
O
Note:
i. `yB Gi AwaK e‡ji jwä wbY©‡qi †ÿ‡Î j¤^vsk Dccv`¨ e¨envi Ki‡Z n‡e|
ii. P, Q, R gv‡bi wZbwU mgZjxq ej GKwU we›`y‡Z wµqv K‡i| P I Q, Q I R Ges R I P Gi ga¨eZ©x †KvY h_vµ‡g , Ges n‡j ej·qi jwä,
F= P2 + Q2 + R2 + 2PQcos + 2QRcos + 2PRcos
iii. ejÎq mgvšÍi avivq Ges mgvb †Kv‡Y wµqviZ _vK‡j jwä R = 3 mgvšÍi avivq mvaviY AšÍi|
iv. wZbwU mggv‡bi ej Ggbfv‡e wµqv K‡i †hb †h †Kvb `yBwU e‡ji ga¨eZ©x †KvY 120 nq, Z‡e jwä k~b¨|
360
v. PviwU mggv‡bi ej †Kvb we›`y‡Z Ggb fv‡e wµqv K‡i †hb ci ci `yBwU e‡ji ga¨eZ©x †KvY = 90 nq Z‡e jwä k~b¨|
4
beg Aa¨vq: MwZwe`¨v
MvwYwZK mgm¨vmg~n mgvav‡bi Rb¨ cÖ‡qvRbxq m~Î I cÖwµqvmg~n:
i. Vmax2+Vmin2=2v2 ii. Vmin = u – v iii. V = u 2 v 2
MwZ‡eM I `ªywZ 2uv
Mo †eM wbY©q: †Kvb ¯’v‡b hvevi †eM = u, Avmvi †eM = v n‡j Mo †eM =
uv
i. f~wgi D‡aŸ© h D”PZv n‡Z u †e‡M Dj¤^fv‡e Dc‡i wbwÿß GKwU e¯‘ KYv t mg‡q f~wg‡Z v †e‡M cwZZ n‡j-
wbw¶ß/cošÍ 1
h = ut + gt2 ; v = u + gt
2
e¯Íyi MwZ
ii. cošÍ e¯‘ cZ‡bi †kl t sec G h `~iZ¡ AwZµg Ki‡j cZ‡bi †gvU mgq = + sec
t h
2 gt
u2sin2
i. m‡e©v”P D”PZv, H = ii. AbyfzwgK `~iZ¡ d = ucos.t
2g
1 2 u sin
iii. Dj¤^ `~iZ¡, h = u sin.t – gt iv. cZbKvj ev DÌvbKvj, t = U
2 g
cÖ‡¶cK usin
2u sin
u 2 sin 2
v. ågbKvj, T = vi. Avbyf~wgK cvjøv, R= ucos
g g
u2 4H
vii. me©vwaK Avbyf~wgKcvjøv, Rmax = viii. = tan
g R
i. †¯ªv‡Zi AbyKz‡j †eM = u+v ii. b~¨bZg `~i‡Z¡ b`x cvi n‡Z jwä †eM= u2 + v2
s ‡¯ªv‡Zi †eM
†mªvZ msµvšÍ iii. b`x cvi n‡Z mgq t = iv. Zx‡ii mv‡_ mvuZviæ ev †b․Kvi ‡e‡Mi ga¨eZx© †KvY = cos–1 mvuZviæi †eM
u + v2
2
mgm¨vejx
s
v. †¯ªv‡Zi cÖwZK~‡j †eM = u v vi. ¯^íZg mg‡q b`x cvi nIqvq mgq, t =
u sin
†e‡Mi gvb = ‡f`K…Z Z³vi msL¨v :
Z³v msµvšÍ 1 S
i. hLb nviv‡bv †eM = 2 ev Avw`‡e‡Mi A‡a©K ZLb `~iZ¡ ev miY = [S = Avw` miY]
3
mgm¨vejx
1 S(n – 1)2
ii. hLb nviv‡bv †eM = Ask| ZLb `~iZ¡/miY = [S = Avw` miY]
n 2n 1
i. `ywU e¯‘ hLb GKB w`‡K Mgb K‡i, ZLb A e¯‘i mv‡c‡ÿ B e¯‘i Av‡cwÿK †eM = VB VA Ges B e¯‘i mv‡c‡ÿ A e¯‘i
Av‡cwÿK †eM Av‡cwÿK †eM = VA VB
msµvšÍ ii. `ywU e¯‘ hLb ci¯úi wecixZ w`‡K Mgb K‡i, ZLbÑ A e¯‘i mv‡c‡ÿ B e¯‘i Avt †eM = VA + VB ; B e¯‘i mv‡c‡ÿ A e¯‘i
Avt‡eM = VA + VB hr
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 43
Aa¨vq g¨vwUª· I wbY©vqK
1g cÎ
[MATRIX & DETERMINATION]
01
cÖ_g Ask: g¨vwUª· [MATRIX]
SURVEY TABLE Kx coe? // †Kb coe? // †Kv_v n‡Z coe? // KZUzKz coe?
VVI For
MAKING DECISION [†h Kvi‡Y co‡e]
TOPICS MAGNETIC DECISION [hv co‡e] This Year
DU JU RU CU GST Engr. HSC Written MCQ
CONCEPT-01 g¨vwUª‡·i cÖKvi‡f` I gvb wbY©q msµvšÍ 40% 30% 60% 40% 50% 30% 60% -
CONCEPT-02 g¨vwUª‡·i •ewkó¨ [Properties of Matrics] 30% 40% 30% 50% 30% 20% 25%
CONCEPT-03 g¨vwUª‡·i gvÎv, †hvM, we‡qvM, ¸Y I mgZv 25% 60% 60% 40% 50% 20% 15%
CONCEPT-04 e¨wZµgx g¨vwUª· 90% 30% 60% 30% 30% 20% 30%
CONCEPT-05 AbyeÜx g¨vwU· Ges wecixZ g¨vwUª· 40% 30% 45% 40% 40% 40% 30%
DU. = Dhaka University, JU. = Jahangirnagar University, RU. = Rajshahi University,
CU = Chittagong University, GST = General, Science & Technology, Engr. = Engineering.
g¨vwUª·: msL¨v ev exRMwYZxq ivwki AvqvZvKvi ev eM©vKv‡i mvRv‡bv e¨e¯’vB n‡jv g¨vwUª·| g¨vwUª·‡K mvaviYZ [ ] ev ( ) Øviv cÖKvk Kiv nq|
a b c a b c
A= p q r ev p q r
x y z x y z
mvaviY AvKv‡ii g¨vwUª· (General Form of matrix)
a11 a12 a13
3 3 µ‡gi †h‡Kv‡bv g¨vwUª· A n‡j, A = [aij]3 3 = a21 a22 a23
a31 a32 a33
m msL¨K mvwi I n msL¨K Kjvgwewkó GKwU g¨vwUª· A Gi fzw³ Arthur Cayley
aij (i = 1, 2, ....., m; j = 1, 2 ......, n) n‡j, g¨vwUª·-Gi Avwe®‹viK
1850 wLª÷v‡ã James Joseph Sylvester
a11 a12 ... a1n a11 a12 ... a1n
cÖ_g g¨vwUª‡·i aviYv †`b| ZviB mnKg©x
a21 a22 ... a2n a21 a22 ... a2n
A= A = [aij]m n A_ev, A = A = (aij)m n Arthur Cayley wecixZ g¨vwUª‡·i aviYvmn
... ... ... .... ... ... ... ....
am1 am2 ... amn
am1 am2 ... amn g¨vwUª‡·i Zvrch© Zz‡j a‡ib Ges cÖ_g
we‡kølYg~jKfv‡e g¨vwUª‡· cÖKvk K‡ib|
Concept-01 g¨vwUª‡·i cÖKvi‡f` I gvb wbY©q msµvšÍ ¸iæZ¡:
01. mvwi g¨vwUª· (Row Matrix): †h g¨vwUª‡·i GKwU gvÎ mvwi _v‡K| †hgb- A = [a b c] hvi GKwU gvÎ mvwi i‡q‡Q| mvwi g¨vwUª‡·i mvaviY gvÎv = 1 n
a
02. Kjvg g¨vwUª· (Column Matrix): †h g¨vwUª‡·i GKwU Kjvg _v‡K| †hgb- b hvi GKwU gvÎ Kjvg i‡q‡Q| Kjvg g¨vwUª‡·i mvaviY gvÎv = n 1
c
03. eM© g¨vwUª· (Square Matrix): hLb mvwi msL¨v = Kjvg msL¨v| A_©vr [Ai j]mn g¨vwUª·wU eM© g¨vwUª· n‡j m = n n‡e|
a11 a12
†hgb- GKwU 2 2 µ‡gi ev 2 µ‡gi eM© g¨vwUª·
a21 a22
04. AvqZ g¨vwUª· (Rectangular Matrix): mvwii msL¨v Kjvg msL¨vi mgvb bv n‡j AvqZvKvi g¨vwUª· nq| A_©vr [Ai j]mn AvqZ g¨vwUª· n‡j m n n‡e|
1 2 3 1 2
Ex: (i) 4 5 6 , gvÎv = 2 3 ; mvwii msL¨v (2) Kjvg msL¨v (3)| (ii) 4 5, mvwi = 3wU, Kjvg = 2wU, mvwi Kjvg|
6 7
05. g~L¨ ev cÖavb KY© (Principal or main Matrix): [Ai j]nn µ‡gi eM© g¨vwUª· n‡j `ywU KY© cvIqv hv‡e| g¨vwUª‡·i cÖ_g mvwi I cÖ_g Kjv‡g †h Dcv`vb
_v‡K Zv‡K wb‡q †h KY© MwVZ nq ZvB g~L¨ KY© Ges Aci KY©wU †MŠY KY©| †hgbÑ
a11 b12 c13 †MŠY KY©
x21 y22 z23
p31 q32 r33
g~L¨ KY©
Note: g~L¨ K‡Y©i Dcv`vb ¸‡jvi †hvMdj‡K ‡Uªm (a + y + r) e‡j| g~L¨ K‡Y©i Dcv`vb ¸‡jvi ¸Ydj‡K g~L¨c` (ayr) e‡j|
†MŠY K‡Y©i Dcv`vb ¸‡jvi ¸Ydj‡K (pyc) †MŠYc` e‡j|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
44 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
06. DaŸ© wÎfzRvKvi g¨vwUª· (Upper Triangular Matrix): †Kvb eM© g¨vwUª· A = [Ai j]nn Gi cÖavb K‡Y©i wb¤œ¯’ FocusKwi
jÿ¨ Point:
5 2 1 fwZ© cixÿv‡Z mivmwi msÁv Lye Kg
Dcv`vb ¸‡jv 0 n‡j, (A_©vr Ai j = 0 hLb i > j) Zv‡K DaŸ© wÎfzRvKvi g¨vwUª· e‡j| †hgbÑ U = 0 4 –3
0 0 8 Av‡m| cixÿvi Av‡M G‡Zv cÖKvi‡f`
07. wb¤œ wÎfzRvKvi g¨vwUª· (Lower Traingular Matrix): †Kvb eM© g¨vwUª· A = [Ai j]nn Gi cÖavb K‡Y©i Dc‡ii g‡b ivLvI KwVb| GB Aby‡”Q‡`
2 0 0 Avgiv †`L‡ev cixÿv cÖkœ wKfv‡e nq
Dcv`vb¸‡jv 0 n‡j (A_©vr Ai j = 0 hLb i < j) Zv‡K wb¤œ wÎfzRvKvi g¨vwUª· e‡j| †hgbÑ L = 6 5 0 Ges mn‡R wKfv‡e Zv wiwfkb Ki‡Z
7 9 12 cvwi|
08. KY© g¨vwUª· (Diagonal Matrix): †Kvb eM© g¨vwUª· [Ai j]nn †K n µ‡gi KY© g¨vwUª· ejv n‡e hw` mvaviYZ msÁv wn‡m‡e KY©
a 0 0 g¨vwUª·, †¯‹jvi g¨vwUª·, A‡f`K
[Ai j = 0 hLb i j A_©vr cÖavb K‡Y©i Dcv`vb e¨ZxZ mKj Dcv`vb Ô0Õ n‡e| †hgbÑ A = 0 b 0 g¨vwUª· G‡m _v‡K|
0 0 c
a 0 0 a 0 0
09. †¯‹jvi g¨vwUª· (Scalar Matrix): †h KY© g¨vwUª‡·i Ak~b¨ Dcv`vb ¸‡jv mgvb| †hgbÑ A = 0 a 0 0 b 0 [KY© g¨vwUª·]
0 0 a 0 0 c
10. GKK ev A‡f`K g¨vwUª· (Unit or Identity Matrix): †Kvb eM© g¨vwUª· [Ai j]nn †K n µ‡gi GKK g¨vwUª· ejv
n‡e hw` Ai j = 0 hLb i j Ges |Ai j| = 1 hLb i = j nq| I Øviv cÖKvk Kiv nq| eM©/KY©/‡¯‹jvi n‡e| a 0 0
0 a 0 [†¯‹jvi g¨vwUª·]
1 0 1 0 0 0 0 a
†hgbÑ I2 = ; I3 = 0 1 0
0 1 0 0 1
1 0 0
11. k~b¨ g¨vwUª· (Zero or Null Matrix): †h g¨vwUª‡·i mKj Dcv`vb k~b¨| †hgb- A =
0 0 0 1 0 [A‡f`K g¨vwUª·]
0 0 0 0 1
2 i. KY© g¨vwUª‡·i a = b = c n‡j,
12. mgNvwZ g¨vwUª· (Idempotent Matrix): eM©vKvi †Kvb g¨vwUª· A †K mgNvwZ g¨vwUª· ejv n‡e hLb A = A n‡e|
Dnv †¯‹jvi|
†hgb: A =
2 1
n‡j, A2 = A.A =
2 1 2 1 2 1
= =A ii. KY© g¨vwUª‡·i a = b = c = 1
–2 –1 –2 –1 –2 –1 –2 –1 n‡j, Dnv A‡f`K g¨vwUª·|
n
13. k~b¨NvwZ g¨vwUª· (Nilpotant Matrix): GKwU eM© g¨vwUª· A †K k~b¨NvwZ g¨vwUª· ejv n‡e| hw` A = 0 nq iii. A‡f`K g¨vwUª‡·i gvb = 1
2 –2
†hLv‡b n N †hgb: A = GLv‡b A =
2 0 0 iv. KY© g¨vwUª‡·i gvb/wbY©vqK = abc
=0
2 –2 0 0 v. †¯‹jvi g¨vwUª‡·i gvb = a3
2
14. A‡f`NvwZ g¨vwUª· (Involuntary Matrix): GKwU eM© g¨vwUª· A †K A‡f`NvwZ g¨vwUª· ejv n‡e hw` A = I nq| GQvov Ab¨ g¨vwUª‡· ag© †_‡KB †ewk
cÖkœ Av‡m| GK bR‡i ¸iæZ¡c~Y©
†hgb- A =
2 3
; GLv‡b A =
2 1 0
=I ag©¸‡jv †`‡L †bqv hvK|
–1 –2 0 1
Nv‡Zi nvZvnvwZ:
15. Uªv݇cvR g¨vwUª· (Transpose Matrix): mvwi Ges Kjvg hLb ¯’vb wewbgq K‡i| †hgbÑ
i. A2 = A mgNvZx|
a b c T a x p
A = x y z Gi Uªv݇cvR g¨vwUª· A ev A = b y q ii. An = 0 k~b¨NvZx|
p q r c z r iii. A2 = I A‡f`NvZx|
T
16. cÖwZmg g¨vwUª· (Symmetric Matrix): GKwU eM© g¨vwUª· A ‡K cÖwZmg g¨vwUª· ejv n‡e hw` A = A nq A_©vr eµcÖwZmg-
2 0 –1 T 2 0 –1 i. cÖavb K‡Y©i f~w³ 0 nq|
Ai j = Aj i nq| †hgbÑ A = 0 3 4 n‡j A = 0 3 4 n‡e ; A n‡e cÖwZmg g¨vwUª·|
–1 4 5 –1 4 5 ii. aij = aji A_©vr K‡Y©i Dc‡i
17. wecÖwZmg/eµ/AcÖwZmg g¨vwUª· (Skew – Symmetric Matrix): GKwU eM© g¨vwUª· A = [Ai j]nn †K wecÖwZmg I wb‡P GKB Ae¯’v‡b f~w³i
T gvb GKB n‡e wKš‘ wPý
g¨vwUª· ejv n‡e hw` A = – A nq
wecixZ n‡e|
0 1 –4 0 1 4 0 1 –4 0 1 –4
T
A_©vr Ai j = – Aj i nq| ‡hgbÑ A = –1 0 3 n‡j A = 1 0 3 = – –1 0 3 = – A
4 –3 0 4 3 0 4 –3 0 –1 0 3
T 4 –3 0
Note: wecÖwZmg g¨vwUª· n‡j A = –A n‡e Ges cÖavb K‡Y©i Dcv`vb k~b¨ n‡e (A_©vr Ai j = 0 hLb i = j) RwUj g¨vwUª‡·i Kvwnbx:
T T
18. j¤^ g¨vwUª· (Orthogonal Matrix): GKwU eM© g¨vwUª· A j¤^ g¨vwUª· ejv n‡e hw` AA = A A = I nq| i. AbyeÜx g¨vwUª· cÖavb
1 1 1 K‡Y©i f~w³¸‡jv RwUj n‡e|
†hgb A = GKwU j¤^ g¨vwUª·
2 1 –1 ii. nv‡g©wkqvb g¨vwUª· cÖavb
i 2 3 K‡Y©i f~w³¸‡jv ev¯Íe n‡e|
19. RwUj g¨vwUª· (Complex Matrix): RwUj Dcv`vb wewkó g¨vwUª·‡K RwUj g¨vwUª· e‡j| †hgb : A = 4 –i 2 iii. wenv‡g©wkqvb g¨vwUª· cÖavb
1 2i 3 cÖavb K‡Y©i Dcv`vb k~b¨
20. AbyeÜx g¨vwUª· (Conjugate Matrix): †Kvb RwUj g¨vwUª· Gi RwUj Dcv`vb ¸‡jvi AYyeÜx Dcv`vb Øviv MwVZ g¨vwUª·‡K H g¨vwUª‡·i AYyeÜx g¨vwUª·
3 + 4i 1 3 – 4i
–
2 2 1
e‡j| †hgb : A = 4 2 – i 0 Gi AYyeÜx RwUj g¨vwUª·, A = 4 2 + i 0
1 2 i 1 2 –i
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 45
21. nviwgwkqvb g¨vwUª· (Hermitian Matrix): †Kvb RwUj eM© g¨vwUª· Gi AbyeÜx g¨vwUª·‡K Uªv݇cvR K‡i hw` cÖ`Ë g¨vwUª‡·i mgvb nq Z‡e cÖ`Ë g¨vwUª·‡K
3 1+i i 1–i –i
–
3
– T
( )
nviwgwkqvb g¨vwUª· e‡j| A_©vr A = A †hgb : A = 1 – i 2 3 + i n‡j A = 1 + i 2 3 – i
–i 3 – i –1 i 3 + i –1
– T
3 1 + i i
( )
A = 1 – i 2 3 + i = A
–i 3 – i –1
3 1 + i i
A = 1 – i 2 3 + i GKwU nviwgwkqvb g¨vwUª·
–i 3 – i –1
()
T
22. wenviwgwkqvb g¨vwUª· (Skew Hermitian Matrix): †Kvb GKwU g¨vwUª· A †K wenviwgwkqvb g¨vwUª· ejv n‡e hw` A = – A nq| †hgb :
0 –3i
A = 3i 0 n‡j A = –3i 0 = – 3i 0 = – A A GKwU wenviwgwkqvb g¨vwUª·
0 3i – 0 3i
Note : wenviwgwkqvb g¨vwUª‡·i cÖavb K‡Y©i Dcv`vb ¸‡jv memgq k~b¨ ev m¤ú~Y© KvíwbK msL¨v n‡e|
GKbR‡i wewfbœ g¨vwUª‡·i gvb wbY©q
g¨vwUª‡·i †Kvb gvb †bB| Z‡e eM© g¨vwUª·‡K wbY©vqK AvKv‡i cÖKvk K‡i †mB
A = ac b [ ]
d eM© g¨vwUª‡·i wbY©vq‡Ki gvb A = Det(A) = ad – bc
wbY©vq‡Ki gvb wbY©q Kiv hvq|
n
n-µ‡gi g¨vwUª· A Gi Rb¨ |MA| = M |A|
gvb wbY©q msµvšÍ kU©KvU© ag©
[†hLv‡b M = †¯‹jvi ivwk, |A| = wbY©vq‡Ki gvb]
1 3 1
MEx 01 4 4 4 g¨vwUª·wUi †Uªm (Trace) Gi gvb 8 n‡j a Gi gvb †KvbwU? [JU. 19-20]
3 1 a
General Rules & Tips
kZ©g‡Z, 1 + 4 + a = 8; a = 3
0; i j
MEx 02 †Kvb eM© g¨vwUª‡·i Dcv`vb¸‡jv aij = n‡j, †mwU †Kvb ai‡bi g¨vwUª‡·i- [IU-F. 2012-13]
1 ; i = j
General Rules & Tips
0; i j
aij = n‡j, cÖ_g fzw³ a11 (GLv‡b i = j) Gi gvb 1 n‡e| GKBfv‡e a22, a33 = 1 n‡e|
1 ; i = j
Aciw`‡K, a12 (GLv‡b i j) a12 = 0 n‡e| GKBfv‡e evwK gvb¸‡jv 0 n‡e|
1 0 0
g¨vwUª·wU nq 0 1 0 hv A‡f`K g¨vwUª·|
0 0 1
–1
MEx 03 3 3 AvKv‡ii GKwU KY© g¨vwUª· D Gi Rb¨ |D| = 20 n‡j |(2D) | Gi gvb KZ? [BUET. 12-13; SUST. 15-16]
General Rules 3 in 1 Shortcut Tricks & Tips
a 0 0
awi, 3 3 AvKv‡ii KY© g¨vwUª·, D = 0 b 0
0 0 c
2a 0 0 a 0 0
2D = 0 2b 0 Ges |D| = 0 b 0= 20 1 1 1 1
0 0 2c 0 0 c 3 µ‡gi g¨vwUª· D Gi Rb¨ |(2D)–1| = = = =
|2D| 23|D| 8 20 160
2a 0 0 0 0
a
|2D| = 0 2b 0 = 23 0 b 0= 8 20 = 160
0 0 2c 0 0 c
–1 1 1
Zvn‡j, |(2D) | = |2D| = 160
MEx 04 3 3 AvKv‡ii KY© g¨vwUª· A Gi KY© Dcv`vb¸wji ¸Ydj 2 2 n‡j |( 2I – A) | Gi gvb KZ?
3
[SAU. 16-17]
General Rules & Tips
3 3 AvKv‡ii KY© g¨vwUª‡·i K‡Y©i Dcv`vb ¸bdj = KY© g¨vwUª‡·i gvb A = 2 2
2 0 0
Avevi, 2I = 0 2 0 ( 2I) = 2 2 2 = 2 2
0 0 2
|( 2I A) | = |(2 2 2 2)3| = 0
3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
46 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
S C Sol n g¨vwUª‡·i †Uªm = gyL¨ K‡Y©i Dcv`vb ¸‡jvi †hvMdj
olve
Dhaka University
3 –4 †Uªm = 1 + 4 + 3 = 8
01. A = n‡j det (2A–1) Gi gvb n‡jv- [DU. 19-20; DU-cÖhyw³: 19-20]
2 –3 05. †Kvb g¨vwUª‡·i mvwi¸‡jv‡K Kjv‡g Ges Kjvg¸‡jv‡K mvwi‡Z cwieZ©b
1 1
A. 4 B. –4 C. D. – Ki‡j †h g¨vwUª· cvIqv hvq Zv‡K e‡j| [JU. 18-19]
4 4
A. mvwi g¨vwUª· B. Kjvg g¨vwUª·
3 –4
S B Sol n A = 2 –3
Aspect Special:
C. eM© g¨vwUª· D. iƒcvšÍwiZ g¨vwUª·
olve
Ans D
GLv‡b, A GKwU 2 2 m¨vwUª·
06. g¨vwUª· X = cÖwZmg n‡j r Gi gvb †KvbwU?
1 –3 4 3 –4 hvi gvb |A| = 9 + 8 = 1 x 4
A–1 = r 6
[JU. 17-18]
– 9 + 8 –2 3 2 –3
=
det (2A ) = |2A | =
2
6 –8
1 1
A. 4 B. 6 C. 4 D. 6
2A–1 = 4 –6 A
S A Sol n X = r 6
x 4
22
olve
4
det (2A–1) = – 36 + 32 = – 4 = = = 4
|A| 1
∵ cÖwZmg X g¨vwUª· ZvB X = XT r 6 = 4 6 r = 4
x 4 x r
a 2 5
02. A = 2 b 3 GKwU eµ cÖwZmg g¨vwU· n‡j a, b, c Gi gvb¸‡jv-
a11 a12 ... ... a1n
5 3 c a23 a22 ... ... a2n
A. –2, –5, 3 B. 0, 0, 0 C. 1, 1, 1 D. 2, 5, 3
[DU. 17-18]
07. ... ... ... ... ... GB g¨vwUª·wU †Kvb cÖK…wZ? [JU. 09-10]
... ... ... ... ...
T
S B Sol n hw` A = –A nq ZLb Zv‡K eµ cÖwZmg g¨vwUª· e‡j| †h
olve
an1 an2 ... ... ann
†Kvb g¨vwUª‡·i eµ cÖwZmg g¨vwUª· †c‡Z n‡j g~L¨ KY© eivei Dcv`vb¸‡jv A. KY© g¨vwUª· B. eM© g¨vwUª·
me k~b¨ n‡Z n‡e| myZivs a = 0, b = 0 Ges c = 0| C. †¯‹jvi g¨vwUª· D. A‡f`K g¨vwUª·
Jahangirnagar University S B Sol g¨vwUª‡·i me©‡kl Dcv`vb ann A_©vr g¨vwUª‡·i mvwii msL¨v
olve n
01. 2 4 g¨vwUª·wUi wecixZ g¨vwUª· Gi †Uªm †KvbwU?
5 6
[JU-A, SetB. 20-21]
= n Ges g¨vwUª‡·i Kjv‡gi msL¨v = n eM© g¨vwUª·|
Rajshahi University
A. 9/8 B. 8/9 C. 8 D. 9
– T
5 6 Aspect Special: ()
S A Sol n 2 4 Gi wecixZ g¨vwUª· n‡jv, wecixZ g¨vwUª‡·i †Uªm 01. A = A n‡j, A g¨vwU‡·i Trace †Kgb n‡e? [RU-Uranus-1, Set-1. 21-22]
olve
A. abvZ¥K B. ev¯Íe msL¨v
1 4 –6 = 1 4 6 g~j g¨vwUª‡·i †Uªm c~ Y© msL¨v D. FbvZ¥K
20 –12 – 2 5 8 – 2 5 = C.
wbY©vq‡Ki gvb – T
1 –3 ( )
S B Sol n A = A n‡j GwU GKwU nviwgwkqvb g¨vwUª· Ges
olve
5+4 9
= =
= 1 5 †Uªm = + =
2 4 1 5 9 20 12 8 nviwgwkqvb g¨vwUª‡·i KY© eivei f‚w³ ev¯Íe msL¨v|
– 2 8 8 †Uªm I ev¯Íe msL¨v n‡e|
4 8
02. wb‡Pi †KvbwU mgNvwZ g¨vwUª·? [RU. Astrazeneca, Set-1. 20-21]
02. wb‡Pi †KvbwU mgNvwZ g¨vwUª·? [JU-H. 19-20]
A. B.
2 1 2 1
2 –2 –4 2 –2 4 2 1 2 1
A. –1 3 4 B. –1 3 4
D.
1
1 –2 –3 1 –2 –3 C.
2 1 2
2 1 2 1
2 –2 5 2 –2 –4
C. –1 4 D. –1 4
S B Sol n –2 –1 –2 –1 = –2 –1 [A = A n‡j mgNvZx]
3 3 2 1 2 1 2 1 2
olve
1 –2 –3 1 –2 3
2 03. A g¨vwUª·wU cÖwZmg g¨vwUª· n‡j wb‡Pi †KvbwU mwVK? [RU. 17-18]
S A Sol n mgNvwZ g¨vwUª‡·i †ÿ‡Î A = A n‡Z nq|
olve
A. AT = –A B. A2 = A
option (A) Gi †ÿ‡Î
C. AT = A D. A 2 = I
2 –2 –4 2 –2 –4 2 –2 –4 n GKwU eM© g¨vwUª· A cÖwZmg n‡j AT = A n‡e|
olve
–1 3 4 –1 3 4 = –1 3 4 S C Sol
1 –2 –3 1 –2 –3 1 –2 –3 Chittagong University
a 2 d 4 0 1
03. 2 b 3 g¨vwUª·wU wecÖwZmg n‡j a + b + c + d = ? [JU-A. 2019-20] 01. a Gi †Kvb gv‡bi Rb¨ 0 3 4 g¨vwUª·wU cÖwZmg (symmetric)
7 3 c 1 a 4
A. 3 B. 2 C. 7 D. 5 g¨vwUª· n‡e? [CU. 20-21]
n wecÖwZmg g¨vwUª‡·i †ÿ‡Î a = b = c = 0
olve
S C Sol A. 4 B. 0 C. – 1 D. – 4
Avevi, d = ( 7) = 7 a + b + c + d = 7 S A Sol n cÖwZmg g¨vwUª‡·i Rb¨ A = A
T
olve
Aspect Special:
cÖ wZmg g¨vwUª‡·i Rb¨
4 0 1 4 0 1
1 3 1
04. 4 4 4 g¨vwUª·wUi †Uªm †KvbwU? [JU-A. 19-20]
0 3 4 = 0 3 a a=4 a ij = aji
3 1 3 a23 = a32
1 a 4 1 4 0
A. 5 B. 12 C. 8 D. 7 A=4
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 47
1 2 + i 1 + 3i S B Sol n †h eM© g¨vwUª‡·i wbY©vq‡Ki gvb Ak~b¨ Zv‡K e‡j Ae¨wZµgx
olve
02. 2 – i 2 –i †Kvb ai‡bi g¨vwUª·? [CU-J. 16-17] g¨vwUª·|
1 – 3i i 0
07. wb‡Pi †KvbwU Complex Matrix? [IU. 17-18]
A. e¨wZµgx B. A¨vWR‡q›U C. nviwgwkqvb
A. A = B. A =
2 3+i 2 3
D. D`NvwZK E. †KvbwUB bq
n nv‡g©wkqvb g¨vwUª‡·i Rb¨ cÖavb K‡Y©i f~w³ ev¯Íe nq| evwK
i 6 –2 6
olve
S C Sol
C. A =
0 1
D. A =
1 1
f~w³¸‡jv AYyeÜx wn‡m‡e _v‡K| 1 0 1 1
a b Gi gvbÑ n †h g¨vwUª‡·i GK ev GKvwaK Dcv`vb RwUj msL¨v Zv‡K
olve
03. c –d [CU-C3. 16-17] S A Sol
A. ad – bc B. bc – ad C. bc + ad RwUj g¨vwUª· e‡j|
D. †bB E. 0 1 2 4
08. 3 a 2g¨vwU·wUi Trace-Gi gvb 7 n‡j a Gi gvb KZ? [IU. 17-18]
S D Sol n c –d = – ad – bc
a b
1 0 4
olve
GST (¸”Q) A. 0 B. 1 C. 2 D. 4
n g¨vwUª‡·i †Uªm = 1 + a + 4 = 5 + a
olve
01. 3 2 Ges 2 3 µg wewkó `ywU g¨vwUª· h_vµ‡g A Ges B Gi fzw³ 0 ev
S C Sol
kZ©vbymv‡i, 5 + a = 7 a = 2
1 n‡j tr(BA) Gi m‡e©v”P gvb n‡e [GST-A. 21-22]
09. KY© g¨vwUª‡·i Ak~b¨ fzw³mg~n mgvb n‡j Zv‡K ejv nqÑ [IU. 16-17]
A. 0 B. 1 C. 6 D. 9
A . mvwi g¨vwUª · B . †¯‹j vi g¨vwUª ·
S C Sol n m‡ev©”P gv‡bi Rb¨,
olve
C. k~b¨ g¨vwUª· D. Kjvg g¨vwUª·
1 1 n †h KY© g¨vwUª‡·i g~L¨ K‡Y©i Dcv`vb¸‡jv mgvb Zv‡K †¯‹jvi
olve
A g¨vwUª· n‡e 1 1 ; B g¨vwUª· n‡e 1 1 1
1 1 1 S B Sol
1 13×2 2×3 g¨vwUª
· e‡j|
10. bx‡Pi †KvbwU AcÖwZmg Matrix? [IU. 16-17]
1 1 1
1 1
3 3 –b –b
BA = 1 1 1 A. –b 0 B. 0 –b C. 0 b D. 0 b
1 1 †Uªm = (3 + 3) = 6 0 b b 0 0 0
1 1 3 3
S A Sol n wecÖZxmg g¨vwUª‡·i cÖavb KY© eivei f~w³¸‡jv k~b¨ nq|
olve
02. eM©vKvi †Kvb g¨vwUª· A-Gi †¶‡Î hw` A2 = A nq, Z‡e †mB g¨vwUª·wU-
[IU-A. 19-20] 11. †h eM© g¨vwUª‡·i K‡Y©i Dcv`vb¸‡jv Ak~b¨ I mgvb Ges Ab¨vb¨ Dcv`vb¸‡jv
A. mgNvwZ B. cÖwZmg C. ch©vqe„Ë D. A‡f`NvwZ k~b¨ Zv‡K ejv nq? [IU. 10-11, 08-09, 04-05; JnU. 12-13]
2
S A Sol n eM©vKvi g¨vwUª· A Gi †ÿ‡Î A = A n‡j A GKwU mgNvwZ g¨vwUª·|
olve
A. KY© B. BDwbU g¨vwUª·
03. GKwU g¨vwUª· A Dj¤^ nIqvi kZ©- [IU. 19-20] C. †¯‹jvi g¨vwUª· D. eM© g¨vwUª·
A. A2 = I B. A2 = 1 n †h eM© g¨vwUª‡·i K‡Y©i Dcv`vb¸‡jv Ak~b¨ I mgvb Ges
olve
S C Sol
C. AA = AA = I D. AA = AA = 1 Ab¨vb¨ Dcv`vb¸‡jv k~b¨ Zv‡K †¯‹jvi g¨vwU· e‡j|
S C Sol n †Kv‡bv eM© g¨vwUª· A †K j¤^ g¨vwUª· ejv n‡e,
olve
12. wb‡Pi †KvbwU cÖwZmg g¨vwUª·? [BSMRSTU-A. 19-20]
hLb AA = AA = I nq|
1 0 1 2 0 1 0 1 4
04. hw` A =
1 2
nq, Z‡e AT = ? A. 2 3 0 B. 0 3 4 C. 1 0 3 D. †KvbwUB bq
2 3 4 1 3 1 4 5 4 3 0
[JKKNIU. 19-20]
A. 2 3 B. 1
1 2 2 3 n GKwU eM© g¨vwUª· A †K cÖwZmg g¨vwUª· ejv hv‡e hw`
olve
S B Sol
3
AT = A nq|
C. 3 2 D. 2
1 2 1 3
2 a 5 4
13. hw` A = –5 b 3 GKwU wecÖwZmg g¨vwUª· nq, Zvn‡j a, b, c Gi
S A Sol n A = 2 3 A = 2
1 2 1 2
T
–4 3 c
olve
3
gvb¸‡jv n‡eÑ [BSFMSTU. 19-20]
05.
1 0
0 1
wU [KU. 17-18] A. –5, –4, –3 B. 0, 0, 0 C. 1, 1, 1 D. 5, 4,
i. KY© g¨vwUª· ii. †¯‹jvi g¨vwUª· iii. A‡f`K g¨vwUª· a 5 4
B Sol n 5 b 3 GKwU wecÖwZmg g¨vwUª· n‡j a = 0, b = 0 Ges c = 0 n‡e|
olve
Dc‡ii Z‡_¨i Av‡jv‡K wb‡Pi †KvbwU mZ¨? S
A. i, iii B. i, iii C. ii, iii D. i, ii, iii 4 3 c
n Z‡_¨i Av‡jv‡K g¨vwUª·- DU Affiliated College Question
olve
S D Sol
i. KY© g¨vwUª· (Diagonal Matrix) : †h eM© g¨vwUª· cÖavb K‡Y©i (aij; i = j) 01. A = 1 2 n‡j |4A1| = ?
fyw³¸‡jv e¨wZZ Ab¨me fyw³ k~b¨| 2 0 [DU-7clg. 22-23]
A. 4 B. 2
ii. †¯‹jvi g¨vwUª· (Scalar Matrix): †h KY© g¨vwUª‡·i Ak~b¨ fzw³¸‡jv mgvb|
C. 2 D. 4
iii. A‡f`K g¨vwUª· ev GKK g¨vwUª· (Identity or unit Matrix) : †h KY©
S D Sol n A = 2 0
1 2
olve
g¨vwUª‡·i KY©w¯’Z mKj fzw³ 1|
06. †h eM© g¨vwUª‡·i wbY©vq‡Ki gvb Ak~b¨ Zv‡K e‡j- [IU. 17-18] |A| = 4
A. e¨vwZµgx g¨vwUª· B. Ae¨wZµgx g¨vwU· 42 4 2
Now, |4A | = =
1 4
= =4
C. cÖwZmg g¨vwUª· D. GKK g¨vwUª· A |A| 4
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
48 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
02. A GKwU 3 3 g¨vwUª· Ges |A| = 7 n‡j, |(2A)1| = ? S D Sol n A = [aij]33
olve
[DU-Tech. 22-23; BRU-E. 19-20]
2 8 1 1 a11 a12 a13 aij = 2i j
A. B. C. D. A = a21 a22 a23
7 7 56 14 a11 = 1 a12 = 0
–1 3 –1 –1 –1 1 a31 a32 a33 a13 = 1 a21 = 3
S C Sol n (2A) = {(2) |A|} = {8 (–7)} = {–56} = – 56
olve
1 0 1 a22 = 2 a23 =1
Engineering = 3 2 1 a31 = 5 a32 = 4
5 4 3 a33 = 3
01. hw` A = [aij]33 GKwU eM© g¨vwUª· nq, †hLv‡b aij = 2i j ; i, j = 1, 2, 3.
Zvn‡j A g¨vwUª·wU GKwU- [CKRuet. 22-23] 1 0 1
|A| = 3 2 1 = 1 (6 4) (12 10) = 2 2 = 0
A. Involutory matrix B. Idempotent matrix 5 4 3
C. Nilpotent matrix D. Singular matrix
E. Orthogonal matrix A g¨vwUª·wU GKwU e¨wZµgx (Singular) g¨vwUª·|
【
? 】 QUICK PRACTICE CONCEPT TEST ?
01. A GKwU 3 × 3 eM©g¨vwUª· Ges |A| = 8 n‡j |2A| Gi gvb †KvbwU? 03. †h g¨vwUª‡·i mvwi I Kjv‡gi msL¨v mgvb Zv‡K ejv nq-
A.16 B.4 C. 64 D.8 A. eM© g¨vwUª· B. Kjvg g¨vwUª·
02. wb‡Pi †KvbwU eµ cÖwZmg g¨vwUª·? C. KY© g¨vwUª· D. A‡f`K g¨vwUª·
0 5
A.
1 5 0
B. C.
1 5 0
D.
1 5 0 Answer 01.C 02.B 03.A
2 0 5 5 0 2 1 5 2 1 1
Concept-02 g¨vwUª‡·i •ewkó¨ [Properties of Matrics] ¸iæZ¡:
g¨vwUª‡· †hvMwewa I ¸Ywewa:
i. A (B + C) = AB + AC ii. (B + C)A = BA + CA iii. A(BC) = (AB)C iv. A + B = B + A
Uªv݇cvR g¨vwUª‡·i KwZcq •ewkó¨:
i. (AT)T = A ii. (AB)T = BTAT (†hLv‡b A I B Gi gvÎv h_vµ‡g m n I n p)
iii. (A ± B)T = AT ± BT iv. (ABC)T = CTBTAT
A‡f`K g¨vwUª‡·i •ewkó¨:
i. AI = IA ii. A.A–1 = A–1A = I
3 2 –1
MEx 01 hw` GKwU eM© g¨vwUª· A Ggb nq †h, 3A – 2A + 5AI + I = 0 nq, Z‡e A = ?
General Rules & Tips
I = – (3A3 – 2A2 + 5AI)
I = 2A2 –3A3 –5AI I = A(2A – 3A2 – 5I) A . A–1 = A(2A – 3A2 – 5I) A–1 = 2A – 3A2 – 5I
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
2 2
S D Sol n A B weKí
olve
Dhaka University
2
= A (I – A) 2 A Ges B = I – A mgNvZx g¨vwU·ª|
01. A = n‡j det (AA1) Gi gvb KZ?
1 2
2 5 [DU-A. 2021-22] = A2 (I2 – 2IA + A2) A2 = A
1 = A2I2 – 2A.A2 + A2.A2 B2 = B = I – A
A. 1 B. 1 C. 0 D. = A – 2A.A + A.A A2B2 = A (I – A)
2
= A – 2A2 + A2 = AI – A2 = A – A
–2 –2
S A Sol n A = 2 5 A = 5–4 –2 1 = –2 1
1 2 1 5 5
–1 = A – A2
olve
=0
1 2 5 –2 = A – A [∵A2 = A] = 0
GLb, A.A–1 = 2 5 –2 1 =0 1 d (A.A–1) = (1–0) = 1
1 0
02. A I B `yBwU cÖwZmg g¨vwUª· n‡j AB BA GKwUÑ [RU-C. 19-20]
–1
Aspect Special: A.A = I Ges det(I) = 1 A. cÖwZmg g¨vwUª· B. KY© g¨vwUª· C. wecÖwZmg g¨vwUª· D. k~b¨ g¨vwUª·
T T
S C Sol n hw` A I B cÖwZmg nq Z‡e A = A Ges B = B
olve
Jahangirnagar University
GLb, (AB BA)T = (AB)T (BA)T
01. hw` g¨vwUª· A=[2 1 3] nq Ges I GKwU 3 3 BDwbU g¨vwU· nq
= BT AT AT BT
Zvn‡j AI = ? [JU. 11-12]
= BA AB
A. 0 B. [0 0 0] C. [2 1 3] D. AmsÁvwqZ = (AB BA)
n A I = A = [2 1 3]
olve
S C Sol ∵ (AB BA) = (AB BA)
T
Rajshahi University GwU wecÖwZmg n‡e|
01. hw` A Ges B = I – A mgNvwZ g¨vwUª· nq, Z‡e A2B2 = KZ? 03. hw` A GKwU eM© g¨vwUª· nq, Z‡e IAI = ? [BRUR. 2015-16; RU. 2014-15]
[RU-C, Corundum-1. 22-23] A. I B. A
A. – I B. I C. A1 D. IA
S B Sol n IAI = IA = A
olve
C. 2I D. 0
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 49
04. A GKwU eM© g¨vwUª· n‡j |A| = ? [RU-H. 14-15] ¸”Q (GST) fwZ© cixÿv
A. 1 B. †Kv‡bvwUB bq C. A1 D. A Ans B 01. A GKwU A‡f`K g¨vwUª· n‡j A–1 + A + A2 = ? [BSMRSTU. 18-19]
05. hw` A, B I C g¨vwUª·¸‡jv †hvM I ¸Y‡bi Rb¨ †hvM¨ nq; Z‡e wb‡Pi A. A2 B. 2A
†KvbwU mwVK? [RU-H. 14-15] C. 3A D. 1 + A2 + A2
A. A(B +C) = AB + AC B. (A + B)C = AC + BC
S C Sol n awi, A = 0 1
1 0 Aspect Special:
olve
C. A(BC) = (AB)C D. me¸‡jvB mwVK A‡f`K g¨vwUª‡·i gvb 1
S D Sol n According to basic properties of Matrices. A–1 = 0 1 = A
olve
1 0 A–1 + A + A2
06. g¨vwUª‡·i †ÿ‡Î †KvbwU wg_¨v bq? [RU-H. 13-14] = I1 + I + I2 = 3
A. †¯‹jvi g¨vwUª‡· Ak~b¨ Dcv`vb¸wj Amgvb n‡Z cv‡i
A2 = A.A = 0 1 0 1 = 0 1 = A
1 0 1 0 1 0
B. mgvb msL¨K Dcv`vbwewkó Kjvg g¨vwUª· A Ges mvwi g¨vwUª· B n‡j
AB n‡Z cv‡i bv A–1 + A + A2 = A + A + A = 3A
C. `ywU mgvb mvB‡Ri BDwbU g¨vwUª· †hvM Ki‡j †¯‹jvi g¨vwUª· cvIqv hvq bv 02. A = 1 2 n‡j, A1 A = ?
D. `ywU ¸Y‡hvM¨ g¨vwUª· me †ÿ‡ÎB †hv‡Mi Rb¨ †hvM¨ Ans B 2 1 [MBSTU-A. 2012-13]
A. I B. 0
Chittagong University
1 1
2 –3 1 –1
01. M = I N = – 1 3 n‡j (MN)–1 Gi gvb KZ? D.
1 0
2 2
0 1 C.
1 1 2 1
A. N M –1 –1 –1 –1
B. M N
[CU-A, Set-3. 20-21]
2 2
1
S A Sol n †h‡Kv‡bv g¨vwUª‡·i †ÿ‡Î, A A = I
olve
C. MN D. †Kv‡bvwUB bq Ans A
【
? 】 QUICK PRACTICE CONCEPT TEST ?
01. hw` X Ges Y g¨vwUª· nq, Z‡e wb‡Pi †KvbwU mwVK? 02. I GKwU 3 3 GKK g¨vwUª· Ges A GKwU 3 3 g¨vwUª· n‡j Al5 = KZ?
A. Xt = (X1)1 B. Y X = X Y A. A B. I C. A D. AIA
C. X + Y Y + X D. (Yt)t = Y Answer 01.D 02.A
Concept-03 g¨vwUª‡·i gvÎv, †hvM, we‡qvM, ¸Y I mgZv ¸iæZ¡:
g¨vwUª‡·i gvÎv/ µg (size/Dimension): g¨vwUª‡·i gvÎv/ µg‡K mvwi Kjvg AvKv‡i †jLv nq|
Ex: (i) 4 5 6 2wU mvwi, 3wU Kjvg gvÎv n‡jv 2 3
1 2 3
1 2 3
(iii) 4 5 6 gvÎv = 3 3
7 8 9
`ywU g¨vwUª‡·i gvÎv mgvb bv n‡j †hvM ev we‡qvM Kiv hvq bv| FocusKwi
jÿ¨ Point: wcÖq wkÿv_©x e„›` P‡jv wP‡Î wP‡Î ¸Y wkwL:
`ywU g¨vwUª· mgvb n‡j Zv‡`i Abyiƒc Dcv`vb¸‡jv mgvb n‡e|
`ywU g¨vwUª· A I B Gi ¸Ydj AB wbY©q Kiv hv‡e hw` A g¨vwUª‡·i Kjvg msL¨v A = 4 7 Ges B = 0 1
1 3 1 0
B g¨vwUª‡·i mvwi msL¨vi mgvb nq|
A = [aij]mn Ges B = [bij]nr n‡j hw` AB = [abij]mr nq Z‡e (1 1) + (3 0) (1 0) + (3 1)
AB = 4 7 0 1 =
1 3 1 0
AB g¨vwUª‡·i gvÎv m r n‡e (4 1) + (7 0) (4 0) + (7 1)
g¨vwUª· ¸Y‡bi †ÿ‡Î †h‡Kv‡bv GKwU mvwi ev Kjvg Gi ¸Y †ei K‡i †`L‡jB
Ans. Ack‡b cvIqv hvq|
(1 1) + (3 0) (1 0) + (3 1)
AB = 4 7 0 1 =
1 3 1 0
(4 1) + (7 0) (4 0) + (7 1)
MEx 01 A = 2 3 Ges B = 3 4 n‡j, AB Gi gvÎv KZ?
General Rules & Tips
1g g¨vwUª‡·i Kjvg msL¨v 3 = 2q g¨vwUª‡·i mvwii msL¨v 3
G‡`i ¸Y m¤¢e
AB Gi gvÎv = 2 4
MEx 02 A, B, C wZbwU g¨vwUª· †hLv‡b A34, B45 Ges C[ci j]mn = A[ai j]34 B[bi j]45 n‡j (m, n) Gi gvb KZ? [JU. 17-18]
General Rules & Tips
A g¨vwUª‡·i gvÎv 3 4
B g¨vwUª‡·i gvÎv 4 5
C g¨vwUª‡·i gvÎv 3 5 [∵A3 4, B4 5 Ges C[ci j]mn = A[ai j]34 B[bi j]45]
m = 3; n = 5
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
50 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
4 –4 8 4
MEx 03 hw` 1 A = –1 2 1 nq, Zvn‡j A g¨vwUª·wU wbY©q Ki| [BUET. 17-18]
6 3 –3 6 3
General Rules & Tips
4 –4 8 4
awi, B = 1 ; C = –1 2 1 myZivs B A = C
3 –3 6 3
GLv‡b B Gi µg = 3 1; C Gi µg = 3 3 ; myZivs A Gi µg = 1 3 [∵ B(31) A(13) = C(33)]
4 –4 8 4
awi, A = [x y z] ; 1 [x y z] = –1 2 1 .......... (i)
3 –3 6 3
(i) bs mgxKiY †_‡K cvB, 4x = – 4 x = –1; 4y = 8 y = 2; 4z = 4 z = 1
myZivs wb‡Y©q A = [x y z] = [–1 2 1]
MEx 04 hw` A = Ges A2 = 2 – 3
cos –sin 1 1
; Gi gvb wbY©q Ki|
sin cos 3 1
[RU, Neptune-2. 2021-22; KUET. 09-10]
General Rules & Tips
–sin cos2 – sin2 –2sin cos
A2 = AA =
cos –sin cos
sin cos sin cos 2cossin
=
cos2 – sin2
1 3
2 – 2
1
Avevi, A = 2
cos2 – sin2 = [g¨vwUª‡·i mgZv †_‡K]
2
3 1
2 2
ev, cos2 = cos60 = cos 3 ; ev, 2 = 2n 3 = n 6
0 0 2i
MEx 05 A = 0 –2i 0 , †`LvI †h, A + 4I = 0, I GKwU GKK g¨vwUª·|
2
[KUET. 03-04]
2i 0 0
General Rules & Tips
0 0 2i 0 0 2i 0 0 2i –4 0 0
A = 0 –2i 0 ; A2 = 0 –2i 0 0 –2i 0 = 0 –4 0
2i 0 0 2i 0 0 2i 0 0 0 0 –4
–4 0 0 1 0 0
A2 + 4I = 0 –4 0 + 40 1 0 A2 + 4I = 0 (Showed)
0 0 –4 0 0 1
4 –6
MEx 06 hw` P = Ges P Q = 0 nq Z‡e g¨vwUª· Q KZ?
5
8
[RUET. 14-15]
–2
General Rules & Tips
P GKwU 2 2 gvÎvi g¨vwUª· Ges P Q g¨vwUª‡·i gvÎv 2 1
Q g¨vwUª‡·i gvÎv n‡e 2 1.
4 –6 x 4x – 6y 4x – 6y
awi, Q = y PQ = –2 8 y = –2x + 8y ; kZ© n‡e, –2x + 8y = 0
x 5
4x – 6y = 5 .......... (i); – 2x + 8y = 0 ......... (ii)
(i) & (ii) mgvavb K‡i cvB, x = 2; y = 0.5 Q = 0.5
2
MEx 07 I A‡f`K g¨vwUª· n‡j B g¨vwUª· wbY©q Ki :
4 3
B = I; I =
1 0
2 1 0 1
[KUET. 03-04]
General Rules & Tips
4w + 3y = 1
Let, B = y z ;
w x
2w + y = 0 ; 4x + 3z = 0; 2x + z = 1
2 1 B = 2 1 y z = 2w + y
4 3 4 3 w x 4w + 3y 4x + 3z 1 0
2x + z = 0 1
–1 3 – 1 3
A_©vr, w = 2 , x = 2, y = 1; z = –2 [mgvavb K‡i] ; B = 2 2
1 –2
[we:`ª: 2 1 g¨vwUª‡·i Inverse Matrix B n‡”Q B| Gfv‡eI AsKwU Kiv hvq|]
4 3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 51
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
Dhaka University Jahangirnagar University
01. hw` A, B, C g¨vwUª· wZbwUi AvKvi h_vµ‡g 4 5, 5 4 Ges 4 2 nq,
01. hw` A =
1 0
Ges B 2 1 n‡j AB Gi gvb †KvbwU?
5 0
Z‡e (AT + B)C g¨vwUª·wUi AvKvi wK? [DU-20-21; CU-D, Set-1. 20-21; DU- 0 5
[JU-A, Set-G. 22-23]
7Clg. 19-20; BUET. 10-11; IU.16-17; BRU-E. 19-20; MBSTU-C. 19-20; e.‡ev. 2019]
A. 2 5 B. 10 5 C. 2 6 D. 12 5
5 0 5 0 6 0 8 1
A. 4 2 B. 5 4
C. 5 2 D. 2 5
S B Sol n AB = 0 5 2 1 = 0 + 10 0 + 5 = 10 5
1 0 5 0 5+0 0+0 5 0
olve
T
S C Sol n (A + B) Gi gvÎv 5 × 4 ; C Gi gvÎv 4 × 2
olve
(A VT + B) C Gi gvÎv 5 × 2 3 5 1
2 2 02. A = 4 0 2 n‡j, A-2I Gi gvb †KvbwU?
02. A = 2 2, AB = ?
[JU-A, Set-N. 22-23]
2 2
Ges B = 3 3 [DU. 14-15; RU. 18-19] 1 6 4
2 2 2 5 5 1 1 5 1
A. 0 0 0
2 0
2 2 B. 0 0 C. 3 0 D. 2 2 A. 4 2 2 B. 4 2 2
1 6 6 1 6 2
2 2 2 2
S D Sol n AB = 2 2 3 3 1 1 3 1
olve
1 3
C. 4 0 2 D. 2 2 0
2.2 + (2).3 2.2 + (2).3 4 6 4 6 2 2
= 1 6 4 1 4 2
(2).2 + 2.3 (2).2 + 2.3= 4 + 6 4 + 6 = 2 2
3 5 1 2 0 0 1 5 1
03. hw` A =
2 0 3 0 nq, Z‡e AB mgvb-
n A 2I = 4 0 2 0 2 0 = 4 2 2
olve
0 3
B = 5 1 S B Sol
1 6 4 0 0 2 1 6 2
[DU. 05-06; JnU. 05-06; JU. 19-20]
1 03. A g¨vwUª‡·i µg 2 3 Ges B g¨vwUª‡·i µg 3 2 n‡j AB Gi µg
A.
6 0 B. 3 1 0 D. 1 2 †KvbwU?
15 3 2 5 2 15 0 5
C. [JU-A, Set-N. 22-23]
A. 2 2 B. 2 3 C. 3 2 D. 3 3
S A Sol n AB = 0 3 5 1
2 0 3 0
olve
n AB Gi gvÎv = A Gi gvÎv B Gi gvÎv
olve
S A Sol
=
23 + 05 20 + 01 6 0 =2 33 2=22
03 35 00 31 = 15 3
2 2
04. hw` A = , Ges B =
2 3 2 2
04. hw` A = nq, Z‡e 2
mgvb- 2 3 3
, n‡j AB Gi gvb †KvbwU?
3 2
A [DU. 04-05, DU. 03-04; RU. 06-07; JU.18-19] 2
[JU-A, Set-R. 22-23]
5 12 5 12 C. 5 12 D. 5 12
A. 12 5 12 5 A.
2 2 B. 0 0
0 0
12 5 B.
12 5 2 2
2 3 2 3
S D Sol n A = 3 2 3 2 C. 3 0 D.
2 2
2 0 0
olve
2 2
22 33 23 32 5 12
= 2
S D Sol n AB = 2 2 3 3
2 2 2
32 + 23 33 + 22 12 5
=
olve
✍ Written =
46 4 6 2 2
4 + 6 4 + 6 2 2
=
2 4 6
01. A = [1 2 3], BA = 3 6 9 n‡j B g¨vwUª·wU wbY©q Ki| [DU-A. 2021-22] 05. A = 3 6 , B = 2 , n‡j AB Gi µg †KvbwU? [JU-A, Set-S. 22-23]
1 2 3 2 2 1
2 4 6 A. 1 2 B. 21 C. 1 1 D. 2 2
Solve †`Iqv Av‡Q, A = [1 2 3]; BA = 3 6 9 n A Gi µg 2 2; B Gi µg 2 1
olve
S B Sol
1 2 3 AB Gi µg 2 2 2 1 = 2 1
2 4 6 1
awi, 3 6 9 = C
06. A = 2 Ges B = (4 5 6) n‡j AB = ?
1 2 3 3
[JU-A, Set-I. 2021-22]
GLv‡b, A g¨vwUª‡·i gvÎv n‡jv 1×3 g¨vwUª· Ges C g¨vwUª‡·i gvÎv 3×3
myZivs B g¨vwUª‡·i gvÎv n‡e 3×1
4
A. (4 10 18) B. 10
a 18
Zvn‡j awi, B = b 4 5 6
c C. 8 10 12 D. Am¤¢e
a 2 4 6 a 2a 3a 2 4 6 12 15 18
kZ©g‡Z, b [1 2 3] = 3 6 9 b 2b 3b = 3 6 9 1
c 1 2 3 c 2c 3c 1 2 3
n Option test: A = 2 Ges B = (4 5 6)
olve
S C Sol
2 3
a = 2, b = 3 Ges C = 1 B = 3 A Gi gvÎv 3 × 1 Ges B Gi gvÎv 1×3
1 AB Gi gvÎv n‡e 3×3 hv Ackb C †Z Av‡Q|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
52 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
–1 0 –1 3 4 x 5
07. A = Ges B = 5 0 n‡j, AB Gi gvb KZ? [JU-A, Set-Q. 2021-22] 14.
2
5 2 4 6 y 8
= n‡j x I y Gi gvb KZ? [JU. 14-15]
5 –5 –2 –5 –5
B. 10 –5 C. 10 4 D. 6 5
0 2 2 A. 1, – 2 B. – 1, 2 C. –1, – 2 D. 2, – 1
A. 1 2 5 5
S A Sol n 4 6 y = 8 4x + 6y = 8
3 4 x 3x + 4y
olve
n AB =
2 –1 0 –1 –5 –2
olve
S B Sol 5 2 5 0 = 10 –5 3x + 4y = –5 Ges 4x + 6y = –8
1 2 3 0 1 myZivs mgxKiYØq mgvavb K‡i cvB, x = 1 Ges y = – 2
08. A = 4 5 6 Ges B = 2 3 n‡j A + B = ? [JU-A, Set-F. 2021-22]
7 8 9 4 5 2 3
15. A = Ges B = 3 5 n‡j AB = ?
4 7
0 1 [JU. 09-10]
1 3 3 1 2 3 1 29
A. 6 8 6 B. 4 7 9 A.
1 29
B.
1 29
C. 1 29
11 13 9 7 12 14 3 5 3 5 3 5 D. 3 5
2 3
1 3 S A Sol n A = 0 1 Ges B = 3 5
4 7
olve
C. 6 8 D. Am¤¢e
7 8 8 9 14 + 15 1 29
AB =
+ 3 0 5 3 5
n A Gi gvÎv 3×3 Ges B Gi gvÎv 3 × 2|
=
0
olve
S D Sol
†h‡nZz gvÎv mgvb bq ZvB †hvM Kiv Am¤¢e| 16. hw` A GKwU m n AvKv‡ii g¨vwU· Ges B GKwU n p AvKv‡ii g¨vwUª·
nq Zvn‡j Zv‡`i ¸Ydj AB Gi AvKvi n‡e| [JU. 09-10; iv.‡ev. 2019]
09. X
1 2
3 4
= (5 6) n‡j g¨vwUª · X = KZ? [JU-A, Set-M. 2021-22] A. n n B. n p C. m p D. m m
1
C.
n
olve
S C Sol A M n n p B
A. (2 1) B. (1 2) D. Am¤¢e AB Gi AvKvi m p
2
S B Sol n X3 4 = (5 6)
1 2 Rajshahi University
olve
3 1
01. P = Ges Q = 2 3 n‡j, P Q g¨vwUª·wU †Kvb ai‡bi?
2 2
awi, X Gi gvÎv 1 × 2 x = [a b]
2 2
(a b) 3 2 = (5 6) (a + 3b 2a + 4b) = (5 6)
1 4 [RU-C, Corundum-1. 22-23]
A. k~b¨NvwZ B. A‡f`NvwZ
a + 3b = 5 ..... (i); 2a + 4b = 6 ...... (ii) C. k~b¨ g¨vwUª· D. †KvbwUB bq
(i) n‡Z a = 5 – 3b; (ii) n‡Z, 2 (5 – 3b) + 4b = 6
3 1
S D Sol n P – Q = 2 2 – 2 3
2 2
10 – 6b + 4b = 6 b = 2 a = – 1 X = (–1 2)
olve
1 2 3 – 2 1 2 1 – 3
10. hw` A = = 2 – 2
1 2 3
4 5 6
Ges B = 1 2 nq Z‡e AB = ? [JU. 18-19] – 2 + 3 0 1
=
0 1 i – 1
02. A =
1 i
3 3 3 3 , B = Ges i = – 1 n‡j, AB = KZ?
3 3
A. 9 12
B. 9
3 3
C. 9 –i 1 –1 – i
12 12 D. 9 12 [RU-C, Quartz-2. 22-23; CU. 18-19; DU. 12-13; RU. 17-18; CU. 14-15]
1 2 A. 0 0
1 0
B. 0 0
0 0
C. 0 1
1 0
D. 0 i
1 0
S A Sol n A = 4 5 6 Ges B = 1 2
1 2 3
olve
0 1 n AB =
1 i i 1 i i 1 i2 0 0
i 1 1 i = i2 1 i i = 0 0
olve
1 + 2 + 0 2 + 4 3 3 3 S B Sol
AB = 4 + 5 + 0 = 9 12
8 + 10 6
03. P + Q =
1 2
Ges P Q = 7 8 n‡j, P = KZ?
5 6
11. g¨vwUª· M Gi AvKvi 4 3 Ges N Gi AvKvi 3 5 n‡j MN Gi 3 4
[RU-C, Topaz-3. 22-23]
AvKvi †KvbwU? [JU. 17-18, 11-12, 09-10]
4 3
A. B. 5 6
3 4
A. 4 4 B. 4 5 C. 3 5 D. 3 4 6 5
n M43 3 5 N MN = 4 5
olve
S B Sol 1 2 5 6
C. 5 D. 5
7 8 9 6 8
12. hw` A = 2 1 7 Ges C = A + I33 Zvn‡j C23 = ?
S B Sol n P + Q + P Q = 3 4 + 7 8
[JU. 16-17] 1 2 5 6
6 5 2
olve
A. 6 B. 7 C. 8 D. 3
2P = 10 12 P = 5 6
6 8 3 4
7 8 9 1 0 0
S B Sol n A = 2 1 7 Ges I3 3 = 0 1 0 04. A = (aij)mn I B = (bij)nn n‡j, ABm Gi Kjvg msL¨v KZ?
olve
6 5 2 0 0 1 [RU-Neptune-2, Set-1. 21-22]
8 8 9 A. m B. n
C = A + I33 = 2 2 7 C23 = 7 C. m + n D. ‡KvbwUB bq
6 5 3 n A = (a )
olve
S B Sol ij m n ; B = (b ij)n n
2x y 5 6 5
13. hw` nq Z‡e x Gi gvb KZ? Bm = (bij)n n ABm = m n n n
y 3 2
= [JU. 15-16]
3
A. 0 B. 1 C. 2 D. 3 ABm = m n
n 2x – y = 6 ; y = –2 2x + 2 = 6 [∵y = –2] x = 2
Kjvg msL¨v = n
olve
S C Sol
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 53
05. A = 11. A = , B = , X = Ges AX = B n‡j (x, y) = ? [RU. 17-18]
3 1
, B = , X = Ges AX = B n‡j, (x, y) = KZ?
6 x 0 1 1 y
2 0 4 y 1 0 2 x
[RU-C, Jupitar-1, Set-1. 2021-22] A. (0, 0) B. (1, 2) C. (2, 1) D. (1, 1)
A. (2, 2) B. (2, 0) C. (2, 4) D. (4, – 3)
n AX = B = = 1 x = 1, y = 2
0 1 y 1 x
olve
n AX = B
3 1 x 6 3x + y
6 S B Sol 1 0 x 2 y 2
2 0 y = 4 2x = 4
olve
S B Sol
12. hw` A =
cos2 –sin2
sin2 cos2 Ges A = I nq, Z‡e Gi gvb KZ? [RU. 17-18]
2
kZ©g‡Z, 2x = 4 x = 2 Ges 3x + y = 6
(x, y) = (2, 0) 32+y=6y=0 A. 0, 30 B. 30, 45 C. 0, 45 D. 45, 60
hw` A = 3 4 Z‡e A2 + 3A – 10I n‡e GKwUÑ [RU. Moderna, Set-2. 20-21] S C Sol n A = sin2 cos2
–sin2
1 2 cos2
06.
olve
A. A‡f`K g¨vwÆ· B. cÖwZmg g¨vwÆ· C. k~b¨ g¨vwÆ· D. †KvbwUB bq
A2 =
cos2 –sin2 cos2 –sin2
sin2 cos2 sin2 cos2
S C Sol n A = 3 4 . 3 4 †Kv‡bv g¨vwUª· A Gi Rb¨,
2 1 2 1 2 Aspect Special:
olve
2 2
2 – sin 2 –2sin2
cos cos2 cos4 –sin4
28 6
=
2sin2 cos2 cos 2 – sin22 sin4 cos4
=
=
2
1 + 6 7 A †Uªm A + |A| I = 0
2
3 12 6 + 16 9 22
=
cos4 –sin4 1 0
GLv‡b A = 3 –4
1 2 2
A2 + 3A10I sin4 cos4 = 0 1 [ A = 1]
7 6 cos4 = 1 4 = 0, 2 Ges sin4 = 0 4 = 0,
= 1 2 101 0 A2 – (1 – 4) A + (– 4 – 6) I = 0
0 1 A2 + 3A – 10 I = 0
9 22 3 4
+ 3
= 0, = 0, 90 = 0, ev, 0, 45
7 + 3 10 6 + 6 0 0 0 2 4
= = 0 0 3, 4 2, 5 4 n‡j,
9 + 9 0 22 12 10 13 . wZbwU g¨vwUª· A, B I C Gi order h_vµ‡g 2
CBA g¨vwUª · wUi order KZ n‡e?
hw` A = 3 4 Ges A2 kA 5I = 0 n‡j k Gi gvb KZ?
1 3 [RU. 17-18]
07. A. 5 3 B. 5 4 C. 2 5 D. 4 3
[RU. Moderna, Set-2. 20-21] C B A
S A Sol n
A. 5 B. 3 C. 7 D. †KvbwUB bq olve 54 42 23
S A Sol n Aspect Special: A = 3 4
1 3
olve
53
14. [a b] Ges g¨vwUª·Ø‡qi ¸Ydj n‡eÑ
A – (1 + 4) A + (4 – 9) I = 0 A – 5A – 5I = 0
2 2 a
[RU. 17-18]
b
†`Iqv Av‡Q, A2 – kA – 5I = 0 k = 5 2
C. b2
a
A. [a2b2] B. [a2 + b2] D. gvb †bB
hw` A = x x Ges A1 = 1 2 nq, Zvn‡j x Gi gvb KZ?
2x 0 1 0
08.
S B Sol n [a b] b = [a + b ]
a 2 2
olve
[RU. Astrazeneca, Set-1. 20-21]
1 1
A. 2 B. C. 1 D. 0 2
15. hw` A =
2 2 0 1 0
1 2 3
Ges B = 1 2 nq Z‡e wb‡Pi †KvbwU mZ¨?
n A = 2x 0 , A–1 = 1 0 0 1
olve
S D Sol x x –1 2 [RU-H. 16-17]
–1 2x 0 1 0 1 A. AB = BA B. AB = B2 C. A = 2B D. AB BA
AA = I 0 2x = 0 1 2x = 1 x =
2 S D Sol n A 2 3 3 2 B AB = 2 2
olve
09. 3 2 x = 5 n‡j x Ges y Gi gvb KZ? B 3 2 2 3 A BA = 3 3
2 –2 y 7 [RU. 18-19]
AB BA
A. x = –2, y = 3 B. x = 2, y = –3 1 2 0
16. hw` A = [3 4] Ges B = nq Z‡e AB wbY©q Ki- [RU. 16-17]
C. x = 2, y = 3 D. ‡KvbwUB bq 4 5 3
S D Sol n 2 –2 y = 7 2x – 2y = 7
3 2 x 5 3x + 2y 5 A. [17 14 –13] B. [19 14 –12]
olve
C. [18 14 –11] D. [20 14 –10]
3x + 2y = 5 ........... (i) ; 2x – 2y = 7 ........... (ii) 2
S B Sol n AB = [3 4].4 5 3 = [3 + 16 6 + 20 012]
1 0
olve
12
= 19 14 12
(i) + (ii) K‡i cvB, 5x = 12 ; x =
5
12
x Gi gvb (i) bs G cvB, 3. + 2y = 5 y = –
11 17. hw` A GKwU 2 5 gvÎvi Ges B GKwU 5 2 gvÎvi g¨vwUª· nq, Z‡e
5 10 BA-Gi gvÎv KZ n‡e? [RU. 15-16; IU. 13-14, 05-06, 02-03]
2 –2 –4 A . 2 2 B . 5 5 C . 25 D. 5 2
10. g¨vwUª· A = –1 3 4 n‡j A2 Gi gvb KZ? [RU. 17-18; CU. 18-19] n B5 2 2 5A
olve
1 –2 –3 S B Sol
BA-Gi gvÎv = 5 5
A. –A B. 0 C. 2A D. A
18. hw` A I B `yBwU g¨vwUª· nq Ges A + I2 = B nq, Z‡e B-Gi gvÎv KZ?
2 –2 –4 2 –2 –4 [RU. 15-16]
S D Sol n A = –1 3 4 –1 3 4
2
olve
A. 2 3 B. n n C. 3 3 D. 2 2
1 –2 –3 1 –2 –3
S D Sol n Avgiv Rvwb, ïaygvÎ GKB gvÎvi g¨vwUª‡·i g‡a¨ †hvM m¤¢e
olve
4 + 2 – 4 –4 – 6 + 8 –8 – 8 + 12 2 –2 –4
= –2 – 3 + 4 2+9–8 4 + 12 – 12 = –1 3 4 = A Ges †hvMdjI GKB gvÎvi nq| myZivs I2 Gi gvÎv †h‡nZz 2 2, ZvB
2 + 2 – 3 –2 – 6 + 6 –4 – 8 + 9 1 –2 –3 A + I2 Gi gvÎvI 2 2 n‡e| B-Gi gvÎv 2 2
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
54 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
19. x y 1 = 8 1 n‡j, (x, y) = ? 06. M =
2 0
Ges N = 0 x n‡j x-Gi gvb KZ n‡e MN = 0 0
0 0 00
7 x + y 7 2 [RU. 12-13; DU. 02-03]
0 0
A. (5, 3) B. (5, 3) [CU-C3. 16-17]
C. (5, 3) D. (5, 3) A. 1 B. 2 C. 3
D. †h‡Kv‡bv ev¯Íe msL¨v E. 4
S B Sol n 7 x + y = 7 2 n‡j, x y = 8 ...... (i)
x y 1 8 1
olve
S D Sol n MN = 0 0 0 x = 0 0 x Gi gvb †h‡Kv‡bv ev¯Íe
2 0 0 0 0 0
olve
Ges x + y = 2 ...... (ii)
msL¨vi Rb¨ MN = 0 n‡e|
GLb (i) I (ii) bs mgxKiY mgvavb K‡i cvB- (x, y) = (5, 3)
0 2 3 1 2 3
20. A =
0 1
[RU. 09-10] 07. X = 4 0 5 Ges Y = 4 1 5 `yÕwU g¨vwUª· n‡j |X – Y| = ?
2
1 0
n‡j A Gi gvb KZ?
6 7 0 6 7 1
C. 1 0 D. 0 1
0 1 1 0
A. 1 B. +1 [CU-C3. 16-17]
A. 1 B. –1 C. 0
S D Sol n A = 1 0 ; A = A.A = 1 0 1 0 = 0 1
0 1 2 0 1 0 1 1 0 D. 60 E. †ei Kiv hv‡e bv
olve
0–1 2–2 3–3
–1 0 0
Chittagong University S B Sol n X – Y = 4 – 4 0 – 1 5 – 5 = 0 –1 0
olve
1 6 – 6 7 – 7 0 – 1 0 0 –1
01. P = 1 Ges Q = [1 1 1] n‡j, PQ = ? [CU-A, Shift-2. 22-23] |X – Y| = – 1 – 1 – 1 = – 1
1 08.
1 + i2 0 x 0
n‡j x I y Gi gvb KZ? [CU-C3. 16-17]
0 1 – i2 y 1
=
1 1 1 1 1 1 1
B. 1 1 1 C. [3] D. 1 1 1
1
A. 1 A. x = y = 0 B. x 0, (1 + i2)
1 5
1 1 1 1 1 1 1
1 1 1 C. x 0, (1 – i2) D. x 0, y 0
1
n PQ = –1 × [1 1 1] = 1 1 1
5
olve
S D Sol (1 + i2)
1 1 1 1 E. x = 0, y =
5
1 0 0
S E Sol 0 1 – i2 y = 1 y(1 – i2) = 1
n
1 + i2 0 x 0 x(1 + i2) 0
olve
02. hw` P = 0 1 0 , Z‡e P2 + 2P = ? [CU-A, Set-1. 20-21]
0 0 1 x (1 + i2) = 0 x = 0
A. P B. 2P C. 3P D. 4P 1 (1 + i2) (1 + i2)
y(1 + i2) = 1 y = = =
1 0 0 1 – i2 (1 + i2) (1 – i2) 5
S C Sol n P = 0 1 0 = I P2 + 2P = I + 2I = 3I = 3P
olve
0 0 1 x –a –1 0 0
09. X = y, Y = b Ges Z = 0 2 0 g¨vwUª·¸‡jv hw` ZX = Y
03. g¨vwUª‡·i ¸Yb bxwZi Rb¨ wb‡Pi †Kvb e³e¨wU mwVK? [CU-A, Set-2. 20-21] z c 0 0 –3
A. cÖ_g g¨vwƇ·i Kjvg msL¨v wØZxq g¨vwƇ·i mvwi msL¨vi mgvb n‡j mgxKiYwU wm× K‡i, Zvn‡j X = ? [CU-C3. 16-17]
`ywU g¨vwÆ· ¸Yb Kiv hv‡e| –a –a
a
B. cÖ_g g¨vwƇ·i mvwi msL¨v wØZxq g¨vwƇ·i Kjvg msL¨vi mgvb n‡j
`ywU g¨vwÆ· ¸Yb Kiv hv‡e| A.
b
2 B.
b
2 C.
– b
2
–a
D. –b
–c
C. cÖ_g g¨vwƇ·i mvwi msL¨v wØZxq g¨vwƇ·i mvwi msL¨vi mgvb n‡j `ywU c c c
–
g¨vwÆ· ¸Yb Kiv hv‡e| 3 3 3
D. cÖ_g g¨vwƇ·i Kjvg msL¨v wØZxq g¨vwƇ·i Kjvg msL¨v mgvb n‡j –x –x –a
n ZX =
2y = b
olve
S A Sol 2y ; ZX = Y
`ywU g¨vwÆ· ¸Yb Kiv hv‡e| Ans A
–3z –3z c
–14
04. x-Gi gvb KZ n‡j, = nq- [CU. 18-19]
2 5 1 3 12 b
4 3 3 x 10 0 – x = – a x = a ; 2y = b y =
2
A. 0 B. 4 C. –4 D. 3
a
–14
S C Sol n 4 3 3 x = 10 0
2 5 1 3 12
x
b
olve
c
– 3z = c z = – X = y = 2
12 –14 12 –14
4 + 9 12 + 3x = 10 0 13 12 + 3x = 10 0
2 + 10 6 + 5x 12 6 + 5x 3 z –c
12 + 3x = 0 ev, x = –4
3
5 0 –1
2 3
1 0 0 1 2 3 10. hw` A = I B = 3 0 nq, Z‡e AB = ?
2 1 3
[CU-G. 16-17]
05. X = 0 1 0 Ges Y = 3 2 1 n‡j XY = KZ? [CU-C3. 16-17] 1 2
0 0 1 2 1 3
A. 12 10 B. 10 12
9 13 9 13
6 0 0 1 0 0
A. 0 6 0 B. 0 2 0
C. 12 13 D. 10 14
9 10 9 13
0 0 6 0 0 3
C. Y D. †ei Kiv m¤¢e bq 10 + 0 – 1 15 + 0 – 2
S B Sol n AB = 4 + 3 + 3 6 + 0 + 6 = 10 12
9 13
olve
S C Sol n X g¨vwUª·wU A‡f` e‡j XY = Y
olve
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 55
1 1 1 0 0 –1 0 0
11. A = n‡j (A2 –2I) Gi gvb nq-
2 3 17. S = 0 1 0 Avi = 0 –1 0 n‡j S
[CU. 16-17]
[CU. 08-09]
3 4 3 4 3 4 3 4 0 0 1 0 0 –1
A. 5 B. C. 8 D.
8 8 5 5 8 5 A. = 2S B. > 2S
1 D. = –2S2
S D Sol A = 2 3
n 1 C. < S E. †Kv‡bvwUB bq
olve
1 0 0 1 0 0
A2 –2I = 2
1 1 1 1
– 2 0 1 =
1 –4 2 0 3 4 S D Sol n S = 0 1 0 = – 0 1 0 S = I, = –I
olve
1 0
–
3 2 3 8 7 0 2 8 5 0 0 1 0 0 1
=
1 1 AZGe, S = ev, S = –I = 2 = S2
12. P = n‡j P2 = ?
2 3
[CU. 15-16]
GST (¸”Q)
1 4 1 4 1 4
A. B. C. 01. A Ges (AT + B)C g¨vwUª· `yBwUi µg h_vµ‡g 4 5 Ges 5 2 n‡j C
8 7 8 7 7 8
1 4 1 4 g¨vwUª· Gi µg Kx n‡e? [GST-A. 22-23]
D. E. A. 4 2 B. 4 3 C. 4 4 D. 4 5
8 7 8 7
n (AT + B)C Gi µg 5 2
olve
1 1 1 1 1–2 1–3 1 4
S E Sol n P = 2 3 2 3 = 2 + 6 – 2 + 9 = 8 7
2 S A Sol
olve
Avevi, A Gi µg 4 5
a 0 1 (AT + B) Gi µg 5 4 C n‡e 4 2
13. hw` 3 2 5 = 4 nq, Z‡e ÔaÕ Gi gvb KZ? [CU-A. 15-16]
A =
1 2
4 0 3 02.
4 3
Ges A2+2A 11X = 0 n‡j X Gi gvb KZ? [KU. 19-20]
A. 2 B. 3 C. 5
A. 0 11 B. 0 1
11 0 1 0
D. 6 E. 7
a 0 1
C. 0 2 D. 0 13
2 0 13 0
S A Sol n 3 2 5 = 4 a(6 – 0) + 1(0 – 8) = 4 6a = 12 a = 2
olve
4 0 3
S B Sol n A = 4 3
1 2
14. A = Ges B = –3 2 n‡j, g¨vwUª· C wbY©q Ki hv‡Z
olve
3 7
2 5 4 –1 1 2 1 2 9 4
5C + 2B = A nq| A2 = A.A =
4 3 4 3 8 17
[CU. 14-15] =
1 9 3
B. –6 7 C. 0 1
9 3 1 0
9 4
A2 + 2A 11X = 0 1 2
A. –6 7
8 17 + 24 3 11X = 0
5
–3 –7
D. –2 –5 9 4 2 4
E. †Kv‡bvwUB bq
8 17 + 8 6 11X = 0
S A Sol n 5C + 2B = A
olve
0 11 11X = 0 X = 0 11 = 0 1
11 0 1 11 0 1 0
–3 2 –6 4
5C = A – 2B = 2 5 – 2 4 –1 = 2 5 – 8 –2 = –6 7
3 7 3 7 9 3 11
2x – y 5
hw` 3 y = 3 2 nq Z‡e x = ?
6 5
03.
C = –6 7
1 9 3 [CoU-A. 18-19]
5 A. 0 B. 1 C. 2 D. 3
–1
15. S =
0
, =
i 0
n‡j S2 n‡eÑ n 2x – y 5 = 6 5
0 –1 0 i S Sol 3 y 3 2
[CU. 13-14]
olve
A. S2 B. 3S 2x – y = 6 Ges y = 2 2x – 2 = 6 2x = 8 x = 4
C. –S D. E. S
A =
0 1
, X = , B Ges AX = B n‡j (x, y) = ? [IU. 15-16]
x 1
2 0
–1 04.
S C Sol n S = 0 –1, = 0 i
0 i 0 y 2
olve
A. (0, 0) B. (1, 2)
i2 0 –1 0
2 = . = 0 i2 = 0 –1 C. (2, 1) D. (1, 1)
S D Sol n A = 2 0, X = y, B 2 Ges AX = B
0 1 x 1
–1 0 –1 0 –1 0
S2 = 0 –1 0 –1 = 0 1 = – 0 –1 = – S
olve
1 0
AX = 2x = 2 y = 1 Ges 2x = 2 x = 1
y 1
1 2 3 –1 –2 –3
16. 4 5 6 = –4 –5 –6 n‡j Gi gvb n‡”Q [CU. 08-09]
7 8 9 –7 –8 –9 4 0 x = 12 n‡j x I y Gi gvb-
A. 1 B. –1
05.
–2 3 y 3 [IU. 15-16]
–3
A. B. C. D.
2 3 4
C. 1 D. 0
3 3 3 3
E. c~e©eZ©x †KvbwUB bq
4 0
x 12 4x 12
1 2 3 –1 –2 –3 S C Sol n –2 3 y = 3 –2x + 3y = 3
olve
S B Sol n 4 5 6 = –4 –5 –6
olve
7 8 9 –7 –8 –9 4x = 12 x = 3
Ges –2x + 3y = 3 –6 + 3y = 3 ; [∵ x = 3]
1 2 3 1 2 3
ev, 4 5 6 = –1 4 5 6 = –1 3y = 9 y = 3 =
x 3
7 8 9 7 8 9 y 3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
56 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
13. g¨vwUª· ¸Y‡bi †ÿ‡Î, wb‡Pi †KvbwU mZ¨?
06. A =
2
1 [PUST. 17-18]
3
Ges B = [2 3] n‡j Zv‡`i †hvMdj KZ? [KU. 12-13]
A. –1 (1 – 1) = (0 0) B. –1 (1 – 1) = 0
4 1 1 0
1
A.
5 1 2
3
B. 2 0 1 –1
C. –1 (1 – 1) = –1 D. –1 (1 – 1) = (2)
4 1 1
1
C.
5 1
0 D. †hvM Kiv hv‡e bv 1 1 1 (–1)
S C Sol n –1 (1 – 1) = (–1) 1 (–1) (–1) = –1 1
2 1 1 –1
olve
n A g¨vwUª‡·i gvÎv 2 2 Ges B g¨vwUª‡·i gvÎv 1 2
olve
S D Sol
14. A Ges B g¨vwUª· `ywUi gvÎv (Order) h_vµ‡g 2 3 Ges 3 5 n‡j
†h‡nZz g¨vwUª·Ø‡qi gvÎv mgvb bq ZvB †hvM Kiv hv‡e bv|
wb‡Pi †KvbwU mZ¨Ñ [SUST. 07-08]
0 1
07. hw` A = Ges X = y nq Zvn‡j AX mgvb-
x
1 0
[JnU. 09-10] A. BA msÁvwqZ n‡e B. AB msÁvwqZ n‡e
x C. AB Gi gvÎv n‡e 52 D. AB Gi gvÎv n‡e 33
A. AX = B. AX =
y
n GLv‡b, A g¨vwUª‡·i gvÎv = 2 3; B g¨vwUª‡·i gvÎv = 3 5
x
olve
y S B Sol
y
C. AX = D. AX =
x AB ¸Yb‡hvM¨ Ges Gi gvÎv = 3 5 Avevi, BA Am¤¢e
y x Engineering
1 y
S D Sol n AX = 1 0 .y = 1.x + 0.y = x
0 x 0.x 1.y
01. X g¨vwUª·wU †ei Ki hLb 2X +
1 2 3 8
olve
3 4 7 2
= [CKRUET. 2021-22]
2 1 –3
A. 2 – 1 B. 2 – 1 C. 4 – 2
1 3 2 6
08. gvb wbY©q Ki- [4 5 6] 3 [KU. 09-10]
1 2 –6 –2 –6
8 D. 4 – 2 E. –4 2
A. [8 15 6] B. [17] C. 15 D. [ 18 ]
S A Sol n 2X + 3 4 = 7 2 2X = 7 2 – 3 4
1 2 3 8 3 8 1 2
6
2 olve
2X = 4 –2 X = 2 –1
2 6 1 3
n [4 5 6] 3 = [8 + 15 6] = [17]
olve
S B Sol
1
02. wZbwU g¨vwU· [x y]
a h x
h b y
Gi ¸Yd‡ji gvb n‡e-
09. hw` A =
3 0 , B = 0 nq Z‡e AB mgvb-
2
5 1 0 3 [JnU. 08-09]
[BUET. 12-13; RU. 17-18; CU. 13-14; BRU-E. 19-20]
3 1 A. [x2a+xyh xyh+y2b] B. [x2a+2xyh+y2b]
A. 15 B. 2
6 0
C. 10
6 0
D. 10 3
6 0
3 3
2
5 C. xyh + y2b
x a + xyh
D. [x2a+xyh+2y2b]
3.2 + 0.0 3.0 0.3
S C Sol n AB = 5 1.0 3= 5.2 + 1.0 5.0 1.3 = 10 3
3 0 2 0 6 0
olve
S B Sol n [x y] h b y = [ax + hy hx + by] y
a h x x
olve
5
– 2 = [ a x2 + hxy + hxy + b y2] = [x2a + 2xyh + y2b]
,B=
2 2 4
10. A = 3 5 n‡j, AB = ? [JUST-A. 19-20]
03. hw` P =
4 6
2 8
Ges P Q = 0 nq Z‡e g¨vwUª· Q KZ? [RUET. 14-15]
5
3
–1
2 16 2 16 2
A. 0.5 B. 0.5 D.
2
C. –1 C. [0.5 2]
A. 0 B. 1 D. 2 3 3 0.5
5 n P GKwU 2 2 gvÎvi g¨vwUª· Ges P Q g¨vwUª‡·i gvÎv
olve
– 2 S A Sol
,B=
2 2 4 1 0
n A=
2 1 ; Q g¨vwUª‡·i gvÎv n‡e 2 1.
olve
S B Sol 3 5 ; AB = 0 1 = 1
3
–1 4 –6x 4x – 6y
awi, Q = y PQ =
x
2
–2 8y = –2x + 8y
11. A =
4 3
, B =
4 2 4x – 6y
kZ© n‡e, –2x + 8y = 0 ; 4x – 6y = 5 ........ (i); – 2x + 8y = 0 ......... (ii)
n‡j- 5
2 1 3 1
[BSFMSTU. 19-20]
(i) A – B = –1 0
0 1
(ii) AT = B (iii)AB BA (i) & (ii) mgvavb K‡i cvB, x = 2; y = 0.5 Q = 0.5
2
A. (i) I (ii) B. (ii) I (iii)
C. (i) I (iii) D. (i), (ii) I (iii)
0 0 2i
04. A = 0 2i 0 n‡j A2 + 4I = ? (I GKwU GKK g¨vwUª·) [KUET. 03-04]
2i 0 0
S D Sol n A = 2 1 ; B = 3 1 ; A B = 1 0
4 3 4 2 0 1
olve
A. 0 B. 1 C. –8i D. 8i
AT = 3 1 = B, AB = 11 5 ; BA = 14 10
4 2 25 11 20 14 0 0 2i
S A Sol n A = 0 2i 0
olve
AB BA 2i 0 0
12. P GKwU 2 3 g¨vwUª· Ges Q GKwU 3 4 g¨vwUª· n‡j QP- [MBSTU-A. 19-20] 0 0 2i 0 0 2i 4 0 0
A. 2 4 g¨vwUª· B. 4 2 g¨vwUª· A2 = 0 2i 0 0 2i 0 = 0 4 0
C. 3 3 g¨vwUª· D. A¸bb †hvM¨ 2i 0 0 2i 0 0 0 0 4
n P g¨vwUª· Gi AvKvi 2 3, Q g¨vwUª‡·i AvKvi 3 4 4 0 0 1 0 0
olve
S D Sol A2 + 4I = 0 4 0 + 40 1 0 A2 +4I = 0
QP g¨vwUª‡·i AvKvi (3 4) (2 3) [hv A¸Yb‡hvM¨] 0 0 4 0 0 1
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 57
【
? 】 QUICK PRACTICE CONCEPT TEST ?
01. hw` A = 1 2 nq, Z‡e A2 + 2A-11I Gi gvb KZ? 03. A = [aij]m n Ges B = [bij]m n n‡j A + B g¨vwUª·wUi µg KZ n‡e?
4 –3 A. m n B. m m C. n m D. n m
A. 1 B. –1
1 1
04. hw` A = IB=
2 0
C. 2 D. 0 5 2 5 0 nq, Z‡e AB Gi gvb †KvbwU? [JU. 16-17]
02.
3 0 x 30
5 2 5 2 5 2 5 2
0 5 y = 30 n‡j x Gi gvb KZ? A. B. D. 5
1 2 10 5 C. 10 4 6
A. 4 B. 6
C. 10 D. 16 Answer 01.D 02.C 03.A 04.B
Concept-04 e¨wZµgx g¨vwUª· ¸iæZ¡:
†h g¨vwUª‡·i wbY©vq‡Ki gvb k~b¨ Zv‡K Singular ev e¨wZµgx g¨vwUª· e‡j| †hgb: 8 4
2 1
a3
MEx 01
5
4 a 4
g¨vwUª·wU e¨wZµgx n‡j a Gi gvb n‡e- [IU. 16-17]
General Rules & Tips
a – 3 5
4 a – 4 g¨vwUª·wU e¨wZµgx e‡j wbY©vq‡Ki gvb k~b¨|
(a – 3) (a – 4) – 20 = 0 a2 – 7a – 8 = 0 a2 – 8a + a – 8 = 0 (a – 8) (a + 1) = 0 a = 8, –1
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
S C Sol n GLv‡b, C Ack‡bi g¨vwUª·wU Ae¨wZµgx KviY D³ g¨vwUª‡·i
olve
Dhaka University
m – 2 6 g¨vwUª·wU e¨wZµgx n‡e hw` m Gi gvbÑ wbY©vq‡Ki gvb (10 + 4) = 14 0
01. 2 m – 3
04. x-Gi gvb KZ n‡j
x + 3 2
[DU. 11-12; RU.17-18; DU-cÖhyw³: 19-20] 5 x
g¨vwUª·wU e¨wZµgx n‡e? [JU. 17-18]
A. 6, –1 B. –4, 6 C. –6, 4 D. 1, –6 A. 2, 5 B. –2, 5 C. –2, –5 D. 2, –5
–
S A Sol n
m 2 6
S D Sol g¨vwUª·wU e¨wZµgx e‡j, (x + 3) (x) 10 = 0
n
olve
2 m – 3 = 0 m – 3m – 2m + 6 – 12 = 0
2
olve
x2 + 3x 10 = 0 (x + 5) (x – 2) = 0 x = 2, 5
m2 – 5m – 6 = 0 (m – 6) (m + 1) = 0 m = 6, 1
4 2
05. A =
3 1
02.
p4 8
g¨vwUª
· wU e¨wZµgx n‡e hw` Gi gvb- 4 I B = 2 1 Gi g‡a¨ †KvbwU e¨wZµgx (Singular)
p + 2
p 1
2
[DU. 09-10, 07-08, 05-06; RU. 17-18; BSMRSTU-B. 19-20] g¨vwUª·- [JU. 13-14]
A. 4, 6 B. 6, 4 C. 4, 6 D. 6, 4 A. A B. B
n g¨vwUª·wU e¨wZµgx e‡j, (p 4) (p + 2) 16 = 0 C. A I B DfqB D. †KvbwUB bq
olve
S A Sol
n †h g¨vwUª‡·i wbY©vq‡Ki gvb k~b¨ Zv‡K e¨wZµgx g¨vwUª·
p2 2p 24 = 0 (p 6) (p + 4) = 0 p = 6, 4
olve
S B Sol
Jahangirnagar University e‡j| †h‡nZz A I B Gi g‡a¨ B Gi wbY©vq‡Ki gvb k~b¨ ZvB B g¨vwUª·wUB
e¨wZµgx g¨vwUª·|
K – 2 4
01. K Gi †Kvb gv‡bi Rb¨ g¨vwUª·wU Ae¨wZµgx bq? [JU-A, Set-I. 2021-22] Rajshahi University
3 9
01.
10 9 x + 4 5
A. B. 30 C. 3 D. 4 3
GKwU e¨wZµgx g¨vwUª· n‡j, x Gi gvb KZ?
3 4
[RU-C, Jupitar-1, Set-1. 2021-22]
S A Sol n g¨vwUª·wU Ae¨wZµgx bq A_©vr e¨wZµgx n‡j 3 9 = 0
K–2 4
olve
A. 0 B. 12 C. 14 D.
(K – 2) × 9 – 12 = 0 9K – 18 – 12 = 0
S D Sol n 4
10 x + 4 5
3 e¨wZµgx n‡j,
olve
9K = 30 K =
3
2 1 x + 4 5 = 3(x + 4) 20 = 0
02. hw` = 0 n‡j, Gi gvb †KvbwU? 4 3
5 + 4
[JU-A. 19-20]
8
A. 5 or 0 B. 6 or 2 3x + 12 20 = 0 3x 8 = 0 x =
3
C. 5 or –3 D. 1 or –3
02. g¨vwUª·wU e¨wZµg n‡j, a Gi gvb KZ?
a 4 8
2 1 a + 2
S D Sol n 5 + 4 = 0 + 4 – 2 – 8 + 5 = 0 2
2
olve
[RU. Sinovac, Set-1. 20-21]
2 + 2 – 3 = 0 2 + 3 – – 3 = 0 A. 4, 6 B. 4, 6
( + 3) – 1( + 3) = 0 = 1, – 3 C. –6, 8 D. 4, –8
03. wb‡Pi †KvbwU Ae¨wZµgx g¨vwUª·? [JU-H. 19-20] n a – 4 8 g¨vwUª· e¨wZµgx n‡j a – 4 8 = 0
olve
–4 4 2 –1 S B Sol 2 a + 2 2 a + 2
A. 8 4 B. –5 5 C. 4 5 D. 0 0
2 1 1 1
a2 – 2a – 8 – 16 = 0 a2 – 2a – 24 = 0 a = 6, – 4
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
58 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
p–4 7
Chittagong University S A Sol n 8 p – 5 g¨vwUª·wU singular ZvB wbY©vq‡Ki gvb k~b¨
olve
01. wb‡Pi †KvbwU e¨wZµgx g¨vwUª·? [CU-A, Set-1. 20-21]
(p – 4) (p – 5) – 56 = 0 p2 – 4p – 5p + 20 – 56 = 0
–2 –3 –2
A. 3 4 B. 1 2 C. 2 1 D. 3 6 Ans D
1 2 3 2 4
p2 – 9p – 36 = 0 p2 – 12p + 3p – 36 = 0
p(p – 12) + 3(p – 12) = 0 (p – 12) (p + 3) = 0 p = 12, –3
02. a 3 GKwU wm½yjvi g¨vwUª· n‡j ‘a’ Gi gvb KZ?
–4 1 [CU. 15-16] DU Affiliated College Question
A. 0 B. 1 C. –1 2 4 1
D. 12 E. –12 01. k Gi †Kvb gv‡bi Rb¨ 2 k 3 g¨vwUª·wU e¨wZµgx n‡e? [DU-Tech. 22-23]
0 0 2
S E Sol n –4 1 GKwU wm½yjvi g¨vwUª· ZvB wbY©vq‡Ki gvb k~b¨
a 3
olve
A. 3 B. 4 C. 3 D. 4
a + 12 = 0; a = –12 2 4 1
GST (¸”Q) S B Sol n 2 k 3
olve
0 0 2
k Gi †Kvb gv‡bi Rb¨
k 2
01.
8 k k g¨vwUª ·wU GKwU e¨wZµgx g¨vwUª ·- [IU-D. 19-20]
2 4 1
A. –4 B. 4 C. 2 D. 4 e¨wZµg n‡e hw`, 2 k 3= 0 2(2k 8) = 0 k = 4
0 0 2
S B Sol n e¨wZµgx g¨vwUª‡·i wbY©q‡Ki gvb k~b¨
olve
02. wb‡Pi †KvbwU e¨wZµgx g¨vwUª·? [DU Tech. 2020-21]
k (k k) – 16 = 0 k2 – 16 = 0; k = 4 k = 4 –1 –1 –1
A. –2 2 B. –6 2 C. –5 2 D. 3 4
3 3 3 1 2
02. Singular Matrix Gi Rb¨ cÖ‡hvR¨- [IU. 17-18]
–2 4 3 –1 3 1
A. A = S B Sol n –6 2 e¨wZµgx g¨vwUª· KviY 6 2 = 0
5 3 5 3 2 4
olve
3 1 B. A = 3 3 C. A = 3 6 D. A = –3 4
S C Sol n Singular Matrix Gi wbY©vq‡Ki gvb k~b¨| Engineering
olve
5 + k –2 3 1 9
03. –4 –8 GKwU e¨wZµgx g¨vwUª· n‡j k Gi gvb n‡eÑ [IU. 14-15]
01. 2x 2 6 GKwU e¨wZµgx g¨vwU· n‡j x Gi gvb wbY©q Ki| [CKRuet: 20-21]
A. –6 B. –4 C. –7 D. 6 x2 3 3
S A Sol n g¨vwUª· e¨wZµgx n‡j Matrix Gi wbY©q‡Ki gvb k~b¨ A. 1,3 B. – 1, – 3 C. 2,3
olve
D. – 2, 3 E. – 1,3
(5 + k) . (–8) – 8 = 0 k = –6 k = –6 2 2
S A Sol n 3(6 – 18) – 1(6x – 6x ) + 9 (6x –2x ) = 0
olve
p 4 7 g¨vwU·wU singular n‡e hw` P Gi gvb nq- [KU. 11-12]
04. 8 p 5 – 36 – 6x + 6x + 54x – 18x = 0
2 2
A. 3, 12 B. 3, 12 C. 4, 5 D. 4, 5 – 12x2 + 48x – 36 = 0 x2 – 4x + 3 = 0 x = 3,1
【
? 】 QUICK PRACTICE CONCEPT TEST ?
01. wb‡Pi †KvbwU e¨wZµgx g¨vwUª·? 1 + 2
A. 12 8
3 2
B. 6
3 5 02. 3
1 g¨vwUª·wU e¨wZµgx (Singular) n‡j Gi gvb KZ?
8 2 2
C. 5 8 D. 9
1 2 5 2 A. = 1 B. 2 C. 5/4 D. 5/2
8 Answer 01.A 02.C
Concept-05 AbyeÜx g¨vwUª· Ges wecixZ g¨vwUª· ¸iæZ¡:
Adjoint g¨vwU·: †Kvb eM© g¨vwUª· A Gi wbY©vqK |A| Gi mn¸YK Øviv MwVZ g¨vwUª‡·i (fzw³¸‡jvi µg Abymv‡i) Transpose g¨vwUª·‡K cÖ`Ë g¨vwUª· A Gi
Adjoint matrix ejv nq| GwU‡K m~wPZ Kiv nq Adj (A) ; A = c
a b d –b
d g¨vwUª· Gi AbyeÜx g¨vwUª· ev adjoint of A / Adj (A) = –c a
cÖvBgvwi K‡Y©i Dcv`vb¸‡jvi ¯’vb cwieZ©b Ges †m‡KÛvwi K‡Y©i Dcv`vb¸‡jvi wPý cwieZ©b Ki‡j hv cvIqv hvq ZvB Adjoint.
wecixZ g¨vwUª·: A g¨vwUª‡·i wecixZ ev Inverse g¨vwUª· =
1 1 d –b
ad – bc –c a
Adj (A) =
Det (A)
FocusKwi
jÿ¨ Point: wecixZ g¨vwUª‡·i •ewkó¨:
i. g¨vwUª·wU Aek¨B eM© g¨vwUª· n‡Z n‡e v. (AB)1 = B1 A1
ii. g¨vwUª·wUi wbY©vq‡Ki gvb k~b¨ nIqv hv‡e bv vi. (BA)A1 = B (AA1) = B
iii. (A1)1 = A vii. I = I1 = In
iv. (AT)1 = (A1)T viii. AB = C n‡j, A = CB1 Ges B = A1 C
MEx 01 hw` A =
2
1
4
3
nq Z‡e A–1 = ? [DU. 06-07; RU. 15-16; JnU. 06-07; CU. 13 -14]
General Rules 3 in 1 Shortcut Tricks & Tips
Det (A) = 4 – 6 = – 2
By Using Tricks,
Gi mn¸YK¸wj n‡”Q, A11 = 4, A12 = – 3, A21 = – 2, A22 = 1
1 1 d –b 1 4 –2
–2 A–1 =
Adj(A) = – –3 ad – bc –c a = – 2 –3 1
1 1 4 Adj(A) =
A–1 = A
A 2 1
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 59
k k 2
GKwU ev¯Íe g¨vwUª·| k Gi †Kvb gv‡bi Rb¨ g¨vwUª·wUi wecixZ g¨vwUª· cvIqv hv‡e bv?
MEx 02
2 k
[KUET. 12-13; SUST. 17-18]
General Rules & Tips
wecixZ g¨vwUª· bv _vK‡j |A| = 0 k.k – 4 = 0 k2 = 4 k = 2 k = 2
–1
MEx 03 `ywU g¨vwUª· A Ges B †`qv Av‡Q| AB I BA Gi g‡a¨ †Kvb m¤úK© _vK‡j Zv wbY©q Ki| B †K x I A Gi gva¨‡g cÖKvk Ki| [BUET. 19-20]
3x –4x 2x x 2x –2x
A=–2x x 0 Ges B = 2x 5x –4x
–x –x x 3x 7x –5x
General Rules & Tips
2
3x –4x 2x x 2x –2x 3 –4 2 1 2 –2 2 1 0 0 x 02 0
AB = –2x x 0 2x 5x –4x = x –2 1 0 . x 2 5 –4 = x 0 1 0 = 0 x 02
–x –x x 3x 7x –5x –1 –1 1 3 7 –5 0 0 1 0 0 x
3 4 2
2 x x x
x 02 0 2 1
Abyiƒcfv‡e, BA = 0 x 0 AB = BA AB = x2I AB = x2BB1 B1 = A2 = 0
0 0 x2 x x x
1 1 1
x
x x
3 4 –1
MEx 04 A = 1 0 3 n‡j A–1 = ? [Ref-†KZve DwÏb]
2 5 –4
General Rules 3 in 1 Shortcut Tricks & Tips
3 4 –1
A = 1 0 3
2 5 –4
A = 3(0 – 15) – 4(–4 – 6) – 1(5 – 0) = –45 + 40 – 5 = – 10
A Gi mn¸YK¸wj n‡”Q,
A11 =
0 3 1 3
5 –4 = –15; A12 = – 2 –4 = 10 a b c 1 q r
A = p q r n‡j A–1 Gi 1st Element =
z
–1 A
A13 = A21 = – y
1 0 4 x y z
2 5 = 5; 5 –4 = 11
–1 3 4 –1
3 3 4
Zvn‡j A = 1 0 3 Gi wecixZ g¨vwUª‡·i 1g Element
A22 = 2 –4 = – 10; A23 = – 2 5 = –7 2 5 –4
–1 3 –1
A31 =
4
0 3 = 12; A32 = – 1 3 = – 10 n‡e= A y z = –10 5 –4 = 2
1 q r 1 0 3 3
A33 =
3 4
1 0 = – 4
–15 5 T
1
10
1
–10
A–1 = Adj(A) = 11 –10 –7
A 12 –10 –4
–15 3/2 –11/10 –6/5
1
11 12
=
–10 10 –10 –10 = –1 1 1 (Ans.)
5 –7 –4 –1/2 7/10 2/5
cos a sin a
MEx 05 A = nq, Zvn‡j a Gi gvb KZ n‡j, |A + A–1| = 1 n‡e?
sin a cos a
[RU. 19-20]
General Rules & Tips
cos a sin a
; A = cos2a + sin2a – sin a cos a = – sin a cos a
–1 1 cos a sin a cos a sin a
A = sin a
cos a
cos a sin a cos a sin a
A + A–1 = sin a
cos a – sin a cos a
+
= 0
2cos a 0 2
2cos a = 4cos a
|A + A–1| = 4cos2a 1 = 4cos2a 2cos a = 1; a = 2n +
3
a Gi gvb n‡j |A + A–1| = 1 n‡e|
3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
60 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
MEx 06 hw` A = 2 1 Ges AB = 4
4 3 10 17
7
nq Z‡e B g¨vwUª· Gi Dcv`vbmg~n †ei Ki| [RU-Uranus-1, Set-1. 21-22;MIST;2021 RUET. 09-10; BUET. 16-17]
General Rules 3 in 1 Shortcut Tricks & Tips
awi, B = c
a b – 1 3
d A = 2 1 A–1 = 2
4 3
2
1 –2
GLb, AB = 4 7 2 1 c d = 4 7
10 17 4 3 a b 10 17
GLb, AB = 4 7 A–1.AB = A–1. 4 7
10 17 10 17
2a + c 2b + d = 4
4a + 3c 4b + 3d 10 17
7
I.B = B = A–1. 4
10 17 1
4a + 3c = 10 4b + 3d = 17 7 ; [ †h‡nZz A . A = I Ges I. B = B]
2a + c = 4 2b + d = 7 – 1 3 10 17 1 2
= 2 2 . 4
mgvavb K‡i cvB c = 2 d = 3 B = 2 3
a=1 b=2 1 2
7 = 2 3
1 –2
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
4 2 1 4 2
S D Sol n A = 4 6 3 1 = 2 3 1
Dhaka University 1 1
olve
2 1 1 3
01. hw` A = 3 1 nq, Z‡e A1 mgvb- [DU. 10-11; BRU-D. 19-20] 03. hw` A = nq, Zvn‡j |Adj(A)| KZ n‡e? [JU-A, Set-O. 2021-22]
2 2 4 2
A. 10 B. 1000
A. 2 4 B. 0 1 C. 1 2 D. 3 4
1 3 1 0 3 4 1 2
C. 100 D. 110
–1 –3
S A Sol n A = 4 2 n‡j Adj A = –4 –1 |AdjA| = (–2 + 12) = 10
2 3
1
olve
1
=
1 1 2 1 2
S D Sol n A = 1 3
olve
3 3 3 4 04. A =
1 2 2 4 Gi wecixZ g¨vwUª· wb‡Pi †KvbwU? [JU-A, Set-O. 2021-22]
2 2
A.
1 4 3
B.
1 1 3
02. A =
7 6
8 7
n‡j A 1
= ? [DU. 08-09] 2 2 1 22 4
7 6 7 8 C. 7 6 D. 7 8 C.
1 4 2 1 4 3
A. 3 1 D.
3 2
8 7 6 7 8 7 6 7
B. 2 10
–1 –3
S C Sol n A = 49 48 8 7 = 8 7
6 6
S A Sol n A = 2 4
1 1 7 7
olve
olve
1 1 4 3 14 3
03. hw` A =
1 2 A–1 = Adj (A) A–1 =
–4 + 6 –2 –1 2 –2 –1
nq Z‡e –1 =
3 4
A [DU. 06-07; RU. 15-16, 12-13,18-19; JnU. 06-07] |A|
A. –2 1 B. –3 1 C. – –2 1 D. – –3 1 05. A = 7 9 n‡j A KZ n‡e?
1 4 –3 1 4 –2 1 4 –3 1 4 –2 4 5 –1
[JU-A, Set-Q. 2021-22]
2 2 2 2
–4 7 9 –5
S D Sol n A = 3 4
1 2 A. 5 –9 B. –7 4
olve
4 –7
C. 5 –4 D. –5 9
1 1 4 –2 1 4 –2 9 7
A–1 = 1 = – 2 –3
4 – 6 –3 1
adj(A) =
Det(A)
9 –5 9 –5
S B Sol n A = 7 9; A = 36 – 35 –7 4 = –7 4
4 5 –1 1
Jahangirnagar University
olve
01. hw` A =
1 4 k1 2
2 6
nq, Z‡e A1 Gi gvb †KvbwU? [JU-A, Set-H. 22-23]
06. k Gi †Kvb gv‡bi Rb¨
2 k 2 g¨vwUª·wU wecixZKiY‡hvM¨ bq?
1 6 4 1 6 4 [JU-A, Set-F. 2021-22]
14 2 1 14 2 1
A. B.
3 ± 17 3 ± 15
A. 2 B. C. 1 D.
1 6 4 1 6 4 2 2
14 2 1 14 2 1
C. D.
k–1 –2
|
S B Sol n kZ©g‡Z, –2 k – 2 = 0 |
olve
n A = 1
4 1 4
olve
S C Sol 2 6 ; |A| = 2 6 = 6 8 = 14
(k – 1) (k – 2) – { – 2 × (– 2)} = 0 k2 – 2k – k + 2 – 2 = 0
1 6 4 1 6 4 3 17
A1 k2 – 3k – 2 = 0 k =
14 2 1 14 2 1
=
2
02. hw` A = nq, Z‡e A1 Gi gvb †KvbwU?
1 2
[JU-A, Set-G. 22-23] 07. A =
4 3
3 4 3 2 Gi wecixZ g¨vwUª· wb‡Pi †KvbwU? [JU-A, Set-M. 2021-22]
1 4 3 1 4 2
A. B.
2 3 2 3 4 3
A. B.
2 3
2 2 1 2 3 1 C. D.
3 4 3 4 3 4 3 2
1 4 3 1 4 2 –3 –2 3
S C Sol n A = 3 2 A = 8 – 9 – 3 4 3 –4
4 3 –1 1 2
C. D.
olve
2 2 1 2 3 1
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 61
m–2 6
08. A = S C Sol n 2 m – 3 = 0 m – 3m – 2m + 6 – 12 = 0
2
Ges AB = 11 24 n‡j, B = ? [JU-A, Set-M. 2021-22]
1 3 12 2
olve
3 4
3 5 1 m2 – 5m – 6 = 0 (m – 6) (m + 1) = 0 m = 6, 1
A. 14 B. 0 C.
12 5 0
D. †KvbwUB bq
1 6 1 6
02. A =
2 3
1 3
Gi wecixZ g¨vwUª· †KvbwU? [RU-C, Feldspar-1. 22-23]
S C Sol n A = 3 4 Ges AB = 11 24
1 2 3 12
olve
1 3 3 1 3 1
A. B.
3 1 2 3 3 2
GLb, AB = 11 24 A–1 AB = A–1 11 24
3 12 3 12
1 3 3 1 3 1
C. D. 3
–2 1 3 1 2 3 2
1 4 –2 3 12 3 1
3 12 5 0
B=
4 – 6 – 3 1 11 24 2 – 2 11 24 –1 6 S A Sol n A = 1 3
= 2 3
olve
–1 –3 1 3 3 1 3 3
09. A = n‡j, A1 KZ? A–1 =
4 6 – 3 1 2 3 1 2
[JU. 18-19; JnU. 14-15] =
2
A.
3
B. 2
1 4 1 1 3 2 1
03. A = I B = 3 5 n‡j, (BA)1 = KZ? [RU-C, Quartz-2. 22-23]
2 3
2 2 1 2 4 5 7
44 1 1 44 1
C. 1 3
A. B.
1 4 2 4
2 3 1 D.
10 2 1 31 1 13 31 1
–1 –3 1 44 1 1 31 1
S A Sol n A = 2 4
olve
13 31 1 13 44 1
C. D.
1 1 4 3 1 4 3 2 1 2 3 4 + 5 – 6 + 7
S B Sol n BA = 3 5 5 7 = 6 + 25 9 + 35 = 31 44
A–1 = 1 1
– 4 + 6 –2 –1 2 –2 –1
adj(A) = =
olve
Det(A)
1 3
hw` A = |BA| = 44 – 31 = 13
4 2 nq, Zvn‡j |adj (A)| KZ n‡e? [JU-A. 2018-19]
10.
1 44 – 1
(BA)–1 =
A. 10 B. 1000 13 – 31 1
04. j2 = – 1 n‡j,
C. 100 D. 110 j j
1 3 2j j
Gi wecixZ g¨vwUª· †KvbwU?
2 3
S A Sol n A = 4 2 Adj A = 4 1
olve
[RU-Neptune-2, Set-1. 21-22]
j –j –j 2j
A. 2j –j B. j –j
|AdjA| =
2 3
4 1 = 2 + 12 = 10 j –j
C. j j D.
j 2j
11. g¨vwUª· M =
a 0
n‡j, M1 I †KvbwU? 2j j
0 b
[JU-A. 2017-18]
S D Sol n awi, 2j j = A
j j
olve
1
A.
0
B.
a 0 a
0 b 1
b 0 1 j j 1 j j j j
A1 = 2 A1 =
j 2j2 2j j 1 + 2 2j j 2j j
=
a
C. D.
0 0 b
0 b a 0 05. A =
2 1 1
4 3 n‡j A = ? [RU. Sinovac, Set-1. 20-21]
S A Sol n M = 0 b |M| = 0 b = ab
a 0 a 0
1 2 1
A.
1 3 1
B.
olve
2 4 2 2 4 3
1 1
1 b 0 a 0 a 1 3 1 3 1
D.
1
0
M = C.
2 4 2 4 2
= =
ab 0 a 0 1 0 b1
b
S C Sol n A = 4 3
2 1
olve
1
M1 I = M1 =
a 0
[†h‡nZz I GKK g¨vwUª·]
0 b1 V
A–1 =
1 3 – 1 = 1 3 – 1
2 × 3 – 4 × 1 – 4 2 2 – 4 2
A =
5 2
12.
3 1
n‡j A1 Gi gvb-
06.
[JU. 15-16; JnU. 07-08] 4 2
1 3 1 1 1 2 3 2 -Gi wecixZ g¨vwUª· †KvbwU? [RU. 17-18]
A. 1
2 11 3 5
B.
5 1 –1 –1 – 3
1 1 2 1 1 2 A. 3 B. 2
C. D. 3 –
2 2
2 1
11 5 3 3 1
2 –1 3 2
1 2 1 1 2
S B Sol n A = 5 + 6 3 5 = 11 3 5
1 1
C. 3 D. 2
olve
– 2 1 –1 1
Rajshahi University
S A Sol n A = 3 2
4 2
olve
01. m – 2 6 g¨vwUª·wUi wecixZ g¨vwUª· we`¨gvb bv _vK‡j m = KZ?
2 m – 3
1 2 –2 1 2 –2
1 –1
1
[RU-C, Feldspar-1. 22-23] A–1 =
Det(A)
adj(A) =
8 – 6 –3 4 2 –3 4 –
= = 3
A. – 1, 6 B. 1, – 6 C. 6, – 1 D. – 1, – 6 2 2
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
62 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
07. †KvbwU wecixZ g¨vwUª‡·i •ewkó¨ bq? [RU-H. 2017-18] 3
2 3
A. (A1)1 = A B. (AB)1 = A1B1 04. g‡b Ki, A = Ges x –1
2 . A = B n‡j x = KZ?
4 5
B = [CU. 17-18]
C. (AT)1 = (A1)T D. (BA)A1 = B 2 1
1 1 1
S B Sol n Option B †Z Av‡Q, (AB) = A B | hv mwVK bq,
olve
5
A. B. 1
(AB)1 = B1A1| mwVK myZivs GwU g¨vwUª‡·i •ewkó¨ bq| 2
2 2
08.
3 2 C. D.
2 1
Gi wecixZ g¨vwUª· †KvbwU? [RU. 16-17; CU. 15-16] 3 5
–3
–3 2
x – 3
1
A. 2 –1 B. 2 2
1 3
1 1 – 2
3
2 (x – 3)
n B–1 =
x–3 = –2
olve
1 –2 –1 2 S A Sol
C. –2 3 D. 2 –3
x – 3
x
–2 x
x–3
1 1 2
S D Sol n A = 3 – 4 2 3
–1 1 5
†h‡nZz A = B–1; †m‡nZz –2 = x – 3 ev, x = 2
olve
1 –2 –1 2
= – –2 3 = 2 –3 05. hw` A =
2 –3
nq, Z‡e A–1 †KvbwU?
4 –1
[CU. 17-18]
09. A =
3 1 –1 –3 3 –1 1 –1 3
A. –4 2 B. 2 –4 C. 0 1
1 0
9 3
n‡j, A Gi wecixZ g¨vwUª· †KvbwU? D.
10 –4 2
[RU-C. 16-17]
3 –1 –3 1
A. –9 3 B. 9 –3 –1 1 –1 3 1 –1 3
S D Sol n GLv‡b, A = 2(–1) – (–3).4 –4 2 = 10 –4 2
olve
3 –9
C. –1 –3 D. wbY©q‡hvM¨ bq 5 3 2
06. If A = 0 4 1 then |A| = KZ? [CU-I. 16-17]
3 –1 0 0 3
S D Sol n A = 9 – 9 –9 3, hv wbY©q‡hvM¨ bq|
–1 1
olve
A. 30 B. 40 C. 50
10.
cos sin D. 60 E. 66
sin cos Gi wecixZ g¨vwUª·- [RU. 15-16; DU. 13-14]
S D Sol n |A| = 5(12 – 0) + 0 + 0 = 60
olve
cos sin cos sin
A. B. 3 0 0
sin cos sin cos 07. X = 4 4 6 , P Gi gvb KZ n‡j X g¨vwUª‡·i wecixZ g¨vwUª· †ei
cos sin
C. D.
cos sin 7 2 p
sin cos sin cos Kiv hv‡e bv? [CU-C3. 16-17]
cos sin
S C Sol n wecixZ g¨vwUª· = cos2 + sin2 sin cos
1 A. 0 B. 2 C. 3
olve
D. –3 E. 7
cos sin
= S C Sol n g¨vwUª·wUi wecixZ g¨vwUª· cvIqv hv‡e bv hw` g¨vwUª·wUi
olve
sin cos wbY©vq‡Ki gvb k~Y¨ nq| 3(4p – 12) = 0 p = 3
Chittagong University 1 –2 4
08. M = –
2 –1 3
n‡j M–1 = KZ? [CU-C3. 16-17]
01. †Kvb GKwU g¨vwUª‡·i Inverse g¨vwUª· †c‡Z n‡j †Kvb kZ© n‡Z n‡e?
3 –4 –4 –3
A. 1 –2 B. –2 1 C. 1 2
[CU-A, Set-3. 20-21] 3 4
A. Ae¨wZµgx B. e¨wZµgx
1 –3 1
D. 3 4 E. – –4 2
C. iæcvšÍwiZ D. mgNvwZ 1 2
n Inverse g¨vwUª‡·i Rb¨ Ae¨wZµgx n‡Z n‡e| 2
olve
S A Sol
1 –2
–1 1 1 –2 4
02. P = n‡j –1
Gi gvb †KvbwU? n M=
–
1 1
1 3
olve
2 –1 3
P [CU-D, Set-2. 20-21] S A Sol =
2 2
A.
0 1 1 1
1 0 B. –1 1
3
2
3 –4
=
–1/2 1
C. D.
1/2 1 1/2 M = –1
2
1 1/2 1/2 1/2 3
– + 1
1 –2
1
1
1 2 2
1
n P = 1
1
2 2 –b
09. hw` A = nq, Z‡e A–1 = KZ?
1 a
1 1 P = 1 1
olve
S D Sol c d
[CU. 15-16]
2 2 A. ad – bc B. ad + bc
–2
03. M = Gi wecixZ g¨vwUª· bv _vK‡j x Gi gvb KZ? [CU. 19-20, 16-17]
1 1 a b 1 a b
x 4 C. D.
ad – bc –c d ad – bc –c d
A. 2 B. 1 1 d b
ad + bc –c a
C. –2 D. –1 E.
1 –2
S C Sol n M = x 4 Gi wecixZ g¨vwUª· bv _vK‡j g¨vwUª·wUi S E Sol n A = c d
a –b
olve
olve
wbY©vq‡Ki gvb k~b¨ 1 d b
A–1 =
4 + 2x = 0; x = – 2 ad + bc –c a
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 63
AB =
GST (¸”Q) 2 1
01. wb‡¤œi †Kvb g¨vwUª‡·i wecixZ †bB? [JKKNIU. 19-20] 1 –1
3 3 2 1
4 –2 –3 1 1 1
A. 1 6 B. –1 6 C. 3 2 D. 2 4
2 4 6 4
B = A–1
1
2 1 1 –1
2
=
1 –1
S C Sol n wecixZ g¨vwUª· _vKvi kZ©:
– 3 3
olve
i. eM© g¨vwUª· n‡Z n‡e| ii. g¨vwUª‡·i wbY©vq‡Ki gvb k~b¨ nIqv hv‡e bv|
3 3 1 0
2 1 1 1
6 4 g¨vwUª‡·i wbY©vq‡Ki gvb k~b¨ ZvB wecixZ g¨vwUª· wbY©q Kiv hv‡e bv| + –
3 2 3 3
= –2 1 –1 –1
=
3 3
4 1
hw` A =
9 4 – + –
02. nq, Z‡e A(AdjA) = ?, †hLv‡b Adj A n‡jv A Gi
6 3 3 3
mnvqK (adjoint) g¨vwUª·| [JnU. 17-18] 08. g¨vwUª· A = + 4 6
4 3
Gi wecixZ g¨vwUª· _vK‡e bv, hw` Gi gvb nq-
3 –4 3 0 3 4 1 0
A.
–6 9 B. 0 3 C. D. [MBSTU-C. 19-20]
–6 9 0 1 A. 0 B. 4
3 –4 9 4 3 –4
S B Sol n Adj(A) = 6 9 A.Adj(A) = 6 3 –6 9 = 0 3
3 0 C. 4 D. 12
olve
n wbY©vq‡Ki gvb k~b¨ n‡j wecixZ g¨vwUª· _vK‡e bv|
olve
S B Sol
A =
4 7
n‡j T –1
KZ? ( + 4)3 24 = 0 n‡j wecixZ g¨vwUª· _vK‡e bv|
–5 –9
03. (A ) [JKKNIU. 17-18]
9 –5 –9 5 9 –5 12
A. 7 –4 B. –7 4 C. –7 4 D. 7 4
9 5 3 + 12 24 = 0 ; =
3
=4
4 –5 x –1 –1
S A Sol n A = –5 –9 A = 7 –9
4 7
09. x Gi †Kvb gvb¸wji Rb¨ 0 x –3 g¨vwUª‡·i †Kvb wecixZ
T
olve
x – 4 –1 1
(AT)–1 =
1 –9 5 = 9 –5
–36 + 35 –7 4 7 –4 g¨vwUª· cvIqv hv‡e bv? [SUST. 15-16]
1 2 A . 6, – 2 B . –6, –2 C . –6, 2
04. M = n‡j M–1 mgvb KZ? D. –6, 2
3 5 E. 6, 4
[JnU. 16-17]
5 2 5 2 x –1 –1 x –1 –1
A. 3 1
5 2
B. C. 1 3
3 1 3 1
D. 2 5 S A Sol n –3 0 x –3= 0
olve
0 x = 0
1 –2
x – 4 –1 1 –4 0 1
S B Sol n M = –3 5 x(x) – 4(3 + x) = 0 x2 – 4x – 12 = 0
olve
1 1 5 2 –5 –2 x2 – 6x + 2x – 12 = 0 x(x – 6) + 2(x – 6) = 0
–1
M = (x – 6) (x + 2) = 0 x = 6, –2
5 – 6 3 1 –3 –1
adj(M) = =
Det(M)
2 –1
A =
8 4 10. X = n‡j X–1 = ?
3 –1
[SUST. 14-15]
05. n‡j n‡e-
3 2
Adj(A) [IU. 14-15]
–2 –1 1 –1 1
A. 3 1 B. –3 2 C. –1 –1
2 4 2 3
A. 3 2
8 4
B. 3 8
2 4
C. 8 3
3 8 4 2
D. 5
–1 1
–4 D. –3 2
S C Sol n Adj(A) = –3 8
2 E. AmsÁvwqZ
olve
–1 1 –1 1
S D Sol n X = –2 + 3 –3 2 = –3 2
1
06. †Kvb g¨vwUª·wUi wecixZ g¨vwUª· †bB- –1
olve
[JnU. 12-13; RU. 16-17]
2 2
A. 1 2 B. 4 2 C. 1 4 D. 4
2 1 2 1 3 2
4 Engineering
2 1
01. hw` A = IB=
S B Sol n Singular matrix does not have any inverse matrix. 2 3
olve
nq, Z‡e (BA)1 Gi gvb KZ?
–1 5 7 3 5
hw` g¨vwUª· A = Ges g¨vwUª· AB =
1 2 1
07.
2 1 1 –1 nq, Z‡e g¨vwUª· [KUET. 2015-16]
1 31
A. B.
44 1 44
B †KvbwU? [SUST-A. 19-20]
31 1 13 1 1
A.
1 0
B.
0 1
C.
2 1
0 1 1 0 1 –1 1 44 1 1 31 1
C. D.
1 –1 13 31 1 13 44 1
D. E.
1 0
44
2 1 –1 –1 E.
1 31
–1 13 1 1
S E Sol n A = 2 1
1 Aspect Special:
olve
2 1 2 3
Ackb †U÷ Ki‡jB n‡e| AL©vr
S B Sol n BA = 3 5 5 7
olve
A–1 =
1
adj (A)
†h Ack‡bi mv‡_ A g¨vwUª· ¸Y
4 + 5 6 + 7 1 1
=
Det(A) Ki‡j AB g¨vwUª· cvIqv hvq †m
6 + 25 9 + 35 = 31 44
1 1 AckbwUB mwVK Ans.
|BA| = 44 31 = 13
=
1 1 1 3 3
3 –2 1
= 44 1
(BA)1 =
2 1 1
–
3 3 13 31 1
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
64 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
S D Sol n A = 2
4
02. A1 wbY©q Ki: A =
1 4 1
olve
2 6 6
[RUET. 13-14]
1 6 4 1 6 4 1 6 4
A. A1 =
14 2 1 14 2 1
B.
|A| 2 1
C.
1 6 4 1 6 4 1 6 4
14 2 1
D. None
14 2 1 14 2 1
= =
【
? 】 QUICK PRACTICE CONCEPT TEST ?
01. wb‡Pi †Kvb g¨vwUª‡·i wecixZ g¨vwUª· cvIqv hv‡e bv|
03. A =
[BUP. 20-21] 1 2
3 n‡j Adj (A) wb‡¤œi †KvbwU?
4
A. B. C. D.
2 1 1 10 1 5 10 1
5 10 2 5 2 10 5 2 2 4 2 1 2 4 2
B. C. D.
4
A.
02. A GKwU Ae¨wZµgx eM© g¨vwUª· n‡j A-1 Gi wecixZ g¨vwUª· †KvbwU? 3 1 3 1 3 4 3 1
A. A B. -A C. A D.(O) Answer 01.C 02.A 03.A
HSC BOARD QUESTIONS ANALYSIS
01. KY© g¨vwUª‡·i †ÿ‡ÎÑ [XvKv †evW©-2019] wb‡Pi Z‡_¨i Av‡jv‡K 05-06 bs cÖ‡kœi DËi `vI:
(i) aij 0, i = j
A = 2
x+4 8
(ii) aij = 0, i > j x – 2 GKwU g¨vwUª·|
(iii) aij = 0, i < j 05. hw` A g¨vwUª·wU e¨vwZµgx nq, Z‡e x Gi gvb wb‡Pi †KvbwU? [iv.†ev. 22]
wb‡Pi †KvbwU mwVK? A. – 4, 2 B. – 2, 4
A. i I ii B. i I iii C. – 4, 6 D. – 6, 4
S D Sol n †Kv‡bv g¨vwUª· e¨wZµgx n‡e hw` H g¨vwUª‡·i wbY©vq‡Ki gvb
olve
C. ii I iii D. i, ii I iii
n KY© g¨vwUª· (Diagonal Matrix): †h eM© g¨vwUª· cÖavb k~b¨ nq|
olve
S D Sol
K‡Y©i (aij 0; i = j) f~w³¸‡jv e¨wZZ Ab¨me f~w³ k~b¨| x + 4 8 = 0
2 1 2 x 2
02. A = 1
5 3 n‡j, wb‡Pi †KvbwUi gvb A ? [XvKv †evW©-2019] (x + 4) (x 2) 16 = 0 x2 + 4x 2x 8 16 = 0
x2 + 2x 24 = 0
A.
3 1 3 1 x (x + 6) (x 4) = 0 x = 6, 4
5 2 5 2
B.
3 1 3 1 06. cÖ`Ë g¨vwUª‡· x = 3 n‡j A2 wb‡Pi †KvbwU? [iv.†ev. 22]
2
C. 5 D.
5 2 A. 16 17
65 64
B. 41 43
49 46
–1 1
S C Sol n A = Det(A)
olve
C. 52 64 D. –2 4
40 48 64 49
1 –3 1 3 1
–6 + 5 –5 2 5 2 S A Sol n x = 3 n‡j, A = 2 3 2 = 2 1
Adj (A) = = 3+4 8 7 8
olve
2
03. A = A n‡j A g¨vwUª·wU- [g.†ev. 22]
A2 = 2 1 2 1 = 14 + 2 16 + 1 = 16 17
7 8 7 8 49 + 16 56 + 8 65 64
A. mgNvZx B. e¨wZµgx
C. cÖwZmg D. Ae¨wZµgx
0 5 – 3
n hw` A2 = A n‡j A g¨vwUª· †K mgNvZx g¨vwUª· ejv nq|
07. – 5 0 y wecÖwZmg g¨vwUª· n‡j x, y = ?
olve
S A Sol [w`.†ev. 22]
x 4 0
2 0 0 A. (–3, –4)
04. A = 0 3 0 Gi A1 †KvbwU? [ivRkvnx †evW©-2019] B. (–3, 4)
0 0 4 C. (3, –4) D. (3, 4)
T
1 0 3 0
S C Sol GKwU eM© g¨vwUª· A †K wecÖwZmg ejv n‡e hw` A = – A
n
olve
2 0 0
1
A. B.
24 0 0 4
nq|
2 0 0
0 3 0 0 5 – 3T 0 5 – 3
0 0 4 –5 0 y =––5 0 y
x 4 0 x 4 0
1 0 0
2 0 0 0 – 5 x 0 –5 3
D. 3 1
2 1 0
C. 24 0 3 0 5 0 4 = 5 0 – y
0 0 4 0 – – x – 4 0
0 0 4 3 y 0
1 x = 3, y = –4
1
S D Sol n A = Det(A)
olve
08. k Gi gvb KZ n‡j,
3 6
5 k
g¨vwUª·wU e¨wZµgx n‡e? [w`bvRcyi †evW©-2019]
1 0 0
1 0 8 0 2 1 0
12 0 0 A. 10 B. 0
Adj (A) = =
24 0 0 6 0 3 1
C. 3 D. 10
0 0 4 n 3k – 30 = 0 k = 10
olve
S D Sol
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (1g Ask: g¨vwUª·)
© Medistry 65
2 4 6 4 3 g¨vwUª‡·i wecixZ g¨vwUª· †KvbwU?
09. 3 x 5 = 0 n‡j, x-Gi gvb †KvbwU? [w`bvRcyi †evW©-2019]
14. 3 2 [P.†ev. 22]
5 10 9 2 –3
A. – 3 4 B. – 3
2 3
A. 4 B. 6 – 4
–2 3 –4
C. 3 – 4 D. 3
C. 5 D. 6 3
– 2
S B Sol n 2(9x – 50) + 4(27 + 25) + 6(–30 – 5x) = 0
olve
S C Sol n awi, A = 3 2
4 3
–12x – 72 = 0 –12x = 72 x = –6
olve
2 4
10. 1 2 – 3
4 8 GKwUÑ [Kzwgjøv †evW©-2019] A–1 =
8 – 9 – 3 4
(i) eM© g¨vwUª· 1 2 – 3 – 2 3
– 1 – 3 4 3 – 4
(ii) e¨wZµgx g¨vwUª· = =
(iii) cÖwZmg g¨vwUª· 15. A GKwU eM© g¨vwUª· Ges K GKwU ‡¯‹jvi n‡j- [wm.†ev. 22]
wb‡Pi †KvbwU mwVK? i. (At)t = A
A. i I ii B. ii I iii ii. (KA)t = KAt
C. i I iii D. i, ii I iii iii. hw` |A| 0 nq, Z‡e |A–1| =
1
n i. eM© g¨vwUª· (Diagonal Matrix) : †h g¨vwUª‡·i mvwi |A|
olve
S D Sol
wb‡Pi †KvbwU mwVK?
msL¨v Kjvg msL¨vi mgvb Zv‡K eM© g¨vwUª· e‡j|
A. i I ii B. i I iii
ii. e¨wZµgx g¨vwUª· : †h eM© g¨vwUª‡·i wbY©q‡Ki gvb k~b¨|
T C. ii I iii D. i, ii I iii
iii. eM© g¨vwUª· A ‡K cÖwZmg g¨vwUª· ejv n‡e hw` A = A nq| 2 3
S A Sol n awi, A = 1 6
olve
1
11. P =
1 2 3 4 Ges Q = 2 n‡j PQ Gi µg KZ? 2 3
At = 1 6 = 3 6
2 1
2 3 4 5
[Kzwgjøv †evW©-2019]
3
4 2 1t 2 3
A. 1 2 B. 2 1 C. 4 1 D. 4 4 (At)t =
3 6 1 6
= =A
S B Sol n P24 41Q
olve
(i) bs mwVK|
= 2 3 2K 3K
PQ Gi AvKvi 2 1 KA = K1 6 = K 6K [K GKwU †¯‹jvi]
2 7
12. B = 2K 3K
t
n‡j B1 †KvbwU? [Kzwgjøv †evW©-2019] t
1 4 (KA) = K 6K
4 7
A. B. 1 =
2K K
= K
7 4 2 1
1 2 2 = KAt
3K 6K 3 6
4 1 2 1 (ii) bs mwVK|
C. 7 D. 7
2 4 Ans B
2 3
– |A| = 1 6 = 12 + 3 = 15 = 0
13. hw` A = nq, Z‡e-
2 3
1 6
[ P.†ev. 22]
2 1
i. A GKwU wecÖwZmg g¨vwUª·
A1 =
1 6 3 = 4 5
ii. |A| = 15 15 1 2 1 2
iii. A GKwU A‡f`NvwZ g¨vwUª· bq 15 15
wb‡Pi †KvbwU mwVK? 2 1
A. i I ii B. i I iii C. ii I iii D. i, ii I iii 1
|A | = 5 5 4
= +
1
= =
5 1
=
1
S C Sol n wecÖwZmg g¨vwUª· (SAkew Symmetric Matrix): GKwU 1 2 75 75 75 15 |A|
olve
15 15
eM© g¨vwUª· A †K wecÖwZmg g¨vwUª· ejv n‡e hw` Ar = A nq|
(iii) bs mwVK bq|
2 3T 2 1 2 1
= + 2
3 6 A
T
A = 1 1 3
6 3 6
=
16. 2 4 8 GKwU e¨wZµgx g¨vwUª· n‡j, -Gi gvb- [wm.†ev. 22]
(i) bs mwVK bq| 3 5 10
2 3
|A| = 1 6 = 12 + 3 = 15 (ii) bs mwVK|
A. –2 B. 2
C. 4 D. –4
A‡f`NvwZ g¨vwUª· (Involutory Matrix): GKwU eM© g¨vwUª· A †K n
g¨vwUª· wU e¨wZµgx n‡j Gi wbY©vq‡Ki gvb k~b¨|
olve
S C Sol
A‡f`NvwZ g¨vwUª· ejv n‡e hw` A2 = 1 nq|
2 3
1 3 + 2
A = 1 n‡j, 2 4 8 = 0
6 3 5 10
2 3 2 3 4 3 6 18 1 24 1 (40 – 40) – 3 (20 – 24) + ( + 2) (10 – 12) = 0
2
A = 1 1
6 1 6 2 + 6 3 + 36 8 33
= =
0 + 12 – 2 – 4 = 0
myZivs, A GKwU A‡f`NvwZ g¨vwUª· bq| – 2 = – 8
(iii) bs mwVK| =4
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
66 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
8 5
17. A–1 =
2 0 –1 0 1
n‡j (AB)–1 Gi gvb KZ? [wm.†ev. 22] 22. A =
0 2
,B =
1 0 7 2 n‡j, adj.A = †KvbwU? [e.†ev. 22]
A. 2 0 B. 0 2
–2 5
A. – 7 – 8 B. – 7 8
0 2 2 0 2 5
C. 0 1 D. 1 0
1 0 0 1
C. – 5 2 D. – 7 8
8 7 2 5
S A Sol n (AB) = B A = 1 0 0 2
0 1 2 0 8 5
S D Sol n A = 7 2
–1 –1 –1
olve
olve
= 2 + 0 0 + 0 = 2 0
0+0 0+2 0 2
Adj A = – 7 8
2 5
18. hw` A GKwU eM© g¨vwUª· Ges A2 = I nq, Z‡e A †K e‡j- [h.†ev. 22] 2 3
23. A + B = 4 5
A. k~b¨NvwZ g¨vwUª· B. A‡f`NvwZ g¨vwUª· 4 1 Ges A B = 2 7 n‡j wb‡¤œ †KvbwU B g¨vwUª·?
C. k~b¨ g¨vwUª· D. wecÖwZmg g¨vwUª· [ewikvj †evW©-2019]
1 4 1 1
A. B.
n k~b¨NvZ g¨vwUª· (Nilpotent Matrix): GKwU eM© g¨vwUª·
olve
S B Sol
n 3 4 3 3
A Gi Rb¨ A = 0 n‡j (†hLv‡b n ÿz`ªZg ¯^vfvweK msL¨v) g¨vwUª·‡K
C. 2 6 D. 1 3
6 2 3 1
k~b¨NvZ g¨vwUª· e‡j Ges n †K k~b¨NvwZ m~PK e‡j|
K AckbwU mwVK bq| 2 3
S A Sol n A + B – A + B = 4 1 – 2 7
4 5
olve
A‡f`NvwZ g¨vwUª·: GKwU eM© g¨vwUª· A †K A‡f`NvwZ ejv n‡e hw`
A2 = I nq| –2 –8
2B = 6 –8
L AckbwU mwVK|
1 4
B = 6 –8 =
k~b¨ g¨vwUª·: †h g¨vwUª‡·i me¸‡jv fzw³ k~b¨| 1 –2 –8
2 3 4
M AckbwU mwVK bq|
24. hw` I3 GKwU wZb µ‡gi g¨vwUª· nq Z‡e (I3)1 = ? [e.‡ev. 2017]
GKwU eM© g¨vwUª· A †K wecÖwZmg g¨vwUª· ejv n‡e hw` AT = – A nq|
A. 1 B . I3
N AckbwU mwVK bq| 1
C. I3 D. 3I3
0 3
19. P = [1 2 3] I Q = 1 n‡j PQ Gi gvbÑ [h‡kvi †evW©-2019]
S B Sol n AB = 1 n‡j,
olve
2
A. [8] B. [1476] A1 = B ev B1 = A
0 GLb, I3.I3 = I3
C. 2 D. [0 2 6] (I3)1 = I3
6
5 0 0
0
25. 0 5 0 g¨vwUª·wU GKwUÑ [mKj †evW©-2018]
S A Sol n PQ = [1 2 3] 1 = [8] 0 0 5
olve
2 (i) eM© g¨vwUª·
20. A =
2 3
n‡j (ii) A‡f`K g¨vwUª·
4 1
Adj(A) = ? [h‡kvi †evW©-2019]
(iii) †¯‹jvi g¨vwUª·
A. 4 2 B. 3 1
1 3 2 4
wb‡Pi †KvbwU mwVK?
1 3 1 3 A. i I ii B. i I iii
C. D.
4 2 4 2 C. ii I iii D. i, ii I iii
S C Sol n A = 4 1
2 3
S B Sol n Z‡_¨i Av‡jv‡K g¨vwUª·:
olve
olve
1 3 i. eM© g¨vwUª· (Diagonal Matrix) : †h g¨vwUª‡·i †iv msL¨v Kjvg
Adj(A) =
4 2 msL¨vi mgvb Zv‡K eM© g¨vwUª· e‡j|
ii. A‡f`K g¨vwUª· ev GKK g¨vwUª· (Identity or unit Matrix) : †h
7
21. A =
1 2 3
4 5 6
, B = 8 n‡j, AB Gi µg KZ? [e.†ev. 22] KY© g¨vwUª‡·i KY©w¯’Z mKj fzw³ 1|
9 iii. †¯‹jvi g¨vwUª· (Scalar Matrix) : †h KY© g¨vwUª‡·i Ak~b¨ fzw³¸‡jv
A. 2 × 1 B. 1 × 2 mgvb|
C. 3 × 1 D. 2 × 3
p + 1 6 g¨vwUª·wU e¨wZµgx n‡j p Gi gvbÑ
S A Sol n GLv‡b, A = 4 5 6
1 2 3 [mKj †evW©-2018]
26. 4 8
olve
A. 8 B. 4
7 C. 4 D. 6
B = 8
9 n (p + 1) (–8) – 24 = 0
olve
S B Sol
A g¨vwUª‡·i µg 2 × 3, B g¨vwUª‡·i µg 3 × 1 – 8p = 32
AB g¨vwUª‡·i µg 2 × 1 p=–4
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 67
Aa¨vq g¨vwUª· I wbY©vqK
1g cÎ
[MATRIX & DETERMINATION]
01
wØZxq Ask: wbY©vqK [DETERMINATION]
SURVEY TABLE Kx coe? // †Kb coe? // †Kv_v n‡Z coe? // KZUzKz coe?
VVI For
MAGNETIC DECISION [hv co‡e] MAKING DECISION [†h Kvi‡Y co‡e]
TOPICS This Year
DU JU RU CU GST Engr. HSC Written MCQ
CONCEPT-01 Abyivwk I mn¸YK wbY©q 15% 5% 5% 10% 20% 5% 5%
CONCEPT-02 wbY©vq‡Ki gvb wbY©q 40% 30% 40% 40% 50% 30% 10%
CONCEPT-03 GKwU ARvbv PjK _vK‡j gvb wbY©q 70% 70% 40% 60% 50% 70% 60%
CONCEPT-04 me¸‡jv ARvbv PjK _vK‡j gvb wbY©q 60% 40% 30% 20% 20% 90% 80%
DU. = Dhaka University, JU. = Jahangirnagar University, RU. = Rajshahi University,
CU = Chittagong University, GST = General, Science & Technology, Engr. = Engineering.
? k‡ãi Drm
cÖv_wgK Z_¨
a11 a12 a13
wbY©vqK: wbY©vqK GKwU we‡kl AvKv‡i wjwLZ wbw`©ó GK cÖKv‡ii ivwk| wbY©vqK‡K mvaviYZ || Øviv cÖKvk Kiv nq| †hgb- A = a21 a22 a23
a31 a32 a33
wbY©vq‡Ki cÖwZwU Dcv`v‡bi GKwU K‡i wPý Av‡Q| Dcv`v‡bi Ae¯’v‡bi msL¨v¸‡jvi †hvMdj †Rvo n‡j wPý abvZ¥K nq, wKš‘ we‡Rvo n‡j wPý FYvZ¥K nq|
†hgb- ' a 11 ' Gi wPý abvZ¥K, wKš‘ ' a 12 ' Gi wPý FYvZ¥K|
g¨vwUª· I wbY©vq‡Ki cv_©K¨:
g¨vwUª· wbY©vqK
mvwi I Kjvg mgvb n‡ZI cv‡i bvI n‡Z cv‡i| mvwi I Kjvg msL¨v Aek¨B mgvb|
g¨vwUª· Gi †Kvb mywbw`©ó gvb †bB, ïay Acv‡iUi wnmv‡e KvR K‡i| wbY©vq‡Ki mywbw`©ó exRMwYZxq gvb Av‡Q|
µg m n n‡j, fzw³ m.n µg n n‡j, †gvU fzw³ n2
g¨vwUª·‡K †Kvb ayªe ivwk Øviv ¸Y Ki‡j Zvi cÖ‡Z¨KwU fzw³‡K H ayªe wbY©vqK‡K †Kvb ayªe ivwk Øviv ¸Y Ki‡j Zvi cÖ‡Z¨KwU fzw³‡K ¸Y bv K‡i
msL¨v Øviv ¸Y Ki‡Z nq| ïaygvÎ †h †Kvb GKwU mvwi ev Kjv‡gi fzw³¸‡jvi mv‡_ ¸Y Ki‡Z nq|
Concept-01 Abyivwk I mn¸bK wbY©q ¸iæZ¡:
Technique:
(i) cix¶vq †hfv‡e Abyivwk [Minor] †ei Ki‡e:
Step-1: †h ivwk ev msL¨vi Abyivwk †ei Ki‡Z ej‡e wVK †mB ivwk eivei Row Ges Column ev` `vI|
Step-2: evwK Dcv`vb ¸‡jv w`‡q wbY©vqK MVb Ki| †mwUB Abyivwk|
Step-3: gvb ‡ei Ki‡Z ej‡j mvaviY wbq‡g wbY©vq‡Ki gvb †ei Ki‡Z n‡e|
(ii) cix¶vq †hfv‡e mn¸YK [Co-factor] †ei Ki‡e:
Step-1: Abyivwk †ei Kivi c×wZ Aej¤^b K‡i cÖ_‡g Abyivwk †ei Ki
Step-2: Abyivwk mvg‡b (–1)R+C m~G e¨envi K‡i h_vh_ wPý emvI| †mwUB mn¸YK| †hLv‡b R = mvwi, C = Kjvg A_©vr mn¸YK = ( 1)R+C Abyivwk
A_©vr, Abyivwk‡K D³ Dcv`v‡bi wPý w`‡q ¸Y Ki‡j mn¸YK cvIqv hvq|
5 6 7
MEx 01 1 2 3 wbY©vqKwUi 1 Ges 7 Gi Abyivwk †ei Ki |
3 6 9
General Rules [Written] & [MCQ]
Step-1: 1 msL¨vwU 1 bs Column Ges 2 bs Row †Z Av‡Q e‡j 1 Column bs Ges 2 Row bs ev` `vI
Step-2: evwK PviwU Dcv`vb w`‡q (2 2) gvÎvi wbY©vqK MVb Ki| †mwUB Abyivwk| †hgb- 1 Gi Rb¨ Abyivwk
6 7
6 9
Step-3: gvb ‡ei Ki‡Z ej‡j AvovAvwo ¸b K‡i gvb †ei Ki| †hgb- 1 Gi Rb¨ gvb = (6 9 – 7 6) = 12
7 Gi Rb¨ Abyivwk
1 2
3 6 Ges 7 Gi Rb¨ Abyivwki gvb = (1 6 2 3) = 0
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
68 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
2 1 5
MEx 02
4 3 2 wbY©vqKwUi –1 Ges 0 Gi Abyivwk †ei Ki|
1 0 6
General Rules [Written] & [MCQ]
4 2
–1 Gi Rb¨ Abyivwk →
1 6 Ges –1 Gi Rb¨ Abyivwki gvb = (4 6 – 1 (–2)) = 26
0 Gi Rb¨ Abyivwk →
2 5
4 2 Ges 0 Gi Rb¨ Abyivwki gvb = (2 (–2) – 4 5) = –24
a1 b1 c1
MEx 03 a2 b2 c2 G b3 Gi mn¸YK KZ?
a3 b3 c3
General Rules [Written] & [MCQ]
b3 Gi mn¸YK = ( 1)
3+2 a1 c1 a1 c1
=
a2 c2 a2 c2
1 0 2
MEx 04 A = 2 1 3 G A12 Gi mn¸YK Ges A21 Gi Abyivwk KZ?
1 5 0
General Rules [Written] & [MCQ]
A12 Gi mn¸YK = ( 1)1+2
2 3
1 0 = 3
0 2
A21 Gi Abyivwk = (1)2+1 5 0
= (0 5 (–2)) = 10
MEx 05 wb‡¤œi wbY©vq‡Ki (–2a) Gi mn¸YK KZ? [KUET. 11-12; RUET. 14-15]
2 2
1 + a – b 2ab –2b
2ab 1 – a2 + b2 2a
2b –2a 1 – a2 – b2
General Rules [Written] & [MCQ]
1 + a2 – b2 –2b
(–1)3+2
2ab 2a
= (–1) [2a + 2a3 – 2ab2 + 4ab2]
= – 2a(1 + a2 + b2)
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
Dhaka University Jahangirnagar University
2 –1 5 2 3 1
01. D = 4 3 –2 wbY©vq‡Ki 0 Gi mn¸YKÑ [DU. 97-98] 01 . A = 4 5 9 n‡j, 7 Dcv`vbwUi Abyivwk †KvbwU? [JU. 19-20]
1 0 6 6 7 8
A. 18 B. –24 A. –4 B. 14 C. –15 D. 32
S B Sol n 7 Gi Abyivwk =4 9= 18 – 4 = 14
2 1
C. 16 D. 24
olve
S D Sol n 0 Gi mn¸YK = – 4 –2
2 5
5 0 3
olve
02. –2 1 4G 4 Abyivwk †KvbwU? [JU-A. 17-18]
= – (–4 – 20) = 24 7 2 7
1 2 3 A. 10 B. –10 C. 8 D. 5
02. 4 5 6 wbY©vqKwUi Ô6Õ Gi mn¸YK KZ?
S A Sol n 4 Gi Abyivwk =7 2= (10 – 0) = 10
[DU. 95-96] 5 0
olve
7 8 9
B.
7 8
A.
1 2 GST (¸”Q)
7 8 1 2
1 2 1 2 2 3 1
C.
7 8 D. 01. 4 5 9 wbY©vqKwUi 7 Gi cofactor-Gi gvbÑ [IU-F. 12-13]
7 8 6 7 8
S C Sol n Ô6Õ Gi mn¸YK = 7 8
1 2 A. 14 B. –14 C. 23 D. –23
olve
2 3 1
B Sol n 4 5 9 7 Gi Co-factor = (–1) 4 9= – 4 9= –14
3+2 2 1 2 1
1 2
=
olve
S
7 8 6 7 8
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 69
1 –1 5 Engineering & Others
02. 0 1 xwbY©vqKwUi (1, 2) Zg Abyivwk –6 n‡j, x Gi gvb KZ?
2 –3 4 2 3 x
[PUST-B. 17-18] 01. 0 4 x wbY©vq‡Ki (2, 1)th fzw³i mn¸YK 9 n‡j x Gi gvb †KvbwU?
A. –2 B. –3 1 3 1 x
[CKRuet. 22-23]
C. 3 D. 6 1 3
S C Sol n wbY©vq‡Ki (1, 2) Zg Abyivwk = (0 – 2x) = – 2x A. B. C. 2
olve
2 2
cÖkœg‡Z, –2x = – 6 D. 0 E. 2
x=3 2 3 x
C Sol
n 0 4 x
olve
1 0 2 S
1 3 1 x
03. X = 1 2 0 n‡j X Gi (3, 2) Zg mn¸YK KZ? [MBSTU-A. 16-17]
0 –2 3
(2, 1) th fzw³i mn¸YK =
3 x
A. 0 B. –2 3 1 x = 9
C. 2 D. –3 (3 3x 3x) = 9 6x 3 = 9 x = 2
n (3, 2) Zg Dcv`vbwU – 2 Gi mn¸YK 2 –3 7 5
olve
S C Sol
02.
3 205 1 u
wbY©vq‡Ki “1” Gi mn¸YK n‡jvÑ [KUET. 14-15]
= – 1 0 = – (0 – 2) = 2
1 2
–1 97 4
3
0 –7 k 7
04.
1 2 A. u B. k C. 0
3 4
g¨vwUª· G 2 Gi mn¸YK KZ? [JUST. 13-14]
D. –935 E. –297
A. –1 B. –2
S C Sol n 1 Gi (mvwi + Kjvg) = (2 + 3) = 5 we‡Rvo
olve
C. –3 D. –4
E. –5 2 –3 5 2 3 2
1+2 1 Gi mn¸YK = – 3 –1 4= 3 1 3; [C3 = C3 + C2]= 0
S C Sol n 2 Gi mn¸YK = (–1) |3| = –3
olve
0 –7 7 0 7 0
【
? 】 QUICK PRACTICE CONCEPT TEST ?
2 5 1 1 2 3
01. 3 4 2 wbY©vqKwUi (2, 3) Zg fzw³i Abyivwk wb‡¤œi †KvbwU? 02. 2 3 4 wbY©vqKwUi (3, 2) Zg fzw³i mn¸YK KZ?
1 5 7
1 3 2
2 1 2 5
C.
A. 2 B. 3 C. 4 D. 5
A.
3 1
B. D. None
4 3 1 2 1 3 Answer 01.C 02.A
Concept-02 wbY©vq‡Ki gvb wbY©q ¸iæZ¡:
wbY©vq‡Ki gvb : †Kvb wbY©vq‡Ki †h †Kvb mvwi ev Kjv‡gi Gi Dcv`vbmg~n I Zv‡`i wbR wbR mnivwki ¸Yd‡ji FocusKwi
jÿ¨ Point:
mgwóB wbY©vq‡Ki gvb|
AwaKvsk †ÿ‡Î wbY©vq‡Ki gvb k~b¨
a1 b1 c1
a2 b2 c2 wbY©vq‡Ki a1, a2, a3 -Gi mn¸YK h_vµ‡g A1, A2, A3 n‡j wbY©vq‡Ki gvb n‡e Av‡m| P‡jv †`‡L †bqv hvK we‡kl
a3 b 3 c3 †ÿθ‡jv wK wK?
= a1A1 + a2A2 + a3A3 = a1
b2 c2 b1 c1 b1 c1 i. wbY©vq‡Ki mvwi¸‡jv Ges
b3 c3+ a2 – b3 c3+ a3b2 c2
Kjvg¸‡jv mgvšÍi cÖMgb fz³
wbY©vqK‡K †Kvb wKQz w`‡q ¸Y Ki‡Z n‡j †h‡Kv‡bv GKwU Kjvg A_ev †h‡Kv‡bv GKwU mvwi‡Z ¸Y Ki‡Z n‡e|
n‡j|
Example: A =
a b
c d n‡j, Ax = ? 1 2 3
†hgb: 4 5 6
Solve: Ax =
ax bx
c d 7 8 9
=
a b
ii. wbY©vq‡Ki †h‡Kv‡bv `ywU mvwi ev
cx dx
Kjvg ¸‡YvËi m¤úK© MVb
=
ax b
cx d 4 0 8
Ki‡j| †hgb: 2 3 4
=
a bx
c dx 1 5 2
GLv‡b cÖ_g I Z…Zxq KjvgØq
wbY©vq‡Ki gvb †h fv‡e †ei Ki‡Z n‡e- ¸‡YvËi m¤úK© m„wó K‡i‡Q|
Step-1: wbY©vq‡Ki 1g, 2q I 3q mvwi †jLvi ci 1g I 2q mvwi cybtivq wb‡P wjL‡Z n‡e|
iii. †h‡Kv‡bv `ywU mvwi ev Kjvg
Step-2: Zvici Ggb fv‡e Zxi KvU‡Z n‡e †hb cÖwZwU Zx‡i wZbwU K‡i msL¨v _v‡K|
Step-3: cÖwZwU Zx‡ii msL¨v¸‡jv Avjv`v Avjv`v fv‡e ¸b K‡i †hvMKi|
GKB n‡j|
Step-4: AZtci wb‡Pi Zx‡ii †hvMdj n‡Z Dc‡ii Zx‡ii †hvMdj we‡qvM Ki‡e| hv cv‡e ZvB Answer|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
70 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
wbY©vq‡Ki mvnv‡h¨ mgvavb wbY©q : †µgv‡ii m~‡Îi mvnv‡h¨ Lye mn‡RB GKNvZ mgxKiY †Rv‡Ui mgvavb †ei Kiv hvq| †hgb :
`yB PjK wewkó GKNvZ mgxKiY †Rv‡Ui †ÿ‡Î †µgv‡ii m~Î wZb PjKwewkó GKNvZ mgxKiY †Rv‡Ui †ÿ‡Î :
: a1x + b1y + c1z = d1 ......... (i)
a1x + b1y = c1 ......... (i) a2x + b2y + c2z = d2 ......... (ii)
a2x + b2y = c2 ......... (ii) a3x + b3y + c3z = d3 ......... (iii)
Dx Dy x y z 1
x= Ges y= Avgiv cvB, = = =
D D D x D y D z D
GLv‡b, x I y Gi mnM¸wj Øviv MwVZ wbY©vqK, a1 b1 c1
GLv‡b, x, y I z Gi mnM¸wj Øviv MwVZ wbY© v qK, D = a2 b2 c2 0
D=
a1 b1
a2 b2 0 a3 b3 c3
x Gi mn‡Mi cwie‡Z© aªæeK c` wb‡q MwVZ wbY©vqK, d1 b1 c1
x Gi mn‡Mi cwie‡Z© aª æeK c` wb‡q MwVZ wbY© v qK, D = d2 b2 c2
Dx =
c1 b1 x
c2 b2 d3 b3 c3
Ges y Gi mn‡Mi cwie‡Z© aªæeK c` wb‡q MwVZ wbY©vqK, a1 d 1 c1
Avevi, y Gi mn‡Mi cwie‡Z© aªæeK c` wb‡q MwVZ wbY©vqK, Dy =a2 d2 c2
Dy =
a1 c1
a2 c2 a3 d3 c3
1 b1 d1
a
Ges z Gi mn‡Mi cwie‡Z© aªæeK c` wb‡q MwVZ wbY©vqK, Dz = a2 b2 d2
a3 b3 d3
Dx Dy Dz
x= ,y= I z = D n‡Z x, y I z Gi gvb A_©vr, cÖ`Ë mgxKiY †Rv‡Ui
D D
mgvavb wbY©q Kiv hvq|
x y 1 x y 1 x y 1
eRª¸Yb m~Îvbymv‡i, = = = = = =
– b1c2 + b2c1 – c1a2 + c2a1 a1b2 – a2b1 c 1 b 1 a 1 c 1 a 1 b 1 D x D y D
c2 b2 a2 c2 a2 b2
Note : hw` D 0 nq, Z‡e mgxKiY †Rv‡Ui Abb¨ mgvavb we`¨gvb| †Kej D 0 k‡Z©B †µgv‡ii wbqg cÖ‡hvR¨|
1 2 3
MEx 01 4 5 6 wbY©vq‡Ki gvb KZ?
0 8 0
General Rules [Written] & [MCQ]
1(5.0 – 8.6) –2(4.0 – 0.6) + 3(4.8 – 5.0) = – 48 – 0 + 96 = 48
Procedure With Steps and Figure
†h fv‡e AsKwU Ki‡Z n‡e: cv‡ki wPGwU fv‡jv K‡i jÿ Ki| c×wZwUi myweav :
0
†Kvb mvwi Kjvg GK Kivi †Kvb Sv‡gjv/Tension _v‡K bv| 48 †hvMdj
cix¶vi n‡j wM‡q ‡PvL eyu‡S AsK Kiv ïiy Ki‡Z cvi‡e| 1 2 3 0
gy‡L gy‡L Kiv m¤¢e| 40 sec Gi †ewk mgq jvM‡e bv| cix¶vq G ai‡bi 4 5 6
we‡qvM
AsKB †ewk Av‡m | 0 8 0
Step-1: 1st Ges 2nd Row `ywU cv‡ki wP‡Îi gZ wb‡P wb‡P wjL|
2
Step-2: Zvici Ggb fv‡e Zxi KvU‡Z n‡e †hb cÖwZwU Zx‡i wZbwU K‡i msL¨v 1 3
_v‡K | 4 5 6 0
Step-3: cÖwZwU Zx‡ii msL¨v¸‡jv Avjv`v Avjv`v fv‡e ¸b K‡i †hvM Ki | 0
96
†hvMdj
Step-4: AZtci wb‡Pi Zx‡ii †hvMdj n‡Z Dc‡ii Zx‡ii †hvMdj we‡qvM
Ki‡e| hv cv‡e ZvB Answer| wbY©vq‡Ki gvb = (0 + 96 + 0) – (0 + 48 + 0) = 48
–2
0
3 0 0 4
2 0 0 0
MEx 02 gvb wbY©q Ki 0 –1 0 5 –3 [RUET. 2015-16]
–4
0
0 1 0 6
–1 0 3 2
General Rules [Written] & [MCQ]
0 –2 4
3 0
0 2 0 0 0 3 0 –2 4 2 0 0
0
0 2 0
0 –1 0 5 –3 3q Kjvg eivei we¯Ívi K‡i = –1 0 –1 5 –3 = (–1) 3 –1 5 –3
–4 –1 3 2
0 –1 3 2
0 1 0 6
0 –1 0 3 2
–3
= (–1) 3 2 3
5
2 = –6(10 + 9) = –114
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 71
MEx 03 wbY©vq‡Ki mvnv‡h¨ mgvavb Ki : 5x + 2y – 11 = 0
3x + 4y – 1 = 0
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
cÖ`Ë mgxKiY ¸‡jv : 5x + 2y = 11 ; 3x + 4y = 1 5x + 2y = 11 ......... (i)
D=
5 2 11 2 3x + 4y = 1 ......... (ii)
3 4= 20 – 6 = 14; Dx = 1 4= 44 – 2 = 42 x = 3 Ges y = –2 n‡j (i) I (ii) mgxKiY wm× nq|
ZvB mgvavb n‡e (x, y) = (3, –2)
Dy =
5 11
3 1 = 5 – 33 = – 28
Dx 42 Dy –28
†µgv‡ii m~Îvbymv‡i, x = = = 3; y = = = –2
D 14 D 14
wb‡Y©q mgvavb (x, y) = (3, –2)
MEx 04 x + y – z = 3, 2x + 3y + z = 10, 3x – y – 7z = 1 mgxKiY wZbwU‡K †µgv‡ii wbq‡gi mvnv‡h¨ mgvavb Ki|
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
1 1 –1 x + y – z = 3 ............... (i)
GLv‡b, D = 2 3 1= –20 + 17 + 11 = 8 2x + 3y + z = 10 ......... (ii)
3 –1 –7 3x – y – 7z = 1 ............ (iii)
3 1 –1 x = 3; y = 1 Ges z = 1 n‡j (i); (ii) I (iii) mgxKiY wm× nq|
Dx = 10 3 1 = –60 + 71 + 13 = 24 ZvB mgvavb n‡e (x, y. z) (3, 1, 1)
1 –1 –7
1 3 –1
Dy = 2 10 1= –71 + 51 + 28 = 8
3 1 –7
1 1 3
Dz = 2 3 10= 13 + 28 – 33 = 8
3 –1 1
Dx 24 Dy 8 Dz 8
x= = = 3, y = = = 1 Ges z = = =1
D 8 D 8 D 8
wb‡Y©q mgvavb (x, y. z) (3, 1, 1)
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
Dhaka University Jahangirnagar University
10 11 12 2 1 5
01. 20 21 24 Gi gvb KZ? [DU. 98-99] 01. 4 3 2 wbY©qvKwUi "0" Gi mn¸YK †KvbwU? [JU-A, Set-H. 22-23]
10 10 10 1 0 6
A. 10 B. 20 C. 1 D. 0 A. 18 B. 24 C. 16 D. 24
10 11 12 1 5
2
D Sol n 4 3 2
olve
S B Sol n 20 21 24 S
olve
10 10 10
1 0 6
'0' Gi mn¸YK = ( 1)3+2
2 5
1 1 12 c1 = c1 c2 = ( 4 20) = 24
1 1 4 2
= 1 3 24 c = c c = 10 = 10 (3 1) = 20
1 3 02.
a3 1
0 0 10 2 2 3 8 a + 4wbY©vqKwUi gvb k~b¨ n‡j "a" Gi gvb †KvbwU?
5 6 7 [JU-A, Set-S. 22-23; DU. 07-08]
02. 1 2 3 wbY©vqKwUi gvb KZ? [DU. 97-98; RU. 06-07] A. 4 or 5 B. 5 or 4
3 6 9 C. 3 D. 10
A. 2 B. 0 C. 1 D. 12 a 3 1
S A Sol n 8 a + 4 = 0
olve
S B Sol n 2 bs I 3 bs mvwi ¸‡YvËi m¤ú‡K© i‡q‡Q ZvB gvb k~b¨|
olve
(a 3) (a + 4) 8 = 0 a2 + a 20 = 0 a = 4 or 5
10 20 30
03. 40 50 60 wbY©vqKwUi gvb KZ? [DU. 96-97] 13 16 19
03. 14 17 20 wbY©vqKwUi gvb †KvbwU?
50 70 90 15 18 21
[JU-A, Set-R. 22-23]
A. 0 B. 100
A. 1 B. 0
C. 100 D. 140 C. 1 D. 2
10 20 30 10 10 30 +3 +3
c1 = c1 c2
S A Sol 40 50 60 =
n 10 10 60 =
olve
=0
50 70 90 2 = c2 c3
c 13 16 19
20 20 90 S B Sol n 14 17 20 mgvšÍi avivq we`¨gvb _vKvq wbY©vqKwUi gvb 0|
olve
[cvkvcvwk `yBwU Kjv‡gi Dcv`vb GKB|] 15 18 21
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
72 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
1 1 1 04. wbY©vq‡Ki mvwi I Kjvgmg~n ci¯úi ¯’vb wewbgq Ki‡j wbY©vq‡Ki gvb- [RU. 15-16]
04. 1 2 3 wbY©vq‡Ki gvb 2 n‡j, K Gi gvb KZ? [JU-A, Set-F. 2021-22] A. k~b¨ n‡e B. cwieZ©b n‡e bv
1 4 K C. cwieZ©b n‡e D. wecixZ wPýwewkó n‡e
A. 9 B. 8 n wbY©vq‡Ki mvwi I Kjvg mg~n ci¯úi ¯’vb wewbgq Ki‡jI
olve
C. 7 D. 6 S B Sol
wbY©vq‡Ki gv‡bi †Kvb cwieZ©b n‡e bv|
1 1 1 05. †Kvb wbY©vq‡Ki GKwU mvwi ev Kjv‡gi Dcv`vb¸‡jv‡K C Øviv ¸Y K‡i wbY©vqKwUi
S A Sol n 1 2 3 = 2
olve
1 4 K Aci GKwU mvwi †_‡K we‡qvM Kiv n‡j wbY©vqKwUi gvb n‡e- [RU. 13-14]
A. C ¸Y †ekx B. C cwigvY †ekx
1(2K – 12) – 1(K –3) + 1(4 – 2) = 2
C. k~b¨ D. AcwiewZ©Z _vK‡e
2K – 12 – K + 3 + 2 = 2 K = 9 n †Kvb wbY©vq‡Ki GKwU mvwi ev Kjv‡gi Dcv`vb ¸‡jv‡K C
olve
2 4 6 S D Sol
05. 0 8 10 wbY©vq‡Ki gvb wb‡Pi †KvbwU? [JU. 18-19] Øviv ¸Y K‡i wbY©vqKwUi Aci GKwU mvwi †_‡K we‡qvM Kiv n‡j
0 0 12 wbY©vqKwUi gvb AcwiewZ©Z _vK‡e|
A. 48 B. 80 C. 12 D. 192 1 3 4
n 2(8 12 10 0) = 192 06. A = 2 1 0 n‡j bx‡Pi †KvbwU mwVK?
olve
S D Sol
[RU. 12-13]
5 1 3 3 2 5
06. M = –3 2 6 n‡j |M| = †KvbwU?
A. det(A) = 3 B. det(A) = 0
[JU. 17-18]
C. det(A) = 63
2 3 9 D. det(A) wbY©q Kiv m¤¢e bq
A. 1 B. –1 C. 0 D. 2
S C Sol n det (A) = 1(5 – 0) – (–3) (10 – 0) + (– 4) (– 4 – 3)
olve
S C Sol n 2q I 3q Kjvg ¸‡YvËi m¤ú‡K© i‡q‡Q|
olve
= 5 + 30 + 28 = 63
3 4 5
07. 6 7 8 Gi gvb KZ? [JU. 12-13, 11-12, 09-10]
Chittagong University
0 2 0 5 3 2
A. 6 B. 12 C. 6 D. 12 01 . If A = 0 4 1 then |A| = KZ? [CU-I. 16-17]
n wbY©vq‡Ki gvb = 3(0 – 16) – 4(0 – 0) + 5 (12 – 0) 0 0 3
olve
S D Sol
A. 30 B. 40
= 3.(–16) – 4.0 + 5.12 = – 48 – 0 + 60 = 12
C. 50 D. 60
Rajshahi University E. 66
x 0 0 S D Sol n |A| = 5(12 – 0) + 0 + 0 = 60
olve
1 4
01. A =
1 2
I B = 3 4 1 n‡j, x-Gi †Kvb gv‡bi Rb¨ |A| = |B| n‡e?
2 2 1 a 0 1
02. hw` 3 2 5 = 4 nq, Z‡e a Gi gvb KZ? [CU-A, Set-2. 20-21]
A. 1 B. 1
[RU-C, Feldspar-1. 22-23]
4 0 3
A. 2 B. 3
C. 0 D. 2
C. 4 D. 5
4 x 0 0
S A Sol n |A| = |B| 1 2 = 3 4 1
1 a 0 1
olve
S A Sol n 2 2 5 = 4 6a 8 = 4 a = 2
olve
2 2 1 4 0 3
2 – (– 4) = x (4 + 2) 6 = x.6 x = 1 –3i
6i 1
1 2 3 03. hw` 4 3i –1= x + iy nq, Z‡e †KvbwU mwVK? [CU-A, Set-1. 20-21]
02. P Gi ‡Kvb gv‡bi Rb¨ 1 2 P wbY©vqKwUi gvb k~b¨ n‡e? 20 3 i
3 4 0 A. x = 3, y = 1 B. x = 1, y = 3
[RU-C, Topaz-3. 22-23]
A. – 3/5 B. 3 / 5 C. x = 0, y = 3 D. x = 0, y = 0
C. – 3 D. 3 6i 3i 1
Sol 4 3i 1 = x + iy
n
olve
1 2 3 S D
20 3 i
S D Sol n 1 2 p = 0
olve
3 4 0 6i 1 1
3(4 6) P(4 6) = 0 6 + 2P = 0 P = 3 3i 4 1 1 = x + iy
03. wb‡Pi †Kvb wbY©vq‡Ki gvb k~b¨? [RU-Uranus-1, Set-1. 21-22] 20 i i
x + iy = 0 x = 0, y = 0
1 0 2 4 0 8
A. 2 0 1 B. 2 3 4 1 2 3
1 3 0 1 5 2 04. 4 5 6 = KZ? [CU. 14-15, 09-10, 06-07, 03-04; KU. 13-14, 03-04]
1 0 0 0 0 1 7 8 9
C. 0 1 0 D. 1 2 3 A. 0 B. 1 C. 2
0 0 10 0 6 0 D. 3 E. 4
S B Sol n Option C Gi 1g I 3q KjvgØq ¸‡YvËi MVb K‡i ZvB S A Sol n †h‡nZz wbY©vqKwUi mvwi ¸‡jv Ges Kjvg ¸‡jv mgvšÍi cÖMgb
olve
olve
wbY©vq‡Ki gvb k~b¨| fz³ ZvB wbY©vqKwUi gvb k~b¨|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 73
2 0 0 0 (iii) – (ii) x + 3y = 11 .......... (iv)
05.
3 1 0 0 (i) 3 + (ii) 5x + 5y = 25
wbY©vqKwUi gvb n‡eÑ
0
[CU. 12-13]
4 2 1 x + y = 5 .......... (v)
5 3 2 1 (iv) – (v) 2y = 6 y = 3
A. 1 B. 2 03. wZb PjK wewkó mgxKiY †Rv‡Ui mgvav‡bi Rb¨ e¨eüZ wbY©vqK cÖwµqv‡K
C. 3 D. 4
Kx e‡j? [JKKNIU-B. 16-17]
E. 5
A. gvwÎK cÖwµqv B. wbDUb cÖwµqv
2 0 0 0 2 0 0 C. †µgv‡ii cÖwµqv D. _gmb cÖwµqv
3 1 0 0
S B Sol n 4 2 1 0 = 3 1 0 = 3 1= 2
2 0
olve
n wZb PjK wewkó mgxKiY †Rv‡Ui mgvav‡bi Rb¨ e¨eüZ
olve
5 3 2 1 4 2 1
S C Sol
wbY©vqK cÖwµqv‡K †µgvi cÖwµqv e‡j|
10 0 0 04. wbY©vq‡Ki mvnv‡h¨ mgvavb Ki: 5x + 2y –11 = 0 Ges 3x + 4y –1 = 0
06. D = 0 10 0 n‡j D Gi gvb KZ n‡e? [CU. 08-09] [CoU. 15-16]
0 0 10 A. (x, y) = (3, –2) B. (x, y) = (–2, 3)
A. 10 B. 102 C. (x, y) = (5, 4) D. (x, y) = (5, –1)
C. 103 D. 0 n Shortcut: 10x + 4y – 22 = 0 ............... (i)
olve
S A Sol
E. c~e©eZ©x †KvbwUB bq 3x + 4y – 1 = 0 ............... (ii)
10 0 0 1 0 0 {(i) (ii)} 7x – 21 = 0
S C Sol n D = 0 10 0 = 10 0 1 0 = 10
3 3
olve
x=3
0 0 10 0 0 1 15 + 2y – 11 = 0 2y = –4
GST (¸”Q) y = –2 (3, –2)
01. †µgv‡ii cÖwµqvq wb‡Pi mgxKiY‡Rv‡Ui mgvavb Ki‡j △x KZ n‡e? 10 13 16
05. 11 14 17 wbY©vq‡Ki gvb- [JnU. 10-11; RU. 10-11]
mgxKiY‡RvU : 2x + 3y = 5, 5x – 2y = 3
[JKKNIU. 19-20; BRUR-E. 16-17]
12 15 18
A. –19 B. 19 A. 0 B. 1
C. 15 D. –15 C. 10 D. 5
S A Sol †h‡nZz wbY©vqKwUi mvwi ¸‡jv Ges Kjvg ¸‡jv mgvšÍi cÖMgb
n
olve
S A Sol △x = 3 –2= – 10 – 9 = –19
n
5 3
olve
fz³ ZvB wbY©vqKwUi gvb k~b¨|
02. x + 2y – z = 9; 2x – y + 3z = –2; 3x + 2y + 3z = 9 mgxKiY¸”Q 50 60 70
mgvavb Ki‡j y Gi gvb KZ n‡e?- [IU. 17-18] 06. 10 20 30 wbY©vqKwUi gvb- [JnU. 08-09]
A. 0 B. 1 30 60 90
C. 2 D. 3 A. 1 B. 2
S D Sol n x + 2y – z = 9 .......... (i)
olve
C. 3 D. 0
n †h‡nZz 2q I 3q mvwi ¸‡YvËi m¤ú‡K© i‡q‡Q ZvB gvb k~b¨
olve
2x – y + 3z = –2 .......... (ii) S D Sol
3x + 2y + 3z = 9 .......... (iii) n‡e|
Concept-03 GKwU ARvbv PjK _vK‡j gvb wbY©q ¸iæZ¡:
GB wbq‡gi AsK¸‡jv mivmwi wbY©vq‡Ki †gŠwjK wbqg e¨envi K‡i Kiv hvq| h_v:
x + 4 3 3
MEx 01 x-Gi mgvavb Ki : 3 x+4 5 = 0 [RUET. 04-05; KUET. 04-05; CUET. 13-14; BUET. 13-14, 01-02]
5 5 x + 1
General Rules [Written] & [MCQ]
x+1 3 3
– x – 1 x+4 5 = 0 ; [c1 = c1 – c2]
0 5 x + 1
1 3 3
(x + 1) –1 x + 4 5 = 0
0 5 x + 1
1 3 3
(x + 1) 0 x + 7 8 = 0 ; [r2 = r1 + r2]
0 5 x + 1
(x + 1) [(x + 7) (x + 1) – 40] = 0
(x + 1) (x2 + 8x – 33) = 0
(x + 1) (x + 11)(x – 3) = 0
x = –1, –11, 3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
74 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
1 1 1
MEx 02 gvb wbY©q Ki : 1 p p2 [BUET. 07-08; RUET. 13-14, 12-13, 08-09, 07-08; Xv.†ev: 12, 07; iv.†ev: 11-10; Kz.†ev: 15]
1 p2 p4
General Rules [Written] & [MCQ]
1 1 1 0 0 1
1 p2 p2=1 – p p – p2 p2[c1= c1 – c2;c2= c2 – c3]
1 p p4 1 – p2 p2 – p4 p4
0 0 1
= (1 – p)21 + p (1 + p)p2
1 p
= (1 – p) (1 – p)
2
1 p p
1 + p (1 + p)p p
2 4
= (1 – p)2 (p2 + p3 – p – p2) = (1– p)2 (p3 – p) = p(1– p)2 (p2 –1)
0 3 2x +7
MEx 03 2 7x 9 + 5x = 0 n‡j, x Gi gvb- [RU-C, Jupitar-1, Set-1. 2021-22,DU. 13-14]
0 0 2x + 5
General Rules [Written] & [MCQ]
5
2{3(2x+5) – 0 (2x+7)} = 0 2(6x+15) = 0 6x = – 15 x =
2
3 + x 4 2
MEx 04 4 2+x 3 = 0 n‡j, x Gi gvb- [CUET. 13-14; Xv.†ev: 05; Kz.†ev: 07; P.†ev: 16,07]
2 3 4 + x
General Rules [Written] & [MCQ]
3 + x 4 2 3 + x + 4 + 2 4 2
4 2+x 3 = 0 4 + 2 + x + 3 2 + x 3 = 0 [∵ c1 = c2 + c2 + c3]
2 3 4 + x 2 + 3 + 4 + x 3 4 + x
x + 9 4 2 1 4 2 0 2 – x –1
x + 9 2 + x 3 = 0 (x + 9) 1 2 + x 3 = 0 (x + 9) 0 x – 1 –1– x= 0 [r1 = r1 – r2 Ges r2 = r2 – r3]
x + 9 3 4 + x 1 3 4 + x 1 3 4 + x
– – –1
(x + 9)
(x 2)
x – 1 –1– x= 0 (x + 9) (x – x – 2 + x – 1) = 0 (x + 9) (x – 3) = 0
2 2
nq, x + 9 = 0 x = – 9 A_ev, x2 – 3 = 0 x2 = 3 x = 3 wb‡Y©q mgvavb : x = – 9, 3
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
Dhaka University 2 1
S D Sol n 5 + 4 = 0
olve
1 1 1
01. k Gi †Kvb gv‡bi Rb¨ 1 k k wbY©vqKwUi gvb k~b¨ n‡e bv? [DU. 17-18]
2 ( 2) ( + 4) + 5 = 0 2 + 2 3 = 0 = 1 or 3
Chittagong University
1 k2 k4
B. k = –1
A. k = 1
a 0 1
C. k = 3 D. k = 0 01. hw` 3 2 5 = 4 nq, Z‡e „a‟ Gi gvb KZ? [CU. 15-16]
1 1 1
2
Aspect Special: 4 0 3
S C Sol n 1 k k = 0
olve
k = 0 n‡j cÖ`Ë wbY©vqKwUi A. 3 B. 2 C. 5
1 k k4
2
2q I 3q mvwi `ywU GKB D. 6 E. 7
1 0 0 nq Ges wbY©vq‡Ki gvb k~b¨ a 0 1
S B Sol 3 2 5 = 4
n
olve
1 k–1 k –k = 0
2
nq|
2 4 0 3
1 k –1 k –k
2 4
k = 1 n‡j cÖ`Ë
2(3a – 4) = 4 6a – 8 = 4
1 0 0 wbY©vqKwUi 1g I 3q Kjvg
(k–1) (k –1) 1 1 k = 0
2 `ywU GKB nq Ges 6a = 12 a = 2
2 1 1 1
1 1 k wbY©vq‡Ki gvb k~b¨ nq|
(k–1) (k2–1) (k2–k) = 0 k = 3 n‡j †Kv‡bv fv‡eB
02 . 1 2 3 wbY©vqKwUi gvb 2; k Gi gvb KZ?
k = 1, –1, 0 wbY©vq‡Ki †h †Kvb `ywU Kjvg 1 4 k
[CU. 12-13; RU. 12-13; DU. 00-01; SUST. 04-05]
k = 3 Gi Rb¨ wbY©vqKwUi gvb k~b¨ n‡e bv ev mvwi GKB n‡e bv| A. 9 B. 8
02. wbY©vqKwUi gvb 0 n‡j, Gi gvb KZ?
2 1 C. 7 D. 6
5 + 4 1 1 1
S A Sol n 1 2 3 = 2
olve
[DU. 01-02; JU. 19-20; JnU. 05-06]
A. 5 A_ev 0 B. 6 A_ev 2 1 4 k
C. 5 A_ev 3 D. 1 A_ev 3 1(2k 12)1(k 3) + 1(4 2) = 2 k = 9
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 75
1 2 a 5 6 x
03. 4 5 6 ; a Gi gvb KZ n‡j wbY©vqKwU singular n‡e? S C Sol 1 2 3 = 4
n
olve
[CU. 10-11]
7 8 9 3 2 1
A. 1 B. 2 C. 3 D. 4 E. 5 5 6 x
1 2 3 = 4 ; [r3 = r3 r2]
S C Sol wbY©vqKwU Singular n‡j wbY©vq‡Ki gvb k~b¨ n‡e
n
olve
2 0 2
1(45 – 48) – 2(36 – 42) + a(32 – 35) = 0
– 3 + 12 – 3a = 0 a = 3
5+x 6 z
4 2 3 = 4 ; [c1 = c1 + c3]
1 1 x 0 0 2
04. x Gi gvb KZ n‡j 2 2 2 = 0 n‡e? [CU. 07-08] 2(10 + 2x 24) = 4
3 4 5 2x 14 = 2 x = 8
A. 2 B. 5 C. 2 Engineering
D. 3 E. 1 4 2
1 1 x 0 1 x x x a
01 . 3 1 b = 0 n‡j a Gi gvb wbY©q Ki| [CKRUET 2021-22]
S E Sol n 2 2 2 = 0 0 2 2 = 0 [c1 = c1 c2] 0 0 c
olve
3 4 5 1 4 5 A. 0 a B. 0 2 C. 0, b
(2 2x) = 0 x =1 D. 0, 3 E. 0, 3
hw` wbY©vqK
3 4
S D Sol n cÖ‡kœ a Gi RvqMvq x n‡e| A_©vr x Gi gvb †ei Ki‡Z n‡e|
olve
05.
5 2a Gi gvb k~b¨ nq, Zv n‡j a Gi gvb n‡e- [CU. 05-06] 4 2
A.
10
B.
6
C.
15 x x a
3 5 8 3 1 b= 0 c (x4 – 3x2) = 0
10 5
0 0 c
D. E. x2 (x2 – 3) = 0 x = 0, 3
3 6
02. x-Gi †Kvb †Kvb gv‡bi Rb¨ wb¤œwjwLZ wbY©vq‡Ki gvb k~b¨ n‡e?
S D Sol n 5 2a = 0 6a 20 = 0 a =
3 4 10
olve
2
x x 2
3 2 1 1
GST (¸”Q)
[BUET. 05-06; KUET. 10-11]
3 3 0 0 5
i3 i5 i3 + i 5 A. x=0, -2 B. x =1,2
01. i 1 n‡j, i i i + i = ?
2
[GST-A. 20-21] C. x= 0,1 D. x =0, 2
i3 i7 i5 + i7 2
x x 2
n 2 1 1 = 5
A. –1 B. 0 C. 1 D. i x2 x
= 0 (x2 2x) = 0
olve
i3
3 5 3 5 3 5 3
i i + i 3
i i3
i S D Sol 2 1
0 0 5
S B Sol n i i i + i i i i ; [c3 = c3 c2] = 0
olve
x (x 2) = 0 x = 0, 2
i3 i7 i5 + i7 i3 i7 i5
2
x x 2 ✍ Written
x Gi †Kvb gv‡bi Rb¨ 3 1 1 = 0 n‡e? x + 4 3 3
02.
[IU. 17-18]
01. x Gi mgvavb Ki 3 x + 4 5 = 0
0 0 –5 5 5 x+1
A. 0, – 3 B. 0, 3 C. 2, 0 D. 0, –2 [BUET. 01-02, 13-14; RUET. 04-05; KUET. 04-05]
2
S B Sol n x (–5 – 0) – x(–15 – 0) + 2(0 – 0) = 0
olve
x+1 3 3
ev, –5x + 15x + 0 = 0 ev, x2 – 3x = 0 ev, x(x – 3) = 0 ev, x = 0, 3
2
Solve
x 1 x + 4 5 = 0 [C1 = C1 – C2]
03. 3 x wbY©vqKwUi gvb (–1) n‡j x Gi gvb KZ? 0 5 x+1
2 1 [KU. 16-17]
1 3 3
A. –2 B. 0 C. 2 D. 4
(x + 1) 1 x + 4 5 = 0
S C Sol n 2 1 = –1 3 – 2x = –1 2x = 4 x = 2
3 x
olve
0 5 x+1
DU Affiliated College Question 1 3 3
(x + 1) 0 x + 7 8 = 0 [r1= r1+r2]
5 6 x 0 5 x+1
01. 1 2 3 g¨vwU·wUi wbY©vqK –4 n‡j, z Gi gvb KZ? [DU Tech. 2020-21]
3 2 1 (x + 1) [(x + 7) (x + 1) – 40] = 0 (x + 1) (x2 + 8x – 33) = 0
A. 7 B. 6 C. 8 D. 1 (x + 1) (x + 11) (x – 3) = 0 x= – 1, – 11, 3
【
? 】 QUICK PRACTICE CONCEPT TEST ?
x 1 3 = 0 n‡j x Gi gvb KZ? 5 6 7
01. 5 x+1 [BUP. 20-21]
02. 1 2 3= ?
A. 16 B. 16 3 6 9
C. 4 A. 0 B. 1 C. 2 D. 3
D. 2 Answer 01. C 02. A
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
76 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
Concept-04 me¸‡jv ARvbv PjK _vK‡j gvb wbY©q ¸iæZ¡:
GB wbq‡gi AsK¸‡jv wewfbœ Dcv‡q Kiv hvq| h_v:
1. wbY©vq‡Ki we¯Íw… Z K‡i| 2. wbY©vq‡Ki †gŠwjK ¸Yvejx e¨envi K‡i| 3. Short Tricks Apply K‡iI Kiv hvq MCQ Gi Rb¨ |
Aspect Special: wbY©vq‡Ki me¸‡jv Dcv`vb Pj‡Ki gva¨‡g w`‡q wbY©vqKwUi gvb wbY©q Ki‡Z ej‡j,
Step-1: PjK¸‡jvi Rb¨ wfbœ& wfbœ Ak~b¨ †QvU gvb we‡ePbv K‡i wbY©vqKwU‡K mivmwi we¯Ívi K‡i gvb †ei Kie
Step-2: wbY©vqKwUi gvb †ei Kivi mgq PjK¸‡jvi †h gvb we‡ePbv Kiv n‡q‡Q, †mB gvb¸‡jv Option ¸‡jv‡Z ewm‡q wbY©vqKwUi cÖvß gv‡bi mgZzj¨ Luy‡R
†ei Ki†e| †h Option Gi mv‡_ wg‡j hv‡e ZvB Answer|
wbY©vq‡Ki †gŠwjK ¸Yvejx:
(i) †Kvb wbY©vq‡Ki mvwi¸wj Zv‡`i Abyiƒc Kjvg mg~‡n cwiewZ©Z n‡j Ges Kjvg¸wj Zv‡`i Abyiƒc mvwimg~‡n cwiewZ©Z n‡j, wbY©vq‡Ki gv‡bi †Kvb
a1 b 1 c1 a1 a2 a3
cwieZ©b nq bv| †hgb, D = 2 2 2 Ges D = b1 b2 b3 D = D .
a b c
a3 b 3 c3 c1 c2 c3
(ii) †Kvb wbY©vq‡Ki †h †Kvb `yBwU mvwi ev Kjvg ci¯úi ¯’vb wewbgq Ki‡j, wbY©vqKwUi wPý e`‡j hvq wKš‘ wbY©vq‡Ki gv‡bi †Kvb cwieZ©b nq bv|
a1 b 1 c1 b 1 a1 c1 a1 b 1 c 1 a2 b 2 c2
†hgb, D = a2 b2 c2 = b2 a2 c2 Ges D = a2 b2 c2 = a1 b1 c1
a3 b 3 c3 b 3 a3 c3 a3 b 3 c 3 a3 b 3 c3
a1 b1 0
(iii) †Kvb wbY©vq‡Ki GKwU mvwi ev Kjv‡gi me¸wj Dcv`vb k~b¨ n‡j, wbY©vqKwUi gvb k~b¨ nq| †hgb, D = a2 b2 0 = 0
a3 b3 0
a1 a1 c1
(iv) †Kvb wbY©vq‡Ki `yBwU mvwi ev Kjv‡gi Abyiƒc Dcv`vb¸wj Awfbœ n‡j, Zvi gvb k~b¨ nq| †hgb, D = = 0, D = a2 a2 c2 = 0
a3 a3 c3
(v) hw` GKwU wbY©vq‡Ki †Kvb mvwi ev Kjvg Gi Dcv`vb¸wj‡K Aci GKwU mvwi ev Kjv‡gi Abyiƒc Dcv`v‡bi mn¸YK Øviv ¸Y Kiv nq, Z‡e ¸Yd‡ji
mgwó k~b¨ nq| †hgb- a2A1 + b2B1 + c2C1 = 0
(vi) †Kvb wbY©vq‡Ki †h †Kvb mvwi ev Kjv‡gi cÖ‡Z¨K Dcv`vb‡K GKB ivwk Øviv ¸Y Ki‡j wbY©vqKwUi gvb‡KI H ivwk Øviv ¸Y Ki‡Z nq| †hgb,
a1 b1 c1 ka1 kb1 kc1 ka1 b1 c1
D = k a2 b2 c2 = a2 b2 c2 = ka2 b2 c2
a3 b3 c3 a3 b3 c3 ka3 b3 c3
(vii) †Kvb wbY©vq‡Ki †Kvb mvwi ev Kjv‡gi cªwZwU Dcv`vb `yBwU ivwki mgwó wnmv‡e cÖKvwkZ n‡j, wbY©vqKwU†K GKB µ‡gi `yBwU c„_K wbY©vq‡Ki †hvMdj
a1 + 1 b 1 c1 a1 b 1 c1 1 b 1 c1
wnmv‡e cÖKvk Kiv hvq| †hgb, D = a2 + 1 b2 c2 = a2 b2 c2 + 1 b2 c2
a3 + 1 b 3 c3 a3 b 3 c3 1 b 3 c3
(viii) †Kvb wbY©vq‡Ki GKwU mvwi ev Kjv‡gi f~w³¸‡jv Aci `yBwU mvwi ev Kjv‡gi Abyiƒc f~w³¸‡jvi mv‡_ mgvšÍi aviv MVb Ki‡j wbY©vq‡Ki gvb k~b¨
n‡e| A_©vr G‡ÿ‡Î `yB †Rvov mgvšÍi aviv cvIqv hv‡e|
(ix) †Kvb wbY©vq‡Ki GKwU mvwi ev Kjv‡gi f~w³¸‡jv Aci †Kvb mvwi ev Kjv‡gi Abyiƒc f~w³¸‡jvi mv‡_ ¸‡YvËi aviv MVb Ki‡j wbY©vq‡Ki gvb k~b¨ n‡e|
x+y x y
MEx 01 x x + z z wbY©vq‡Ki gvb- [JU-A, Set-O. 2021-22,DU. 08-09; BAU. 08-09; JnU. 06-07; RU. 05-06; P.†ev: 14; Kz.†ev: 11; h.†ev: 05]
y z y+z
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
x + y x y
x x+z z
y z y+z wbY©vqKwU‡Z x = 1, y = 2, z = 3 we‡ePbv Ki‡j wbY©vqKwUi AvKvi nq:
x + y x y x y 3 1 2
= x x z z x + z z ; [c1 = c1 (c2 + c3) cÖ‡qvM K‡i] 1 4 3 = 3(20 9) 1(5 6) + 2(3 8) = 24
y – y z z z y + z 2 3 5
0 x y 0 x y GLb †h Option ¸‡jv Av‡Q, Zv‡`i g‡a¨ †hB Option G x=1, y=2,
= 2z x + z z = 2z 1 x + z z
z = 3 emv‡j 24 cvIqv hv‡e †mwUB Correct Answer.
2z z y + z 1 z y + z
GLv‡b, x = 1, y = 2, z = 3 a‡i,
0 x y 4xyz = 4.1.2.3 = 24 ZvB Correct Answer (4xyz)
= 2z 0 x y ; [r2 = r2 r3]
1 z y + z
= 2z
x y
x y = 2z( xy xy) = 2z( 2xy) = 4xyz
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 77
1 x y+z
MEx 02 1 y z + x Gi gvb-
1 z x+y
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
1 x y + z 1 x x+y+z x = 1, y = 2, z = 3 a‡i,
1 y z + x = 1 y x + z + x ; [c3 = c2 + c3] 1 x y + z 1 1 5
1 z x+y 1 z x+y+z 1 y z + x = 1 2 4
1 z x + y 1 3 3
1 x 1
= (x + y + z) 1 y 1 = 1(–6) –1(–1) + 5(1) = 0
weKí Element ¸‡jv PµvKvi
1 z 1
= (x + y + z) 0 = 0 [x y + z; y z + x; z x + y]
ZvB Ans: 0
2 2
a ab b
MEx 03 2a a + b 2b Gi gvb- [wm.†ev: 03]
1 1 1
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
2 2
a ab b a(a b) b(a b) b 2
a = 1, b = 2, c = 3 a‡i,
2a a + b 2b = a b a b 2b ; 1 2 4
1 1 1 0 0 1 2 3 4= 1(3 – 4) – 2(2 – 4) + 4 (2 – 3) = –1
[c1 = c1 + c2] Ges [c2 = c2 + c3] cÖ‡qvM K‡i] 1 1 1
2 Verification: a = 1, b = 2, c = 3 a‡i Option Gi gvb wbY©q Ki‡Z
a b b n‡e| Zvici †hB Option †_‡K –1 cvIqv hv‡e †mB Option wU Answer
= (a b) (a b) 1 1 2b
0 0 1 n‡e| †hgb: (a – b)3 = (1 – 2)3 = –1
= (a b) (a b)
a b
1 1
= (a b) (a b) (a b) = (a b)3
1 a b + c
MEx 04 1 b c + a wbY©q‡Ki gvb KZ? [BUTex. 15-16]
1 c a + b
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
1 a b + c 1 a a + b + c a = 1, b = 2, c = 3 a‡i,
1 b c + a = 1 b a + b + c; [c3 = c3 + c2] 1 a b + c 1 1 5
1 c a + b 1 c a + b + c 1 b c + a = 1 2 4
1 a 1 1 c a + b 1 3 3
= (a + b + c) 1 b 1 = 1(–6) –1(–1) + 5(1) = 0
1 c 1 weKí Element ¸‡jv PµvKvi [a b + c; b c + a; c a + b]
= (a + b + c) 0 = 0 ZvB Ans: 0
logx logy logz
MEx 05 The value of log2x log2y log2z is: [BUET. 09-10, 11-12; CUET. 07-08;RUET. 11-12; e.†ev: 13]
log3x log3y log3y
General Rules [Written] 3 in 1 ASPECT Tricks & Tips [MCQ]
logx logy logz
log2x log2y log2z
log3x log3y log3z
log log logz
x y
y z
†Kvb wbY©vq‡Ki GKwU mvwi ev Kjv‡gi f~w³¸‡jv Aci `yBwU mvwi ev
log log log2z
x y
= ; [c1' = c1 – c2, c2' = c2 – c3] Kjv‡gi Abyiƒc f~w³¸‡jvi mv‡_ mgvšÍi aviv MVb Ki‡j wbY©vq‡Ki gvb
y z
k~b¨ n‡e| A_©vr G‡ÿ‡Î `yB †Rvov mgvšÍi aviv cvIqv hv‡e|
log log log3z
x
y
y
z
y
1 1 logz
= log log 1 1 log2z = 0
x
y z
1 1 log3z
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
78 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
a b ax + by
MEx 06 cÖgvY Ki: b c bx + cy = (b2 – ac) (ax2 + 2bxy + cy2)
ax + by bx + cy 0
[Xv.†ev:15,10; Pv.†ev:15,13,12,09 ; Kz.†ev: w`.†ev: iv.†ev:16,12,09 ; e.†ev:14,07; wm.†ev:12,05; h.†ev:14,10]
General Rules [Written] & [MCQ]
a b ax + by a b 0
L.H.S = b c bx + cy= b c 0 ; [∵ c3 = c3 – (c1x + c2y)]
ax + by bx + cy 0 ax + by bx + cy –(ax2 + 2bxy + cy2)
= –(ax2 + 2bxy + cy2)
a b
b c [3q Kjv‡gi mv‡c‡ÿ we¯Ívi K‡i]
= – (ax2 + 2bxy + cy2) (ac – b2) = (b2 – ac) (ax2 + 2bxy + cy2) = R.H.S (Proved)
a – b – c 2a 2a
MEx 07 cÖgvY Ki: 2b b–c–a 2b = (a + b + c)3 [BUET. 11-12; Xv.†ev: 13; h.†ev: 11, 08; P.†ev: 08; Kz.†ev: 13,06]
2c 2c c – b – c
General Rules [Written] & [MCQ]
a + b + c a + b + c a + b + c
L.H.S = 2b b–c–a 2b ; [r1 = r1+ r2 + r3 cÖ‡qvM K‡i]
2c 2c c – a – b
1 1 1 0 0 1
= (a + b + c) 2b b – c – a 2b = (a + b + c) a + b + c –(a + b + c) 2b ; [c1 = c1 – c2 Ges c2 = c2 – c3 cÖ‡qvM K‡i]
2c 2c c – a – b 0 a + b + c c – a – b
–(a
= (a + b + c)
a + b + c + b + c)
0 a + b + c [1g mvwi mv‡c‡ÿ we¯Ívi K‡i]
= (a + b + c) {(a + b + c)2 – 0} = (a + b + c)3 = R.H.S (Proved)
x2 y z
MEx 08 cÖgvY Ki: z2 = (xyz – 1) (x – y) (y – z) (z – x)
2
x y
x – 1 y – 1 z – 1
3 3 3
[Xv.†ev:, w`.†ev :, wm.†ev: I h.†ev:2018 Gi m„Rbkxj-1(M); Xv.†ev:14,11,06,03; e.†ev:16,15,08; h.†ev:06; wm.†ev:15,14,11,04; iv.†ev:06; w`.†ev:16,13,10]
General Rules [Written] & [MCQ]
2 3
x x x – 1
L.H.S = y y y – 1 [1g, 2q I 3q mvwi‡K h_vµ‡g 1g, 2q I 3q Kjv‡g ¯’vcb K‡i]
2 3
z z2 z3 – 1
2 3 2 2 2
x x2 x3 x x2 1 1 x x2 1 x x2
= y y y – y y 1= xyz 1 y y – 1 y y
z z2 z3 z z2 1 1 z z2 1 z z2
[2q wbY©vq‡K 2q I 3q Kjvg ¯’vb wewbgq Kivi ci 1g I 2q Kjvg ¯’vb wewbgq Kiv n‡q‡Q|]
2
– x2 – y2
2
1 x x 0 x y
= (xyz – 1) 1 y y = (xyz – 1) 0 y – z y – z ; [r1 = r1 – r2 Ges r2 = r2 – r3 cÖ‡qvM K‡i]
2 2
1 z z 2
1 z z
2
2 2
– –
= (xyz – 1)
(x y) (x y )
(y – z) (y2 – z2); [cÖ_g Kjvg mv‡c‡ÿ we¯Ívi K‡i|]
(x – y) (x – y)(x + y)
= (xyz – 1) (y – z) (y – z)(y + z) = (xyz – 1) (x – y) (y – z)1 y + z
1 x + y
= (xyz – 1) (x – y) (y – z) (y + z – x – y) = (xyz – 1) (x – y) (y – z) (z – x) = R.H.S (Proved)
2 2
1 + a – b 2ab –2b
MEx 09 cÖgvY Ki: 2ab 1 – a 2 + b2 2a = (1 + a2 + b2)3 [Kz‡qU:03-04,11-12; iv.†ev:09; h.†ev:16; w`.†ev:14; P.†ev:16; wm.†ev:16,13,10]
2b –2a 1–a –b
2 2
General Rules [Written] & [MCQ]
1 2ab –2b
L.H.S = (1 + a2 + b2) 0 1 – a + b ; [c1 = c1 – b c3 cÖ‡qvM K‡i]
2 2
2a
b –2a 1 – a2 – b2
1 0 –2b
= (1 + a2 + b2) 0 1 + a2 + b2 2a ; [c2 = c2 + ac3 cÖ‡qvM K‡i]
b –a(1 + a2 + b2) 1 – a2 – b2
1 0 –2b
= (1 + a2 + b2)20 1 2a ; [cÖ_g mvwi mv‡c‡ÿ we¯Ívi K‡i]
b –a 1 – a – b 2 2
= (1 + a2 + b2)2 [1{1 – a2 – b2 – a(–2a)} – 0 – 2b {0 – b}]
= (1 + a2 + b2)2 {1 – a2 – b2 + 2a2 + 2b2} = (1 + a2 + b2)2 (1 – a2 + b2) = (1 + a2 + b2)3 = R.H.S (Proved)
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 79
REAL TEST ANALYSIS OF PREVIOUS YEAR QUESTIONS
Dhaka University x+y x y
05. wbY©vqK x x + z z Gi gvb-
x yz
01. = 0, x = ? [DU. 14-15] z y+z
x [DU. 08-09; RU. 18-19, 05-06; IU. 14-15; JnU. 06-07; RUET. 12-13; KUET.
08-09, 10-11, 17-18]
A. , , B. , A. 4xyz B. 3xyz
C. , D. , C. 2xyz D. xyz
n Aspect Special: wbY©vqKwU‡Z x = 1, y = 2, z = 3
x
olve
Aspect Special: S A Sol
S B Sol = 0
n
†h †Kvb wbY©vq‡Ki `ywU mvwi we‡ePbv Ki‡j wbY©vqKwUi AvKvi nq,
olve
x ev Kjvg Awfbœ n‡j 3 1 2
0 x wbY©vq‡Ki gvb k~b¨ n‡e| 1 4 3 = 3(20 9) 1(5 6) + 2(3 8) = 24
0 = 0; [c1 = c1 – c2] 2 3 5
x = ev x = n‡j
– x x GLb †h Option ¸‡jv Av‡Q, Zv‡`i g‡a¨ †hB Option G x=1, y=2 ,
( – x) (x – ) = 0 wbY©vq‡Ki `ywU mvwi Awfbœ n‡e
z = 3 emv‡j 24 cvIqv hv‡e †mwUB Correct Answer.
x = , Ges wbY©vq‡Ki gvb k~b¨ n‡e| Verification: x = 1, y = 2, z = 3:
a 1 b+c A. 4xyz = 4.1.2.3 = 24;
2 2
B. x2yz = 12.2.3 = 6
02. b 1 c + a Gi gvb KZ? [DU. 12-13; JnU. 11-12,MIST. 20-21] C. xy z = 1.2 .3 = 12; D. xyz2 = 1.2.32 = 18
c 1 a+b 1 1 1
A. abc B. 0 06. x a b = 0, x = ? [DU. 03, 04; CU. 02-03]
C. abc (a+b) (b+c) (c+a) D. (a+b) (b+c) (c+a) x2 a2 b2
S B Sol n a = 1, b = 2, c = 3 n‡j = 0 A. a ev b B. a ev b C. a ev b D. a ev b
olve
1 bc bc (b + c) 1 1 1 Aspect Special:
S D Sol 2 2 2
n
olve
x a b = 0 x = a n‡j wbY©vq‡Ki
03. wbY©vqK 1 ca ca (c + a) Gi gvb KZ? [DU. 10-11] x a b
1 ab ab (a + b) 1g I 2q Kjvg `ywU
0 0 1
x – a a – b b = 0
c1 = c 1 – c2 GKB nq Ges
A. abc(a+b) (b+c) (c+a) B. abc (a+b+c)
C. 1 D. 0 x2 – a2 a2 – b2 b2 2 2 3 c = c – c wbY© vq‡Ki gvb k~ b¨ nq|
Abyiƒc fv‡e x = b n‡j
1 bc bc(b + c) 1 a abc abc(b + c) 0 0 1
S D Sol 1 ca ca(c + a) = abc b aba abc(c + a)
n (x – a) (a – b) 1 1 b = 0 1g I 3q Kjvg `ywU
olve
1 ab ab(a + b) 1 abc abc(a + b) x + a a + b b 2
GKB nq Ges
(x – a) (a – b) (a + b – x – a) = 0 wbY© v q‡Ki gvb k~b¨
a 1 b+c (x – a) (a – b) (b – x) = 0 x = a, b n‡e| x = a, b
= abc b 1 c + a ; [c1=c1+c3]
c 1 a+b Jahangirnagar University
a + b + c 1 b + c 1 1 1
= abc a + b+ c 1 c + a 01. 2 a b c wbY©vq‡Ki gvb KZ?
a – bc b2 – ca c2 – ab
a+b+c 1 a+b [JU-A, Set-I. 2021-22, JU. 13-14, KU. 06-07; RU. 06-07]
1 1 b+c A. 2 B. abc
= abc (a + b + c) 1 1 c + a C. 0 D. a2b2c2
1 1 a+b 1 1 1
= 0 [†h‡nZz wbY©vq‡Ki `yBwU Kjv‡gi Dcv`vb GKB|] S C Sol 2 n c
olve
a b
– 2
– 2
– ab
1 a bc b ca c
2
0 0 1
c 1
04.
2
1 hw` 1 Gi GKwU RwUj Nbg~j nq, Z‡e cÖ`Ë c = c1 – c2
2 = a–b b–c
1 a2 – bc – b2 + ca b2 – ca – c2 + ab c2 – ab c2 = c2 – c3
wbY©vqKwUi gvb- [DU. 09-10; JU. 09-10; SUST. 09-10; CU. 07-08] = 1{(a – b) (b2 – c2 + ab – ca) – (b – c) (a2 – b2 + ca – bc)}
A. 0 B. 1 = (a – b){(b – c)(b + c) + a(b – c)} – (b – c){(a – b)(a + b) + c(a – b)}
C. D. 2 = (a – b) (b – c) (a + b + c) – (a – b) (b – c) (a + b + c) = 0
weKí a = 1, b = 2, c = 3 ai‡j
1 2 1 + + 2 2
2 2 2 Aspect Special:
S A Sol n 2 1 = 1 + + 2 1 ; [c1 = c1 + c2 + c3] 1 1 1 wbY©vq‡Ki Element ¸‡jv
olve
1 1 + + 1 2 a b c PµvKv‡i i‡q‡Q
a – bc b2 – ca c2 – ab [a a2 – bc;
0 2
2
= 0 1 ; [∵ 1 + + 2 = 0] 1 1 1 b b2 – ca; c c2 – ab]
= 1 2 3= 0
0 1 –5 1 7 ∵ Element ¸‡jv PµvKv‡i
=0 i‡q‡Q ZvB wbY©vq‡Ki gvb k~b¨|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
80 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
1 1 1 0 b – a c – a Aspect Special:
02. 1 1 + x 1 wbY©vq‡Ki gvb †KvbwU? S D Sol a – b 0 c – b
n
olve
[JU-A, Set-Q. 2021-22]
1 1 1 + y a – c b – c 0 Element ¸‡jv
A. x + y B. – xy a – b b – c c – a c1 = c1 – c2 PµvKv‡i Av‡Q ZvB
C. xy D. 1 – xy = a – b b – c c – b; wbY©vqKwUi gvb k~b¨|
1 0 0 1 a – b b – c 0 c2 = c2 – c3
1 1 c1 = c1 c2
S C Sol 1 1 + x 1 = x x 1 c2 = c2 c3
n
1 1 c – a
olve
1 1 1 + y 0 y 1 + y = (a – b) (b – c) 1 1 c – b= 0
1 1 0
x x
= = xy
0 y Chittagong University
03. †Kvb wbY©vq‡Ki `ywU mvwi ev Kjvg m`„k n‡j, H wbY©vq‡Ki gvb n‡e- [JU. 15-16]
a + b + 2c a b
A. 1 B. 0 01. c b + c + 2a b wbY©vqKwUi gvb KZ?
C. 2 D. 3 c a c + a + 2b
Sol †Kvb wbY©vq‡Ki `ywU mvwi ev Kjvg m`„k n‡j H wbY©vq‡Ki gvb k~b¨|
n
olve
[CU. 13-14]
S B
A. (a + b + c)3 B. 2(a +b + c)2
1 1 y+z
C. (a +b + c) D. 2(a + b + c)3
04. 1 1 z + x wbY©vqKwU gvb KZ? [JU. 09-10]
E. (a + b + c) 2
1 1 x+y
A. 3 + x + y + z B. 3 (x+y+z) a + b + 2c a b
S D Sol n c
olve
b + c + 2a b
C. 3 D. †KvbwUB bq c a c + a + 2b
1 1 y + z 2(a + b + c) a b
S D Sol n 1 1 z + x = 0 = ; [c1 = c1 + c2 + c3]
olve
2(a + b + c) b + c + 2a b
1 1 x+y 2(a + b + c) a c + a + 2b
[†h‡nZz wbY©vq‡Ki cvkvcvwk `yBwU Kjv‡gi Dcv`vb GKB|] 1 a b
= 2(a + b + c) 1 b + c + 2a b
Rajshahi University 1 a c + a + 2b
1 1 1 0 –(a + b + c) 0
01. 1 1 + a 1 Gi gvb- [RU. 15-16; KU. 03-04] = 2(a + b + c) 0 a + b + c –(a + b + c) ;
1 1 1+b 1 a c + a + 2b
A. ab B. 1–ab [r1 = r1 – r2, r2 = r2 – r3]
C. a + b + 1 D. 0 = 2(a + b + c) . 1({– (a + b + c) – (a + b + c)} = 2(a + b + c)3
1 0 0 1 Aspect Special: a = 1; b = 2; c = 3 a‡i,
1 1
S A Sol n 1 1 + a 1 = a a 1 ; c2 = c2 c3
c = c c
a + b + 2c
1 1 2 a b
olve
1 1 1 + b 0 b 1 + b c b + c + 2a b
a a
c a c + a + 2b
= = ab 9 1 2
0 b = 3 7 2
1 1 1 3 1 8
02. wbY©vqK a b c Gi gvb KZ? [RU. 15-16, 07-08] = 9(56 – 2) – 1(24 – 6) + 2(3 – 21) = 432
a2 b2 c2
Option D †Z a = 1; b = 2; c = 3
A. (a b) (b c) (c a) B. (a b ) (b c) (c a)
2 2
C. (a b) (b c ) (c a)
2 2
D. (a b) (b c) (c a )
2 2 a‡i, 2(a + b + c)3 = 2(1 + 2 + 3)3 = 2 63 = 432
1 1 1 1 x y + z
S A Sol 2 2 2
n 02. 1 y z + x Gi gvb n‡e- [CU. 12-13, 05-06]
olve
a b c
a b c 1 z x + y
A. (x + y + z)3 B. (1+ x + y + z)3
0 0 1
c ; 1 2
c = c – c
= a – b
1
b–c C. 0 D. (y + z) (z +x) (x+y)
a – b b – c c
2 2 2 2 2 c 2 = c 2 – c 3 E. y(x+y)
0 0 1 1 x y + z Aspect Special:
= (a – b) (b – c) 1 c S C Sol 1 y z + x
n
olve
1
a + b b + c c 2 1 z x + y wbY©vq‡Ki Element ¸‡jv
= (a – b) (b – c) (b + c – a – b) 1 x x + y + z PµvKv‡i i‡q‡Q|
= (a – b) (b – c) (c – a) = 1 y x + y + z; [c3 = c2 + c3] [x y + z; y z + x;
0 b – a c – a 1 z x + y + z
z x + y] ZvB wbY©vq‡Ki
03. a – b 0 c – b Gi gvb KZ? [RU. 08-09] 1 x 1
a – c b – c 0 = (x + y + z) 1 y 1 gvb k~b¨|
A. (a + b + c) 3
B. (a b) 3 1 z 1
C. (a b) (b c) (c a) D. 0 = (x + y + z) 0 = 0
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 81
1 0 0 1 0 0 GST (¸”Q)
03. a 0 1 0 + b 0 1 0 = 0 n‡j- [CU. 12-13, 08-09]
2
0 0 1 0 0 1 01. GK‡Ki GKwU RwUj Nbg~j n‡j, 1 wbY©vq‡Ki gvb KZ?
2 2
3
A. a + b = 0 B. a b = 0 C. ab = 0 D.
a
= 1 1
b [SUST. 08-09]
1 0 0 1 0 0 A. 1 B. C. 2 D. 0
n a
+ b 0 1 0 = 0 2
1 2
olve
S B Sol 0 1 0
0 0 1 0 0 1 2 2
S D Sol n 1 = 1
2
olve
a b = 0 a = b 3 1 2 1
1+ +
2 2 2 2
02 ab c a2
04. D = a b 0 bc Gi gvb- = 1 + + 1 ; [c1 = c1 + c2 + c3]
[CU. 11-12] 2 2
a2c b2c 0
1+ + 1 2
A.0 B. a+b + c
0
2
D. a2c ab2 + a2b
= (1 + + 2) 0 1 = (1 + + 2) 0 = 0
C. 2a3b3c3 2
0 ab2 c2a
S C Sol n a2b 02 bc
2 2 0 1
olve
a c b c 0 a a x
0 a a 2 2 2 0 0 a 02. hw` = m m nnq, Z‡e Δ = 0 mgxKi‡Yi g~j n‡”Q- [SUST. 05-06]
= a b c b 0 b= a b c b –b b; [c2 = c2 – c3]
2 2 2 b x b
c c 0 c c 0 A . x = a, x=m B. x = b, x = m
2 2 2 2 2 2
= a b c {a(bc + bc)}= a b c .2abc = 2a b c 3 3 3
C . x = a, x = b D. †KvbwUB bq
Aspect Special: a = 1; b = 2; c = 3 a a x 0 a x
S D Sol = m m n = 0 m n ; [c1 = c1 + c3]
2 2
02 ab c a2 0 4 9 olve
n
a2b 02 bc = 2 0 18 b x b bx x b
a c b c 0 3 12 0
= (b x)
a x
= 0 – 4(0 – 54) + 9(24 – 0) = 216 + 216 = 432 m n = (b x) (an mx)
Option C †Z a = 1; b = 2; c = 3 a‡i an
2a3b3c3 = 2 13 23 33 = 2 8 27 = 432 ∵ = 0 (b x) (an mx) = 0 x = b,
m
2(y + z) 1 2x Engineering
05. 2(z + x) 1 2y= ? [CU. 07-08]
2(x + y) 1 2z a ab ac
2
01. D = ab b bc n‡j D Gi gvb KZ?
2
A. 6(x + y + z) B. 0 C. 2(x + y + z)
[BUTex. 16-17]
2 2 2
D. 8(x + y + z ) E. 1 ac bc c2
2(y + z) 1 2x A. 4abc B. abc
S B Sol n 2(z + x) 1 2y
olve
C. 4a2b2c2 D. a2b2c2
2(x + y) 1 2z
a ab ac
2
2(x + y + z) 1 2x n D = ab b2 bc
olve
= 2(x + y + z) 1 2y; [c1 = c1 + c3] S C Sol
2(x + y + z) 1 2z ac bc c2
1 1 2x a b c
= 2(x + y + z) 1 1 2y = 2(x + y + z) 0 = 0 = abc a b c
1 1 2z
a b c
Aspect Special: wbY©vq‡Ki Element ¸‡jv PµvKv‡i i‡q‡Q|
1 1 1
[2(y+z) 2x; 2(z+x) 2y; 2(x+y) 2z] ZvB wbY©vq‡Ki gvb k~b¨| = a b c 1 1 1 = 4a2b2c2
2 2 2
b+c a
1 1 1
06. wbY©vqK D = c +a b Gi gvb n‡e-
[CU. 05-06]
1 a b+c
a+b c 02. 1 b c + a wbY©q‡Ki gvb KZ? [BUTex. 15-16]
A. abc + B. 0 C. (bc) (ca) (ab) 1 c a+b
A. a + b + c B. 0 C. 1 D. abc
D. E. a b c
abc 1 a b + c 1 a a + b + c
b+c a a+b+c a S B Sol n 1 b c + a = 1 b a + b + c ; [c3 = c3 + c2]
olve
1 c a+b 1 c a+b+c
S B Sol n D = c + a b = a + b + c b ; [c1 = c1+c3]
olve
a+b c a+b+c c 1 a 1
1 1 a = (a + b + c) 1 b 1
= (a + b + c) 1 1 b = (a + b + c) 0 = 0 1 c 1
1 1 c = (a + b + c) 0 = 0 = (a + b + c)
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
82 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
03. hw` 1 Gi GKwU RwUj Nbg~j nq, Z‡e wb‡Pi wbY©vqKwUi gvb KZ? logx logy logz
1 S B Sol log2x log2y log2z
2 n
olve
1 2 log3x log3y log3z
2
[BUET. 10-11; KUET. 06-07; SUST. 08-09]
1 log log logz
x
x y
A. 4 B. 2 C. 3 D. None of the above y z
1
logy
2
log log2z
y
= z [c1 = c1 c2, c2 = c2 c3]
S D Sol n 1 = 1 + ( ) + ( )
2 3 2 2 2 4
olve
2 1
= 2 + 0 + 2 ( 3.) = 2 23 = 4 logxy log log3z
y
z
logx logy logz y
1 1 logz
= log log = 1 1 log2z = 0
x
04. The value of log2x log2y log2z is- [BUET. 09-10; RUET. 11-12; KUET. 07-08]
log3x log3y log3z y z
1 1 log3z
2 3 [wbY©vq‡Ki `yBwU Kjvg GKB nIqvq wbY©vqKwUi gvb k~b¨]
A. log B. 0 C. log D. 1
3 2
【
? 】 QUICK PRACTICE CONCEPT TEST ?
a + b + 2c a b 0 (x y) (x z)3
3
01. =? 02. (y x) (y z)3 =?
c b + c + 2a b 3
0
c + a + 2b
c a (z x) (z y)
3 3
0
A. 2(a + b + c)3 A. x+y + z B. x3+ y3 + z3
B. 2(a + b + c) C. 0 D. 1
C. 2(a + b + c)
D. a3 + b3 + c3 3abc Answer 01. A 02. C
HSC BOARD QUESTIONS ANALYSIS
5 0 1 1 2 3
01. –3 2 3wbY©vq‡K (2, 1) Zg fzw³i mn¸YK KZ? [Xv.†ev: 2017] 03. p Gi †Kvb gv‡bi Rb¨ 1 2 pwbY©vq‡Ki gvb k~b¨ n‡e? [iv.†ev: 2017]
6 7 –1 3 4 0
A. –7 B. –3 3 3
A. – B.
C. 3 D. 7 5 5
5 0 1 C. –3 D. 3
S D Sol –3 2 3wbY©vq‡K (2,1) Zg fzw³i mn¸YK
n
1 2 3
olve
6 7 –1 S D Sol n 1 2 p wbY©vq‡Ki Rb¨ p = 3 n‡j 1g I 2q mvwii
olve
3 4 0
= (–1)2+1
0 1
7 –1 = –1(0 – 7) = 7 AYyiƒc Dcv`vb¸‡jv mgvb n‡e Ges wbY©vq‡Ki gvb n‡e k~b¨|
3 2 4 p = 3 n‡e|
02. 0 3 6 wbY©vqKwUi- [iv.†ev: 2019] 1 4 – 3
1 1 2 04. 2 – 1 x Gi (1, 1) Zg fzw³i Abyivwk 4 n‡j x Gi gvb KZ?
(i) gvb 0 6 2 8
(ii) (2, 3) Zg fzw³i Abyivwk 5 [w`.†ev. 22]
(iii) (2, 1) Zg fzw³i mn¸YK 0 A. 6 B. 2 C. – 2 D. – 6
–1 x
wb‡Pi †KvbwU mwVK? S C Sol n (1,1) Zg fzw³i Abyivwk 2 8= – 4
olve
A. i I ii B. ii I iii – 8 – 2x = – 4 – 2x = 8 – 4 x = – 2
C. i I iii D. i, ii I iii = 54 – 72 + 18 = 72 – 72 = 0
3 2 4 2 4 6
S C Sol 0 3 6 = 3(– 6 + 6) – 2(0 – 6) + 4(0 – 3)
n
05. 3 x 5 = 0 n‡j, x Gi gvb †KvbwU?
olve
i. [w`.†ev: 2019]
1 1 2 5 10 9
= 0 + 12 – 12 = 0 A. 4 B. 6
3 2 4 C. 5 D. 6
ii. 0 3 6 wbY©vqKwUi (2, 3) Zg f~w³i AYyivwk =
3 2
1 –1 2 4 6
1 1 2 S B Sol n 3 x 5 = 0
olve
= – 3 – 2 = –5 5 10 9
3 2 4 2(9x – 50) + 4(27 + 25) + 6(– 30 – 5x) = 0
iii. 0 3 6wbY©vqKwUi (2,1) Zg f~w³i mn¸YK = (–1)2+1
2 4
–1 –2 18x – 100 + 208 – 180 – 30x = 0
1 1 2 – 12x – 72 = 0
= – (– 4 + 4) = 0 12x = – 72 x = – 6
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 83
2 10 x 1 1
1
06. 23 12 y wbY©vq‡Ki mgvb †KvbwU? [w`.†ev: 2019] 09. A = 2 – 2 2 Gi gvbÑ [Kz.†ev: 2017]
4 14 z –3 3 –3
4 10 x 4 10 2x 1 2 –3
A. 6 12 y B. 6 12 2y (i) 1 –2 3 Gi gv‡bi mgvb
8 14 z 8 14 2z 1 2 –3
4 7 x + 2 2 10 x 1 + c2 2 –3
C. 6 8 y + 2 D. 3 12 y
8 9 z + 2 4 14 z (ii) 1 – c2 –2 3 Gi gv‡bi mgvb
1 + c2 2 –3
2 10 x 1 1 1
S A Sol n 23 12 y
olve
4 14 z (iii) –3 3 –3 Gi gv‡bi mgvb
2 –2 2
4 10 x wb‡Pi †KvbwU mwVK?
= 6 12 y
8 14 z A. i B. ii C. i I iii D. i, ii I iii
1 –3 1
p 2 q + r 2 1 1
07. q 2 r + p wbY©vqKwUi gvb KZ? C Sol n (i) 1 –2 3 = 2 –2 2
olve
[w`.†ev: 2017] S
r 2 p + q 1 2 –3 –3 3 –3
A. 0 B. 1 wbY©vq‡Ki †iv¸‡jv Kjvg Ges Kjvg¸‡jv †iv n‡j gv‡bi cwieZ©b nq bv|
C. pqr D. p + q + r 1 + c2 2 –3 1 + c2 c2 –3
n p=1
(ii) 1 – c2 –2 3= c 1 – c2 –c2 3
olve
S A Sol
q=2 1 + c2 2 –3 1 + c2 c2 –3
r=3 1 c2 –3 1 2 –3
1 2 5 =c 1 –c 2 3 [c1 = c1 – c2] = c. c 1 –2 3
2 2 4= 1(6 – 8) –2(6 – 12) + 5(4 – 6) 1 c2 –3 1 2 –3
3 2 3 1 2 –3 1 1 1
= – 2 + 12 – 10 = 0 = c2 1 –2 3 = c2 2 –2 2
1 2 3 1 2 –3 –3 3 –3
08. 1 2 3 Gi gvb| [Ky.‡ev. 2017] 1 1 1 1 1 1
1 2 3 (iii) –3 3 –3= – 2 –2 2
2 –2 2 –3 3 –3
1 1 1 \ †Kvb wbY©vq‡Ki †h †Kvb `yBwU mvwi ev Kjvg ci¯úi ¯’vb wewbgq Ki‡j,
i. 2 2 2 Gi gv‡bi mgvb|
3 3 3 wbY©vqKwUi wPý e`‡j hvq wKš‘ wbY©vq‡Ki gv‡bi †Kvb cwieZ©b nq bv|
10. x Gi †Kvb gv‡bi Rb¨ |A| = D n‡e? [P.†ev: 2019]
1 + c2 2 3 A. 5 B. 1 C. 1 D. 5
ii. 1 c2 2 3 Gi gv‡bi mgvb| 3 2
S D Sol n A = 2 2
x 0 0
1 + c2 2 3
olve
kZ©g‡Z, 2 4 1 = 10
1 1 1 |A| = 6 + 4 = 10 3 2 0
iii. 3 3 3 Gi gv‡bi mgvb| x(0 + 2) = 10 2x = 10 x = 5
2 2 2 1 –2 3
wb‡Pi †KvbwU mwVK? 11. 0 1 –2G (1, 2) Zg fzw³i mn¸YK †KvbwU? [P.†ev: 2017]
–1 0 2
A. i I ii B. ii I iii
A. –4 B. –2 C. 2 D. 4
C. i I iii D. i, ii I iii –2
1 3
1 2 3 S C Sol n 0 1 –2G (1, 2) Zg fzw³i mn¸YK
olve
S A Sol n (i) mZ¨ KviY, 1 2 3 –1 0 2
olve
1 2 3 –2
= (–1)2+1
0
1 1 1 –1 2 = – (0 – 2) = 2
= 2 2 2 [r1 c1, r2 c2, r3 c3] 1 1 1
3 3 3 12. A = e π 3 n‡j, |A| = ? [wm.†ev. 22]
1 2 3 3 3 3
(ii) mZ¨ KviY, A = 1 2 3 A. e B. π C. 2 (e – π + 3) D. 0
1 2 3 1 1 1
1 + c2 2 3 S D Sol n |A| = e π 3
olve
= 1 – c2 2 3 [c1 = c2 + c2] 3 3 3
1 + c2 2 3 = 1 (3π – 3 3) – (3e – 3 3) + (3e – 3π)
(iii) mZ¨ bq| = 3π – 3 3 – 3e + 3 3 + 3e – 3π = 0
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
84 An Exclusive Parallel Text Book Of Mathematics
© Medistry ASPECT MATH
17. wb‡Pi †Kvb wbY©vq‡Ki gvb k~b¨? [e.†ev: 2017]
13. A =
2 3
5 4
n‡jÑ [wm.†ev: 2017]
1 0 2 1 0 0 4 0 8 0 0 1
(i) |A| = –7 A. 2 0 1 B. 0 1 0 C. 2 3 4 D. 1 2 3
(ii) (1, 2) Zg fzw³i mn¸YK 5 1 3 0 0 0 1 1 5 2 0 6 0
(iii) (2, 1) Zg fzw³i Abyivwk 3 n Option C Gi 1g I 3q KjvgØq ¸‡YvËi MVb K‡i ZvB
olve
S C Sol
wb‡Pi †KvbwU mwVK? wbY©vq‡Ki gvb k~b¨|
A. i I ii B. i I iii C. ii I iii D. i, ii I iii 18. wb‡Pi †Kvb wbY©vq‡Ki gvb k~b¨? [e.‡ev. 2017]
n A=
2 3
n‡j 1 0 2 1 0 0 4 0 8 0 1
0
olve
S B Sol 5 4 |A| = 7
1+2 A. 2 0 1 B. 0 1 0 C. 2 3 4 D. 1 2 3
(1, 2) Zg fzw³i mn¸YK = (–1) |5| = –5 1 3 0 0 0 1 1 5 2 0 6 0
(2, 1) Zg fzw³i Abyivwk = |3| = 3
n i. Gi gvb = 3
1 2
2 1= 3(1 4) = 9
olve
1 –2 3 S C Sol
14. 0 1 – 2 wbY©vqKwUi (1, 2) Zg fzw³i mn¸YK †KvbwU? [h.†ev. 22]
ii. Gi gvb = 1
1 0
–1 0 2 0 1= 1(1 0) = 1
A. 4 B. 2 C. – 2 D. – 4
– 4 0 8 4 0 4
1 2 3 iii. Gi gvb = 2 3 4= 22 3 2= 2 0 = 0 [∵ c1 = c3]
S B Sol n cÖ`Ë wbY©vqK, 0 1 – 2 1 5 2 1 5 1
olve
–1 0 2 [C1 †_‡K 2 Kgb wb‡q]
0 –2
(1, 2) Zg fzw³i mn¸YK = (– 1)1 + 2 – 1 2
iv. Gi gvb = 1
1 2
= (– 1)3 × (0 – 2) = – 1 × (– 2) = 2
0 6 = 1 6 = 6
wb‡Pi DÏxc‡Ki Av‡jv‡K 19 bs cÖ‡kœi DËi `vI:
1 1 3
15. 0 1 x wbY©vqKwUi (1, 2) Zg Abyivwk 3 n‡j, x Gi gvbÑ [h.†ev: 2019]
3 2 x 0 0
1 3 3 A = 2 2 , D = 2 4 1
A. 12 B. 3 C. 3 D. 12 3 2 0
1 1 3 2 3 1
S C Sol n 0 1 x wbY©vqKwUi (1, 2)Zg Abyivwk = 1 3= – x 19. 5 61 04 wbY©vqKwUi (2, 3) Zg mn¸YK †KvbwU?
0 x [mKj.†ev: 2018]
olve
1 3 3 2
kZ©g‡Z, – x = –3 x = 3 A. 8 B. 3
C. 8 D. 17
16.
0 1
2 –1 Gi gvb †KvbwU? [h.†ev: 2017] 2 3 1
S A Sol n 5 6 0 wbY©vqKwUi (2, 3) Zg mn¸YK = (–1) –2 1
2+3 2 3
olve
A. –1 B. –2 C. 2 D. 1
2 1 4
S B Sol n 2 –1 Gi gvb = 0 – 2 = –2
0 1
olve
= – (2 + 6) = –8
MCQ CONCEPT TEST: TEST YOUR SKILL Written
1 1 3 06. 3 3 AvKv‡ii GKwU KY© g¨vwUª· D Gi Rb¨ |D| = 20 n‡j |(2D)–1| Gi
01. 0 1 x wbY©vqKwUi (1, 2) Zg Abyivwk 3 n‡j x Gi gvb- gvb KZ?
1 3 3 1 1 1 1
A. 12 B. 3 C. 3 D. 12 A. B. C. D. –
100 40 10 160
i i 1 1
02. hw` A = i i Ges B = 1 1 nq Z‡e A8 = ? 07. A =
1 2
Ges A2 + 2A 11x = 0 n‡j X Gi gvb KZ?
4 3
A. 64B B. 128A C. 128B D. 64A
A. 0 11 B. 0 1 C. 0 2 D. 0 13
11 0 1 0 2 0 13 0
a 11 a 12 a 13
a21 a22 a23 g¨vwUª‡· aij Gi mn¸YK Aij n‡j, a21 A11 + a22 A12 +
08. A1 wbY©q Ki: A =
03. 1 4
a31 a32 a33 2 6
a23 A13 = ? 1 1 6 4
B.
6 4
A. 1 B. a22 a11 a33 A.
14 2 1 14 2 1
C. 0 D. a23 a12 a32
1 6 4
5
– 2
C.
14 2 1 D. None
2 2 4
04. A= ,B=
3 5 n‡j, AB = ? 09. hw` A GKwU 2 5 gvÎvi Ges B GKwU 5 2 gvÎvi g¨vwUª· nq, Z‡e
3
–1 BA-Gi gvÎv KZ n‡e?
2
A. 0 B. 4 A. 2 2 B. 5 5 C. 2 5 D. 5 2
10. A =
C. –1 D. 1 0 40623542
3 1 9 0 n‡j A =?
2x2 2 6 GKwU e¨wZµgx g¨vwU· n‡j x Gi gvb wbY©q Ki| A. I2
05. B. A
x 3 3 C.
A
D. I
A. 1,3 B. – 1, – 3 C. 2,3 D. – 2, 3
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
ASPECT MATH cÖ_g cÎ g¨vwUª· I wbY©vqK (2q Ask: wbY©vqK)
© Medistry 85
0 1 –4 cÖkœ DËi e¨vL¨v
11. A = –1 0 3 n‡j a Gi †Kvb gv‡bi Rb¨ A GKwU wecÖwZmg 1 1 1 1 2 2
a –3 0 B2 =
1 1 1 1 2 2
= = 2B
g¨vwUª· n‡e? i i
A= 1 1
A. a = 1 B. a = – 02 C i i = i 1 1 = iB
C. a = 0 D. a = 4 A8 = (iB)8 = B8 = (2B)4; [∵ i2 = 1
12. 3 3 AvKv‡ii KY© g¨vwUª· A Gi KY© Dcv`vb¸wji ¸Ydj 2 2 n‡j i8 = 1 Ges B2 = 2B] = 16 B4 = 16 (2B)2 = 64 B2 = 128B
3
a21 A11 + a22 A12 + a23 A13
|( 2I – A) | Gi gvb KZ?
= a21
a22 a23 a21 a23 a21 a22
A. 2 2 B. 24 2 a32 a33 + a22 () a31 a33 + a23 a31 a32
D. 0 03 C
C. 12 2 = a21 (a22 a33 a32 a23) a22 (a21 a33 a31 a23) + a23 (a21 a32 a31 a22)
10 11 12 = a21 a22 a33 a21 a23 a32 a21 a33 + a22 a23 a31 + a21 a23 a32 a22 a23 a31
13. 11 12 13= ? =0
12 13 14
5
– 2
A. 2 10 11 12 B. 2 11 12 13
,B=
2 4
; AB =
2 1 0
C. 11 12 13 D. 2 12 13 14
04 D A= 3 5 0 1 = 1
3
–1
2 2
14. gvb wbY©q Ki- [4 5 6] 3 3(6 – 18) – 1(6x – 6x2) + 9 (6x –2x2) = 0
1 05 A – 36 – 6x + 6x2 + 54x – 18x2 = 0 – 12x2 + 48x – 36 = 0
8 x2 – 4x + 3 = 0 x = 3,1
A. [8 15 6] B. [17] C. 15 D. [ 18 ] 1
6 06 A |D| = 20 n‡j |(2D)–1| = (2 2 2 20)–1 =
160
15. eM©vKvi †Kvb g¨vwUª· A-Gi †¶‡Î hw` A2 = A nq, Z‡e †mB g¨vwUª·wU-
A =
1 2
A. mgNvwZ B. cÖwZmg 4 3
C. ch©vqe„Ë D. A‡f`NvwZ 1 2 1 2 9 4
A2 = A.A =
4 3 4 3 8 17
=
OMR SHEET
9 4 1 2
A2 + 2A 11x = 0
01. A B C D 06. A B C D 11. A B C D 07 B
8 17 + 24 3 11x = 0
02. A B C D 07. A B C D 12. A B C D
9 4 2 4
03. A B C D 08. A B C D 13. A B C D 8 17 + 8 6 11x = 0
04. A B C D 09. A B C D 14. A B C D
0 11 11x = 0 x =
11 0 1 11 0 1 0
11 0 11 0 1
05. A B C D 10. A B C D 15. A B C D =
1 6 4 1 6 4 1 6 4
A = 2 A1 =
1 4
✍ Written 08 D 6 |A| 2 1 14 2 1 14 2 1
= =
01. hw` A = 4 3 Ges AB = 10 17 nq Z‡e B g¨vwUª· Gi 09 B B 5 2 2 5 A ; BA-Gi gvÎv = 5 5
2 1 4 7
Dcv`vbmg~n †ei Ki| 3A A
C A = I A = 40623542 I= 3I = 2. I = 2 A =
40623542
10 =
DËi:.......................................................................................
01 –4
02. hw` A =
cos2 –sin2
sin2 cos2 Ges A = I nq, Z‡e Gi gvb KZ? A = –1 3
2
0
DËi:....................................................................................... a
–3 0
11 D
0 –1 4 0 1 –4
b +c c + a a + b a b c a = 4 n‡j AT = 1 0 –3 = – –1 0 3 = – A
03. q + r r + p p + q = 2 p q r –4 3 0 4 –3 0
y+z z+x x+y x y z 3 3
DËi:.......................................................................................
12 D |( |
2I – A) = (2 2 – 2 2) = 0
10 11 10 12 11 10 1 11 132
xa yb 11 12 11 13 12 11= 10 11 12 1 12 156
04. z c x + a = 0 12 13 12 14 13 12 1 13 182
y+b z+c 13 A 1 11 132 r3 = r3 r2
DËi:....................................................................................... = 10 11 12 0 1 24 ;
0 1 26 r2 = r2 r1
Answer Analysis = 2 10 11 12
2
cÖkœ DËi e¨vL¨v 14 B [4 5 6] 3 = [8 + 15 6] = [17]
1
C
0 x
01
1 3 = 3 0 x = 3 x = 3 15
2
A eM©vKvi g¨vwUª· A Gi †ÿ‡Î A = A n‡j A GKwU mgNvwZ g¨vwUª·|
ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES ASPECT SERIES
ASPECT SERIES
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