kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 gƒj eBGqi AwZwiÚ Ask 1

kxlÆÕ©vbxq Õ•zGji wbeÆvPwb cixÞvi cÉk²cò c†Ó¤v 2-56

m†Rbkxj eüwbeÆvPwb
2 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ

347 kwn` exi-Dîg ˆj. AvGbvqvi MvjÆm KGjR, XvKv


MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
. wPGò DE || BC nGj ADE ‰i gvb KZ? 22. ‰KwU wmwj´£vGii DœPZv 8 ˆm.wg. ‰es
1. 
 5.78 msLÅvwUGK mvgvbÅ f™²vsGk cÉKvk
K 50 L 64
Ki| f„wgi eÅvmvaÆ 4 ˆm.wg. nGj
M 66 N 114
78 78 i. ‰i mgMÉZGji ˆÞòdj 301.59 eMÆ ˆm.wg.
K 5 L5 12. ‰KwU wòfzGRi `ywU evüi Š`NÆÅ h^vKÌGg
90 9
4 ‰KK I 6 ‰KK| Z‡Zxq evüi Š`NÆÅ KZ ii. ‰i eKÌZGji ˆÞòdj 201.06 eMÆ ˆm.wg.
71 71 iii. ‰i AvqZb 100.53 Nb ˆm.wg.
M 5 N5 ‰KK nGj wòfzRwU AuvKv hvGe?
90 9
K 11 ‰KK L 7 ‰KK wbGPi ˆKvbwU mwVK?
2. A = {x  ô : 3 ≤ X ≤ 7} nGj
M 2 ‰KK N 1 ‰KK K i I ii L i I iii
i. A ˆmGU ˆgŒwjK msLÅv 3wU
ii. P(A) ‰i Dcv`vb msLÅv 16 13. ˆKvb aiGbi wòfzGRi cwie†Gîi ˆK±`Ê M ii I iii N i, ii I iii
iii. P ˆmGU 3 «¼viv wefvRÅ msLÅv 2wU wòfzRwUi e†nîi evüi Dci AewÕ©Z? 23. 
 ‰KwU mgw«¼evü wòfzGRi cwimxgv
wbGPi ˆKvbwU mwVK? K mgevü L mƒßGKvYx 16 ˆm.wg| ‰i f„wg 6 ˆm.wg., DœPZv KZ?
K i I ii L i I iii M Õ©ƒjGKvYx N mgGKvYx K 12 ˆm.wg. L 8 ˆm.wg.
M ii I iii N i, ii I iii 14. ‰KwU eGMÆi A¯¦e†GÆ îi eÅvmvaÆ 3 ˆm.wg. nGj M 6 ˆm.wg. N 4 ˆm.wg.
eGMÆi evüi Š`NÆÅ KZ? A
 x + = 0 nGj 2  x +
1 1 
3.  ‰i gvb K 3 ˆm.wg. L 6 ˆm.wg. B F N
x  x
M 3 ˆm.wg. N 6 ˆm.wg.
KZ? 15.
C
E M
K 0 L1 S
D
M 2 N4 wPGò ABCDEF ‰KwU mylg lofzR|
DóxcGKi AvGjvGK (4 I 5) bs cÉGk²i Dîi CD = 4 ˆm.wg. ‰es EM = 13 ˆm.wg.
O
`vI: DcGii ZG^Åi AvGjvGK (24 I 25) bs
1 88 Q
P =7 P cÉGk²i Dîi `vI|
P R
wPGò POQ = 88 nGj PRQ = ? 24. 
 mÁ·ƒYÆGÞGòi cwimxgv KZ?
4. P + 1 2 = ? K 44 L 88 K 58 ˆm.wg. L 54 ˆm.wg.
 P
K 53 L 51 M 92 N 136 M 50 ˆm.wg. N 46 ˆm.wg.
M 47 N 45 16.  M ‰i
 NƒYÆb cÉwZmvgÅ ˆKvY KZ? 25. 
 ABCDEF ˆÞGòi ˆÞòdj KZ
5. P
‰i gvb KZ? K 90 L 180 eMÆ ˆm.wg.?
P2  6P  1 M 270 N 360 24
1 1 17. ‰KwU mylg cçfzGRi kxlÆ ˆKvY KZ wWMÉx? K L 24 3
K L 3
12 2 K 60 L 90 M 108 N 120
M 1 N2 4
18. wPGò MN || XY nGj L M 4 3 N
1 3
6. log x = log y nGj log x2 ‰i gvb KZ? wbGPi ˆKvbwU mwVK?
2 X Y wbGPi ZG^Åi AvGjvGK (26 I 27) bs cÉGk²i
K x L y Dîi `vI:
M N
M log y N log y K LM:LN = LX:XY 50 Rb wkÞv^Æxi MwYZ welGqi cÉvµ¦ bÁ¼Gii
7. 2x  5 + 3 = 2 ‰i mwVK mgvavb ˆmU L LM:MX = XY:NY
MYmsLÅv wbGekb mviwY wbÁ²i…c:
ˆKvbwU? M LM:LN = LX:LY
N LM:MX = LN:LY ˆkÉwY 91-
K { 3} L {  3}M {  3}N 41-50 51-60 61-70 71-80 81-90
19. sec2 + tan2 = 3 nGj, cosec ‰i gvb eÅvwµ¦ 100
8. hw` ˆKvb eMÆGÞGòi cÉGZÅK evüi Š`NÆÅ
KZ? (hLb  mƒßGKvY) wkÞv^Æxi
20% e†w«¬ cvq ZGe ˆÞòdj kZKiv KZ 6 7 10 12 8 7
2 msLÅv
fvM e†w«¬ cvGe? K L 2
K 36 L 44 M 72 N 80 3 26. 
 gaÅK wbYÆGq Fc ‰i gvb wbGPi ˆKvbwU?
1 K 12 L 23 M 35 N 43
9. 2x  y = 8, x + y = 4 mgxKiY ˆRvU M 1 N
i. mãwZcƒYÆ ii. wbfÆikxj 2 27. 
 cÉPziK wbGPi ˆKvbwU?
iii. mgvavb ˆbB 20. 
 A = 15 nGj, K 74.33 L 77.67
M 81.33 N 86.67
wbGPi ˆKvbwU mwVK? i. tan 3A = 2 sin 3A
K i I ii Li ii. cot 4A =
1 28. cosec A + cot A = 1 nGj
2
M ii N i, ii I iii 3
iii. sin 4A = cos 2A cosec A  cot A = KZ?
10. 
 2 + a + b + c + 162 àGYvîi avivfzÚ
wbGPi ˆKvbwU mwVK? 1 1
K L2 M N 4
nGj mvaviY AbycvZ KZ? K i I ii L i I iii 2 3
K 3 L4 M5 N 6 1
M ii I iii N i, ii I iii 29. 
 cosec A + cot A = nGj sec A = KZ?
11. A 2
21. n evü «¼viv MwVZ mylg eüfzGRi cÉGZÅKwU 5 5 1 1
ˆKvGYi cwigvY KZ? K L M N
D
50
E 3 3 3 2
180 (n2) 180 (n+2)
K
n
L
n
30. 2x  1 = 2 nGj 3x  1 ‰i gvb KZ?
3x 2x
64 90 (n2) 90 (n+2) 2 3
M N K L1 M N 3
B C n n 3 2

1 M 2 L 3 M 4 K 5 M 6 M 7 N 8 L 9 L 10 K 11 M 12 L 13 N 14 N 15 N
Dîi

16 N 17 M 18 M 19 L 20 N 21 K 22 K 23 N 24 M 25 L 26 L 27 K 28 L 29 K 30 N
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 3
348 gwZwSj gGWj Õ•zj ‰´£ KGjR, XvKv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 (2x + y, 3) = (6, x  y) nGj 11. KqwU Õ¼Z¯¨ Dcvî Rvbv ^vKGj wbw`ÆÓ¡ 20. 
 ‰KwU cZvKvi LuywU ˆfGã, fvãv Ask
(x, y) = KZ? PZzfzÆR AuvKv hvq? f„wgi mvG^ 30 ˆKvY Drc®² KGi| LuywUwUi
K (0, 3) L (3, 0) K `yBwU L wZbwU fvãv AsGki Š`NÆÅ 16 wgUvi nGj `´£vqgvb
M (6, 3) N (1, 4)
M PviwU N cuvPwU AsGki Š`NÆÅ KZ wgUvi?
1
2. (x) = x2  2x + 3 nGj   ‰i gvb 12. `ywU e†î ciÕ·iGK A¯¦tÕ·kÆ KGi| ‰G`i K 4 L 8
2
KZ? ‰KwUi eÅvm 10 ˆm.wg. ‰es AciwUi M 16 N 16 3
7 7 eÅvmvaÆ 4 ˆm.wg. nGj, ‰`i ˆK±`Ê«¼Gqi 21. ‰KwU mgevü wòfzGRi ˆÞòdj 4 3 eMÆwgUvi
K  L nGj cwimxgv KZ wgUvi?
4 4 gaÅeZÆx `ƒiZ½ KZ ˆm.wg.?
11 9 K 1 L 4 K 4 2 L 4 3
M N
4 4 M 5 N 9 M 12 N 12 3
3.  x + y = 5 ‰es x  y = 3 nGj x2 +
 a
13. cosec = b nGj tan ‰i gvb KZ? 22. ‰KwU mylg lofzGRi ˆK±`Ê ˆ^GK ˆKŒwYK
y2 ‰i gvb KZ? we±`yi `ƒiZ½ 6 wg. nGj ‰i ˆÞòdj KZ
K 0 L 1 b 2
a b 2

M 2 N 4 K L
b
eMÆwgUvi?
a2  b 2
4. ABC ‰ C = 30 nGj, A = 2B nGj K 109 3 L 54 3
a2 + b 2 b
B ‰i AGaÆK KZ? M N M 27 3 N 9 3
b a + b2
2

K 45 L 30 23. AebwZ ˆKvGYi gvb KZ wWwMÉ nGj LuywUi

wbGPi ZG^Åi AvGjvGK (14 I 15) bs cÉGk²i
M 25 N 22.5 Š`NÆÅ Qvqvi Š`GNÆÅi 3 àY nGe?
5. wòfzR AuvKGZ cÉGqvRb@ Dîi `vI:
K 30 L 45
PQR mgGKvYx wòfzGR R mgGKvY|
i. wZbwU evü M 60 N 90
ii. `ywU evü ‰es ZvG`i A¯¦fzÆÚ ˆKvY 3 tanP = 1 24. 'H'-‰i NƒYÆb ˆKvY KZ?
iii. `ywU ˆKvY I ‰G`i msj™² ‰KwU evü
14. 
 P-‰i gvb KZ? K 90 L 180
K 30 L 45
wbGPi ˆKvbwU mwVK? M 270 N 360
M 60 N 90
K i I ii L i I iii 25. 
 ˆK±`Êxq cÉeYZvi cwigvc nGjv@
15.  DcGii Z^Å AbymvGi@

MvwYwZK Mo
M ii I iii N i, ii I iii i.
i. sin(P + Q) = 1
6. ˆKvGbv e†Gî AwaPvGci A¯¦wjÆwLZ ˆKvY gaÅK
ii.
ii. PQ = 1 + 3
nGœQ@ iii. cosP + sinQ = tan 60 cÉPziK
iii.
K mƒßGKvY L mgGKvY wbGPi ˆKvbwU mwVK? wbGPi ˆKvbwU mwVK?
M Õ©ƒjGKvY N cÉe†«¬ ˆKvY K i I ii L i I iii K i I ii L i I iii
7. O e†Gîi ˆK±`Ê nGj x A
M ii I iii N i, ii I iii M ii I iii N i, ii I iii
‰i gvb KZ? x+10 16. `ywU msLÅvi AbycvZ 5 : 7 ‰es ZvG`i wbGPi ZG^Åi AvGjvGK (26 I 27) bs cÉGk²i
O
M.mv.à 4 nGj, j.mv.à. KZ? Dîi `vI:
x+80
B C K 60 L 80 X 61-65 66-70 71-75 76-80 81-85
K 20 L 30 M 120 N 140  2 8 20 7 3
M 40 N 60 17. 
 3x  5y = 7, 6x  10y = 15 ‰B 26. mviwYi gaÅK KZ?
8. 
 2x  5 + 3 = 2 ‰i mwVK mgvavb ˆmU mgxKiY ˆRvUwU@ K 68.5 L 68.6
M 73.4 N 73.5
ˆKvbwU? i. Amgém
K {} L {3} 27. cÉPziK ˆkÉwYi gaÅgvb/gaÅwe±`y KZ?
ii. ‰KwU gvò mgvavb AvGQ
M {3} N {3} K 22.28 L 68
iii. ciÕ·i AwbfÆikxj M 73 N 78
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i wbGPi ˆKvbwU mwVK?
Dîi `vI: 28. ˆKvGbv NbGKi c†Ó¤ZGji KGYÆi Š`NÆÅ
K i I ii L i I iii 6 2 ˆm.wg. nGj AvqZb KZ?
`yB AâwewkÓ¡ ‰KwU msLÅvi `kK Õ©vbxq
M ii I iii N i, ii I iii K 112 L 144
Aâ ‰KK Õ©vbxq AGâi wZbàY|
18. 1 + 3 + 5 + ... ... ... + 101 avivwUi c` M 216 N 432
9. ‰KK Õ©vbxq Aâ x nGj msLÅvwU KZ
msLÅv KZ? 29.  wbGPi
 ˆKvbwU wewœQ®² PjGKi D`vniY?
nGe?
K x L 3x
K 51 L 101 K Zvcgvòv L e†wÓ¡cvZ
M 30x N 31x M 201 N 204 M cixÞvi bÁ¼i N evqyPvc
1
10.  A⫼q
 Õ©vb wewbgq KiGj msLÅvwU 19.  1 + 3 ... ... ... avivwUi 8g c` KZ? 30. A = {1, 3, 5, 7} nGj A ‰i cÉK‡Z DcGmU
3
KZ nGe? KqwU?
K 11x L 13x K  27 3 L  27 K 4 L 8
M 30x N 31x M 27 N 27 3 M 15 N 16

1 L 2 N 3 N 4 M 5 N 6 K 7 N 8 K 9 N 10 L 11 N 12 K 13 K 14 K 15 L
Dîi

16 N 17 L 18 K 19 L 20 L 21 M 22 L 23 M 24 L 25 N 26 N 27 M 28 M 29 M 30 M
4 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
349 ˆm´Ÿ ˆhvGmd DœP gvaÅwgK we`Åvjq, XvKv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
.
 0.3  0.6 = KZ?
.
9. 
a a2 + b2
 = 2 nGj, 2 2 ‰i gvb wbGPi ˆKvbwU? 23. 
 avivwUi 2n + 1 msLÅK cG`i mgwÓ¡
b a b
. . . . KZ?
K 0.2 L 0.4 M 0.5 N 0.1 K 2/3 L 4/3 M 5/3 N 3/2
K 0 L4 M 4 N 8
. . 10. cosec(90  ) = 2 nGj, cos = KZ? 24. mgvb f„wgwewkÓ¡ `yBwU wòfzGRi DœPZv
2. 15.3 I 14.7 f™²vsk `yBwU@
1 3 1
i.m`†k I gƒj` K 2 L M
2
N
2
h^vKÌGg 2 ˆm.wg. I 8 ˆm.wg. nGj, ZvG`i
2
ii. àYdj Ave†î `kwgK nGZI cvGi bvI
ˆÞòdj ‰i AbycvZ wbGPi ˆKvbwU?
11. tan245sin260 = KZ? K 2:3 L 1:4 M 3:8N 4:3
nGZ cvGi 3 1 3 3 3
K L M N 25. A
iii. fvM cÉwKÌqvi ˆÞGò fvMdj me mgqB 4 2 2 4
1
Ave†î `kwgK nGe 12. 
 logx  =  2 nGj, x = KZ?
D E

wbGPi ˆKvbwU mwVK?

25 
B C
1 1
K i I ii L i I iii K 5 L M N 5
5 5 AD : BD = AE : CE nGj@
M ii I iii N i, ii I iii 13. 
 wbw`ÆÓ¡ PZzfzÆR AâGb cÉGqvRb@ i. BC = DE
3. 
 NbGKi c†Ó¤ZGji ˆÞòdj 600 eMÆ ‰KK i. PviwU evü I ‰KwU ˆKvY ii. BC || DE
iii. BED = CDE
nGj, ‰i avi KZ ‰KK? ii. PviwU evü `ywU KYÆ
wbGPi ˆKvbwU mwVK?
K 5 L 10 M 15 N 20 iii. wZbwU ˆKvY I `yBwU evü
K i I ii L i I iii
wbGPi ZG^Åi AvGjvGK (4 I 5) bs cÉGk²i Dîi wbGPi ˆKvbwU mwVK?
M ii I iii N i, ii I iii
`vI: K i I ii L i I iii
26. wPGò AOC ˆK Kx ˆKvY ejv nq?
ABC mgevü wòfzGR AB = AC = BC = 2 ˆm.wg., M ii I iii N i, ii I iii
A
AE  BC| 14. 2x  y = 8 ‰es x  2y = 4 nGj, C O
D
A
x + y = KZ?
B
K 1 L2 M4 N 8
15. wbGPi ˆKvb we±`ywU x AÞGiLvi Dci KcÉe†«¬ ˆKvY L Õ©ƒjGKvY
D
B E C
AewͩZ? MmgGKvY N mħGKvY

K (0, 6) L (5, 0) M (0, 4) N (5, 5) 27. O ˆK±`ÊwewkÓ¡ e†Gî@ C

4. 
 ABC-‰i cwimxgvi mgvb cwimxgv
8
wewkÓ¡ eGMÆi KGYÆi Š`NÆÅ KZ ˆm.wg.? 16. p  = 2 nGj, p = ?
p A O B

K 2 3 L 3 2 K 1 L2 M3 N 4 i. ACB = AaÆe†îÕ© ˆKvY

5 3 3 2
17. 
 ( 5)x+1 = ( 5)2x+1 nGj, x = KZ?
3 ii. BAC + ABC = ACB
M N
2 2 iii. AO = BO
5. 
 AE ‰i gvb KZ ˆm.wg.? K 1 L 2 M5 N 5
wbGPi ˆKvbwU mwVK?
18. 3y2 + 2y  1 ‰i Drcv`K ˆKvbwU?
K 3 L 2 K i I ii L i I iii
K y1 L y+1
M 3 2 N 2 3 M ii I iii N i, ii I iii
M 3y  2 N 3y + 1
6. cÉ`î wPGò@ 28. 
 e†Gîi `yBwU RÅv ciÕ·iGK mgvb fvGM
C 19. 500 UvKvi 2 eQGii mij gybvdv 20 nGj,
B D fvM KiGj ˆQ`we±`yi AeÕ©vb e†Gîi@
25 gybvdvi nvi KZ?
A
30
K 2% L 3% M 4% N 5% K IcGi L evBGi
O E
wbGPi ZG^Åi AvGjvGK (20 I 21) bs cÉGk²i M cwiwa N ˆKG±`Ê
i. BOA + BOC = 90 Dîi `vI: 29. 36 wgUvi jÁ¼v MvGQi x wgUvi `ƒiGZ½ D®²wZ
ii. BOC = DOE = 60 7, 10, 12, 11, 8, 14, 9, 6, 7, 13, 10, 8 ˆKvY 30, x-‰i gvb ˆKvbwU?
iii. BOC + COD = 85 20. DcvGîi cwimi KZ? K 30 L 36 M 36 3 N 36 2
wbGPi ˆKvbwU mwVK? K 3 L4 M8 N 9 30. 
 wPGò CD || EF ‰es AB ZvG`i ˆQ`K
K i I ii L i I iii 21. DcvGîi gaÅK KZ? nGj, x = KZ?
M ii I iii N i, ii I iii K 9 L 10 M 9.5 N 10.5 D F
7. hw` (x) = x3  4x + 8 nq, ZGe (2) = KZ? wbGPi ZG^Åi AvGjvGK (22 I 23) bs cÉGk²i x+20

K 2 L0
M4 N 8 Dîi `vI: A
O M
B

8. 3 + 10 + 17 + 24 + ... ... ... avivi AÓ¡g c` 4  4 + 4  4 + ... ... ... ‰KwU àGYvîi aviv| C
x+10
E

KZ nGe? 22. avivwUi mvaviY AbycvZ KZ?

K 36 L 52 M 62 N 86 K 0 L1 M 1 N 4 K 30 L 45 M 60 N 75

1 2 K 3 4 5 6K 7 8 L9 10 11
N 12 13
K 14 M 15 L
L N L M N K N L
Dîi

16 N 17 K 18 L 19 K 20 N 21 M 22 M 23 L 24 L 25 M 26 K 27 N 28 N 29 M 30 N
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 5
350 bÅvkbvj AvBwWqvj Õ•zj, wLjMuvI, XvKv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 0.2̇34̇ ‰i mvgvbÅ f™²vsk ˆKvbwU? 10. 
1
 x2  2 = KZ? 21. `ywU msLÅvi AbycvZ 3 : 4 ‰es ZvG`i
x
K
211
L
234 j.mv.à. 180| msLÅv `ywU KZ?
900 909 K 6 4 L 4 6
K 30, 45 L 45, 60
234 26 M 2 3 N 8 6 M 45, 75 N 45, 70
M N
900 111 11. 
 cwimxgv I ‰KwU ˆKvGYi gvb Rvbv 22. mgevü wòfzGRi DœPZv 6 ˆm.wg. nGj, evüi
2. 
 10% nvi gybvdvq 6000 UvKvi 3 eQGii cÉGqvRb ˆKvbwU AuvKGZ? Š`NÆÅ KZ?
PKÌe†w«¬ gybvdv I mij gybvdvi cv^ÆKÅ K eMÆ L iÁ¼m K 12 3 L 4 3
KZ? M AvqZ N mvgv¯¦wiK M 6 3 N 24 3
K 186 L 1800 12. 60 ˆKvGYi mÁ·ƒiK ˆKvGYi AGaÆK wbGPi 23. 
 ˆKvb eMÆGÞGòi ˆÞòdj Zvi KGYÆi
M 1986 N 6000 ˆKvbwU? Dci AwâZ eMÆGÞGòi ˆÞòdGji KZàY?
3. ‰KwU msLÅvGK a  10n AvKvGi ˆjLvi RbÅ K 30 L 60 1 1
kZÆ ˆKvbwU? M 90 N 120 K L
4 2
K 1 < a < 10 L 1  a  10 13. mgGKvYx wòfzGRi mƒßGKvY«¼Gqi A¯¦i 4 M
1
N 2
3
M 1  a < 10 N 1 < a  10 nGj, ‰i e†nîg ˆKvYwUi gvb KZ?
wbGPi ZG^Åi AvGjvGK (24 I 25) bs cÉGk²i
4. 2x  3 =  3 ‰i mgvavb ˆmU KZ? K 47 L 43
M 76 N 86 Dîi `vI:
K {} L {} B

M {1} N { 3} 14. ˆKvbwU mylg PZzfÆyR? 60

S
A
5. 6x  y = 5 ‰es 5x  2y = 2 nGj, K AvqZGÞò L mvgv¯¦wiK C
O

x + y = KZ? M iÁ¼m N eMÆGÞò

K 2 L 3 15. ˆKvb e†Gîi AwaPvGc A¯¦wbÆwnZ ˆKvYwU Kx OˆK±`ÊwewkÓ¡ e†Gî AC = 12 cm
M 4 N 5 aiGbi ˆKvY? 24. AB PvGci Š`NÆÅ KZ?
K mħGKvY L mgGKvY K 40.48 cm L 12.57 cm
6. 
 a, b, c KÌwgK mgvbycvZx nGj
M 6.28 cm N 3.14 cm
i. b2 = ac M Õ©ƒjGKvY N cËe†«¬ ˆKvY
a+c wbGPi ZG^Åi AvGjvGK (16 I 17) bs cÉGk²i 25. e†îKjvi AOB ‰i ˆÞòdj KZ?
ii. b = K 150.80 cm2 L 75.40 cm2
2 Dîi `vI: A M 40.84 cm2 N 18.85 cm2
a+b b+c
iii. =
b c x 26. 
 ‰KwU NbGKi ‰K c†GÓ¤i KGYÆi Š`NÆÅ
wbGPi ˆKvbwU mwVK? O
8 2 ˆm.wg.| NbKwUi KYÆ KZ ˆm.wg.?
B C
K i I ii L i I iii x + 60
K
8
L
8
3 2
M ii I iii N i, ii I iii
1
wPGò O e†Gîi ˆK±`Ê M 8 3 N 24
7. ,  1, 7,..... AbyKÌgwUi mvaviY AbycvZ 16. 
 BAC = KZ? 27. wbGPi ˆKvbwU wewœQ®² PjGKi D`vniY?
7
K 30 L 45 K eqm L Zvcgvòv
ˆKvbwU?
M 60 N 120
1 1 M RbmsLÅv N IRb
K L  17. 
 cËe†«¬ BOC ‰i gvb KZ?
7 7 28. 1 ˆ^GK 23 chƯ¦ ˆgŒwjK msLÅvàGjvi gaÅK
K 120 L 180
M 7 N  7 KZ?
M 240 N 280
8. BsGiwR S eGYÆi NƒYÆb ˆKvY KZ? K 7 L 11
18. ˆKvb wòfzGR KqwU ewne†î AuvKv hvq?
K 90 L 180 M 13 N 17
K 1 L 2
M 270 N 360 M 3 N 4
29. 
 DcvîmgƒnGK mviwYfzÚ Kiv nGj cÉwZ
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i Dîi ˆkÉwYGZ hZàGjv Dcvî A¯¦fÆyÚ Kiv nq,
19. sec 1  cos2 = KZ?
`vI: K sin L cos
Zvi wbG`ÆkK wbGPi ˆKvbwU?
x2  5  2 6 = 0 M tan N cot K ˆkÉwYmxgv L ˆkÉwYi gaÅwe±`y
9. 
 x ‰i gvb ˆKvbwU? 3 M ˆkÉwYmsLÅv N ˆkÉwYi MYmsLÅv
20. cos = 2 nGj,  ‰i gvb KZ?
K 3+ 2 L 3 2 30. 31  40 ˆkÉwYeÅvwµ¦i DœPmxgv KZ?
K 90 L 60 K 40 L 45.5
M ( 3 + 2)2 N 3 2 M 45 N 30 M 35 N 31

1 N 2 K 3 M 4 L 5 L 6 L 7 N 8 L 9 K 10 L 11 L 12 L 13 K 14 N 15 K
Dîi

16 M 17 M 18 M 19 M 20 N 21 L 22 L 23 L 24 M 25 N 26 M 27 M 28 L 29 N 30 K
6 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
351 ‰m I ‰m nvigÅvb ˆgBbvi KGjR, XvKv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. wbGPi ˆKvbwU gƒj` msLÅv? 11. 5x + 3y = 4 ‰es 2x + 7y = 9 ‰B mgxKiY 21. 
 eGMÆi evüi Š`NÆÅ KZ ˆm.wg.?
K 3 L
3
7 ˆRvUwU@ K 2 2 L4 2 M3 2N 8
i. msMwZcƒYÆ 22. ‰KwU eMÆGÞGòi cwimxgv ‰i KGYÆi
3
M 27 N 19 ii. AmsLÅ mgvavb AvGQ Š`GNÆÅi KZ àY?
1 iii. ciÕ·i AwbfÆikxj K L 2 2
2. 
 x = 5+2 6 nGj, x + x ‰i gvb 2
wbGPi ˆKvbwU mwVK? M 3 N 3 2
KZ?
K i I ii L i I iii 23. mgGKvYx wòfzGRi 30 ˆKvY AâGbi
RbÅ
K 2 3 L2 2
M ii I iii N i, ii I iii wbGPi ˆKvbwU mwVK?
M 3 2 N2 5
12. 
1+
1 1
avivwUi ˆKvb K f„wg > jÁ¼ L f„wg < jÁ¼
wbGPi ZG^Åi AvGjvGK (3 I 4) bs cÉGk²i Dîi  + ............
3 3 M f„wg = jÁ¼ N f„wg = AwZfzR
`vI: 1 24. AaÆe†Gî A¯¦wjÆwLZ ˆKvY@
c` ?
5% nvi gybvdvq 500 UvKv 3 eQGii RbÅ eÅvsGK 9 3 K 60 L 90
ivLv nGjv| K 6 L5 M 120 N 180
3.   PKÌe†w«¬ gybvdv KZ UvKv? M 7 N8
25. 
 wPGò, ECD = KZ?
K 78.81 L 75.81 13. sin4 + sin2 = 1 nGj, wbGPi ˆKvbwU A D
M 500 N 578.81 mwVK?
100
4.  mij gybvdv
 I PKÌe†w«¬ gybvdvi cv^ÆKÅ K tan.cosec = 2 L sin2 = cos
KZ UvKv? M sin = cos2 N cot.sec = 2
K 2.81 L 1.81 1 + x2 E
M 3.81 N 78.81
14. cosec =
x
nGj, sec = KZ? B
C
5. A = {1, 2, 3, 6, 9, 18} ˆmUwUGK
ˆmU MVb K 80 L 90
K x L 1 + x2
M 100 N 120
c«¬wZGZ cÉKvk KiGj wbGPi ˆKvbwU nGe? M
1
N 1
x 26. e†îÕ© UÇvwcwRqvGgi ZxhÆK evü«¼Gqi ŠewkÓ¡Å
K A = {x : x ˆgŒwjK msLÅv}
15. cot (  30) = 3 nGj, sin = KZ? Kxi…c?
L A = {x : x  ô ‰es x > 18}
1 1 K ciÕ·i Amgvb L ciÕ·i jÁ¼
M A = {x : x, 18 ‰i àwYZK} K L
2 2 M ciÕ·i mgv¯¦ivj N ciÕ·i mgvb
N A = {x : x, 18 ‰i àYbxqK}
3 27. 
 e†Gîi ˆK±`Ê ˆ^GK RÅv ‰i `ƒiZ½ 5 ˆm.wg.
6. F(x) = x  5 nGj@ M N1
2 ‰es e†Gîi eÅvmvaÆ 9 ˆm.wg. nGj, RÅv ‰i
i. ˆWvg F = {x  Ñ : x  5} 16. hw` ‰KwU LyuwUi Qvqvi Š`NÆÅ kƒbÅ nq, ZGe Š`NÆÅ KZ?
ii. dvskGbi ˆjLwPò ‰KwU mijGiLv D®²wZ ˆKvY KZ? K 7.48 ˆm.wg. (cÉvq)
iii. ˆié F = {x  Ñ : x  0} K 30 L 45 M 60 N 90 L 6.23 ˆm.wg. (cÉvq)
wbGPi ˆKvbwU mwVK? 17. 
 wPGò M 14.97 ˆm.wg. (cÉvq)
A 60 D
K i I ii L i I iii i. ACB = 60 N 12.46 ˆm.wg. (cÉvq)
M ii I iii N i, ii I iii ii. BC = 18 36
28. 
 A
7. 324 ‰i jM 4 nGj, wfwî KZ? iii. AB = 3 BC B C
K 2 3 L 4 3
wbGPi ˆKvbwU mwVK? 80

M 4 2 N 3 2 K i I ii L i I iii D E
F
50
8. 0.000835 ‰i jGMi cƒYÆK KZ? M ii I iii N i, ii I iii
B
K 4 L4 18. 
 ABC mgevü wòfzGRi evüi Š`NÆÅ C
6 ˆm.wg. nGj, gaÅgvi Š`NÆÅ KZ?
wPGò BD || CF ‰es BC || DE
M 3 N 3
DBC + EFC = KZ?
9. 
 3x  5 + 7 = 2 ‰i mgvavb ˆmU K 3 3 L2 3 M 3 N 2
K 100 L 120
ˆKvbwU? 19. 
 mylg eüfzGRi cÉGZÅKwU kxlÆGKvGYi
M 150 N 180
K 5 L5 cwigvY 108| eüfzGRi evüi msLÅv KqwU? 29. 3, 7, 15, 5, 8, 6, 19, 16, 12, 14 msLÅvàGjvi
M 1 N{ } K 4 L5 M6 N 8
gaÅK KZ?
10. 
x:y=3:2 ‰es y : z = 3 : 2 nGj@ wbGPi ZG^Åi AvGjvGK (20 I 21) bs cÉGk²i K 8 L 10
i. x, y, z KÌwgK mgvbycvZx Dîi `vI : M 12 N 14
ii. y : x = 2 : 3 iii.
x y
= 4 ˆm.wg. eÅvmvaÆ wewkÓ¡ e†Gî ‰KwU eMÆ 30. ˆkÉwY eÅvwµ¦ 21-30 31-40 41-50 51-60
y z
A¯¦wjÆwLZ| MYmsLÅv 4 12 8 10
wbGPi ˆKvbwU mwVK? 20. 
 e†Gîi eÅvm KZ ˆm.wg.? cÉ`î DcvGîi cÉPziK KZ?
K i I ii L i I iii K 6 L7 K 37.67 L 37.67̇
M ii I iii N i, ii I iii M 8 N9 M 41.25 N 42.25

1 M 2 K 3 K 4 M 5 N 6 L 7 N 8 K 9 N 10 N 11 L 12 K 13 L 14 L 15 M
Dîi

16 N 17 N 18 K 19 L 20 M 21 L 22 L 23 K 24 N 25 M 26 N 27 M 28 K 29 L 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 7

352 KÅvgweËqvb Õ•zj ‰´£ KGjR, XvKv


MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. A = {1, 2, 3, 4} nGj, ˆmU A ‰i cÉK‡Z 11. 
 cos2B ‰i gvb KZ? 22. cot = 3 nGj@
DcGmU KqwU? 2 2 10 i. tan =
1
K L 3
K 4 L 14 M 15 N 16 3 3
3x + 1 f(1) + 1 8 10 ii. sec = 2 tan
2. f(x) =
3x  1
nGj, f(1)  1 = KZ? M
9
N
9 iii. 4 sin = sec 2
K 0 L1 12. wZbwU evüi Š`NÆÅ ˆ`Iqv AvGQ| wbGPi ˆKvb wbGPi ˆKvbwU mwVK?
M 2 N3 ˆÞGò ‰KwU wòfzR AuvKv mÁ¿e? K i I ii L i I iii
3. A = {3, 5, 7}, B = {4, 5, 7} nGj@ K 2, 3, 5 L 4, 5, 10 M ii I iii N i, ii I iii
i. AB = {5, 7} M 6, 7, 8 N 7, 5, 2 23.  1 + 3  9 + ...... avivwUi PZz^Æ c` KZ?
ii. P(AB) ‰i Dcv`vb msLÅv 16 13. 
 mgevü ABC ‰i cwiGK±`Ê O nGj, K  27 L  12 M 12 N 27
iii. A\B = {3, 4} AOB ‰i gvb KZ? 4n  1
wbGPi ˆKvbwU mwVK? 24. 2n  1 ‰i gvb wbGPi ˆKvbwU?
K 120 L 90 M 60 N 30
K i I ii L i I iii 14. 
 wPGò, O e†Gîi ˆK±`Ê K 2n + 1 L 2n  1
n+1
M ii I iii N i, ii I iii
O 5cm M 2 N 2n  1
‰es BD = 4 ˆm.wg.|
1
4. x + y = 6, x  y = 4 nGj, xy ‰i gvb AOB ‰i ˆÞòdj KZ
A
D
B 25. 
 logx  =  2 nGj, x ‰i gvb KZ?
25 
wbGPi ˆKvbwU? eMÆ ˆm.wg.? K ±5 L 5
K 2 L 5 K 12 L 20 M 24 N 30 1 1
M 24 N 26 M ± N
15. KgcGÞ KqwU evüi Š`NÆÅ ˆ`Iqv ^vKGj 5 5
5. S = {(3, 1), {3, 2), (4, 2)} A®¼Gqi ˆié eMÆ Aâb Kiv mÁ¿e? 26. (34  40) I (41  47)
ˆKvbwU? K 4wU L 3wU M 2wU N 1wU i. ‰i ˆkÉwY eÅeavb 7
K {3, 4} L {1, 2} 16. 
 O e†Gîi ˆK±`Ê nGj, A ii. ˆkÉwYeÅvwµ¦«¼q AwewœQ®²
M {1, 2} N {3, 4} iii. ‰i cÉ^g ˆkÉwYi Ea»Æmxgv 40
x ‰i gvb KZ? x+10
wbGPi ZG^Åi AvGjvGK (6 I 7) bs cÉGk²i Dîi wbGPi ˆKvbwU mwVK?
K 20 L 30 C
`vI: M 40 N 60
B x+80
K i I ii L i I iii
2
a  5a  1 = 0
17. `yBwU e†î ciÕ·iGK ˆQ` KiGj ZvG`i M ii I iii N i, ii I iii
1
6.  a2 + 2 ‰i gvb KZ?
a gGaÅ mGeÆvœP KqwU mvaviY Õ·kÆK Aâb wbGPi ZG^Åi AvGjvGK (27 I 28) bs cÉGk²i Dîi `vI:
K 23 L 25 M 27 N 29 Kiv mÁ¿e? 10, 9, 8, 6, 11, 12, 9, 14, 7, 9
1 K 1 L2 M3 N 4 27. cÉ`î DcvîàGjvi cÉPziK KZ?
7.  a + ‰i gvb KZ? K 14 L 9
a 18. 
 11 + 7 + 3 + ....  49 avivwUi@
M 7 N 6
K 29 L 27 M 23 N 21 i. mvaviY A¯¦i  4
1 28. cÉ`î DcvîàGjvi gaÅK ˆKvbwU?
8. x + = 5 nGj@ ii. 9 Zg c`  43
x K 11.55 L 11
iii. c`msLÅv 16 M 9 N 8.5
i. x2 + 5x + 1 = 0
1 wbGPi ˆKvbwU mwVK? 29. AwRf ˆiLv AâGb x-AÞ eivei ˆKvbwUGK
ii. x  = 1
x K i I ii L i I iii aiv nq?
1
iii. x2 + 2 = 3 M ii I iii N i, ii I iii K MYmsLÅv
x
19. 
 wPGò, O ˆK±`ÊwewkÓ¡ S L gaÅgvb
wbGPi ˆKvbwU mwVK?
e†Gî PQRS A¯¦wjÆwLZ R
M ˆkÉwYi DœPmxgv
K i I ii L i I iii O 80
nGqGQ| SPQ = KZ? P N KÌgGhvwRZ MYmsLÅv
M ii I iii N i, ii I iii Q
K 80 L 90 M 180 N 360 30. 
 7.5 wg. Š`NÆÅwewkÓ¡ ‰KwU AvqZGÞGòi
9. 2sin = 1 nGj, cot ‰i gvb KZ? cÉÕ© Š`GNÆÅi ‰K-Z‡Zxqvsk nGj@
1
20. 
 ˆKvGbv wòfzGRi `yB evüi Š`NÆÅ 9 ˆm.wg.
K 0 L M 3 N 2 I 10 ˆm.wg. ‰es ‰G`i A¯¦fzÆÚ ˆKvY 60| i. cÉÕ© 2.5 wg.
3
wòfzRwUi ˆÞòdj KZ eMÆ ˆm.wg.? ii. cwimxgv 20 wg.
wbGPi ZG^Åi AvGjvGK (10 I 11) bs cÉGk²i
K 22.5 L 38.97 M 45 N 77.94 iii. KYÆ 55.25 wg.
Dîi `vI:
21. 
 ABCD e†Gî A¯¦wjÆwLZ ‰KwU PZzfzÆR| wbGPi ˆKvbwU mwVK?
tan(2A  45) = 1 = 3sinB
BAD = 95 nGj, BCD = KZ? K i I ii L i I iii
10. 
 A ‰i gvb KZ?
K 30 L 45 M 60 N 90 K 85 L 90 M 95 N 105 M ii I iii N i, ii I iii

1 M 2 N 3 K 4 L 5 L 6 M 7 K 8 M 9 M 10 L 11 M 12 M 13 K 14 K 15 N
Dîi

16 N 17 L 18 L 19 K 20 L 21 K 22 N 23 N 24 K 25 L 26 L 27 L 28 M 29 M 30 K
8 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
353 DBj&m wjU&j dÑvIqvi Õ•zj ‰´£ KGjR, XvKv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
. 11. wòfzGRi evüi Š`NÆÅ wbG`Æk KiGj ˆKvbwU 20. cot 1  cos2 = KZ?
1. 
 2.02 ‰i `kwgK f™²vsk ˆKvbwU?
182 200 182 200
«¼viv wòfzR MVb mÁ¿e bq? K sec L cos
K L M N K 4, 5, 8 L 13, 15, 10 M sin N tan
9 9 90 90
M 7, 11, 4 N 12, 16, 27 21. ˆKvb e†Gîi eÅvm 28 ˆm.wg. nGj, ‰i cwiwa
2. A = {0} nGj, A ‰i cÉK‡Z DcGmU KZwU?
wbGPi ZG^Åi AvGjvGK (12 I 13) bs cÉGk²i KZ ˆm.wg.?
K 1 L0 M2 N bvB
Dîi `vI: K 42.48 L 48.94
3. 
 hw` (x) = x + 9x + 11kx + 5k nq
7 4
C B M 44.43 N 87.96
ZGe k ‰i ˆKvb gvGbi RbÅ (1) = 0 22. 
 3 ˆm.wg. aviwewkÓ¡ NbGKi c†Ó¤ZGji
O
nGe? KGYÆi Š`NÆÅ KZ?
A E D
3 4 2 3 K 6 ˆm.wg. L 3 2 ˆm.wg.
K L M N
4 3 3 2
O ˆK±`ÊwewkÓ¡ ABCD e†Gî AB = 10cm M 6 ˆm.wg. N 9 ˆm.wg.
4. a2  2a + 1 = 0 nGj, a2 + 3a + 4 ‰i gvb
OEAD, OE = 3 ˆm.wg. ACBC 23. H eYÆwUi NƒYÆb cÉwZmgZvi gvòv KZ?
ˆKvbwU nGe?
12. 
 ACB ‰i cwimxgv KZ ˆm.wg.? K 1 L2 M 2 N 2 2
K 2 2 L 3 2 24. 0.0035 ‰i mvaviY jGMi cƒYÆK KZ?
K 20.14 L 22.14
M 4 2 N 2 M 24.14 N 26.14 . .
K 3 L1 M2 N 3
5. hw` x + y = 6 ‰es x2  y2 = 12 nq 13. 
 OED ‰i ˆÞòdj KZ eMÆ ˆm.wg.?
25. ˆKvb kGZÆ ax = bx nGj, a = b nGe?
ZvnGj@ K 20 L 15 M 12 N 6
K a = 0, b = 0, x  0
1 2 3 4 ... ... ...
i. (x  y)2 = 4 14.  +  +
2 3 4 5
AbyKÌGgi mvaviY L a > 0, b > 0, x  1
ii. x = 4 M a > 1, b > 0, x  1
c` ˆKvbwU?
iii. xy = 8 N a > 0, b > 0, x  0
n n
wbGPi ˆKvbwU mwVK? K (1)n+1 L (1)n+1 1
n+1 n1 26. x = 5 + 4 nGj x + x = KZ?
K i I ii Li I iii M (1)n1
n
N (1)n+1
n1
n+1 n+1 K 2 5 L 4
M ii I iii N i, ii I iii
15. mgevü wòfzGRi ˆÞòdj 6 3 eMÆwgUvi M 52 N 5+2
6. (3a1 + 2b1)1 ‰i gvb wbGPi ˆKvbwU?
nGj ‰i cwimxgv KZ? 3
27. x  x = 2 nGj
10
‰i gvb KZ?
ab 2a + 3b x2  2x + 2
K L K 4.89 L 14.69
2a + 3b ab K 2 L3 M5 N 10
M 19.56 N 72
2a + 3b 6ab
M N 16. ˆKvbwU wewœQ®² PjK? 28. Dcvîmgƒn mviwYfzÚ Kiv nGj cÉwZ ˆkÉwYGZ
6ab 3a + 2b
K Zvcgvòv L cvwLi msLÅv
hZàGjv Dcvî A¯¦fzÆÚ nq Zvi wbG`ÆkK
7. 
 ˆKvb wòfzGR ‰KwU ewne†Æî AuvKGj e†îwU
M eqm N DœPZv
wbGPi ˆKvbwU?
KqwU evüGK Õ·kÆ KiGe?
K ˆkÉwY mxgv L ˆkÉwYi gaÅwe±`y
K 1 L2 M3 N 0 17. mgGKvYx wòfzGRi mgGKvY eÅZxZ Aci `yB
M ˆkÉwYmsLÅv N ˆkÉwYi MYmsLÅv
8. sin3 = cos nGj  gvb KZ? ˆKvGYi AbycvZ 3 : 2 nGj, ˆKvb `yBwU
wbGPi ˆKvbwU? 29. 
 AwRf ˆiLvi ˆÞGò wbGPi ˆKvbwU
K 60 L 45 M 30 N 15
K 54, 36 L 55, 35 mwVK?
9. tan(  30) = 3 nGj sin ‰i gvb KZ?
M 50, 40 N 45, 45 K Ea»ÆMvgx L wbÁ²Mvgx
1
K 0 L
2 18. ‰KwU mgevü wòfzGRi evüi Š`NÆÅ 2 wgUvi M mgv¯¦ivj N DjÁ¼

3 nGj, ‰i ˆÞòdj KZ? 30. 1 + 3 + 5 + 7 + ... ... ... avivi cÉ^g

M N1
2 K 2 eMÆwgUvi L 3 eMÆwgUvi n msLÅK cG`i mgwÓ¡ KZ?
10. 
 mgGKvYx wòfzGRi mƒÞGKvY«¼q mgvb M 5 eMÆwgUvi N 7 eMÆwgUvi  n(n + 1) 2
K n2 L  
nGj ˆKvYàwji AbycvZ wbGPi ˆKvbwU? 19. 
 ‰KwU mylg cçfzGRi cÉwZwU kxlÆ  2 
K 2:3:4 L 2:2:3 ˆKvGYi cwigvb KZ? M 
 n(n + 1) 
 N
n
M 1:1:2 N 1:2:4 K 60 L 90 M 108 N 120  2  2

1 M 2 K 3 L 4 K 5 N 6 K 7 L 8 L 9 N 10 M 11 M 12 M 13 N 14 K 15 L
Dîi

16 L 17 K 18 L 19 M 20 L 21 N 22 L 23 L 24 N 25 N 26 K 27 K 28 N 29 K 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 9
354 we‰‰d kvnxb KGjR, KzwgÆGUvjv, XvKv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. `yBwU KÌwgK ˆRvo msLÅvi àYdj KZ «¼viv 11. 
 ‰KwU mgw«¼evü wòfzGRi mgvb mgvb 20. 
 13 + 20 + 27 + ... ... ... + 111 avivwUi
wefvRÅ? evüi Š`NÆÅ 50 ˆm.wg. ‰es ˆÞòdj 1200 c` msLÅv KZ?
K 8 L 7 eMÆ ˆm.wg. nGj, mgvb mgvb evüi gaÅeZÆx K 10 L 13
M 6 N 5 ˆKvY KZ? M 15 N 20
2. nGj,
A = {1, 3, 5, 7, 9}, B = {5, 7} K 63.74 L 65.74 21. ‰KwU àGYvîi avivi w«¼Zxq c` 32 ‰es
P(AB) ‰i Dcv`vb msLÅv wbGPi ˆKvbwU? M 73.74 N 74.73 PZz^Æ c` 8 nGj AÓ¡g c` KZ?
K 3 L 4 12. ˆKvGbv e†Gîi@ 1 1
K L
M 8 N 16 2 4
i. AwaPvGc A¯¦wjÆwLZ ˆKvY mƒßGKvY
3. a2 + b2 = 9 ‰es ab = 3 nGj@ 1 1
ii. DcPvGc A¯¦wjÆwLZ ˆKvY Õ©ƒjGKvY M
8
N
16
i. (a  b)2 = 3
ii. (a + b)2 = 15 iii. A¯¦wjÆwLZ mvgv¯¦wiK ‰KwU AvqZ AB AC BC
22. ABC I DEF ‰i = = nGj@
iii. a2 + b2 + a2b2 = 18 wbGPi ˆKvbwU mwVK? DE DF EF
wbGPi ˆKvbwU mwVK? K i I ii L i I iii K A = E L A =  B
M A = F N A = D
K i I ii L i I iii M ii I iii N i, ii I iii
M ii I iii N i, ii I iii 13. 
 cos9B = sinB ‰es B < 10 nGj cot5B
23. H AÞiwUi NƒYÆb ˆKvY KZ wWMÉx?
K 108 L 120
wbGPi ZG^Åi AvGjvGK (4 I 5) bs cÉGk²i Dîi ‰i gvb KZ?
M 180 N 360
`vI: 1
K 0 L
3
24. 
 ABC ‰KwU mgw«¼evü mgGKvYx wòfzR|
2
y+
y
=3
M 1 N 3
BC ‰i AwZfzR ‰es P, BC ‰i Dci
4 14. sin3A = cos3A nGj tan4A = KZ? ˆhGKvGbv we±`y| ZvnGj 2PA2  PC2 = KZ?
4.  y + 2 = KZ?
 2
y K PB2 L AB2
K 3 L 1
K 5 L 7 M AC2 N BC2
M 11 N 13 3 1
M
2
N 25. ‰KwU mgevü wòfzGRi cwimxgv 18 ˆm.wg.
3 3
5. 
y 3 2
+   ‰i gvb ˆKvbwU? nGj wòfzRwUi DœPZv KZ ˆm.wg.?
y 15. ‰KwU ˆ`qvGj mƒGhÆi AvGjv coGj ˆ`qvGji
K 3 L 3 3
K 9 L 15 cv`G`k nGZ 2 wgUvi `ƒiGZ½ 45 D®²wZ
M 3 5 N 6
M 18 N 45 ˆKvY ŠZwi KGi| ˆ`qvjwUi DœPZv KZ
26. 
 ‰KwU iÁ¼Gmi ˆÞòdj 40 eMÆ wgUvi
6. hw` abvñK msLÅv ô ‰i ŠeævwbK i…c wgUvi?
‰es ‰KwU KYÆ 8 wgUvi nGj, Aci KYÆ KZ
a  10n nq ZGe logN = KZ? K 2 L 4
K a + logn L nloga M 6 N 8 wgUvi?
M n + loga N logan 16. mgGKvYx wòfzGRi@ K 5 L 8
M 10 N 20
7. wbGPi ˆKvb kGZÆ ax2 + bx + c = 0 ‰KwU i. jÁ¼ > f„wg, hLb D®²wZ ˆKvY 60
w«¼NvZ mgxKiY? 27. mylg cçfzGRi ‰KwU kxlÆ ˆKvY KZ wWMÉx?
ii. jÁ¼ > f„wg, hLb D®²wZ ˆKvY 30
K 108 L 120
K a0 L b0 iii. jÁ¼ = f„wg, hLb D®²wZ ˆKvY 45
M a>0 N b<0 M 180 N 360
wbGPi ˆKvbwU mwVK? 28. wbGPi ˆKvbwU wbYÆGqi RbÅ KÌgGhvwRZ
8. `yBwU mÁ·ƒiK ˆKvY mw®²wnZ nGj Drc®² K i I ii L i I iii
nq@ MYmsLÅv mviwY cÉGqvRb?
M ii I iii N i, ii I iii
K cƒiK ˆKvY L wecÉZxc ˆKvY K MvwYwZK Mo L eÅewa
17. 
 ˆKvGbv eGMÆi evüi Š`NÆÅ 10% nËvm ˆcGj
M ŠiwLK hyMj ˆKvY N ‰Kv¯¦i ˆKvY M cÉPziK N gaÅK
‰i ˆÞòdj kZKiv KZ nËvm cvGe?
9. ABC mgevü wòfzGRi B I C ‰i wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
K 10% L 19%
mgw«¼L´£K«¼q O we±`yGZ wgwjZ nGj, M 21% N 30% Dîi `vI:
BOC ‰i gvb wbGPi ˆKvbwU? 18. 6x  y = 5 ‰es 5x  2y = 2 nGj, ˆkÉwYeÅwµ¦ 21-25 26-30 31-35 36-40 41-45
K 60 L 90 x + y = KZ? MYmsLÅv 3 6 6 7 4
M 120 N 150 K 2 L 3 29. cÉPziK ˆkÉwYi gaÅgvb KZ?
10.  ˆKvGbv
 e†Gîi ‰KB PvGci Dci M 4 N 5 K 28 L 33
`´£vqgvb e†îÕ© I ˆK±`ÊÕ© ˆKvGYi gvb 19.
x 2
hw` y = 3 nq, ZGe
6x + y
‰i gvb KZ? M 38 N 43
3x + 2y 30. gaÅK wbYÆGqi RbÅ Fc ‰i gvb wbGPi
h^vKÌGg (2y + 10) ‰es (y + 110) nGj,
4 14
y ‰i gvb KZ? K
5
L
15 ˆKvbwU?
K 30 L 45 5 20 K 6 L 7
M 60 N 90 M N M 9 N 15
4 13

1 K 2 M 3 N 4 K 5 K 6 M 7 K 8 M 9 M 10 K 11 M 12 N 13 M 14 K 15 K
Dîi

16 L 17 L 18 L 19 M 20 M 21 K 22 N 23 M 24 K 25 L 26 M 27 K 28 N 29 M 30 M
10 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
355 eËvBU Õ•zj AÅv´£ KGjR, XvKv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
4
1. cot + cosec = 3 nGj, cot – cosec = KZ? 9. wbGPi ˆKvbwU gƒj` msLÅv wbGPi ˆKvbwU mwVK?
5 27 6 8 K i I ii L i I iii
3 –3 1 2 K L M N
K
4
L
4
M
3
N
3
10 48 3 7 M ii I iii N i, ii I iii

2. 3x – 5y = 7, 6x – 10y = 15 ‰B mgxKiY 10. 8 cm eÅvm I 3 cm eÅvmvaÆwewkÓ¡ `ywU e†î 21. logxa = 3 ‰es logay = 2 nGj, logxy ‰i
ˆmUwU@ ciÕ·iGK A¯¦Õ·kÆ KiGj ZvG`i gvb KZ?
ˆK±`Ê«¼Gqi gaÅeZÆx `ƒiZ½ KZ? K 1 L5 M6 N 9
i. Amgém
K 1 cm L 5 cm M 11 cm N 7 cm wbGPi ZG^Åi AvGjvGK (22 I 23) bs cÉGk²i
ii. ‰KwU gvò mgvavb AvGQ
11. f(x) =x4 + 5x + 3 nGj, f(– 2) ‰i gvb KZ? Dîi `vI:
iii. ciÕ·i AwbfÆikxj K – 23 L – 17 M 1 N 9 7 + 13 + 19 + 25 + ....... ‰KwU aviv|
wbGPi ˆKvbwU mwVK? 12.  ‰KwU AvqZGÞGòi Š`NÆÅ 10% e†w«¬ I
 22. avivwUi 20Zg c` ˆKvbwU?
K i I ii L i I iii cÉÕ© 10% nËvm ˆcGj AvqZGÞGòi ˆÞòdj K 26 L 121 M 133 N 139
M ii I iii N i, ii I iii kZKiv KZ e†w«¬ ev nËvm cvGe? 23. avivwUi 1g 30wU cG`i mgwÓ¡ KZ?
3. 
 A K 1% nËvm L 1% e†w«¬ K 3225 L 3000 M 2820 N 1880

6m 30 M 21% nËvm N 21% e†w«¬ 24. E

F D
B C 13. O
A C
wPGò AB ‰KwU MvQ nGj, MvGQi Qvqvi 3
 B
Š`NÆÅ KZ wgUvi? 1 wPGò, ABCDEF eüfzGRi ˆK±`Ê O|
K 2 3 L6 2 M6 3N 6 eüfzRwUi kxlÆGKvY KZ?
wbGPi ZG^Åi AvGjvGK (4 I 5) bs cÉGk²i Dîi wPò nGZ sin.sec ‰i gvb KZ?
1 2 K 60 L 70 M 110 N 120
`vI: K
3
L 3 M1 N
3 25. ˆKvGbv ˆmU A ‰i cÉK‡Z DcGmU msLÅv
cÉvµ¦ bÁ¼i 51-60 61-70 71-80 81-90 91-100 14. 
 wPGò PQRS iÁ¼Gmi 31 nGj, H ˆmGUi Dcv`vb msLÅv KqwU?
MYmsLÅv 8 12 15 7 8 cwimxgv 20 cm ‰es
P
S K 4wU L 5wU M 6wU N 3wU
4. gaÅK ˆkÉwYi KÌgGhvwRZ MYmsLÅv wbGPi PR = 6 cm nGj@
O 26. b, a, c wZbwU KÌwgK mgvbycvZx nGj@
ˆKvbwU? i. QO = 4 cm i. b : a = a : c
R ii. a2 = bc iii. b2 = ac
K 42 L 35 M 20 N 15 ii. PO = 3 cm Q
iii. iÁ¼mwUi ˆÞòdj = 24 cm
wbGPi ˆKvbwU mwVK?
5. cÉ`î DcvGîi@
K i I ii L i I iii
i. cÉPziK ˆkÉwY (71-80) wbGPi ˆKvbwU mwVK?
M ii I iii N i, ii I iii
ii. cÉPziK = 77.54 (cÉvq) K i I ii L i I iii
27. (x – y + z)2 = KZ?
iii. gaÅK wbYÆGq fm = 15 M ii I iii N i, ii I iii
K x2 – y2 + z2 – 2xy – 2yx +2xz
wbGPi ˆKvbwU mwVK? 15. DcPvGci Abye®¬x PvGc A¯¦wjÆwLZ ˆKvY L x2 + y2 – z2 + 2xy + 2yz – 2zx
K i I ii L ii I iii wbGPi ˆKvbwU? M x2 + y2 + z2 – 2xy – 2yz +2zx
K 180 L 80 M 110 N 120 N x2 + y2 + z2 – 2xy – 2yz – 2zx
M i I iii N i, ii I iii
6. ‰KwU wòfzGRi KqwU ewnteƆî AuvKv hvq? 16. 
 cot( – 60) = 3 nGj, cos = KZ? 28. 
 A D
E
20
1 3
K 2wU L 3wU M 4wU N 1wU K 0 L M1 N 105
2 2
7. 
 kZKiv evwlÆK 7 UvKv mij gybvdvq
17. e†Gîi cÉwZmvgÅ ˆiLv KqwU? B C
ˆKvGbv gƒjab 2 eQGi me†w«¬gƒj 912 UvKv wPGò DCE = KZ?
K 0 L1 M2 N AmsLÅ
nGj, gƒjab KZ? K 60 L 55 M 125 N 65
18. mgGKvYx wòfzGRi cwiGK±`Ê@ wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
K 894.11 UvKv L 852.33 UvKv
K wòfzGRi evBGi L wòfzGRi Afů¦Gi Dîi `vI: P
M 796.57 UvKv N 800.00 UvKv M AwZfzGRi Dci N AwZfzGRi evBGi x
1
8. x + x = 5 nGj@ 19. 0.0000525 ‰i ŠeævwbK i…c wbGPi ˆKvbwU?
O
K 5.25  105 L 5.25  10– 5
i. x2 – 5x + 1 = 0
M 5.25  10– 4 N 52.5  10– 5
1 1
iii. x2 + 2 = 5 wPGò, O e†Gîi ˆK±`Ê| Q x + 55
R
ii. x – = 1
x x 20. – 3 + 3 – 3 + 3 – 3 + .............
29. 
 QPR =
wbGPi ˆKvbwU mwVK? i. ‰wU ‰KwU àGYvîi aviv K 60 L 110 M 120 N 55
K i I ii L i I iii ii. mvaviY AbycvZ 1 30. 
 cÉe†«¬ QOR ‰i gvb KZ?
M ii I iii N i, ii I iii iii. cÉ^g 11wU cG`i mgwÓ¡ – 3 K 250 L 360 M 110 N 260
1 L 2 L 3 K 4 L 5 M 6 L 7 N 8 K 9 L 10 K 11 N 12 K 13 L 14 N 15 L
Dîi

16 K 17 N 18 M 19 L 20 L 21 M 22 L 23 M 24 N 25 L 26 K 27 M 28 L 29 N 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 11
356 ‰UwgK ‰bvwRÆ wimvPÆ ‰Õ¡vweÐmGg´Ÿ Õ•zj ‰´£ KGjR, XvKv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 0.00336 ‰i ŠeævwbK i…c ˆKvbwU? 11. mgGKvYx wòfzGRi 70 ˆKvY AâGbi 20. 3  4 secA.sinA = 0 nGj, tanA KZ?
K 3.36  104 L 33.6  103 ˆÞGò wbGPi ˆKvbwU mwVK? K
3
L
3
7 4
M 3.36  103 N 336  105 K jÁ¼ = f„wg L jÁ¼ > f„wg 4 7
2. (21 + 31)1 ‰i gvb KZ? M N
M f„wg > jÁ¼ N AwZfzR = jÁ¼ 3 3
6 5 21. (90  x) ˆKvGYi cƒiK ˆKvY KZ?
K
5
L
6 12. 
 wPGòi O ˆK±`ÊwewkÓ¡ e†Gî cÉe†«¬
POR ‰i gvb KZ? Q K 90 L x
2 1
M N M x + 90 N x + 180
3 6 K 60
3. A ˆmGUi cÉK‡Z DcGmU 15 wU nGj, L 120
x
22. 
 wòGKvYwgwZK AbycvGZi ˆÞGò
O
P(A) ‰i Dcv`vb KqwU? M 240 i. sin + cos > 1
x + 60 ii. cosec2  cot2 = 1
K 3 L 4 N 360 P R
iii. tan2  sec2 =  1
M 16 N 32 13. 32 x = 8 nGj, x ‰i gvb KZ?
(64) wbGPi ˆKvbwU mwVK?
4. ‰KwU AvqGZi mw®²wnZ evü«¼q 5 ˆm.wg.
1 1 K i I ii L i I iii
‰es 7 ˆm.wg.| evü«¼Gqi A¯¦fzÆÚ ˆKvY KZ K  L
3 3 M ii I iii N i, ii I iii
nGj AvqZwU Aâb mÁ¿e? M 4 N 4 23. ‰KwU mylg lofzGRi cÉwZwU kxlÆGKvY KZ?
K 30 L 45 14. evwlÆK kZKiv 5 UvKv nvi gybvdvq 500 K 60 L 90
M 60 N 90
UvKvi 3 eQGii mij gybvdv KZ UvKv? M 105 N 120
8
5. 
 x2 + 2 = 3x nGj, x3 + 3 ‰i gvb KZ? K 15 L 45 24. sin3A = tan45 nGj, cos2A = KZ?
x
M 60 N 75 1
K 9 L 18 K 0 L
15. 5 ˆm.wg. eÅvmvaÆ ‰es 6 ˆm.wg. eÅvmwewkÓ¡ 2
M 21 N 27
3
wbGPi ZG^Åi AvGjvGK (6 I 7) bs cÉGk²i Dîi `yBwU e†î ciÕ·i A¯¦tÕ·kÆ KiGj, M
2
N 1

`vI: ˆK±`Ê«¼Gqi gaÅeZÆx `ƒiZ½ KZ ˆm.wg.? 25. f(x) = x4  ax2 + 5 ‰es f(1) = 0 nGj,
K 1 L 2
6 + x + y + 162 àGYvîi avivfzÚ| a ‰i gvb KZ?
M 3 N 4
6. avivwUi mvaviY AbycvZ KZ? K 6 L 4
1
16. 
 Õ©ƒjGKvYx wòfzGRi cwiGK±`Ê ˆKv^vq M 1 N 6
K 2 L
3 AewÕ©Z? 26. 
 ‰KwU mgGKvYx wòfzGR KqwU ewne†Æî
M 3 N 6 K wòfzGRi Afů¦Gi AuvKv hvq?
7. (y  x) ‰i gvb ˆKvbwU? L AwZfzGRi Dci K 1 L 2
K 6 L 18 M 3 N 4
M e†nîi evüi Dci
M 36 N 54 27. e†Gîi DcPvGc A¯¦wjÆwLZ ˆKvY Kxi…c?
N wòfzGRi ewnfÆvGM
2sinA
8. 
 A = 30 nGj, ‰i gvb KZ? K mƒßGKvY L Õ©ƒjGKvY
1  sin2A 17. ‰KwU eGMÆi cwimxgv 16 wgUvi nGj, eMÆwUi
M mgGKvY N mij ˆKvY
K 2 2 L
2 KGYÆi Dci AwâZ eGMÆi cwimxgv KZ wgUvi?
3 28. 3 + 3  3 + 3  ... ... ... avivwUi
K 4 2 L 4 3
4 15-Zg c` KZ?
M N 1 M 12 2 N 16 2
3 K 30 L 15
9. ˆKvb kGZÆ a = 1 nq?
0 18. A = {x  ô : x2  5x  6 = 0} ˆmGUi M 3 N 3
K a>0 L a<0 ZvwjKv c«¬wZ ˆKvbwU? 29. wbGPi ˆKvbwU AwewœQ®² PjGKi D`vniY?
M a1 N a0 K {1, 6} L {1, 6} K Qvòx msLÅv L Zvcgvòv
10. 
 1 ˆ^GK 17 ‰i gaÅeZÆx Õ¼vfvweK M {6} N  M RbmsLÅv N cixÞvi bÁ¼i
msLÅvàGjvi gGaÅ 3 ‰i àwYZKmgƒGni 19. 
 ‰KwU NbGKi c†Ó¤ZGji ˆÞòdj 30. 
 ‰KwU ˆkÉwYi DœPmxgv 50 ‰es gaÅgvb
gaÅK KZ? 216 eMÆwgUvi nGj, ‰i AvqZb KZ NbwgUvi? 48 nGj, H ˆkÉwYi wbÁ²mxgv KZ?
K 9 L 12 K 216 L 144 K 41 L 45
M 15 N 18 M 36 N 6 6 M 46 N 49.5

1 M 2 K 3 L 4 N 5 K 6 M 7 M 8 M 9 N 10 K 11 L 12 M 13 L 14 N 15 L
Dîi

16 N 17 N 18 M 19 K 20 L 21 L 22 M 23 N 24 L 25 K 26 M 27 L 28 M 29 L 30 M
12 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
357 KvwbR dvGZgv MvjÆm Õ•zj ‰´£ KGjR, gvwbKMé

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. ˆKvGbv Dcvî mviwYfzÚ KiGZ nGj cÉ^Gg 12. (x2  3)2 = 0 mgxKiGYi KZwU gƒj AvGQ? 22. P

ˆKvbwU wbaÆviY KiGZ nq? K 4 L2 M1 N 3

K cwimi L ˆkÉwY eÅeavb 13. 
 ˆKvGbv eMÆGÞGòi cÉGZÅK evüi Š`NÆÅ O
M
30 M
M
M ˆkÉwY msLÅv N gaÅK 10% e†w«¬ ˆcGj ˆÞòdj kZKiv KZ e†w«¬
Q
wbGPi ZG^Åi AvGjvGK 2 bs cÉGk²i Dîi `vI: cvGe?
wPGò, POQ = KZ?
ˆkÉwYeÅvwµ¦ 41-50 51-60 61-70 71-80 K 10% L 11% M 21% N 44%
1 K 60 L 120
MY msLÅv 7 10 12 11 14. cos2  sin2 = nGj, cos4  sin4
3
‰i M 108 N 150
2. 
 gaÅK ˆkÉwYi wbÁ²mxgv KZ? gvb KZ? 23. Õ©ƒjGKvYx wòfzGRi cwie†Gîi ˆK±`Ê ˆKv^vq
K 71 L 51 M 61 N 70 1
K 3 L2 M1 N AewÕ©Z?
3. 
 cot(  60) = 3 nGj, cos  = KZ? 3
K wòfzGRi Afů¦Gi L wòfzGRi ewnfÆvGM
1 3 15. ˆKvGbv e†Gîi AwaPvGci A¯¦wjÆwLZ ˆKvY@
K 0 L M1 N M jGÁ¼i Dci N AwZfzGRi Dci
2 2 K Õ©ƒjGKvY L mƒßGKvY
4. 0.0035 ‰i mvaviY jGMi cƒYÆK KZ? M mgGKvY N mijGKvY
24. A

K 2̄ L 1 M3 N 3̄ 1 2a
16. a + a = 3 nGj, 3a2  2a + 3 ‰i gvb KZ?
5. kƒbÅ gvòvi mîv ejv nq KvGK?
2 2 108
K ˆiLv L we±`y K L B x d
11 7 C
M ˆKvY N ˆiLvsk 2 2
M  N  O ˆK±`ÊwewkÓ¡ e†Gîi x ‰i gvb KZ?
6. ‰KwU eMÆGÞGòi@ 7 11
K 72 L 126 M 108 N 54
i. NƒYÆb ˆKvY 90 17. A = {x  ô : 3  x  7} nGj, P(A) ‰i
25. wbGPi ˆKvbwU Av`kÆ cÉwZmg wPò?
ii. cÉwZmvgÅ ˆiLvi msLÅv 4 Dcv`vb msLÅv KZ?
K 8 L 16 M 32 N 64 K mgevü wòfzR L eMÆ
iii. NƒYÆb cÉwZmgZvi gvòv 6
18. 
 ‰KwU wòfzGRi `yBwU evüi Š`NÆÅ h^vKÌGg M iÁ¼m N e†î
wbGPi ˆKvbwU mwVK?
K i I ii L i I iii 4 ˆm.wg. I 9 ˆm.wg. nGj, Z‡Zxq evüi Š`NÆÅ 26. 
 mge†îf„wgK wmwj´£vGii f„wgi eÅvmvaÆ 5

M ii I iii N i, ii I iii KZ ˆm.wg.? ˆm.wg. ‰es DœPZv 12 ˆm.wg. nGj,

K 4 L5 M6 N 13 wmwj´£vGii eKÌZGji ˆÞòdj KZ eMÆ
7. 
 `yBwU msLÅvi AbycvZ 5 : 7 ‰es ZvG`i
1 2 3
M.mv.à. 4 nGj, msLÅv `yBwUi j.mv.à. KZ? 19. , , , ... ... ... AbyKÌGgi mvaviY
2 3 4
c` ˆm.wg.?
K 120 L 140 M 35 N 28 K 120 L 60
ˆKvbwU?
8. ˆKvb kGZÆ loga1 = 0 nGe? 1 n M 17 N 24
K L
K a > 0, a  1 L a>1 n n+1 27. mƒGhÆi D®²wZ ˆKvY 90 nGj, ˆKvGbv MvGQi
M a=1 N a>0 1 n1 Qvqvi Š`NÆÅ KZ wgUvi nGe?
M n N
2 n+1
9. a : b = x : y nGj, a : x = b : y K 90 L0 M 30 N 45
mgvbycvGZi ‰B agÆGK eGj@ 20. evwlÆK 10% nvGi 3000 UvKvi 3 eQGii
28. wbGPi ˆKvbwU gƒj` msLÅv?
K AvoàYb L eÅÕ¦KiY
mij gybvdv KZ UvKv?
3 5
K 180 L 600 K L
M ‰Kv¯¦iKiY N ˆhvRb 3 5
M 900 N 1800
10. 
 GKvGbv wòfzGRi `yBwU evüi Š`NÆÅ
21. 
 3x  5y  7 = 0 ‰es 6x  10y  15 = 0
7 9
M N
h^vKÌGg 9 ˆm.wg. I 10 ˆm.wg. ‰es ‰G`i 3 4
mgxKiY ˆRvUwU@
A¯¦fzÆÚ ˆKvY 60| wòfzRwU ˆÞòdj KZ 29. 10 ˆm.wg. eÅvm I 4 ˆm.wg. eÅvmvaÆ wewkÓ¡
i. Amgém
eMÆ ˆm.wg.? `ywU e†î ciÕ·iGK A¯¦tÕ·kÆ KiGj ZvG`i
ii. ciÕ·i AwbfÆikxj
K 38.97 L 22.5 ˆK±`Ê«¼Gqi `ƒiZ½ KZ ˆm.wg. nGe?
iii. ‰i ‰KwU mgvavb AvGQ
M 77.94 N 45 K 1 L9 M 14 N 6
11. 2  2 + 2  2 + ... ... ... avivwUi 17 Zg c` wbGPi ˆKvbwU mwVK?
30. (2x  y, 3) = (6, x  y) nGj, (x, y) = KZ?
K i I ii L i I iii
ˆKvbwU? K (0, 0) L (3, 0)
K 2 L2 M 34 N 34 M ii I iii N i, ii I iii M (0, 3) N (3, 3)

1 K 2 M 3 K 4 N 5 L 6 K 7 L 8 K 9 M 10 K 11 L 12 L 13 M 14 N 15 L
Dîi

16 L 17 M 18 M 19 L 20 M 21 K 22 N 23 L 24 L 25 N 26 K 27 L 28 N 29 K 30 L
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 13
358 mwdDwób miKvi ‰KvGWwg Õ•zj ‰´£ KGjR, MvRxcyi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. (x) = x2  2x + 3 nGj, 2  ‰i gvb
1 11. (21 + 31)1 ‰i gvb KZ? 3
22. sin2 = 2 nGj,  ‰i gvb KZ?
  K
1
L
2
M
5
N
6
KZ? 6 3 6 5 K 30 L 45 M 60 N 90
1 1 + tan2A
K 
7
L
7
M
9
N
11 12.  x = 625 nGj, x ‰i gvb KZ?

5 23. 
4
 tanA = nGj, ‰i gvb
4 4 4 4 3 tan2A
K 4 L 5 KZ?
2. 
 (p + 5,  5) = (5, q  5) hw` nq, ZGe
M 25 N 125 5 4 25 16
(p, q) = KZ? K L M N
13. e†Gîi@ 4 5 16 25
K (10, 10) L (10, 10)
i. eÅvmB e†nîg RÅv 24.  1  1 + 1  1 + 1  1 + ... ... ... àGYvîi

M (0, 0) N (1, 1)
ii. mKj mgvb RÅv ˆK±`Ê ˆ^GK mg`ƒieZÆx avivwUi cÉ^g (2n + 1) msLÅK cG`i mgwÓ¡
3. P = {x, y} ‰es Q = {y, x} nGj, P  Q
iii. ˆK±`Ê ˆ^GK mg`ƒieZÆx mKj RÅv KZ?
ˆKvbwU?
K {} L {0}
M {} N {x, y}
ciÕ·i mgvb K 0 L1 M2 N 4
2 wbGPi ˆKvbwU mwVK? 25. cÉ^g n msLÅK Õ¼vfvweK msLÅvi NGbi
1
4. x  x + 1 = 0 nGj, x +  ‰i gvb
4 2
x K i I ii L i I iii mgwÓ¡ wbGPi ˆKvbwU?
  n2(n + 1)2
M ii I iii N i, ii I iii K Sn =
KZ? 4
K 4 L3 M2 N 1 14. 5, 11, 13, 6, 13, 6, 11, 9, 6 msLÅvàGjvi n2(n + 1)2
gGaÅ cÉPziK ˆKvbwU? L Sn =
5. 
 x = 7 + 4 3 nGj, x = KZ? 8
K 6 L9 M 11 N 13 n(n + 1)(2n + 1)
K 2+ 3 L 2 3 M Sn =
15. 2x+1 = 8 nGj, x ‰i gvb KZ? 6
M 3+ 3 N 3 3 n
K 1 L1 M2 N 3 N Sn = {2a + (n  1)d}
1 6a 2
6. a + a = 5 nGj, a2 + a + 1 = KZ? 16. log2 2 64 ‰i gvb wbGPi ˆKvbwU?
26. ‰KwU e†Gîi eÅvmvaÆ 5 ˆmw´ŸwgUvi ‰es
K 5 L5 M0 N 1 1
K
4
L1 M2 2N 4 ‰KwU e†îPvc ˆKG±`Ê 60 ˆKvY Drc®²
wbGPi ZG^Åi AvGjvGK (7 I 8) bs cÉGk²i Dîi KGi| e†îKjvi ˆÞòdj KZ?
17. 
 log42  log 3 27 = KZ?
`vI: K 13.09 eMÆ ˆmw´ŸwgUvi
2 K 3 L6 M9 N 27
x + 2 = 3x
18. e†Gîi Afů¦iÕ© ‰KwU we±`y ‰es ewntÕ© L 78.54 eMÆ ˆmw´ŸwgUvi
2
7. x + x ‰i gvb KZ? ‰KwU we±`yi msGhvRK ˆiLvsk e†îwUGK M 31.42 eMÆ ˆmw´ŸwgUvi
K 3 L 2 M 3 N 4 KZwU we±`yGZ ˆQ` KGi? N 47124 eMÆ ˆmw´ŸwgUvi
8 K AmsLÅ 27. ‰KwU PvKv 720 wgUvi c^ ˆhGZ 18 evi
8. x + x3 ‰i gvb wbGPi ˆKvbwU?
3 L 1
M 2 N 3 ˆNvGi, PvKvwUi cwiwa KZ?
K 9 L 18 M 21 N 27 K 40 wgUvi L 738 wgUvi
19. 4 ˆmw´ŸwgUvi ‰es 5 ˆmw´ŸwgUvi
9. 
 x2 + 1= 2x nGj@ M 702 wgUvi N 1298 wgUvi
1
eÅvmvaÆwewkÓ¡ `ywU e†î ciÕ·i A¯¦tÕ·kÆ
i. x+ =2
x KiGj ZvG`i ˆK±`Ê«¼Gqi `ƒiZ½ KZ? 28. 35, 40, 42, 50, 56, 42, 50, 64, 42, 35, 40
1 1 K 1 ˆmw´ŸwgUvi L 4 ˆmw´ŸwgUvi bÁ¼iàGjvi Mo KZ?
ii. x + = x2 + 2 K 41.09 L 45.09
x x M 5 ˆmw´ŸwgUvi N 9 ˆmw´ŸwgUvi
1 1 M 49.09 N 50.09
iii. x2 + 2 = x3 + 3 20. mgevü ABC ‰i cwiGK±`Ê O nGj,
x x 29. gaÅK wbYÆGqi mƒò ˆKvbwU?
AOB ‰i gvb KZ?
wbGPi ˆKvbwU mwVK? K
n
gaÅK = L +  2 + Fc  f
h
K 120 L 90
K i I ii L i I iii   m
M 60 N 30 n h
M ii I iii N i, ii I iii 21. wbGPi Z^ÅàGjv jÞÅ Ki: L gaÅK = L +   fm 
1  2  Fc
10. a + a = 5 nGj@ i. e†Gîi ˆhGKvGbv RÅv ‰i jÁ¼w«¼L´£K n h
M gaÅK = L +   Fc 
2 ˆK±`ÊMvgx  2  fm
i. a  1a  = 21 ii. ˆhGKvGbv mijGiLv ‰KwU e†îGK `yGqi n h
  N gaÅK = L    Fc 
ii. a2  5a + 1 = 0 AwaK we±`yGZ ˆQ` KiGZ cvGi bv  2  m f
1 iii. wòfzGRi ˆhGKvGbv `yB evüi Š`GNÆÅi 30. 
 wbGPi ˆKvbwU AwewœQ®² PjK?
iii. a3 + 3 = 25
a A¯¦i Z‡Zxq evüi ˆPGq Þz`ËZi K ˆkÉwYi gaÅgvb
wbGPi ˆKvbwU mwVK? wbGPi ˆKvbwU mwVK? L ˆkÉwYi MYmsLÅv
K i I ii L i I iii K i I ii L i I iii M ˆkÉwYmsLÅv
M ii I iii N i, ii I iii M ii I iii N i, ii I iii N KÌgGhvwRZ MYmsLÅv

1 M 2 M 3 K 4 L 5 K 6 N 7 M 8 K 9 N 10 K 11 N 12 K 13 N 14 K 15 M
Dîi

16 N 17 K 18 L 19 K 20 K 21 N 22 K 23 K 24 L 25 K 26 K 27 K 28 L 29 M 30 K

14\\E:\Data 2024-25\SSC Made Easy 2025\General Math\Link\mthmto25.doc 2nd proof 14.11.24

14 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
359 Uãx cvBjU Õ•zj ‰´£ MvjÆm KGjR, MvRxcyi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. gƒj` msLÅvi ˆÞGò@ 9. 
 H ˆkÉwYi ˆeç msLÅv KZ? 18. 
 sin.cos = KZ?
i. cÉGZÅK cƒYÆ msLÅvB gƒj` msLÅv K 60 L 15 M 12 N 18 4 3 4 1
K L M N
ii. a I b `yBwU gƒj` msLÅv nGj a + b, a  b wbGPi ZG^Åi AvGjvGK (10 I 11) bs cÉGk²i 3 4 3 3
‰es ab gƒj` msLÅv Dîi `vI: 19. Þz`Z
Ë g ˆgŒwjK msLÅv wbGPi ˆKvbwU?
A
a K 1 L2 M3 N 4
iii.
b
gƒj` msLÅv, hLb b  0 ‰es a, b cƒYÆ 20. 18 wgUvi jÁ¼v ‰KwU gB ‰KwU ˆ`qvGji
70
msLÅv Qv` eivei ˆVm w`Gq f„wgi mvG^ 30 ˆKvY
wbGPi ˆKvbwU mwVK? D
E
F
Drc®² KGi| ˆ`qvGji DœPZv KZ wgUvi?
K i I ii L i I iii
40
K 9 2 L 9
B C
M ii I iii N i, ii I iii M 9 3 N 18
2. A I B `yBwU ˆmGUi RbÅ@ DE || BC ‰es BD || CF 21. 1 Nb ˆm.wg.
KvGVi IRb 7
ˆWwm MÉvg|
i. A  B ‰i cÉwZwU m`mÅ KÌgGRvo bq
10. BDE = KZ? KvGVi IRb mgAvqZb cvwbi IRGbi
K 70 L 135 M 100 N 110 kZKiv KZ?
ii. n(A) = a I n(B) = b nGj n(A  B) = ab
11. ABC + ACB = KZ? K 80% L 50%
iii. A  B = {(x, y) : x  A ‰es y  B} K 70 L 135 M 60% N 70%
wbGPi ˆKvbwU mwVK? M 100 N 110 22. 
 wbGPi ˆKvb mgxKiY ˆRvUwUi mgvavb ˆbB?
K i I ii L i I iii 12. ÷ay ‰KwU evüi Š`NÆÅ ˆ`Iqv ^vKGj AuvKv K 3x + 6y = 2 L 2x + y = 13
M ii I iii N i, ii I iii hvq@ 3x + 4y = 2 4x + 2y = 5
3. x+
1
=2
1
nGj, x + x ‰i gvb KZ? i. eMÆGÞò M 3x  5y = 7 N x  2y = 0
x ii. iÁ¼m 6x  10y = 14 2x  4y = 0
K 2 L 2 2 iii. mgevü wòfzR 23. 1 + 3 + 5 + 7 + ... ... ... avivwUi n Zg c`
M 4 N 2 3 wbGPi ˆKvbwU mwVK? KZ nGe?
1 1 K i I ii L i I iii K 2n  1 L 2n + 1
4. 
 2x 
3x
=5 nGj, 8x 3
27x3
‰i gvb M n2 N n2 + 1
M ii I iii N i, ii I iii
KZ? 13. RÅvwgwZK Dccv`Å cÉgvGY mvaviYZ KqwU 24. 4  4 + 4  4 + ... ... ... àGYvîi avivi 2n
K 25 L 135 M 145 N 110
avc ^vGK? msLÅK cG`i mgwÓ¡ KZ?
5. 
 (a) = a3 + 3a + 36 ‰es (a + 3), K 4 L 3 K 0 L 4n2 M n2 N 4
(a) ‰KwU Drcv`K nGj@ M 2 N 1 25. mylg lofzGRi KqwU cÉwZmvgÅ ˆiLv nGqGQ?
i. (3) = 0 nGe 14.  128 + 64 + 32 + ... ... ... avivwUi
 KZ K 0 L3 M 12 N 6

ii. (a  4), (a) ‰i ‰KwU Drcv`K nGe 1 26. 4 + a + b + 32 + ... ... ... avivwUi mvaviY
Zg c` 2|
iii. a2  3a + 12, (a) ‰i ‰KwU Drcv`K
AbycvZ@
K 9 Zg L 8 Zg K 1 L2 M3 N 4
nGe
M 7 Zg N 6 Zg 27. 
 ‰KwU iÁ¼Gmi ˆÞòdj 1944 eMÆ ‰KK|
wbGPi ˆKvbwU mwVK?
15.
3
nGj, tan ‰i gvb KZ? ‰i ‰KwU KYÆ 54 ‰KK nGj, Aci KGYÆi
K i I ii L i I iii sin =
2 Š`NÆÅ KZ ‰KK?
M ii I iii N i, ii I iii 3 1 K 48 L 54 M 72 N 96
K 3 3 L 3 M N
6. ‰KwU `ËeÅ 20% ÞwZGZ weKÌq Kiv nGjv, 7 3 28. AvqZGjL AâGbi RbÅ ˆkÉwYeÅvwµ¦ ˆKgb
KÌqgƒjÅ I weKÌqgƒGjÅi AbycvZ ˆKvbwU? 16.
5
cosec + cot = nGj, nGZ nGe?
6
K 4:5 L 5:4 K wewœQ®² L AwewœQ®²
M 5:6 N 6:5 cosec  cot = KZ?
1 5 6 M abvñK cƒYÆmsLÅv N cƒYÆmsLÅv
7. wbGPi ˆKvbwU msLÅvi ŠeævwbK i…c? K
6
L
6
M1 N
5 wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
K a + 10n L a  10n wbGPi ZG^Åi AvGjvGK (17 I 18) bs cÉGk²i Dîi `vI:
M a  10n N + 10n
Dîi `vI: ˆkÉwY 11-20 21-30 31-40 41-50
wbGPi ZG^Åi AvGjvGK (8 I 9) bs cÉGk²i Dîi A

`vI: 2
MYmsLÅv 4 16 20 25
29. cÉPziK ˆkÉwY ˆKvbwU@
‰KwU ˆkÉwYi cÉwZ ˆeGç 4 Rb KGi emGj 3wU
B  K 11-20 L 21-30
ˆeç Lvwj ^vGK| Avevi cÉwZ ˆeGç 3 Rb KGi 3
C
M 31- 40 N 41-50
emGj 6 Rb QvòGK `uvwoGq ^vKGZ nq| 17. tan = KZ? 30. cÉPziK ˆkÉwYi gaÅgvb ˆKvbwU?
8.   H ˆkÉwYi Qvò msLÅv KZ?
K 3 L
1
M
2
N
1 K 25.5 L 15.5
K 60 L 15 M 12 N 18 2 3 3 M 45.5 N 35.5

1 N 2 M 3 K 4 L 5 L 6 L 7 M 8 K 9 N 10 N 11 N 12 L 13 K 14 K 15 L
Dîi

16 N 17 N 18 L 19 L 20 L 21 N 22 L 23 K 24 K 25 N 26 L 27 M 28 L 29 N 30 M
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 15
360 BKevj wmwóKx Õ•zj AÅv´£ KGjR, MvRxcyi


E:\Data 2024-25\SSC Made Easy 2025\General Math\Link\mthmto25.doc 2nd proof 23.10.24

16 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
361 bvivqYMé miKvwi evwjKv DœP we`Åvjq, bvivqYMé

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. evÕ¦e msLÅvi eMÆ meÆ`vB ˆKvb aiGbi 10. ˆQvU msLÅvi 5 àY ˆ^GK msLÅv«¼Gqi 19. evwowUi DœPZv KZ wgUvi?
msLÅv? mgwÓ¡i A¯¦i wbGPi ˆKvbwU? K 2.31 L 4
K 100 L 105 M 6.93 N 8
K Õ¼vfvweK L ˆgŒwjK
M evÕ¦e N cƒYÆ
M 110 N 120 20. f„wg ˆ^GK AebwZ ˆKvY Drc®²Kvwi we±`yi
.. . 11. `ywU mgv¯¦ivj mijGiLvGK Aci ‰KwU `ƒiZ½ KZ wgUvi?
2. 5.12  3.45 = KZ? ˆiLv ˆQ` KiGj ‰es A¯¦tÕ© ˆKvY«¼Gqi K 14.93 L 9.93
.. .
‰KwU ˆKvY 135 nGj Aci ˆKvGYi M 2.12 N 1.42
K 1.665 L 1.665
.. . wecÉZxc ˆKvY KZ wWMÉx? 21. 10000 UvKv 3 : 4 : 5 : 6 : 7 AbycvGZ fvM
M 1. 67 N 1.68 KiGj e†nîi I Þz`ËZi AsGki cv^ÆKÅ KZ
K 40 L 45 M 50 N 55
3. B ˆmGUi cÉK‡Z DcGmU A nGj- UvKv nGe?
12. ‰KwU wòfzGRi wZbwU ˆKvGYi AbycvZ
i. A  B = A K 1600 L 800 M 400 N 200
1 : 1 : 2| wòfzRwU ˆKvb aiGbi?
ii. A  B = B
K mgevü L mgw«¼evü 22. 
 ˆKvGbv e†Gîi eÅvmvaÆ x nGZ (x + y) Kiv
iii. A  B = 
M mgGKvYx mgw«¼evü N welgevü nGj, e†nîg RÅv ‰i Š`NÆÅ Kx cwigvY e†w«¬
wbGPi ˆKvbwU mwVK?
13. mvgv¯¦wiGKi `ywU mw®²wnZ ˆKvGYi cvGe?
K i I ii L i I iii
K 2x L 2x + y
M ii I iii N i, ii I iii mgw«¼L´£GKi A¯¦fzÆÚ ˆKvY KZ wWMÉx?
M 2y N x + 2y
K 60 L 70 M 80 N 90
4. A = {x  ô : x  1 + 2 = 1} ˆmUwUi 23. `ywU abvñK cƒYÆmsLÅvi eGMÆi A¯¦i 3 ‰es
14. ‰KwU mgGKvYx mgw«¼evü wòfzGRi mgvb
ZvwjKvi…c wbGPi ˆKvbwU? àYdj 2| ‰G`i eGMÆi mgwÓ¡ KZ?
K (1) L {0}
evü«¼Gqi Š`NÆÅ 18 ˆm.wg. nGj, wòfzRwUi
K 2 L3 M4 N 5
M {} N {2} ˆÞòdj KZ eMÆ ˆm.wg.?
K 36 L 81 M 162 N 324
24. 
 3 + 12 + 48 + ... + 768 àGYvîi
5. a3 + b3 = 2, a2  ab + b2 = 4 nGj- avivwUGZ KqwU c` iGqGQ?
15. 1 ‰KK eÅvmvaÆwewkÓ¡ e†Gîi cwiwa I
(a + b)2 ‰i gvb wbGPi ˆKvbwU? K 4 L5 M6 N 8
1 1 1 ˆÞòdGji A¯¦i wbGPi ˆKvbwU? 1 2 3
K 2 L
4
M
5
N
8  3 25. 2 , 3 , 4 , ..... AbyKÌgwUi mvaviY c` wbGPi
K L M N 2
2 2
6. 
 4a + 12a + 7a  3a  2 ‰i
4 3 2
ˆKvbwU?
16. ‰KwU e†Gîi-
Drcv`K- 1 n1 1 n
i. NƒYÆb ˆK±`Ê `ywU eÅvGmi ˆQ`we±`yGZ K L M n N
i. a + 2 ii. 2a  1 n n+1 2 n+1
iii. 2a + 1 AewÕ©Z wbGPi ZG^Åi AvGjvGK (26 I 27) bs cÉGk²i
wbGPi ˆKvbwU mwVK? ii. NƒYÆb cÉwZmgZvi gvòv Amxg Dîi `vI:
K i I ii L i I iii iii. ˆK±`ÊMvgx ˆhGKvGbv ˆiLvB ‰i cÉwZmvgÅ ABC-‰, B = 90, AB = BC, AC = 3 2
M ii I iii N i, ii I iii ˆiLv ‰es AC I BC ‰i gaÅwe±`y h^vKÌGg D I E|
7. 
 0.000289 msLÅvwUi ŠeævwbK i…c wbGPi wbGPi ˆKvbwU mwVK? 26. 
 AB ‰i Š`NÆÅ KZ?
ˆKvbwU? K i I ii L i I iii K 12 L9 M6 N 3
K 0.0289  10 4
L 0.289  10 4 M ii I iii N i, ii I iii 27. 
 DE ‰i Š`NÆÅ KZ?
M 2.89  104 N 28.9  104 17. sin3A = cos3A nGj, A ‰i gvb wbGPi K 1.5 L2
8. 
 x = log232, y = log5625 nGj (x + y) ˆKvbwU? M 2.5 N 3.5

‰i gvb wbGPi ˆKvbwU? K 15 L 20 M 25 N 45 28. 

 PQRS iÁ¼Gmi `ywU KGYÆi AGaÆK
K 7 L 9 1  tan260 h^vKÌGg 5 ˆm.wg. I 12 ˆm.wg. nGj, PQ
18. 

1 + sin260
+ 2sin260 ‰i gvb wbGPi
M 10 N 12 ‰i Š`NÆÅ KZ ˆm.wg.?
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i ˆKvbwU? K 11 L 12
3 5 9 13 M 13 N 14
Dîi `vI: K
14
L
14
M
14
N
14
2 29. 7, 15, 6, 9, 7, 11, 8, 14 DcvîàGjvi gaÅK
‰KwU msLÅv Aci msLÅvi 3 àY| msLÅv `ywUi wbGPi ZG^Åi AvGjvGK (19 I 20) bs cÉGk²i KZ?
mgwÓ¡ 100 Dîi `vI: K 10.5 L 8.5 M 7.5 N 6.5
9. eo msLÅvwU ˆQvU msLÅvwU AGcÞv KZ ‰KwU evwoi ˆ`qvj ˆ^GK f„ZjÕ© 4 wgUvi 30. wZbwU msLÅvi Mo 20 ‰es Aci 5wU
ˆewk? `ƒGii ‰KwU we±`yi D®²wZ ˆKvY 60 ‰es Qv` msLÅvi Mo 24 nGj, 8wU msLÅvi Mo wbGPi
K 15 L 20 ˆ^GK 3 wgUvi DcGii ˆKvGbv we±`yi AebwZ ˆKvbwU?
M 25 N 30 ˆKvY 45| K 21.5 L 22 M 22.5 N 23

1 M 2 K 3 N 4 M 5 L 6 N 7 M 8 L 9 L 10 K 11 L 12 M 13 N 14 M 15 L
Dîi

16 N 17 K 18 L 19 M 20 L 21 K 22 M 23 N 24 L 25 N 26 N 27 K 28 M 29 L 30 M
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 17
362 we±`yevwmbx miKvwi evjK DœP we`Åvjq, UvãvBj

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. a, b, c evÕ¦e msLÅv; a < b ‰es c > 0 nGj, z2 1
11. z  1 = 2  z  1 ‰i mgvavb ˆmU wbGPi 21. ˆKvGbv eMÆGÞGòi ˆÞòdj Zvi KGYÆi Dci
wbGPi ˆKvbwU mZÅ? AwâZ eMÆGÞGòi ˆÞòdGji@
K ac = bc L ac > bc ˆKvbwU? K AGaÆK L mgvb
M ac < bc N ac e bc K {1} L {0} M w«¼àY N PviàY
2. ˆmU C nGZ ˆmU B-‰ ‰KwU mÁ·KÆ R M {} N {2}
wbGPi ZG^Åi AvGjvGK (22 I 23) bs cÉGk²i
nGj, wbGPi ˆKvbwU mwVK? 12. 20 ˆKvGYi mÁ·ƒiK ˆKvGYi AGaÆK KZ? Dîi `vI:
B
K RC L RB K 35 L 70
M RCB N CBR M 80 N 160 60
C A
3. A = {a, b, c, d} nGj, A ‰i cÉK‡Z DcGmU 13. wbGPi ˆKvb ˆÞòwU mvgv¯¦wiK bq? O

KqwU? K AvqZGÞò L iÁ¼m

K 14 L 15
M eMÆGÞò N UÇvwcwRqvg wPGò e†Gîi ˆK±`Ê O ‰es AC = 12 cm
M 16 N 4
14. e†Gîi A¯¦wjÆwLZ ABCD PZzfzÆGRi 22. 
 AB PvGci Š`NÆÅ KZ cm?
4. 
 (x) = x3 + px2  6x  9 nGj, P ‰i
B = 60 nGj, D = KZ? K 4084 L 12.57
ˆKvb gvGbi RbÅ f(3) = 0 nGe? M 6.28 N 3.14
K 4 L 2 K 40 L 90
23. 
 AOB e†îKjvi ˆÞòdj KZ cm2?
M 2 N 6 M 120 N 110
K 150.80 L 75.40
wbGPi ZG^Åi AvGjvGK (5 I 6) bs cÉGk²i Dîi 15. 
 `yBwU e†Gî mGeÆvœP KqwU Õ·kÆK AuvKv M 40.84 N 18.85
`vI: hvq? 24. ˆKw±`Êq cÉeYZvi cwigvc nGjv@
a2  3a + 1 = 0; ˆhLvGb, a > 1 K 1 L 2 i. Mo
1 M 3 N 4 ii. AvqZGjL
5. 
 a2 + 2 ‰i gvb wbGPi ˆKvbwU?
a 16. 
 sec(90  ) = 2 nGj, tan ‰i gvb iii. gaÅK
K 6 L 7
wbGPi ˆKvbwU? wbGPi ˆKvbwU mwVK?
M 9 N 10
1 K i I ii L i I iii
1 K L 1
6. 
 a2  2 ‰i gvb wbGPi ˆKvbwU?
a 3 M ii I iii N i, ii I iii
K 45 L 40 M 3 N AmãvwqZ 25. wbGPi ˆKvbwU wewœQ®² PjK?
M 3 5 N 3 3
17.
1
cos   sin  = nGj, cos4  sin4
2 2
‰i K Zvcgvòv L RbmsLÅv
3
7. y UvKvi y% nvi mij gybvdvq 4 eQGii M eqm N DœPZv
gybvdv y UvKv nGj, y ‰i gvb wbGPi gvb KZ? 26. 
 wbGPi mviwYi cÉPziK KZ?
K 3 L 2
ˆKvbwU? ˆkÉwYeÅvwµ¦ 21-23 24-26 27-29 30-32
1
K 25 L 50 M 1 N MYmsLÅv 3 5 7 5
3
M 75 N 100 K 25.5 L 27.5
18. 
 18 wg. jÁ¼v ‰KwU gB ˆ`qvGji Qv`
8. wbw`ÆÓ¡ PZzfzÆR AuvKvi RbÅ KZwU Õ¼Z¯¨ M 28 N 28.5
DcvGîi cÉGqvRb? eivei ˆVm w`Gq f„wgi mvG^ 30 ˆKvY
27. logaa = 1 wbGPi ˆKvb kGZÆ?
K 2 L 3 Drc®² KGiGQ| ˆ`qvGji DœPZv KZ wgUvi? K a>0 L a1
M 4 N 5 K 9 L 12.72 M a > 0, a  1 N a  0, a > 1
9. e†Gîi@ M 36 N 9.72 28. log5p2 = 2 nGj, P ‰i gvb KZ?
i. mgvb mgvb RÅv ˆK±`Ê nGZ mg`ƒieZÆx 19. 
 `yB AsK wewkÓ¡ ˆKvb msLÅvi AsK«¼Gqi 1
K 5 L
5
ii. AaÆe†îÕ© ˆKvY 180 mgwÓ¡ 13 ‰es àYdj 40 nGj, AsK«¼q
M 2 N 10
iii. AwaPvGc A¯¦wÆ jwLZ ˆKvY mƒßGKvY KZ?
29. 0.0037 ‰i mvaviY jGMi cƒYÆK KZ?
wbGPi ˆKvbwU mwVK? K 7, 6 L 9, 4
K 3 L 2
K i I ii L i I iii M 5, 8 N 3, 10
M 3 N 2
M ii I iii N i, ii I iii 4 4
20. 
 12 + 4  + .... avivwUi ˆKvb c`  ?
3 27
30. 
 26 cm eÅvm wewkÓ¡ e†Gîi ˆK±`Ê ˆ^GK
10. (x  4)2 = 0 mgxKiGYi gƒj KqwU? 5 cm `ƒGi AewÕ©Z RÅv ‰i Š`NÆÅ KZ cm?
K 4 Zg L 5 Zg
K 1 L 2 K 12 L 18
M 3 N 4 M 6 Zg N 7 Zg M 20 N 24

1 M 2 M 3 L 4 M 5 L 6 M 7 K 8 N 9 L 10 L 11 K 12 M 13 N 14 M 15 N
Dîi

16 L 17 N 18 K 19 M 20 L 21 K 22 M 23 N 24 L 25 L 26 N 27 M 28 L 29 M 30 N
18 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
363 NvUvBj KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, UvãvBj

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. f(x) = x3  6x + 3 nGj, f(3) = KZ? 11. 
 mgevü wòfzGRi ‰KwU evüGK Dfqw`GK 21. 
 eMÆGÞGòi ‰K evüi cwigvc x ‰KK
K 6 L  12 ewaÆZ KiGj ˆh ewntÕ© ˆKvY«¼q Drc®² nq nGj, Dnvi cwimxgv I KGYÆi Š`GNÆÅi AbycvZ
M 18 N 42 ZvG`i mgwÓ¡ KZ? KZ?
1
2. x + = 5 nGj
x
K 120 L 180 K 2 2:4 L 2 2:3
M 240 N 270 M 2 2:2 N 2 2:1
1
i. x2  2 = 5 21 12. 
 1 + tan2 = 4 nGj,  ‰i gvb KZ?
x 22. 1 + 4 + 16 + .......... avivi ˆKvb c` 1024?
1 K 0 L 30 K 5g L 6Ó¤ M 10g N 11Zg
ii. x3 + 3 = 110
x M 45 N 60 23. ABC mgGKvYx wòfzR nGe, hw` ‰i evüàGjvi
iii. x2  5x + 1 = 0 wbGPi ZG^Åi AvGjvGK A
wbGPi ˆKvbwU mwVK? x cwigvc nq
(13 I 14) bs cÉGk²i i. 5, 12, 13 ‰KK
K i I ii L i I iii Dîi `vI:
O
ii. 6, 8, 10 ‰KK
M ii I iii N i, ii I iii wPGò O e†Gîi ˆK±`Ê| B
x + 60
C
3. 
 (x  2y, 3x + 2y) = (1, 19) nGj, (x, y) iii. 14, 16, 20 ‰KK
13. BAC = KZ? wbGPi ˆKvbwU mwVK?
‰i gvb ˆKvbwU?
K 30 L 45 K i I ii L i I iii
K (5, 2) L (2, 5) M (9, 4) N (4, 9)
M 60 N 120 M ii I iii N i, ii I iii
4. ˆhGKvGbv abvñK msLÅvi ŠeævwbK i…c a  10n
14. cÉe†«¬ ˆKvY BOC ‰i gvb KZ? 24. ˆKvbwU wewœQ®² PjK?
‰i ˆÞGò a ‰i mxgv wbGPi ˆKvbwU?
K 120 L 180
K 1 < a < 10 L 1 < a  10 K Zvcgvòv L cvwLi msLÅv
M 240 N 280
M 1  a < 10 N 1  a  10 M eqm N DœPZv
15. a = 5 , b = 3 nGj (a + b)2  2ab ‰i
5.   8  3 + 2 + 7 + ... ... ... avivwUi
 wbGPi ZG^Åi AvGjvGK (25 I 26) bs cÉGk²i
gvb KZ?
15 Zg c` KZ? Dîi `vI: C
K 62 L 67 K 2 L 15 2x
M 78 N 83 M 2 15 N 8 4 cm

1 16. log2 2 64 ‰i gvb KZ? x

6. 

25x
= 125 nGj, x ‰i gvb KZ? B
AB = 4 ˆm.wg.
A
K 2 L 3
3 2
K 3 L M1 N M 4 N 8 25. ACB ˆKvGYi gvb KZ?
2 3
17. 1 + 3 + 5 + 7 + ............. avivi cÉ^g n K 15 L 30 M 45 N 60
wbGPi ZG^Åi AvGjvGK (7 I 8) bs cÉGk²i Dîi 26. AC = KZ?
msLÅK cG`i mgwÓ¡ KZ?
`vI: n(n + 1) 2 K 2 2 ˆm.wg. L 2 3 ˆm.wg.
50 Rb wkÞv^Æxi MwYZ welGqi cÉvµ¦ bÁ¼Gii K n2 L 
 2 
  M 4 2 ˆm.wg. N 4 3 ˆm.wg.
MYmsLÅv wbGekb mviwY wbGÁ²i…c: n(n + 1) n2 27. `yBwU e†Gî mGeÆvœP KqwU mvaviY Õ·kÆK
M N
ˆkÉwYeÅvwµ¦ 41-50 51-60 61-70 71-80 81-90 91-100 2 2
AuvKv hvq?
wkÞv^Æxi 6 7 10 12 8 7 wbGPi ZG^Åi AvGjvGK (18 I 19)bs cÉGk²i
K 1wU L 2wU M 3wU N 4wU
msLÅv Dîi `vI: C 28. 
 tan(  30) = 3 nGj sin ‰i gvb KZ?
7. 
 gaÅK wbYÆGq Fc ‰i gvb wbGPi ˆKvbwU? O ˆK±`ÊwewkÓ¡ ACBD e†Gî B 1 3
K 12 L 23 K 0 L M N1
AB = 10 ˆm.wg., OE  AD, O 10
2 2
M 35 N 43 3
OE = 3 ˆm.wg., AC  BC. A D 29. `yBwU e†î ciÕ·iGK ewntÕ·kÆ KGi| ‰G`i
8. 
 cÉPziK wbGPi ˆKvbwU? E

K 74.33 L 77.67 ‰KwUi eÅvm 10 ˆm.wg. ‰es AciwUi

M 81.33 N 86.67
18. 
 OAE ‰i ˆÞòdj KZ?
eÅvmvaÆ 4 ˆm.wg.| e†î«¼Gqi ˆKG±`Êi gaÅeZÆx
K 6 eMÆ ˆm.wg. L 12 eMÆ ˆm.wg. `ƒiZ½ KZ?
9. A = (a, b, c) ‰es B = (a, b) nGj
i. B  (AB) = A  B M 15 eMÆ ˆm.wg. N 20 eMÆ ˆm.wg. K 1 ˆm.wg. L 6 ˆm.wg.
ii. (A  B)  B = A 19. 
 ACB ‰i cwimxgv KZ?
M 9 ˆm.wg. N 14 ˆm.wg.
iii. A = A  (A  B) K 15.14 ˆm.wg. (cÉvq) 30. mge†îf„wgK ˆejGbi f„wgi eÅvmvaÆ 3 ˆm.wg.
wbGPi ˆKvbwU mwVK? L 20.14 ˆm.wg. (cÉvq) ‰es DœPZv 7 ˆm.wg. ‰i eKÌZGji ˆÞòdj
K i I ii L i I iii M 24.14 ˆm.wg. (cÉvq) KZ?
M ii I iii N i, ii I iii N 30.14 ˆm.wg. (cÉvq) K 131.95 eMÆ ˆm.wg.
1 a b 2
10. sec2A  1
= KZ? 20. 
 = = nGj, a : c ‰i gvb KZ?
b c 3
L 188.50 eMÆ ˆm.wg.
K cotA L tanA K 2:3 L 3:4 M 197.95 eMÆ ˆm.wg.
M cosA N sinA M 4:9 N 9:4 N 395.84 eMÆ ˆm.wg.

1 K 2 N 3 K 4 M 5 K 6 L 7 L 8 K 9 M 10 K 11 M 12 N 13 M 14 M 15 N
Dîi

16 M 17 K 18 K 19 M 20 M 21 N 22 L 23 K 24 L 25 N 26 M 27 N 28 N 29 M 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 19
364 Avéygvb Av`kÆ miKvwi DœP we`Åvjq, ˆbòGKvYv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 wbGPi ˆKvbwU gyj` msLÅv bq? 11. 
5
 hw` sec + tan = nq, ZGe 21. A
2
K 0.4̇ L 9
M 5. 6̇39̇ N 11
sec  tan ‰i gvb KZ? B

5 3 D
2. {x  ô : x ˆgŒwjK msLÅv ‰es x ≤ 5) K
2
L
5
C

ˆmUwUi ZvwjKv c«¬wZ wbGPi ˆKvbwU? ADC : BDC = ?

5 2 i. AD : BD ii. CD : CD
K {1, 3, 5} L {5, 7, 11} M N
3 5 iii. AD2 : CD2
M {2, 3, 5} N {3, 5, 7} 12. A = 15 nGj wbGPi ˆKvbwU mwVK?
1 y K i L iii
3.  y + = 5 nGj, 2
 = KZ? i. tan 3A = 2 sin 3A
y y - 3y + 1 M i I iii N ii I iii
1 1 1
K L ii. cot 4A =
3
22. ‰KwU eGMÆi cwimxgv 16 wgUvi nGj ‰i
8 2
KYÆ KZ wgUvi?
M 2 N 8 iii. sin 4A = cos 2A
K 4 2 L 4 3
1 1 wbGPi ˆKvbwU mwVK?
4. x + = 2 nGj, x3 + 3 = ? M 8 2 N 8 3
x x K i I ii L i I iii 23. 5 ˆm.wg. avi wewkÓ¡ NbGKi KGYÆi Š`NÆÅ
K 3 L 2 M ii I iii N i, ii I iii KZ ˆm.wg.?
M 1 N 0
13. AebwZ ˆKvGYi gvb KZ wWMÉx nGj, K 3.87 L 7.07
5.  wòfzGRi
 wZbwU evüi Š`NÆÅ ˆm.wg. ‰KGK M 8.66 N 15.03
1wU Lywu Ui Š`NÆÅ I Qvqvi Š`NÆÅ mgvb nGe?
ˆ`Iqv nGjv| wbGPi ˆKvb ˆÞGò wòfzR 24. wbGPi ˆKvbwU AwewœQ®² PjK?
K 30 L 45
AsKb Kiv hvq? K ˆkÉwYi gaÅgvb
M 60 N 90 L ˆkÉwYi MYmsLÅv
K 5, 6, 18 L 6, 7, 19
M 7, 8, 17 N 9, 6, 13
14. 3 : 8 :: y : 32 nGj, y ‰i gvb KZ? M ˆkÉwY msLÅv
K 3 L 12 N KÌgGhvwRZ MYmsLÅv
6. M 4cm Q M 24 N 48 25. 10g ˆkÉwYi wkÞv^ÆxG`i MwYGZ meÆwbÁ² bÁ¼i
3cm 15. ax  cy = 0, ay  cx = a2  c2 mgxKiYwUi 35 I cwimi 56 nGj mGeÆvœP bÁ¼i KZ?
mgvavb ˆKvbwU? K 80 L 85
N P M 96 N 90
K (a, c) L (2, y)
‰i cwimxgv KZ cm? 26. KÌgGhvwRZ MYmsLÅv cÉGqvRb
MNPQ M (c, a) N ( c,  a)
i. cÉPziK wbYÆGq
K 7 L 8 wbGPi ZG^Åi AvGjvGK (16 I 17) bs cÉGk²i
M 12 N 14 ii. gaÅK wbYÆGq
Dîi `vI: iii. AwRf ˆiLv wbYÆGq
wbGPi ZG^Åi AvGjvGK (7 I 8) bs cÉGk²i Dîi
2 + 1 + 4 + 7 +  ‰KwU aviv| wbGPi ˆKvbwU mwVK?
`vI:
E 16. avivwUi mvaviY c` KZ? K i I ii L i I iii
A
K 3n  1 L 3n + 1 M ii I iii N i, ii I iii
4cm M 3n  5 N 3n + 5
27. ˆkÉwY webÅÕ¦ DcvGîi cÉPziK wbYÆGqi mƒò :
f1 f1
17. 
 avivwUi 1g `k cG`i mgwÓ¡ KZ? K L+ h L L h
60 50 (f1 + f2) (f1  f2)
B
6cm C
D K 115 L 125 f1 f1
M L+ h N L h
M 145 N 155 (f1  f2) (f1 + f2)
7. BA||CE nGj, ACE ‰i gvb KZ? 18. ABC I DEF m`†k ‰es 28. e†Gîi eÅvm I cwiwai AbycvZ KZ?
K 50 L 60 K :1 L 1:
AB : DE = 2 : 3 nGj, DEF : ABC = KZ?
M 70 N 110 M 2: N :2
K 4:9 L 9:4 29. avivwUi 1g n msLÅK
8. ABC ‰i ˆÞòdj KZ eMÆ ˆm.wg.? 1 + 3 + 5 +…..
M 2:3 N 3:2 cG`i ˆhvMdj ˆKvbwU?
K 6 3 L 12
19. ‰KwU AvqZGÞGòi NƒbÆb ˆKvGYi cwigvY n(n + 1) n(n + 1)
2
M 24 N 12 3 K L 
 2 
KZ? 2  
9. `yBwU KGYÆi Š`NÆÅ ˆ`Iqv ^vKGj wbGPi
K 360 L 180 n2
ˆKvbwU AuvKv mÁ¿e? M
2
N n2
M 90 N 45
K UÇvwcwRqvg L iÁ¼m 3
20. ‰KwU mgevü wòfzGRi cÉGZÅK evüi Š`NÆÅ 30. sin = 2 nGj, tan ‰i gvb KZ?
M AvqZ N mvgv¯¦wiK
10 ˆm.wg. nGj ‰i gaÅgvi Š`NÆÅ KZ ˆm.wg.? K L 3 3
10. eGMÆ A¯¦wjÆwLZ e†Gîi Õ·kÆGKi msLÅv KZ? 3
K 1 L 2 K 5 3 L 3 5 3 1
M N
M 3 N 4 M 75 N 125 7 3

1 N 2 M 3 L 4 L 5 N 6 N 7 M 8 K 9 L 10 N 11 N 12 N 13 L 14 L 15 M
Dîi

16 M 17 K 18 L 19 L 20 K 21 K 22 K 23 M 24 K 25 N 26 M 27 K 28 L 29 N 30 K
20 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
365 Rvgvjcyi miKvwi evwjKv DœP we`Åvjq, Rvgvjcyi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 ˆKvb ˆmGUi kwÚ ˆmGUi Dcv`vb msLÅv ii. (x) = 1 +
1 1
nGj, x  = x + 1 20. ‰KwU eGMÆ A¯¦wjÆwLZ e†Gîi KqwU Õ·kÆK
64 nGj, H ˆmGUi Dcv`vb msLÅv KZ? x   AvGQ?
K 5 L6 M7 N 8 iii. mKj A®¼qB dvskb K 1 L2 M3 N 4
2. A = {x  ô : x, 42 ‰i àYbxqK} nGj, wbGPi ˆKvbwU mwVK? 21. 
 A D
i. A = {1, 2, 3, 6, 7, 14, 21, 42} K i I ii L i I iii
ii. B = {x  ô : x, 21 ‰i àYbxqK} M ii I iii N i, ii I iii 40 15 40 15
6 6
A ‰i ‰KwU cÉK‡Z DcGmU 11. 
 4x + y = 2 ‰es 2x + 3y = 4 mgxKiY
iii. A/B =  ˆRvGUi mgvavb (x, y) ˆKvbwU? 70 70
wbGPi ˆKvbwU mwVK? B 70 70 C E F
K (2, 1) L (1, 2) 4 10
K i I ii L i I iii M (1, 3) N (1, 2)
M ii I iii N i, ii I iii wbGPi ZG^Åi AvGjvGK (12 I 14) bs cÉGk²i DcGii wPGò ABC ‰i ˆÞòdj: DEF
2
Dîi `vI: ‰i ˆÞòdj = KZ?
3. p + p1  = 3 nGj,
1
p3 + 3 KZ?
p log3 + log9 + log27 + ... ... ...
K 6 : 15 L 4 : 10
  M 15 : 6 N 4 : 25
K 2 L0 M2 N 3 12. avivwUi mvaviY A¯¦i ˆKvbwU?
K 3 L9 M log3 N 2 log3
22. ˆKvYGKi eKÌZj I f„wgi ˆÞòdj mgvb
4. (x) = x3 + kx2  4x  8 nGj, k ‰i ˆKvb nGj, mgMÉZGji I eKÌZGji ˆÞòdGji
gvGbi RbÅ (2) = 0 nGe? 13. avivwUi mµ¦g c` KZ?
K log81 L log243 AbycvZ KZ?
K 2 L1 M2 N 3 M log729 N log2187 K 2:1 L 1:2 M 2:3N 3:2
5. wbGPi Z^ÅàGjv jÞÅ Ki: 14. avivwUi cÉ^g 7wU cG`i mgwÓ¡ KZ? 23. sin230 + cos2 = 1 nGj,  ‰i gvb KZ?
i. jGMi AskK meÆ`v abvñK K 4 log3 L 7 log3 K 15 L 30 M 45 N 90
ii. Ave†î `kwgGKi àYdj meÆ`v Ave†î M 28 log3 N 32 log3 24. ‰KwU UvIqvGii cv`G`k ˆ^GK 75 wg. `ƒGi
iii. mKj exRMwYZxq mƒòB AGf` 15. a, b, c evÕ¦e msLÅv nGjv@ f„ZjÕ© ˆKvb we±`yGZ UvIqvGii kxGlÆi
wbGPi ˆKvbwU mwVK? i. (a + b) + c = a + (b + c) D®²wZ ˆKvY 30 nGj, UvIqvGii DœPZv KZ?
ii. (a + b) + c = ac + bc
K i I ii L ii I iii K 43.30 wg. (cÉvq) L 44.30 wg. (cÉvq)
iii. (ab)c = a(bc)
M i I iii N i, ii I iii wbGPi ˆKvbwU mwVK? M 45.30 wg. (cÉvq) N 46.30 wg. (cÉvq)
6. mvgv¯¦wiGKi NƒYÆb cÉwZmgZvi gvòv KZ? K i I ii L i I iii
25. D®²wZ ˆKvY@
i. 60 nGj jÁ¼ < f„wg
K 1 L2 M3 N 4 M ii I iii N i, ii I iii
7. 
 16. 
 ii. 30 nGj f„wg > jÁ¼
A F
P iii. 45 nGj f„wg = jÁ¼
B
A
135 wbGPi ˆKvbwU mwVK?
C Q D K i I ii L i I iii
B D C
E M ii I iii N i, ii I iii
ABC ‰ D, BC ‰i gaÅ we±`y| ABC wPGò, CQE = KZ? wbGPi ZG^Åi AvGjvGK (26-28) bs cÉGk²i Dîi
K 60 L 30 M 45 N 35 `vI:
‰i ˆÞòdj 60 eMÆ ˆm.wg. nGj, ABD 17. ˆh wòfzGRi@
‰i ˆÞòdj KZ? AvqZvKvi ‰KwU NGii ˆgGSi Š`NÆÅ, cÉÕ©
i. wZbwU ˆKvY mgvb Zv ‰KwU mgevü AGcÞv 2 wg. ˆewk ‰es ˆgGSi cwimxgv 20
K 60 eMÆ ˆm.wg. L 45 eMÆ ˆm.wg. wòfzR
M 30 eMÆ ˆm.wg. N 25 eMÆ ˆm.wg.
wg.| NiwUi ˆgGS ˆgvRvBK KiGZ cÉwZ eMÆ wg.
ii. wZbwU ˆKvY mƒßGKvY ZvGK mƒßGKvYx 900 UvKv LiP nq|
wbGPi ZG^Åi AvGjvGK (8 I 9) bs cÉGk²i Dîi wòfzR eGj 26.  NiwUi ˆgGSi Š`NÆÅ KZ wg.?
`vI: iii. ‰KwU ˆKvY mgGKvY ZvGK mgGKvYx K 4 L6 M8 N 10
p, q, r, s PviwU ivwk ˆhLvGb p I q ‰K RvZxq wòfzR eGj 27. 
 NiwUi ˆgGSi ˆÞòdj KZ eMÆ wg.?
‰es r I s AbÅ ‰K RvZxq ‰es p : q = m : n wbGPi ˆKvbwU mwVK? K 80 L 48 M 32 N 24
8. cÉ`î ivwkàGjv mgvbycvZ MVb KiGj K i I ii L i I iii 28.  NiwUi ˆgGS ˆgvRvBK KiGZ ˆgvU KZ
ˆKvbwU mwVK? M ii I iii N i, ii I iii LiP nGe?
K
r m
= L
r 2m
=
18. 
 D C K 21600 UvKv L 28800 UvKv
s n s 3n
p r 75
M 43200 UvKv N 72000 UvKv
M 
q s
N pm = qr A B wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
q wPGò, ABCD ‰KwU mvgv¯¦wiK| A = 75 Dîi `vI:
9. m ‰i gvb n ‰i wZbàY nGj ‰i gvb
p nGj, B + D = KZ? ˆkÉwYeÅvwµ¦ 30-39 40-49 50-59 60-69
ˆKvbwU? K 150 L 180 M 210 N 220 MYmsLÅv 6 16 30 14
K
1
L
1
M3 N 32 19. e†îÕ© UÇvwcwRqvGgi wZhÆK evü«¼Gqi ŠewkÓ¡Å 29. gaÅK ˆkÉwYi MYmsLÅv KZ?
32 3
wKi…c? K 8 L 14 M 16 N 30
10.  (i) `yB ev ZGZvwaK PjGKi gaÅKvi
 K ciÕ·i mgv¯¦ivj L ciÕ·i Amgvb 30. DcvGîi cÉPziK KZ?
mÁ·KÆB dvskb M ciÕ·i mgvb N ciÕ·i jÁ¼ K 45.33 L 50.53 M 54.67N 55.33

1 L 2 K 3 L 4 M 5 M 6 K 7 M 8 K 9 L 10 K 11 L 12 M 13 N 14 M 15 L
Dîi

16 M 17 N 18 M 19 M 20 N 21 N 22 K 23 L 24 K 25 M 26 L 27 N 28 K 29 N 30 M
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2022 21
366 Mft jÅveGiUix nvB Õ•zj, ivRkvnx

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. wbGPi ˆKvbwU gƒj` msLÅv? 12. a1x + b1y = c1 ‰es a2x + b2y = c2 23. 
1
 128 + 64 + 32 + ............. + avivwUGZ
2
K 0.4 L 0.9 mgxKiY ˆRvGUi Amgém I ciÕ·i
KZwU c` AvGQ?
M 0.04 N 0.025 AwbifÆkxj nIqvi kZÆ ˆKvbwU?
K 10 L 9
2. ( 3x  2y) ‰i eMÆ wbGPi ˆKvbwU? a1 b 1 c1 a1 b 1 c1
K   L  = M 8 N 7
K  9x2  12xy  4y2 a2 b 2 c2 a2 b 2 c2
L 9x2  12xy  4y2 a1 b 1 c1 a1 b 1 c1 24. a, b, c KÌwgK mgvbycvZx nGj
M = = N =  i. b2 = ac
M 9x2  12xy + 4y2 a2 b 2 c2 a2 b 2 c2
ii. 2b = a + c
N 9x2 + 12xy + 4y2 13. x ‰i ˆKvb gvGbi RbÅ lnx msævwqZ? a+b b+c
× × K x0 L x<0 iii. =
3. 0.3  0.6 = KZ? b c
× ×× M x>0 N x0 wbGPi ˆKvbwU mwVK?
K 0.18 L 0.18
×
14.  = 90 ‰i RbÅ wbGPi ˆKvbàwj msævwqZ? K i I ii L i I iii
M 0.2 N 0.2 K cot, sec L sin, tan M ii I iii N i, ii I iii
wbGPi ZG^Åi AvGjvGK (4 I 5) bs cÉGk²i Dîi M cosec, sec N cot, cos
wbGPi ZG^Åi AvGjvGK (25 I 26) bs cÉGk²i
`vI: 15. ˆKvb kGZÆ logaa = 1 nGe? Dîi `vI:
A = {1, 1, 2, 3} ‰es B = {x : x2  x  2 = 0} K a>0 L a1
4. wbGPi ˆKvb ˆmUwU A  B ˆmUGK wbG`Æk ˆkÉwY 31-40 41-50 51-60 61-70 71-60
M a > 0, a  1 N a  0, a >  1
KGi? MYmsLÅv 5 8 12 9 6
3
K {1, 2} L {1, 3} 16. hw` 2sinAcosA = 2 nq, ZGe A ‰i gvb 25. 
 cÉPziK ˆkÉwYi gaÅgvb KZ?
M {1, 3} N {1, 2} K 45.5 L 51.5
KZ?
5. A  B ‰i Dcv`vb msLÅv KZ? M 55.5 N 65.5
K 30 L 45
K 4 L 5
M 60 N 90
26. 
 gaÅK wbYÆGqi ˆÞGò
M 6 N 8 i. Fc = 13
17.  r eÅvmvaÆ
 wewkÓ¡ ‰KwU e†î ‰es H
6. mKj cƒYÆ ‰es f™²vsk msLÅvGK ejv nq ii. fm = 12
e†îÕ© eGMÆi ˆÞòdGji AbycvZ ˆKvbwU? n
K Agƒj` msLÅv L gƒj` msLÅv iii. = 20
2
K 4: L :2
M AFYvñK msLÅv N Õ¼vfvweK msLÅv
M 2:r N r:2 wbGPi ˆKvbwU mwVK?
7. (x  5)2 = x2  10x + 25 nGj 18. 
 AebwZ ˆKvGYi gvb KZ wWMÉx nGj, K i I ii L i I iii
i. ‰KwU AGf` M ii I iii N i, ii I iii
LuywUi Š`NÆÅ, Zvi Qvqvi Š`GNÆÅi 3 àY nGe?
ii. ‰KwU mgxKiY 27. ˆKvGbv e†Gîi eÅvmvaÆ r nGZ r + x Kiv nGj,
K 30 L 45
iii. x ‰i wbw`ÆÓ¡ gvGbi RbÅ mZÅ
M 60 N 90 e†nîg RÅv ‰i Š`NÆÅ Kx cwigvY evoGZ
wbGPi ˆKvbwU mwVK? 19. 
 Z eYÆwUi NƒYÆb cÉwZmgZvi gvòv ˆKvbwU? cvGi?
K i I ii L i I iii K 1 L 2 K 2r + x L x
M ii I iii N i, ii I iii M 3 N 4 M 2r N 2x
8. (21 + 31) = KZ? 20. 
 ABC ‰i AB I AC
evüi gaÅwe±`y 28. 
 5  2cosecAcosA = 0 nGj, tanA = ?
K 1/6 L 5/6 2 2
M 6/5 N 6
h^vKÌGg D I E nGj BDE: ABC = K L
7 5
9.  4x2 + 2
 ‰i mvG^ KZ ˆhvM KiGj KZ? 5 7
M N
ˆhvMdj cƒYÆeMÆ nGe? K 1: 2 L 1:2 2 2
1 2
M 1:3 N 1:4 29. 
 log3 + log9 + log27 + ...........
K  L  4x
4x 21. cvGki wPGòi Mvp wPwn×Z AsGki ˆÞòdj i. avivi cieZÆx c` log81
1 KZ eMÆ ˆm.wg.?
M x 2
N
4x2
ii. avivwU ‰KwU mgv¯¦i aviv
K 4.2 7 ˆm.wg. iii. avivi mvaviY A¯¦i log6
10. ‰KwU eGMÆi evüGK wZbàY Kiv nGj L 2.15 wbGPi ˆKvbwU mwVK?
4 ˆm.wg.

cwiewZÆZ ˆÞòdj cƒeÆvGcÞv KZàY e†w«¬ M 11.49

cvGe? K i L i I ii
N 15.43
K 3 L 6 M i I iii N i, ii I iii
M 8 N 9 22. RwgwUi KZ eMÆ ˆm.wg. RvqMv w`Gq ivÕ¦v 30. 
 wPGò BCD ‰i gvb KZ wWwMÉ?
C
11. 3x + 2y = 1 mgxKiGY x =  3emvGj ˆMGQ? 20 ˆm.wg. K 70
D
mgxKiGYi ˆjGLi we±`ywU ˆKvb PZzfÆvGM K L
8 ˆm.wg.

20 60
10 ˆm.wg.

cGo? L 16 M 55 20

O
K 1g L 2q M 10 N 50 E 110
N 5 15 ˆm.wg. ivÕ¦v B
M 3q N 4^Æ A

1 M 2 N 3 M 4 N 5 N 6 L 7 K 8 L 9 N 10 M 11 L 12 N 13 M 14 N 15 M
Dîi

16 K 17 L 18 M 19 L 20 N 21 N 22 N 23 L 24 L 25 M 26 N 27 N 28 L 29 L 30 N
22 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
367 miKvwi cÉg^bv^ evwjKv DœP we`Åvjq, ivRkvnx

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
1
 p2  3p + 4 ‰i Drcv`GK weGkÏwlZ 10. x + y = 0 ‰es x  y = 2 mgxKiY ˆRvGUi wbGPi ZG^Åi AvGjvGK (22 I 23) bs cÉGk²i
2
mgvavb we±`y ˆjLwPGòi ˆKvb PZzfÆvGM Dîi `vI:
i…c wbGPi ˆKvbwU? AewÕ©Z? ˆkÉwYeÅvwµ¦ 15-19 20-24 25-29 30-34
1 1
K
2
(p  4)(p + 2) L
2
(p + 4)(p  2) K cÉ^g fvGM L w«¼Zxq fvGM MYmsLÅv 2 8 10 6
1 M Z‡Zxq fvGM N PZz^Æ fvGM 22. cÉPziK wbYÆGq (f1 + f2) ‰i gvb KZ?
M (p  4)(p  2) N (p  4)(p + 2)
2 11. ( 2)x+1 = 16 nGj, x ‰i gvb KZ? K 4 L6 M8 N 10
2. x2 = x 3 mgxKiYwUi mgvavb ˆmU wbYÆq K 7 L8 M9 N 16 23. Dcvî mgƒGni gaÅK ˆKvbwU?
Ki| 12. P = {3, 2, 1, 0, 1, 2}, K 36.5 L 31.0 M 26.5 N 25.0
K {0, 3} L {0,  3} Q = {3, 2, 0, 1, 3} nGj Q  P = KZ? 24. 128 + 64 + 32 + ............... avivwUi KZ
M {0, 3} N { 3} K {3, 2, 1, 0, 1, 3} 1
Zg c` 2 ?
3. wPGò BC || DE ‰es AB = 8 ˆm.wg.‰es L {3, 2, 0, 1}
M {1, 2} N {3} K 9 Zg L 8 Zg
BC = 6 ˆm.wg. nGj A
13. 35  2y  y2 ‰i Drcv`K ˆKvbwU? M 7 Zg N 6 Zg
i. DE = 3 ˆm.wg. D E
K 5+y L y5 25. 
 wPGò AB : AD = AC : AE ‰es ADE
ii. AD = 4 ˆm.wg.
M 7+y N 7y ‰i ˆÞòdj 16 eMÆ ‰KK nGj
iii. ABC I ADE m†`k B C
14.  wPGò QR = KZ ˆm.wg.?
 i. DE ‰i `ƒiZ½ 4 ‰KK A
wbGPi ˆKvbwU mwVK? K 1 P ii. ABC I ADE m`†k
K i I ii L i I iii L 2 8
AF AB D E
1 ˆm.wg.

60
M ii I iii N i, ii I iii iii. = G
2
M 3 AG AD
4. wPGò ABCD AvqZGÞGòi KYÆ«¼Gqi N 2 R
wbGPi ˆKvbwU mwVK? B F C
Q
ˆQ`we±`y O, OP = 4 ˆm.wg. ‰es OA = 5 15.
1
, 1, 3 ......... ‰i cieZÆx c`wU KZ K i I ii L i I iii
ˆm.wg. nGj BC KZ? 3 M ii I iii N i, ii I iii
K 2 ˆm.wg. A B nGe? 26. 
 ‰KwU e†Gîi KZwU NƒYÆvqb cÉwZmvgÅ
L 4 ˆm.wg. O P K 3 3L3 3 M 3 N 3 ˆiLv ^vGK?
M 6 ˆm.wg. 3m + n
D C
16. n  m = 9 nGj, m : n = ? K 2 L 3
N 8 ˆm.wg. M 8 N AmsLÅ
K 1:5 L 5:1
5.  2x  y = 13 ‰es 5x + 6y = 7
 M 2:3 N 3:2 wbGPi ZG^Åi AvGjvGK (27 I 28) bs cÉGk²i
mgxKiY«¼q 17. ‰KwU UÇvwcwRqvGgi mgv¯¦ivj evü«¼Gqi Dîi `vI: A
Q

i. ciÕ·i wbfÆikxj Š`NÆÅ 18 ˆm.wg. I 14 ˆm.wg. ‰es ˆÞòdj wPGò AB > AC y y

ii. ‰i ‰KwU mgvavb AvGQ 128 eMÆ ˆm.wg. nGj mgv¯¦ivj evü«¼Gqi ‰es AP || CQ B
P
C

iii. ciÕ·i mgém gaÅeZÆx `ƒiZ½ KZ? 27. wbGPi ˆKvbwU mwVK?
wbGPi ˆKvbwU mwVK? K 8 L 16 M 32 N 64 K AC : AB = PC : PB
L BP = PC
K i I ii L i I iii 18. 
x x
 + 3 =  2 nGj, x ‰i gvb KZ?
4 3 M AP : BP = AP : PC
M ii I iii N i, ii I iii
K 120 L 60 M 36 N 6 N ABP @ APC
6. kZKiv evwlÆK 7 UvKv mij gybvdvi ˆKvb 2
19. 39 ‰i Ave†î `kwgK f™²vsGki i…c ˆKvbwU? 28. wPòvbymvGi
gƒjab 2 eQGii me†w«¬gƒj 912 UvKv nGj i. ACQ = CAP
gƒjab KZ? ×× × ×
K 0. 63 L 0.16 M 3.2 N 0.22
× ii. BAP = AQC
K 894.50 UvKv L 852.33 UvKv iii.  ˆÞò APB   ˆÞò ACP
20. wPGò PA I PB `ywU Õ·kÆK ‰es PAB =
M 800 UvKv N 796.57 UvKv wbGPi ˆKvbwU mwVK?
30 nGj AOB = ?
7. a = 3 ‰es b = 12 nGj wbGPi ˆKvbwU K 120 A K i I ii L i I iii
Agƒj` msLÅv? L 90 O 30
P M ii I iii N i, ii I iii
K a + b L ab M
a
N
b M 60 29. 
 ˆKvb mgevü wòfzGRi ‰Kevü a ˆm.wg.
b a B
N 30 nGj, Zvi DœPZv KZ ˆm.wg.?
(1  cot260)
8. 
 ‰i gvb ˆKvbwU? 21. 
 wPGò AvqZGÞòwUi Mvp wPwn×Z AsGki 3 2
(1 + cot260) K a L a
1 4 2 ˆÞòdj KZ eMÆ ˆm.wg.? 2 3
K 2 L M N K 28.27
2 3 3 2 3
6 ˆm.wg.

L 31.73 M a N a
× 3 2
9. 2.05 ˆK mvgvbÅ f™²vsGk cÉKvk Ki? M 33.27
205 203 37 41 N 60
30. log42  log 27 = KZ?
K L M N 10 ˆm.wg. 3
100 90 18 20 K 27 L9 M6 N 3
1 M 2 K 3 N 4 M 5 M 6 M 7 K 8 L 9 M 10 N 11 K 12 N 13 M 14 M 15 N
Dîi

16 M 17 K 18 L 19 M 20 M 21 L 22 L 23 M 24 K 25 N 26 N 27 K 28 N 29 K 30 N
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368 ivRkvnx KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, ivRkvnx

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. a, b, c  Ñ; a > b > 0 ‰es c < 0 nGj, 12. `yBwU e†î ciÕ·iGK ewntÕ·kÆ KiGj ZvG`i 21. 
 sin + cos = 1 nGj, (sin – cos)2 = ?
wbGPi ˆKvbwU mwVK? gGaÅ mGeÆvœP KqwU mvaviY Õ·kÆK Aâb K 2 L1
K ac = bc L ac > bc Kiv mÁ¿e? M 0 N–1
M ac < bc N ab < bc
K 1wU L 5wU 22. sin4 + sin2 = 1 nGj,
2. B = {a} nGj, P(B) ‰i Dcv`vb msLÅv M 3wU N 2wU
i. sin2 = cos
ˆKvbwU? wbGPi ZG^Åi AvGjvGK (13 I 14) bs cÉGk²i
ii. tan = cosec
K 2 L3 M1 N 0 iii. tan.cosec = 2
Dîi `vI: P
wbGPi ˆKvbwU mwVK?
3. f(x) = 2x2 + kx + 3 nGj, k ‰i ˆKvb gvGbi
RbÅ f(– 1) = 0 nGe?
3x K i I ii L i I iii
O
K 2 L3 M5 N –1 M ii I iii N i, ii I iii
Q S
wbGPi ZG^Åi AvGjvGK (4 I 5) bs cÉGk²i Dîi 2x + 60
3
23. sin2 = 2 nGj, tan2 = ?
`vI: R
1 1 1
x– =7 K  L M N 3
2
x 13. 
 QRS ‰i gvb KZ? 3
1 K 60 L 120 24. e†Gîi eÅvm 12 ˆm.wg. nGj ‰i cwimxgv KZ
4. 
 x2 – 2 = ?
x M 45 N 135 ˆm.wg.?
K 53 L 7 53 M 7 N 49 14. 
 OQR + OSR = ? K 37.69 L 36.69
x M 24.69 N 18.69
5. 
 2
x – 6x – 1
‰i gvb KZ? K 120 L 135
M 145 N 45 25. ‰KwU mge†îf„wgK wmwj´£vGii f„wgi eÅvmvaÆ
K 1 L2 M3 N 4
15. mgw«¼evü wòfzGRi cÉwZmvgÅ ˆiLv KqwU? 4 ˆm.wg. ‰es DœPZv 12 ˆm.wg. nGj
6. 33 ‰i 3 wfwîK jM KZ?
3 2 1
K 0 L2 M1 N 3 eKÌZGji ˆÞòdj KZ eMÆ ˆm.wg.?
K 3 L M N 16. 
 P K 201.59 L 301.59
2 3 3
M 80.59 N 48.59
7. 
 log2 + log4 + log8 + ......... avivi 6Ó¤ y y

c` KZ? 26. 
 eGMÆi KGYÆi Š`NÆÅ 6 2 ˆm.wg. nGj, Zvi
K log2 L 3log2 Q R cwimxgv KZ?
S K 24 ˆm.wg. L 12 ˆm.wg.
M 5log2 N 6log2
8.
a b
+ = 2 nGj, a – b ‰i gvb KZ? wbGPi ˆKvbwU mwVK? M 18 ˆm.wg. N 20 ˆm.wg.
b a K QS : RS = PQ : PR 27. e†Gîi cwiwa : eÅvm = ?
K 0 L1 M ab N a + b L QS : RS = PQ : PS  
2 3 M K 1: L :1 M :1N 1:
9. =
x + 1 2x – 1
mgxKiGYi gƒj ˆKvbwU? QS : RS = PR : PS 2 2
N PQ : SR = QS : PQ 28. 
 AebwZ ˆKvGYi gvb KZ nGj LuywUi Š`NÆÅ
K –5 L –3 M 3 N 5
17. 4 ‰i NƒYÆb ˆKvY KZ? Qvqvi Š`GNÆÅi 3 àY nGe?
10. 
 A
K 90 L 120 M 360 N 180 K 60 L 30
18. ‰KwU wòfzRGK Aci ‰KwU wòfzGRi Dci M 45 N 90
D E
(x + 2) cm
Õ©vcb KiGj hw` wòfzR `ywU meÆGZvfvGe wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
B C wgGj hvq ZGe@ Dîi `vI:
(5x – 2) cm
i. wòfzR `ywU meÆmg ˆkÉwY 31-40 41-50 51-60 61-70 71-80
ABC ‰i AB I AC ‰i gaÅwe±`y h^vKÌGg
ii. wòfzR `ywUi Abyi…c evü mgvb MYmsLÅv 5 8 12 9 6
D I E nGj, BC ‰i Š`NÆÅ KZ ˆm.wg.?
iii. Abyi…c ˆKvY mgvb 29. gaÅK wbYÆGqi ˆÞGò?
K 6 L 8
M 4 N 2 wbGPi ˆKvbwU mwVK? n
i. = 20
11. 
 K i I ii L i I iii 2
A ii. fm = 12
D M ii I iii N i, ii I iii
80 E iii. Fc = 15
19. 
 tan 1 – sin2 = ?
wbGPi ˆKvbwU mwVK?
K cos L tan
K i I ii L i I iii
B M sec N sin
C
5 M ii I iii N i, ii I iii
20. 
 cosec(90 – ) = nGj, cot = ?
3 30. cÉPziK ˆkÉwYi DœPmxgv KZ?
wPGò ABC = KZ? 3 4 3 4 K 70 L 60
K 80 L 90 M 100 N 180 K L M N
5 3 4 5 M 50 N 55.5
1 M 2 K 3 M 4 L 5 K 6 L 7 N 8 K 9 N 10 L 11 K 12 M 13 N 14 L 15 M
Dîi

16 K 17 N 18 N 19 N 20 M 21 L 22 K 23 N 24 K 25 L 26 K 27 L 28 K 29 K 30 L
24 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
369 bvGUvi miKvwi evwjKv DœP we`Åvjq, bvGUvi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
.
1. 0.71 ‰i mvaviY f™²vsk ˆKvbwU?
4
11. y2  y2 = 0 mgxKiYwUi- wbGPi ZG^Åi AvGjvGK (20 I 21) bs cÉGk²i
71 64 Dîi `vI:
K L i. PjGKi mGeÆvœP NvZ 4
99 99 ‰KwU avivi cÉ^g c` a, mvaviY AbycvZ r, PZz^Æ
32 71 ii.`yBwU gƒj ( 2  2 )
M
45
N
90 c` 2
iii. a˂e c` 4
2. S = {(5, 3), (5, 6), (7, 6), (8, 9)} ‰i ˆié 20. 
 beg c` 8 2 nGj, r = KZ?
wbGPi ˆKvbwU mwVK?
ˆKvbwU? K  2 L 2
K i I ii L i I iii
K {5, 5, 7, 8} L {5, 7, 8} M 2 N 2
M {3, 6, 6, 9} N {3, 6, 9} M ii I iii N i, ii I iii
21. 
 cÉ^g c`wU KZ?
3.
1
  =
2x + 3
nGj, (2) ‰i gvb ˆKvbwU? 12. 
 8 ˆmw´ŸwgUvi I 10 ˆmw´ŸwgUvi 1
x  6x  1 eÅvmwewkÓ¡ `yBwU e†î ciÕ·i A¯¦:Õ·kÆ K  L 2
2
7 KiGj ZvG`i ˆK±`Ê«¼Gqi `ƒiZ½ KZ nGe?
K L 1 1
11 M N  2
K 1 L 2 2
M 2 N 4
M 4 N 5 22. cçfzGRi ˆKvYàGjvi mgwÓ¡ KZ?
4. x2 + 3x ‰i mvG^ KZ ˆhvM KiGj ˆhvMdj
13. 
 wbGPi wPGò POM = KZ? K 450 L 540
cƒYÆeMÆ nGe? P
9 4 K 60 M 630 N 360
K L L 90
4 9 O 23. cÉwZmvgÅ ˆiLv ˆbB ˆKvbwUi?
3 M 120
M 9x2 N 60 K mgevü wòfzGRi L welg evü wòfzGRi
2 N 150 M N
5. 
 ‰KwU eBGqi gƒjÅ 72 UvKv| ‰B gƒjÅ M e†Gîi N eMÆGÞGòi
eB ŠZwii gƒGjÅi 90%| evwK UvKv miKvi 24. 
 ‰KwU NbGKi mÁ·ƒYÆ c†GÓ¤i ˆÞòdj
wbGPi ZG^Åi AvGjvGK (14 I 15) bs cÉGk²i
fZzwÆ K ˆ`b| miKvi cÉwZ eBGq KZ UvKv 384 eMÆ ˆmw´ŸwgUvi nGj KGYÆi Š`NÆÅ KZ
Dîi `vI:
fZzwÆ K ˆ`b? ˆmw´ŸwgUvi?
O
K 8 UvKv L 10 UvKv K 8 3 L 6
3
M 18 UvKv N 28 UvKv A C B M 64 N 8
1
6. m + m = 2 ˆhLvGb m > 0 nGj- 25. 
 ‰KwU iÁ¼Gmi `yBwU KGYÆi Š`NÆÅ 12
wPGò O e†Gîi ˆK±`Ê
1 ˆmw´ŸwgUvi I 16 ˆmw´ŸwgUvi nGj, evüi
i. m3 + =2 14. 
 OA = 5 ˆmw´ŸwgUvi nGj, e†Gîi AB RÅv
m3 Š`NÆÅ KZ ˆmw´ŸwgUvi?
1 ‰i Š`NÆÅ KZ ˆmw´ŸwgUvi? K 4 L 6
ii. m5  5 = 0
m K 8 L 16
M 8 N 10
1 M 20 N 25
26. 1 ˆ^GK 10 chƯ¦ ˆgŒwjK msLÅvàGjvi Mo
3
iii. m + 4 = 2
m
15. 
 OAB = 50 nGj, AB PvGci Dci
wbGPi ˆKvbwU mwVK? KZ?
ˆK±`ÊÕ© ˆKvGYi gvb KZ?
K i I ii L i I iii K 4.50 L 4.25
K 60 L 80
M ii I iii N i, ii I iii M 5.5 N 10
M 100 N 120
7. 4a2 + 1 ‰i Drcv`K ˆKvbwU? 27. wbGPi ˆKvbwU wewœQ®² PjK?
16. sin45 = 2A nGj, A ‰i gvb KZ?
K (2a2 + 2a + 1) (2a2  2a + 1) K Zvcgvòv L cÉvYxi msLÅv
L (4a2 + 4a + 1) (4a2  4a + 1) 1 1 3
K 1 L
2
M N
2 M eqm N DœPZv
M (2a2  2a  1) (2a2 + 2a + 1) 2
N (4a2 + 4a + 1) (4a2  4a  1) 17. 
 sin3 = cos3 nGj, tan2 ‰i gvb KZ?
28. AebwZ ˆKvGYi gvb KZ wWMÉx nGj, LuywUi
8. 0.00000045-‰i mvaviY log ‰i cƒYÆK KZ? 1 Š`NÆÅ Qvqvi Š`GNÆÅi 3 àY nGe?
K 1 L
  3 K 30 L 45
K 5 L6
  M 3 N2 3 M 60 N 90
M 7 N8
18. 
 `yBwU msLÅvi AbycvZ 5 : 4 ‰es ‰G`i 29. wbGPi ˆKvb evüàGjv «¼viv wòfzR AuvKv
9. log93 = KZ?
j.mv.à 120 nGj, M.mv.à KZ? mÁ¿e?
1
K L3 K 4 L5 K 4, 5, 9 L 3, 5, 10
2
M 6 N9 M 6 N9 M 6, 8, 11 N 5, 5, 12
10. 2logx = log(2x  1) nGj, x ‰i gvb KZ? 19.
a b 2
 = = nGj, a : c ‰i gvb
 KZ? 30. 
 (31 + 21)1 ‰i mij gvb KZ?
K 1 L0 b c 3 5 6
K 2:3 L 3:4 K L
1 6 5
M N1 M 4:9 N 9:4
2 M 5 N 6

1 M 2 N 3 M 4 K 5 K 6 N 7 K 8 M 9 K 10 N 11 K 12 K 13 N 14 K 15 L
Dîi

16 L 17 L 18 M 19 M 20 K 21 M 22 L 23 L 24 K 25 N 26 L 27 L 28 M 29 M 30 L
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 25
370 eàov wRjv Õ•zj, eàov

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. wbGPi ˆKvb Z^Å (ˆmw´ŸwgUvi) «¼viv wòfzR 12. AwRf ˆiLvi ˆÞGò wbGPi ˆKvbwU mwVK? 22. Px = N, (P > 0, P  1) nGj, N ‰i P
Aâb Kiv hvq? K Ea»ÆMvgx L wbÁ²Mvgx wfwîK jM KZ?
K ‰KwU ˆKvY 30, f„wg = 2.4, `yB evüi M mgv¯¦ivj N DjÁ¼ K N = logxP L X = logpN
mgwÓ¡ 2.2 M N = logpX N P = logxN
13. 
 mylg eüfzGRi cÉwZwU A¯¦tÕ© ˆKvY
L f„wg = 2.4, `yB evüi mgwÓ¡ 4.2 23. KÌgGhvwRZ MYmsLÅv cÉGqvRb@
150 nGj, KGYÆi msLÅv KZ?
M ‰KwU ˆKvY 30, f„wg = 2.4, `yB evüi i. gaÅK wbYÆGq ii. Mo wbYÆGq
K 72wU L 54wU
mgwÓ¡ = 4.5 iii. AwRfGiLv AâGb
M 32wU N 16wU
N ‰KwU ˆKvY 30, `yB evüi mgwÓ¡ = 4.2 wbGPi ˆKvbwU mwVK?
14. ˆKvGbv eGMÆ A¯¦e†Æî AuvKGZ KqwU avc
2. ‰KwU Mvwoi PvKvi eÅvm 10cm nGj, ‰i K i I ii L i I iii
AbymiY KiGZ nq? M ii I iii N i, ii I iii
ˆÞòdj KZ?
K 1wU L 2wU 24. {x  N : x3 > 25 ‰es x4 < 264} ‰i
K 25cm L 16cm2
M 32cm 2
N 25cm2 M 3wU N 4wU ZvwjKv c«¬wZ wbGPi ˆKvbwU?
3. 
 45, 46, 36, 35, 43, 33, 38, 43, 40, 50 15. 
 kZKiv evwlÆK 5 UvKv nvi gybvdvq KZ K {2, 3, 4} L {3, 4, 5}
DcvGîi cwimi KZ? UvKvi 12 eQGi me†w«¬gƒj 1280 UvKv nGe? M {4, 5, 6} N ˆKvGbvwUB
bq
K 10 L 12 K 600 UvKv L 800 UvKv wbGPi ZG^Åi AvGjvGK (25 I 26) bs cÉGk²i
M 16 N 18 M 1000 UvKv N 1080 UvKv Dîi `vI: C
4. ˆKvGbv wòfzGRi AuvKv hvq@ 3
16. cosec + cot = 2 nGj, cot  cosec =?
i. ‰KwU A¯¦e†Æî ii. wZbwU ewne†Æî O
iii. wZbwU cwie†î K 
3
L
2
M
2
N
3
2 3 3 2
wbGPi ˆKvbwU mwVK? A P B
K i L i I ii
17. e†Gîi ˆKvGbv Pvc «¼viv Drc®² ˆK±`ÊÕ© ˆKvY
M i I iii N i, ii I iii H e†îPvGci@
O ˆK±`ÊwewkÓ¡ ABC e†Gî OA = 10 cm ‰es
5. 
 ‰KwU mylg mµ¦fzGRi kxlÆ ˆKvGYi gvb K mgvbycvwZK L eÅÕ¦vbycvwZK
OP = 6 cm, OPAB.
cÉvq KZ wWwMÉ? M mgvb N ˆKvGbvwUB bq 25. 
 AB ‰i Š`NÆÅ wbGPi ˆKvbwU?
K 120 L 128.6 18. ABC mgGKvYx wòfzGRi AwZfzR AC = 2, K 6 cm L 8 cm
M 130.5 N 107 AB = 1. M 10 cm N 16 cm
A
wbGPi ZG^Åi AvGjvGK (6 I 7) bs cÉGk²i Dîi 26. 
 e†Gîi cwiwa KZ ˆmw´ŸwgUvi?
`vI: 1 2 K 31.42 L 52.42
ˆkÉwYeÅvwµ¦ 31-40 41-50 51-60 61-70 71-80 81-90 91-100 M 62.83 N 125.64
MYmsLÅv 6 12 16 24 12 8 2 27. ‰KwU AbyKÌGgi mvaviY c`
6. cÉPziK ˆkÉwYi gaÅgvb ˆKvbwU? B C
(1)n1 
1 
K 55.5 L 65.5 i. cosA = sinC  n + 1  nGj@
M 75.5 N 85.5 5 1 1 1 1
7. cÉPziK KZ? ii. cosA + secA = i. AbyKÌgwU 2 ,  3 , 4 ,  5 ......
2
K 55 L 65 1 1
M 75 N 85 iii. tanC = ii. `kg c` = 20
3
8. 1 + 3 + 5 + 7 + .... avivwUi cÉ^g n msLÅK wbGPi ˆKvbwU mwVK? iii. 11 I 12 Zg cG`i ˆhvMdj = 156
1
cG`i mgwÓ¡ KZ? K i L i I iii
 n(n + 1) 2 n(n + 1) wbGPi ˆKvbwU mwVK?
K   L M ii I iii N i, ii I iii K i L i I ii
 2  2
19. KqwU Dcvî ˆ`Iqv ^vKGj wòfzR AuvKv
n2 M i I iii N ii I iii
M
2
N n2 hvq? 28. wbGPi ˆKvbwU wewœQ®² PjK?
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i K 1 L 2 K Zvcgvòv L Qvò-Qvòxi msLÅv
M 3 N 4
Dîi `vI: M eqm N DœPZv
2cos(A  B) = 1, 2sin(A + B) = 3 nGj,
20. ‰KwU wòfzGRi ˆKvYàGjvi AbycvZ 1 : 1 : 2 1
nGj, wòfzRwU wK aiGbi? 29. sin2  cos2 = 2 nGj,
9.  (A + B) = KZ?
K 15 L 30 K Õ©ƒjGKvYx L mgGKvYx sin4  cos4 = KZ?
M 45 N 60 M mƒßGKvYx N welgevü 1 3
K L
10.  A ‰i gvb KZ?
 21. `yBwU e†î ciÕ·iGK ewntÕ·kÆ KGi| ‰G`i 2 2
3 1
K 7
1
L 23
1 1
M 52 N 77
1 ‰KwUi eÅvm 10 cm ‰es AciwUi eÅvmvaÆ M N
2 2 2 2 2 4
4 cm nGj, ˆK±`Ê«¼Gqi gaÅeZÆx `ƒiZ½ KZ 12
11. 0.00573 msLÅvwUGZ jGMi cƒYÆK KZ?
cm nGe? 30. A ‰i ˆKvb gvGbi RbÅ, secA = 5 ?
K 3 L2
K 9 L 10 K 65.37 L 55.37
 
M 3 N2 M 14 N 7 M 45.37 N ˆKvGbvwUB bq
1 2 M 3 4 N 5 6 7
N 8 9
L 10 11
L 12 13
L 14 L15 L N N M M K L M
Dîi

16 L 17 K 18 N 19 M 20 L 21 K 22 L 23 L 24 N 25 N 26 M 27 M 28 L 29 K 30 K
26 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
371 eàov KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, eàov

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 0.00000456 ‰i jGMi cƒYÆK KZ? 11. e†Gîi cÉwZmgZvi gvòv KZ? wbGPi ZG^Åi AvGjvGK (23 I 24) bs cÉGk²i
K 6 L
5 K 2 L6 Dîi `vI:
M 2 N
4 M AmsLÅ N4 1 + 3 + 5 + 7 + ...... ‰KwU mgv¯¦i aviv|
wbGPi ZG^Åi AvGjvGK (2 I 3) bs cÉGk²i Dîi 12. 
 ‰KwU LuywUi DœPZv I Qvqvi Š`GNÆÅi 23.  avivwUi r-Zg c` KZ?
`vI: AbycvZ 3 : 3 nGj, mƒGhÆi D®²wZ ˆKvY KZ K 2r – 1 L 2r + 1
ˆkÉwY 11-20 21-30 31-40 41-50
wWwMÉ? M 2r N 2r – 3
MYmsLÅv 4 18 22 16 K 50 L 45 24. 
 avivwUi cÉ^g 10wU cG`i mgwÓ¡ KZ?
2. DcvGîi gaÅK ˆkÉwY ˆKvbwU? M 60 N 30
K 50 L 100
K 41- 50 L 31- 40 13. 
 ‰KwU eGMÆi cÉwZ evüi Š`NÆÅ 10% e†w«¬
M 21- 30 N 11- 20 M 110 N 90
ˆcGj ‰i ˆÞòdj kZKiv KZ e†w«¬ cvGe?
3. gaÅK wbYÆGq fm = ? 25. a3 + 3 3 ‰i Drcv`K ˆKvbwU?
K 20% L 23%
K 4 L 16 K a2 + 3a + 9 L a2 + a 3 + 3
M 19% N 21%
M 22 N 18
14. ˆKvbwU mgGKvYx wòfzGRi ˆKvYàGjvi AbycvZ? M a+ 3 N a– 3
x–3
4. 

x
= x – 3 mgxKiGYi mgvavb ˆmU K 4:5:9 L 4:6:7 26. 
 tan( – 30) = 3 nGj, sin ‰i gvb KZ?
KZ? M 1:5:9 N 6 : 13 : 18 3
15. `yBwU e†Gî mGeÆvœP KqwU mvaviY Õ·kÆK K 0 L
K {1, 0} L {1} 2
M {3} N {1, 3} Aâb Kiv hvq? 1
M N1
5. RÅvwgwZGZ Dccv`Å cÉgvGYi avc KqwU? K 4 L2
2
K 4 L2
M 3 N1
27. 
 `yBwU e†î ciÕ·iGK A¯¦tÕ·kÆ KGi ‰es
M 3 N5 ‰G`i eÅvm h^vKÌGg 12 cm, 8 cm| ‰G`i
3
6. 5% nvi gybvdvq 1500 UvKvi 3 eQGii 16. log5( 5. 5) ‰i gvb KZ?
ˆK±`Ê«¼Gqi gaÅeZÆx `ƒiZ½ KZ ˆm.wg.?
mij gybvdv KZ? K
6
L
5
K 2 L1
K 235 L 225 5 6
4 3 M 3 N4
M 275 N 220 M N
7. cwimxgv ˆ`Iqv ^vKGj Aâb Kiv hvq?
3 4 28. ‰KwU mge†îf„wgK wmwj´£vGii eÅvmvaÆ r cm
17. 
 – 8 – 3 + 2 + 7 + ....... avivwUi 15-Zg ‰es DœPZv h cm nGj@
i. eMÆ
c` KZ?
ii. mgevü wòfzR i. f„wgi ˆÞòdj r2 eMÆ ˆm.wg.
K 67 L 78
iii. UÇvwcwRqvg ii. AvqZb = 2r2h Nb ˆm.wg.
M 62 N 83
wbGPi ˆKvbwU mwVK? 18. (a + b, 2) = (4, a – b) nGj (a, b) ‰i gvb KZ? iii. eKÌZGji ˆÞòdj 2r(r + h) eMÆ ˆm.wg.
K i I iii L i I ii K (2, 4) L (4, 2) wbGPi ˆKvbwU mwVK?
M i, ii I iii N ii I iii M (1, 3) N (3, 1)
K i I iii L ii
8. 
 19. 13, 17, 14, 11, 9, 14 msLÅvàGjvi gaÅK KZ?
M i N i I ii
K 13.5 L 13
2 cm
M 12.5 N 14 29. p(x) = x3 – 4x + 3 nGj p(– 1) = ?
20. A = {a, b, c} ‰es B = {a, b} nGj, K 2 L5
Mvp wPwn×Z AsGki ˆÞòdj KZ eMÆ ˆm.wg.?
i. A\B = {c} M 6 N7
K 2.434 L 3.03
M 3.13 N 3.434 ii. A ‰i cÉK‡Z DcGmU B 30. 

iii. n(B) = 3
9. 2x + y = 1 ‰es x = – 4 mgxKiY«¼Gqi P
mgvavb we±`y ˆKvb PZzfÆvGM AeÕ©vb KGi? wbGPi ˆKvbwU mwVK? 3x + 1
K i I ii L i I iii
K 2q L 1g O A
M 3q N 4^Æ M ii I iii N i, ii I iii
. 21. 
 cosec sec2 – 1 = ? 2x + 7
10. 1.13 ‰i mvaviY f™²vsk KZ? K tan L sin M sec N cos
Q
19 17 wPòvbyhvqx, x ‰i gvb KZ?
K
15
L
15 22. e†Gîi eÅvm I cwiwai AbycvZ KZ?
15 18 K 2: L:1 K 4 L7
M N M 5 N6
17 17 M  N1:
1 K 2 L 3 M 4 N 5 K 6 L 7 L 8 N 9 K 10 L 11 M 12 M 13 N 14 K 15 K
Dîi

16 L 17 M 18 N 19 K 20 K 21 M 22 N 23 K 24 L 25 M 26 N 27 K 28 M 29 M 30 N
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 27
372 AvgÆW cywjk eÅvUvwjqb cvewjK Õ•zj I KGjR, eàov

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
wbGPi ZG^Åi AvGjvGK (1 I 2) bs cÉGk²i Dîi 11. ‰KwU eGMÆ A¯¦wjÆwLZ e†Gîi eÅvmvaÆ 22. 5 ˆm.wg. I 7 ˆm.wg. eÅvmvGaÆi `ywU e†î
`vI: 3 ˆm.wg. nGj, eGMÆi KGYÆi Š`NÆÅ KZ? ciÕ·iGK A¯¦tÕ©fvGe Õ·kÆ KiGj ‰G`i
a b
+ =2 K 3 ˆm.wg. L 3 ˆm.wg. ˆKG±`Êi `ƒiZ½ KZ?
b a
M 6 ˆm.wg. N 6 2 ˆm.wg. K 2 ˆm.wg. L 5 ˆm.wg.
1. a  b ‰i gvb KZ?
12. ‰KwU mgw«¼evü wòfzGRi cÉwZmvgÅ ˆiLvi M 7 ˆm.wg. N 12 ˆm.wg.
K 0 L1 M ab N ab
msLÅv KqwU? 23. ‰KwU wòfzGRi `ywU evüi Š`NÆÅ 9 ˆm.wg. I
a 3 b 3
2. b  + a  ‰i gvb KZ? 10 ˆm.wg. ‰es evü«¼Gqi A¯¦fzÆÚ ˆKvY 60
    K 4 L3 M2 N 1
K 8 L6 M4 N 2 13. cos3A = sin3A nGj, A ‰i gvb KZ? nGj, ‰i ˆÞòdj KZ?
1 K 0 L 15 M 30 N 60 K 22.5 eMÆ ˆm.wg. L 38.97 eMÆ ˆm.wg.
3. 
 hw` logx  =  6 nq, x ‰i gvb
64  M 45 eMÆ ˆm.wg. N 77.94 eMÆ ˆm.wg.
4 1  tan2A
KZ? 14. tanA = 5 nGj, tan2A
‰i gvb 24. 
 ‰KwU mge†îf„wgK ˆejGbi eÅvmvaÆ
1 1 KZ? 3 ˆm.wg. ‰es DœPZv 7 ˆm.wg. nGj, ‰i
K 2 L M N 2
2 2
5 4 3 16 mgMÉZGji ˆÞòdj KZ?
4. 
 ‰KwU eGMÆi evüi Š`NÆÅ x ‰KK nGj, K L M N
4 5 4 25 K 131.95 eMÆ ˆm.wg.
‰i cwimxgv I KGYÆi Š`GNÆÅi AbycvZ KZ? wbGPi ZG^Åi AvGjvGK (15 I 16) bs cÉGk²i L 188.50 eMÆ ˆm.wg.
K 2 2:4 L 2 2:3 Dîi `vI: M 197.85 eMÆ ˆm.wg.
M 2 2:2 N 2 2:1 R
OS  PQ, PR = 10cm, N 395.84 eMÆ ˆm.wg.
2x + 1 x1 O
5. hw` ( 3) =  3
3 nq, ZvnGj x PQ = 8cm, PQO = 55
25. tan(x + 30) = 3 nGj, x = ?
  S K 0 L 30 M 60 N 90
P Q
‰i gvb KZ? 26. ‰KwU ˆejGbi DœPZv 13 ˆm.wg. ‰es f„wgi
15. 
 QOR ‰i gvb KZ?
5 4 4 5
K 
4
L
5
M
5
N
4 K 60 L 90 M 110 N 145 eÅvmvaÆ 6 ˆm.wg. nGj, ‰i 
1 16. 
 OS ‰i Š`NÆÅ KZ? i. f„wgi ˆÞòdj 113.10 eMÆ ˆm.wg.
6. hw` a2 + a2 = 11 nq, ZvnGj
K 3 ˆm.wg. L 4 ˆm.wg. ii. eKÌZGji ˆÞòdj 490.09 eMÆ ˆm.wg.
1
i. a  = 3 M 5 ˆm.wg. N 6 ˆm.wg. iii. AvqZb 1470.27 Nb ˆm.wg.
a
1 2 17. ABC-‰ C = 1 mgGKvY, B = 2A ‰es wbGPi ˆKvbwU mwVK?
ii. a +  = 13
 a BC = 4 ˆm.wg. nGj, AB ‰i Š`NÆÅ KZ? K i I ii L i I iii
1 M ii I iii N i, ii I iii
iii. a3 
a3
= 18 K 2 ˆm.wg. L 4 ˆm.wg.
27. ‰KwU MvGQi Š`NÆÅ ‰es ‰i Qvqvi Š`GNÆÅi
wbGPi ˆKvbwU mwVK? M 6 ˆm.wg. N 8 ˆm.wg.
AbycvZ 3 : 3 nGj, D®²wZ ˆKvY KZ?
K i I ii L i I iii 18. sin = 21 nGj, sec = ?
K 60 L 45 M 30 N 15
M ii I iii N i, ii I iii K 0 L1 M
2
N 2 28. 
 ‰KwU mylg lofzGRi ˆK±`Ê nGZ
3
7. hw` P = {a, b, c}, Q = {b, d} nq, ZvnGj ˆhGKvGbv kxGlÆi `ƒiZ½ 6 ˆm.wg. nGj, ‰i
P\Q ‰i KqwU cÉK‡Z DcGmU ^vKGe?
19. ‰KwU wbw`ÆÓ¡ PZzfzÆR AuvKGZ KqwU Õ¼Z¯¨
ˆÞòdj KZ eMÆ ˆm.wg.?
K 2 L 3 Dcvî cÉGqvRb?
K 108 3 L 54 3
M 4 N 7 K 5 L4 M3 N 2
M 27 3 N 9 3
8. 
 4 + p + q + 32 ............ ‰KwU àGYvîi 20. 
 ‰KwU mylg cçfzGRi ewntÕ©
wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
aviv nGj, p2 + q2 ‰i gvb KZ? ˆKvYàGjvi ˆhvMdj KZ?
Dîi `vI:
K 80 L 264 M 320 N 576 K 72 L 108
ˆkÉwY eÅeavb 14-18 19-23 24-28 29-33 34-38
9. wbGPi ˆKvbwU gƒj` msLÅv? M 252 N 360
MYmsLÅv 15 20 35 22 18
K
5
L
27
M
6
N
8 21. ‰KwU mgevü wòfzGRi cwimxgv 18 ˆm.wg.
10 48 3 7 29. cÉPziK ˆkÉwYi gaÅgvb KZ?
nGj, ‰i DœPZv KZ?
1 1 K 16 L 21 M 26 N 31
10. = 3 + 2 2 nGj, a  ‰i gvb KZ? K 3 ˆm.wg. L 3 3 ˆm.wg.
a a 30. gaÅK ˆkÉwYi KÌgGhvwRZ MYmsLÅv KZ?
K  4 2 L 4 M0 N 4 2 M 3 5 ˆm.wg. N 6 ˆm.wg. K 35 L 70 M 92 N 110

1 K 2 N 3 N 4 N 5 L 6 K 7 L 8 M 9 L 10 K 11 N 12 N 13 L 14 M 15 M
Dîi

16 K 17 N 18 N 19 K 20 N 21 L 22 K 23 L 24 L 25 L 26 N 27 K 28 L 29 M 30 L
28 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
373 cvebv wRjv Õ•zj, cvebv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. O ˆK±`ÊwewkÓ¡ e†Gî OM ‰KwU eÅvmvaÆ| 10. 
 6 + 12 + 24 + 48 + ........ + 384 avivwUi 21. 
 ‰KwU mgGKvYx wòfzGRi AwZfzR
M we±`yGZ KqwU Õ·kÆK AuvKv hvGe? c`msLÅv KZ? 5 ˆmw´ŸwgUvi ‰es Aci ‰KwU evü 4 ˆmw´ŸwgUvi|
K 2 L 3 K 6 L 7 wòfzRwUi ˆÞòdj KZ eMÆ ˆmw´ŸwgUvi?
M 4 N 1 M 32 N 64 K 6 L 12
2.  3 ˆmw´ŸwgUvi
 I 4 ˆmw´ŸwgUvi 1 1 1 M 10 N 20
11. 1, , , , ....... AbyKÌGgi mvaviY c`
eÅvmvaÆwewkÓ¡ `yBwU e†î ciÕ·iGK A¯¦tÕ·kÆ 3 7 15 22. log625  2log5 = ?
KiGj, e†î«¼Gqi ˆK±`Ê«¼Gqi gaÅeZÆx `ƒiZ½ ˆKvbwU? K log5 L 2log5
KZ ˆmw´ŸwgUvi? 1 1 M log600 N log125
K n L n
K 5 L 7 2 1 2 +1 23. wbGPi ˆKvbwU Agƒj` msLÅv?
M 1 N 12 1 1 5 32
M n N
2 n K 0.5̇ L
3. ˆKvb e†Gîi AwaPvGci A¯¦tÕ© ˆKvY@ 243
K mħGKvY L mgGKvY
12. mgw«¼evü wòfzGR KqwU cÉwZmvgÅ ˆiLv AvGQ?
3 27
M Õ©ƒjGKvY N cƒiK ˆKvY K kƒbÅwU L ‰KwU M N
3
64 5
wbGPi ZG^Åi AvGjvGK (4 I 5) bs cÉGk²i Dîi M `yBwU N wZbwU
24. 
 wbGPi ˆKvbwU {x  ô : 11 < x < 13
`vI: 13. 
 ‰KwU eGMÆi KGYÆi Š`NÆÅ 3 2
‰es x ˆgŒwjK msLÅv} ˆmUwUGK ZvwjKv
P ˆmw´ŸwgUvi nGj, ‰i ˆÞòdj KZ eMÆ c«¬wZGZ cÉKvk KGi@
ˆmw´ŸwgUvi? K  L {}
K 18 L 9 M {0} N {11, 12, 13}
M 27 N 36 25. f(x) = x2  12x + 20 ‰es f(x) = 0 nGj,
Q

R 14. 
 ‰KwU e†Gîi cwiwa 6 ‰KK nGj, ‰i x = KZ?
tan = 3 ˆÞòdj KZ eMÆ ‰KK? K  10, 2 L  10,  2
4. 
 PR ‰i gvb KZ? K 3 L 6 M 2, 10 N 10,  2
K 3 L 4 M 9 N 18 26. i. a3  b3 = (a + b) (a2  ab + b2)
M 2 N 1 15. 
 ‰KwU mgevü wòfzGRi cwimxgv 18 ˆmw´ŸwgUvi ii. (a + b)2 = (a  b)2 + 4ab
5. cosec ‰i gvb KZ? nGj, ‰i ˆÞòdj KZ eMÆ ˆmw´ŸwgUvi? iii. 4ab = (a + b)2  (a  b)2
3 2 wbGPi ˆKvbwU mwVK?
K L K 3 3 L 4 3
2 3 K i I ii L ii I iii
M 9 3 N 18 3
1 M i I iii N i, ii I iii
M
2
N 2 16. wbGPi ˆKvbwU AwewœQ®² PjK?
27. 0.0374 ‰i mvaviY jGMi cƒYÆK KZ?
2 K R¯Ãmvj L eqm
6. 
 sec(  60) = nGj, sin(90  ) 
K 3

L 2
3 M wkÞv^Æxi msLÅv N RbmsLÅv
‰i gvb KZ? M 3 N 2
wbGPi ZG^Åi AvGjvGK (17 I 18) bs cÉGk²i
K 0 L 1 28. iÁ¼Gmi@
Dîi `vI:
3 1 i. mw®²wnZ evüàGjv ciÕ·i mgvb
M N ˆkÉwYeÅvwµ¦ 51-60 61-70 71-80 81-90 91-100
2 2 ii. mw®²wnZ ˆKvYàGjv ciÕ·i mÁ·ƒiK
MYmsLÅv 5 10 8 20 5
7. mƒGhÆi D®²wZ ˆKvY 60 nGj, 6 3 wgUvi iii. KYÆ«¼q ciÕ·iGK mgGKvGY ˆQ` KGi
Š`GNÆÅi ‰KwU LyuwUi Qvqvi Š`NÆÅ KZ wgUvi 17. gaÅK ˆkÉwYi DœPmxgv KZ? wbGPi ˆKvbwU mwVK?
K 70 L 71
nGe? K i I ii L ii I iii
M 80 N 90
K 3 3 L 2 3 M i I iii N i, ii I iii
18. cÉPziK ˆkÉwYi wbÁ²mxgv KZ?
M 9 N 6 29. D
K 61 L 71
8. a, b, c, d KÌwgK mgvbycvZx nGj@ C
M 81 N 91
i. c2 = bd
ii. a : b :: c : d 19. 
 `ywU msLÅvi AbycvZ 3 : 5 ‰es ‰G`i
iii. ad = bc M.mv.à. 6| msLÅv `ywUi Mo KZ? 30 30
A B
wbGPi ˆKvbwU mwVK? K 18 L 24
K i I ii L ii I iii M 30 N 36 wPGò ACD = KZ?
M i I iii N i, ii I iii 20. 
 mgevü wòfzGRi ‰KwU evüGK Dfq K 60 L 70
9. 2p  q = 8 ‰es p  2q = 4 nGj, w`GK ewaÆZ KiGj Drc®² ewntÕ© ˆKvY«¼Gqi M 80 N 90
p + q = KZ? ˆhvMdj KZ? 30. ‰KwU wòfzGRi KZwU ewnte†î AuvKv hvq?
K 8 L 4 K 240 L 180 K ‰KwU L `yBwU
M 12 N 0 M wZbwU N PviwU
M 120 N 360
1 N 2 M 3 K 4 M 5 L 6 K 7 N 8 N 9 L 10 L 11 K 12 L 13 L 14 M 15 M
Dîi

16 L 17 N 18 M 19 L 20 K 21 K 22 L 23 N 24 K 25 M 26 L 27 L 28 N 29 K 30 M
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 29
374 w`bvRcyi wRjv Õ•zj, w`bvRcyi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 0.1̇6̇  1.3̇ = KZ? 10. gaÅK ˆkÉwYi DœPmxgv KZ? wbGPi ˆKvbwU mwVK?
K 12.12̇ L 0.12̇ M 1.2 N 0.1̇2̇ K 45 L 50 M 55 N 60 K i I ii L i I iii
1 M ii I iii N i, ii I iii
2. A = {x  ô : 2 < x < 6} nGj 11. ‰KwU `ËeÅ 12 2 % ÞwZGZ weKÌq KiGj
i. A ˆmGU ˆgŒwjK msLÅv 2wU 20.  2 + 1 + 4 + 7 + ... avivi mvaviY c` ˆKvbwU?
KÌqgƒjÅ I weKÌqgƒGjÅi AbycvZ KZ?
ii. P(A) ‰i Dcv`vb msLÅv 8wU K 3n + 1 L 3n  5
K 7:8 L 8:9
M 3n  1 N 3n + 5
iii. A ˆmGU 2 «¼viv wefvRÅ msLÅv 1wU M 9:8 N 8:7
21.  cot(B + 30) = 0 nGj, cosecB =
 KZ?
wbGPi ˆKvbwU mwVK? 12. 35  2x  x2 ‰i Drcv`GK weGkÏlY KiGj
2 3 1
K i I ii L i I iii wbGPi ˆKvbwU mwVK nGe? K L
2
M N 0
3 2
M ii I iii N i, ii I iii K (5  x) (x + 7) L (7 + x)
wbGPi ZG^Åi AvGjvGK (22-24) bs cÉGk²i Dîi
3. ˆKvb msLÅvi PviàGYi mvG^ 3 ˆhvM KiGj M (5 + x) (7  x) N (5 + x) (7 + x)
`vI:
ˆhvMdj H msLÅvi 3 àY nGZ 5 ˆewk nGe? wbGPi ZG^Åi AvGjvGK (13 I 14) bs cÉGk²i
P ‰KwU wZbwU evüi Š`GNÆÅi AbycvZ 1 : 2 : 2 ‰es
K 1 L2 M3 N 4 Dîi `vI:
cwimxgv 60 wgUvi|
wbGPi ZG^Åi AvGjvGK (4 I 5) bs cÉGk²i Dîi 1 22.  wòfzRwU Kx aiGbi?
`vI: E
Q R K mgGKvYx L mgevü
B
M mgw«¼evü N welgevü
D

1
45 C 13. 2 P ‰i gvb ˆKvbwU? 23.  wòfzRwUi Þz`ËZg evüi Š`NÆÅ KZ wgUvi?
K 5 L 12 M 24 N 30
A K 30 L 45 M 60 N 90
24. 
 wòfzRwUi ˆÞòdj KZ eMÆwgUvi?
75 14. PQR ‰i ˆÞGò
K 36 15 L 36 5
5
B i. secP = cosecP ii. cosP + secP = M 34 15 N 34 5
2
4. 
 CDE ‰i gvb KZ wWMÉx? 1 25. ˆKvGbv e†Gîi AwaPvGc A¯¦wjÆwLZ ˆKvY
K 105 L 95 M 85 N 75 iii. tanR =
3 ˆKvb aiGbi?
5. 
 CAD ‰i gvb KZ wWMÉx? wbGPi ˆKvbwU mwVK? K mƒßGKvY L Õ©ƒjGKvY
K 30 L 35 M 40 N 45
K i I ii L i I iii M mgGKvY N cƒiK ˆKvY
6. 
 ‰KwU mylg lofzGRi ˆÞòdj 18 3 eMÆ M ii I iii N i, ii I iii 26. wbGPi ˆKvb A®¼qwU dvskb bq?
‰KK nGj, ‰i evüi Š`NÆÅ KZ ‰KK? 15. 
 2logx  log(4x  1) + log3 = 0 nGj, K {(5, 7), (1, 3), (2, 9)}
K 9 L6 M2 3N 3 x ‰i gvb KZ? L {(3, 1), (1, 6), (3, 3)}
x y z M {(2, 8), (1, 2), (2, 7)}
7. = = nGj, wbGPi ˆKvbwU mwVK?
2 3 4
K 1 L2 M3 N 4
N {(2, 3), (1, 2), (4, 5)}
K x :y:z=2 :3 :4 L x:y:z=4:3:2 16. wbw`ÆÓ¡ ‰KwU PZzfzÆR AvuKv mÁ¿e hw` ˆ`qv ^vGK
27. hw` x + x = 0 nq, ZGe 2  x +
1 1
M x :y:z=8 :9 :6 N x:y:z=6:8:9 i. `yBwU evü I wZbwU ˆKvY = KZ?
 x
8. 3 ˆm.wg. eÅvmwewkÓ¡ NbGKi ii. PviwU evü I ‰KwU ˆKvY
K 4 L3 M2 N 1
i. AvqZb 27 Nb ˆm.wg. iii. `yBwU evü I ‰G`i A¯¦fzÆÚ ˆKvY
28. ‰KwU wgbvGii DœPZv 20 3 wgUvi|
ii. mÁ·ƒYÆ ZGji ˆÞòdj 54 eMÆ ˆm.wg. wbGPi ˆKvbwU mwVK? wgbvGii cv`G`k nGZ 20 wgUvi `ƒGii
iii. KGYÆi Š`NÆÅ 3 3 ˆm.wg. K i I ii L i I iii
ˆKvGbv we±`yi AebwZ ˆKvY KZ wWMÉx?
wbGPi ˆKvbwU mwVK? M ii I iii N i, ii I iii K 30 L 45 M 60 N 90
4
K i I ii L i I iii 17. tanA = nGj, secA = KZ? 29. KÌgGhvwRZ MYmsLÅv cÉGqvRb
3
M ii I iii N i, ii I iii 3 5 3 4 i. gaÅK wbYÆGq
K L M N
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i 5 3 4 5 ii. cÉPziK wbYÆGq
Dîi `vI: 18. 
 64 + 32 + 16 + 8 + .... avivi beg c` KZ? iii. AwRf ˆiLv wbYÆGq
ˆkÉwY K 1 L
1
M
1
N
1 wbGPi ˆKvbwU mwVK?
31-35 36-40 41-45 46-50 51-55 56-60 2 4 8
eÅeavb K i I ii L i I iii
MYmsLÅv 5 8 18 15 12 10 19. a > 0, b > 0 ‰es a  1, b  1 nGj M ii I iii N i, ii I iii
i. logaMr = Mlogar
9. cÉPziK ˆkÉwYi cƒGeÆi ˆkÉwYi KÌgGhvwRZ 30. M = {a, b, c}, N = {1, 2, 3} nGj M  N ‰i
5
MYmsLÅv KZ? ( 3
ii. log0 a a = )
6 Dcv`vb msLÅv KZwU?
K 8 L 13 M 18 N 31 iii. logab  logba = 1 K 3 L6 M9 N 12

1 N 2 N 3 L 4 N 5 K 6 M 7 K 8 N 9 L 10 L 11 N 12 K 13 K 14 M 15 K
Dîi

16 K 17 L 18 M 19 M 20 L 21 K 22 M 23 L 24 K 25 K 26 L 27 M 28 M 29 L 30 M
30 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
375 iscyi wRjv Õ•zj, iscyi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 0.012̇ ‰i mvaviY f™²vsk wbGPi ˆKvbwU? 12. 
 EFG Kx aiGbi wòfzR? 22. mgevü wòfzGRi evüi Š`NÆÅ 8 cm nGj,
K
11
L
11
M
11
N
11 K mgevü wòfzR L mgw«¼evü wòfzR DœPZv KZ?
900 990 999 1000 M Õ©ƒjGKvYx wòfzR N welgevü wòfzR K 4 3 L 8 3 M 16 3 N 32 3
1 1
2. (x) = x + nGj,   = ? 13. 
 FEG-‰i gvb KZ? 23. ABCDE ‰KwU mylg cçfzR nGj, ‰i
x x  K 50 L 35 M 30 N 25 A¯¦t Õ© cÉwZwU ˆKvGYi cwigvY KZ?
1 1 14. ABC-‰ AB > AC ‰es D, BC ‰i
K x2 + L 1+ 2 K 92 L 108 M 110 N 112
x2 x
1 gaÅwe±`y nGj 24. x + 3y = 1 ‰es 5x + 15y = 5 mgxKiY
M x+ N x2 + 1 i. ABC < ACB
x
ii. AB + AC > 2AD
ˆRvUwU-
3. A = {2, 3, 5}
iii. ABC < ADC i. mãwZcƒYÆ ii. ciÕ·i wbfÆikxj
R = {(x, y) : x  A, y  A ‰es y = x  1) wbGPi ˆKvbwU mwVK? iii. ‰KwU gvò mgvavb wewkÓ¡
nGj, R ˆK ZvwjKv c«¬wZGZ cÉKvk Ki| K i I ii L i I iii wbGPi ˆKvbwU mwVK?
K {(2, 3)} L {(3, 2)}
M ii I iii N i, ii I iii K i I ii L ii I iii
M {(3, 3)} N {(5, 5)}
15. ABCD ‰KwU mvgv¯¦wiK| M i I iii N i, ii I iii
4.  U ‰i DcGmU msLÅv 64 nGj, U ‰i
 A D
m`mÅ msLÅv = ? (3a+6) 25. wPGòi mylg eüfzRwUi-
K 2 L4 M5 N 6 2a
B C
5. a2  2 a + 1 = 0 nGj,
1
i. a + = 2
1
ii. a2 + 2 = 2
wPGò a ‰i gvb KZ?
a a K 30 L 34.8 M 60 N 74
1 16. 
 O ˆK±`Ê wewkÓ¡ e†Gî OC = 3cm,
iii. a3 + 3 =  2
a i. NƒYÆb gvòv 4 ii. NƒYÆb ˆKvY 60
AB = 8cm ‰es OC AB nGj, OP = KZ? cÉwZwU ˆKvY 120
wbGPi ˆKvbwU mwVK? iii.
K i I ii L ii I iii O wbGPi ˆKvbwU mwVK?
M i I iii N i, ii I iii
C
B K i I ii L ii I iii
A
6. 0.00357 ‰i cƒYÆK KZ? P M i I iii N i, ii I iii
  K 4 L5 M6 N 8 26. ˆhvwRZ MYmsLÅv cÉGqvRb-
K 3 L3 M2 N 2 17. 
 7cm I 5cm eÅvmvGaÆi `ywU e†î ciÕ·i
i. Mo wbYÆGq ii. gaÅK wbYÆGq
7. 
 1600 ‰i jM 4 nGj wfwî KZ? A¯¦tÕ·kÆ KiGj ZvG`i ˆK±`Ê«¼Gqi `ƒiZ½ KZ?
iii. AwRf ˆiLv Aâb KiGZ
K 2 5 L 2 10 K 2 L5 M7 N 12
M 10 2 N 3 2 18. 

wbGPi ˆKvbwU mwVK?
E D
wbGPi ZG^Åi AvGjvGK (8 I 9) bs cÉGk²i Dîi `vI:
C 20 K i I ii L ii I iii
M i I iii N i, ii I iii
84
`yB AâwewkÓ¡ ‰KwU msLÅvi ‰KK Õ©vbxq B A
Aâ `kK Õ©vbxq AGâi AGaÆK ‰es Aâ 27. AwRf ˆiLvi ˆÞGò ˆKvbwU mwVK?
`yBwUi àYdj 25| wPGò ADE ‰i gvb KZ? K Ea»ÆMvgx L wbÁ²Mvgx
8. msLÅvwU KZ? K 64 L 76 M 70 N 36 M mgv¯¦ivj N DjÁ¼
K 84 L 48 M 42 N 24 19. eMÆGÞGòi ‰K evüi cwigvY x ‰KK nGj, 28. wbGPi ˆKvbwU wewœQ®² PjGKi D`vniY?
9. msLÅvwUi ˆgŒwjK àYbxqGKi ˆmU ˆKvbwU? Dnvi cwimxgv I KGYÆi Š`GNÆÅi AbycvZ KZ? K eqm L Zvcgvòv
K {1, 2, 2, 2, 3} L {2, 2, 2, 2, 3} K 2 2:4 L 2 2:1 M RbmsLÅv N IRb
M {2, 3, 7} N {2, 2, 3, 7} M 2:2 N 2 2:2
2(x2 + 1) wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
10.  x2 + 1 =
 ‰i mgvavb ˆmU KZ? 20. A
Dîi `vI: A
x
K {1} L {0} M {2} N {3} 3 x
11.  PQR ‰i Q I R ‰i mgw«¼L´£K«¼q
  C
O we±`yGZ wgwjZ nGqGQ P = 50 nGj, B 4 O

QOR = KZ? tan + cot  sec = KZ? x+60

K 40 L 65 M 115 N 130 5 5 25 5 B C
K L M N
wbGPi ZG^Åi AvGjvGK (12 I 13) bs cÉGk²i 4 12 12 6
Dîi `vI: 21.  sin4A + sin2A = 1 nGj,
 wPGò O e†Gîi ˆK±`Ê|
i. sin2A = cosA ii. tanA = cosecA
C A
D
iii. tanA.cosecA = 1 29. 
 BAC = KZ?
wbGPi ˆKvbwU mwVK? K 30 L 45 M 60 N 120
35 30 K i I ii L i I iii 30. 
 cÉe†«¬ BOC ‰i gvb KZ?
F G K 120 L 180 M 240 N 280
M ii I iii N i, ii I iii
1 K 2 M 3 L 4 N 5 M 6 K 7 L 8 K 9 M 10 M 11 M 12 N 13 K 14 N 15 L
Dîi

16 L 17 K 18 L 19 L 20 N 21 K 22 K 23 L 24 K 25 L 26 L 27 K 28 M 29 M 30 M
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 31
376 KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, iscyi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 0.53̇ ‰i mvaviY f™²vsk ˆKvbwU? 12. wbw`ÆÓ¡ PZzfÆR
z AuvKv mÁ¿e hw` ˆ`qv ^vGK 19. 
 3x  5y  7 = 0 ‰es 6x  10y  15 = 0

K
53
L
50 i. PviwU evü I ‰KwU KYÆ mgxKiY ˆRvUwU
90 90
ii. wZbwU evü I `yBwU KYÆ K mgém L mgvavb AmsLÅ
53 24
M N M ‰KwUgvò mgvavb AvGQ
100 45 iii. `yBwU evü I wZbwU ˆKvY
N ciÕ·i AwbfÆikxj
2. 
 A = {x  ô : 0  x <5} nGj, P(A) ‰i wbGPi ˆKvbwU mwVK? 20. 3x  3x + 3x  3x + 3x  ....... avivwUi 12
Dcv`vb msLÅv KZ? K i I ii L i I iii cG`i mgwÓ¡ KZ?
K 5 L 16
M ii I iii N i, ii I iii K 36x L 6x
M 32 N 64
M 3x N 0
13. 
 `yBwU e†î ciÕ·iGK ewntÕ·kÆ KGi|
3. f(x) = x + 1 nGj, f  x  = KZ?
3 1 21. iÁ¼Gmi cÉwZmvgÅ ˆiLv KqwU?
‰G`i ‰KwUi eÅvm 10 ˆm.wg. ‰es
K ‰KwU L `yBwU
K 3x + 1 L 3+x AciwUi eÅvmvaÆ 4 ˆm.wg.| e†î«¼Gqi M wZbwU N PviwU
3+x x
M N ˆKG±`Êi gaÅeZÆx `ƒiZ½ KZ ˆm.wg.? 22. ar + ar3 + ar5 + ..... avivwUi n-Zg c` KZ?
x 3x + 1
1 K 1 L 6 K arn L arn1
4.  y2  6 y + 1 = 0 nGj, y  KZ?

y M 9 N 14 M ar 2n1
N ar2n2
K L 2 7 23. 
 `yBwU msLÅvi AbycvZ 2 : 5 ‰es ‰G`i
2 14. tan2  sec2 + 3 = KZ?
M 6 N 10
A¯¦i 36 nGj, Þz`ËZg msLÅvwU KZ?
1 4 K 12 L 24
5. wbGPi ˆKvbwU x3  2x  4 eüc`xi ‰KwU K
3
L
3 M 36 N 60
Drcv`K? M 3 N 2 wbGPi ZG^Åi AvGjvGK (24 I 25) bs cÉGk²i
K x4 L x2 25 Dîi `vI:
M x+2 N x+4 15. ‰KwU `ËeÅ % ÞwZGZ weKÌq KiGj
2 ˆkÉwYeÅwÚ 11-20 21-30 31-40 41-50 51-60
6. logaN = x nGj weKÌqgƒjÅ I KÌqgƒGjÅi AbycvZ KZ? MYmsLÅv 4 15 20 10 7
i. N > 0 ii. x > 0
K 7:8 L 8 :9 24.  gaÅK wbYÆGqi ˆÞGò FC ‰i gvb KZ?
iii. a > 0, a  1 K 19 L 20
M 9:8 N 8:7
wbGPi ˆKvbwU mwVK? M 28 N 39
K i I ii L i I iii
wbGPi ZG^Åi AvGjvGK (16 I 17) bs cÉGk²i f1
25. 
 cÉPziK wbYÆGqi ˆÞGò ‰i gvb KZ?
M ii I iii N i, ii I iii Dîi `vI: f1 + f2
D K 0.33 L 0.67
7. 320.42 I 0.0931 ‰i jGMi cƒYÆK ‰i M 0.79 N 0.87
mgwÓ¡ KZ? 2 2
2 60 26. ‰KwU eGMÆi A¯¦e†ÆGîi eÅvmvaÆ 3 ˆm.wg.
K 5 L 3 2 nGj, eGMÆi evüi Š`NÆÅ KZ ˆm.wg.?
E F
M 1 N 0 G
K 3 L 3
8. (x + 3)2 = x2 + bx + c mgxKiGY b I c ‰i 16. 
  ‰i gvb KZ?
M 6 N 6
gvb KZ? K 15 L 30 27. ‰KwU wmwj´£vGii f„wgi eÅvm 6 ˆm.wg. I
K 2, 9 L 6, 9 M 45 N 60 DœPZv 7 ˆm.wg. nGj, eKÌc†GÓ¤i ˆÞòdj
M 3, 6 N 9, 6 17. 
 DGF-‰ KZ eMÆ ˆm.wg.?
9. mgw«¼evü wòfzGRi f„wg msj™² ˆKvY«¼Gqi i. FG = 6 K 9 L 21
cÉGZÅKwU kxlÆGKvGYi w«¼àY nGj kxlÆ 3 M 42 N 63
ii. cosF = 28.  ‰i Q = 90 ‰es P = 30
ˆKvGYi cwigvY KZ? 2  PQR
K 30 L 35 iii. sin(GDF + DFG) = 1 nGj wbGPi ˆKvbwU mwVK?
M 36 N 38 wbGPi ˆKvbwU mwVK? K PR = 2 QR L PR = 2QR
10. ‰KwU mijGKvGYi mgw«¼LíKGK Kx eGj? K i I ii L i I iii M PR = 3 QR N PR = 3QR
K f„wg L jÁ¼ 29.  ˆKvb aiGbi wòfzGRi cwie†Gîi
 ˆK±`Ê
M ii I iii N i, ii I iii
M DœPZv N mƒßGKvY wòfzRwUi e†nîg evüi Dci AewÕ©Z?
18. 
 ‰KwU jvwVi Qvqv ‰i Š`GNÆÅi KZàY
K mgevü L mƒßGKvYx
11. Õ©ƒjGKvYx wòfzGRi Õ©ƒjGKvY Qvov evwK nGj, D®²wZ ˆKvY 60 nGe? M Õ©…jGKvYx N mgGKvYx
ˆKvY `yBwU KZ nGj wòfzR Aâb mÁ¿e? 1 1
K L 30. y2 = 7 y mgxKiGYi gƒj KqwU?
K 30 I 60 L 45 I 45 3 2 K 3 L 2
M 40 I 50 N 50 I 30 M 2 N 3 M 1 N 0

1 N 2 L 3 K 4 K 5 L 6 L 7 N 8 L 9 M 10 L 11 N 12 N 13 M 14 L 15 K
Dîi

16 K 17 N 18 K 19 N 20 N 21 L 22 M 23 L 24 K 25 K 26 M 27 M 28 L 29 N 30 L
32 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
377 KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, jvjgwbinvU

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg
«¼viv mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. wZbwU KÌwgK Õ¼vfvweK msLÅvi àYdj 12. `yBwU e†î ciÕ·iGK ewntÕ·kÆ KiGj e†î 20. ‰KwU wòfzGRi wZbwU ˆKvGYi AbycvZ
meÆ`vB wbGPi ˆKvbwU «¼viv wefvRÅ? `yBwUi gGaÅ mGeÆvœP KZwU mvaviY Õ·kÆK 2: 3 : 5 nGj, Þz`ËZg ˆKvGYi gvb KZ wWwMÉ?
K 5 L6 AuvKv hvq? K 18 L 20
M 7 N 11 K 4 L3 M 25 N 36
2.  3a + 1
 f(a) = 3 nGj, f(1) ‰i gvb wbGPi M 2 N1 21. 2x  y = 8, x  2y = 4 nGj, x + y = KZ?
3a + 3 13. 
 K 0 L4 M8 N 12
ˆKvbwU? B 22. 2 + 4 + 6 + 8 + .. .. avivwUi cÉ^g n msLÅK
2 C
K
3
L3 cG`i mgwÓ¡ KZ?
K n2 L n(n + 1)
M 0 N AmsãvwqZ O
M n(n + 2) N n(n + 3)
3.  A ˆmGUi cÉK‡Z DcGmGUi msLÅv 63
 A D wbGPi ZG^Åi AvGjvGK (23 I 24) bs cÉGk²i
nGj, A ‰i m`mÅ msLÅv KZ? Dîi `vI:
K 2 L4 M5 N 6 wPGò, O e†îwUi ˆK±`Ê ‰es AOD = 90
nGj, ABD + ACD = KZ? ‰KwU mgv¯¦i avivi cÉ^g c` 3 ‰es 10 Zg c` 21
4.  a  b = 6, a + b = 3 nGj,

K 90 L 120 M 180 N 360 23. 
 avivwUi mvaviY A¯¦i KZ?
a  b = KZ? K 8 L 2 M2 N 24
K 2 L 2 M1 N  3
14. ABCD PZzfƃGRi A = 120, C = 60
nGj 24. 
 avivwUi cÉ^g 10wU cG`i mgwÓ¡ KZ?
5. x2  2x + 1 = 0 nGj K 330 L 150
i. B + D = 180
1 1
ii. x2 + 2 = 0 M 120 N 99
i. x+ = 2 ii. BC ˆK E chƯ¦ ewaÆZ KiGj ECD = 120
x x 25.
1 iii. A, B, C, D we±`y PviwU mge†î A
iii. x3 + 3 =  2
x wbGPi ˆKvbwU mwVK?
wbGPi ˆKvbwU mwVK? K i I ii L i I iii M N
K i I ii L i I iii M ii I iii N i, ii I iii O
M ii I iii N i, ii I iii 15. A ‰i ˆKvGbv ‰KwU gvGbi RbÅ B C

6. 

64
x = 16 nGj, x = KZ? i. sin A =
3
ii. cos A =
5 wPGò, BC || MN nGj
(64) 5 7 i. BOC I MON m`†k
1 1 5
K 4 L M N 4 iii. sec A = ii. AM : BM = AN : CN
3 3 4 iii. BO : ON = CO : OM
7. log625  2log5 = KZ? wbGPi ˆKvbwU mwVK? wbGPi ˆKvbwU mwVK?
K log600 L log125 K i I ii L i I iii K i I ii L i I iii
M log25 N log5
M ii I iii N i, ii I iii M ii I iii N i, ii I iii
x2 1
8. =2 nGj mgxKiGYi mgvavb wbGPi ZG^Åi AvGjvGK (16 I 17) bs cÉGk²i 26. Õ©ƒjGKvYx wòfzGRi Õ©…jGKvGYi wecixZ
x1 x1
ˆmU wbGPi ˆKvbwU? Dîi `vI: evüi Dci AwâZ eMÆGÞGòi ˆÞòdj Aci
K {1} L {} M {} N {2} sin cos
= `yB evüi Dci AwâZ eMÆ ˆÞGòi ˆÞòdGji
x y
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i K mgwÓ¡i mgvb L mgvb
x
Dîi `vI: 16. 
 = 1 nGj,  = KZ wWwMÉ? M mgwÓ¡ AGcÞv eo N mgwÓ¡ AGcÞv ˆQvU
y
A
K 30 L 45 27. 
 ‰KwU mvgv¯¦wiK ˆÞGòi ˆÞòdj
M 60 N 90 120 eMÆ ˆm.wg. ‰es ‰KwU ˆKŒwYK we±`y
O 17. 
 sin = KZ? ˆ^GK Zvi wecixZ KGYÆi Dci AwâZ
K
x
L
y jGÁ¼i Š`NÆÅ 5 ˆm.wg. nGj, DÚ KYÆwUi Š`NÆÅ
B C x2 + y2 x2 + y2 KZ ˆm.wg.?
wPGò ABC = 70, AB = AC, OB I OC nGjv M
x
N
y K 20 L 24 M 26 N 28
B I C ‰i mgw«¼LíK| x2  y2 x2  y2 28. ‰KwU mgevü wòfzGRi ˆÞòdj 9 3
9. BAC = KZ wWwMÉ? 18. 
 mƒGhÆi D®²wZ ˆKvY 90 nGj, 90 wgUvi eMÆwgUvi nGj, ‰i cwimxgv KZ wgUvi?
K 70 L 65 M 55 N 40 Š`GNÆÅi ‰KwU UvIqvGii Qvqvi Š`NÆÅ KZ K 18 L 20 M 30 N 45
10. BOC = KZ wWwMÉ? wgUvi nGe? 29. DcvGîi mGeÆvœP gvb 57, cwimi 37 nGj,
K 55 L 100 M 110 N 120 K 0 L 45 M 60 N 90 DcvGîi meÆwbÁ² gvb KZ?
11. ÷ay cwimxgv Rvbv ^vKGjB AuvKv mÁ¿e 19. a, b, c, d KÌwgK mgvbycvwZ nGj K 21 L 22 M 23 N 27
i. eMÆ ii. AvqZ i. c2 = bd ii. a : b = c : d 30. 

iii. mgevü wòfzR iii. ad = bc 21-30 31-40 41-50 51-60 61-70 71-80
wbGPi ˆKvbwU mwVK? wbGPi ˆKvbwU mwVK? 8 10 5 10 12 8
K i I ii L i I iii cÉPziK KZ?
K i I ii L i I iii K 60 L 60.33 M 62 N 63.33
M ii I iii N i, ii I iii M ii I iii N i, ii I iii

1 L 2 N 3 N 4 L 5 N 6 L 7 M 8 M 9 N 10 M 11 L 12 L 13 K 14 N 15 L
Dîi

16 L 17 K 18 K 19 N 20 N 21 L 22 L 23 M 24 M 25 N 26 M 27 L 28 K 29 K 30 N
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 33
378 KzwoMÉvg miKvwi DœP we`Åvjq, KzwoMÉvg

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 4x  3y = 10 ‰es x  y = 1 nGj x ‰i 11. D
5 ˆm.wg. C 21. 
 log 3 27 ‰i gvb KZ?
gvb KZ? F K 1 L 3
K 6 L 7 M 6 N 9

ˆm.wg.

ˆm.wg.
30
M 12 N 13 O E 4y + 1 1
22. f(y) = 4y  1 nGj f   ‰i gvb KZ?

5
A
2. wPGò, DE || BC nGj,  2
50
ADE ‰i gvb KZ? D E A
ˆm.wg.
B 1
K 50 L 64 B 64 5 K 1 L 
3
i. eMÆwUi ˆÞòdj = 25 eMÆ ˆm.wg.
C
M 66 N 114 1
ii. e†Gîi cwiwa 15.71 ˆm.wg. M N 1
3.  cvGki wPòvbyhvqx Z^ÅàGjv jÞÅ Ki@
 3
i. BOE = 2 A iii. EOF e†îvsGki ˆÞòdj = 1.64 eMÆ ˆm.wg. wbGPi ZG^Åi AvGjvGK (23 I 24) bs cÉGk²i
ii. DOE = OAD + ODA wbGPi ˆKvbwU mwVK? Dîi `vI:
O
1
iii. BOD = BAD K i I ii L i I iii x=7+4 3
2 B E D
M ii I iii N i, ii I iii 1
wbGPi ˆKvbwU mwVK? C 23. 
 x2 + 2 ‰i gvb KZ?
12. 0.000045 msLÅvwUGZ jGMi cƒYÆK KZ? x
K i I ii L i I iii   K 190 L 194
M ii I iii N i, ii I iii K 5 L 4 M 198 N 200
4. Pvi cvLv wewkÓ¡ ‰KwU wmwjs dÅvGbi NƒYÆb M 4 N 5 1
13.  `ywU
 msLÅvi AbycvZ 7 : 10
‰es ZvG`i 24. 
 x ‰i gvb KZ?
cÉwZmgZvi gvòv KZ? x
K 1 L 2 j.mv.à 910 nGj, msLÅv `ywUi A¯¦i KZ? K 2 3 L 4
M 3 N 4 K 39 L 91
M 8 3 N 14
5.  p + q = 3 ‰es p  q = 2
 nGj@ M 130 N 221
1
3+ 2 14. y2  5y = 0 mgxKiYwUi mgvavb ˆmU 25. 3 x = 81 nGj, x ‰i gvb KZ?
i. p =
2 wbGPi ˆKvbwU? K 4 L 3
ii. p2  q2 = 6 K {0} L { 5} M 3 N 4
1
iii. pq = M {0, 5} N {0, 5} . .
4 26. 
 x = 0.4 ‰es y = 0.8 nGj@
wbGPi ˆKvbwU mwVK? wbGPi wPò ˆ^GK (15 I 16) bs cÉGk²i Dîi . 32
K i I ii L i I iii `vI: A i. x + y = 1.3 ii. xy =
81
5 wgUvi
M ii I iii N i, ii I iii iii.
x
= 0.5
wbGPi DóxcGKi AvGjvGK (6 I 7) bs cÉGk²i y
Dîi `vI: A C B wbGPi ˆKvbwU mwVK?
12 wgUvi
K i I ii L i I iii
N 15. cosC ‰i gvb KZ?
M ii I iii N i, ii I iii
5 ˆm.wg.

5 12
K L 27. A
13 13
C
B 12 ˆm.wg. 13 13 65
M N
12 5
6. ABC ‰i cwimxgv KZ ˆm.wg.? P
C
K 15.0 L 18.81 16. cotA + tanC = KZ? B 102

M 24.81 N 30.0 K
5
L
3 wPGò P ‰i gvb KZ?
7. BN ‰i Š`NÆÅ KZ ˆm.wg.? 6 2 K 37 L 65
K 4.62 L 6.5 181 169
M N M 78 N 115
M 9.23 N 18.46 65 60
28. ‰KwU mgw«¼evü mgGKvYx wòfzGRi mgvb
8. A 17. 
1+3+5+ .......... + 101 avivwUi c`
mgvb evü«¼Gqi Š`NÆÅ 12 ˆm.wg. nGj,
msLÅv KZ?
C K 51 L 101 wòfzRwUi ˆÞòdj wbGPi ˆKvbwU?
10 wgUvi

60
M 201 N 204 K 24 eMÆ ˆm.wg. L 36 eMÆ ˆm.wg.
D B
18. (5, 3) we±`ywU x-AÞ ˆ^GK KZ `ƒGi M 72 eMÆ ˆm.wg. N 144 eMÆ ˆm.wg.
 wPGò AB ‰i Š`NÆÅ KZ wgUvi?

K 20.0 L 21.55 AewÕ©Z? wbGPi DóxcGKi AvGjvGK (29 I 30) bs cÉGk²i
M 24.14 N 30.0 K 5 ‰KK L  3 ‰KK Dîi `vI:
9. A = {a, b, c, d}, B = {b, c, d, e} nGj M 3 ‰KK N 5 ‰KK ˆkÉwYeÅvwµ¦ 1519 2024 2529 3034
P (AB) ‰i m`mÅ msLÅv wbGPi ˆKvbwU? 19. a : b = c : d nGj, wbGPi ˆKvbwU mwVK? MYmsLÅv 2 8 10 6
K 3 L 5 K bc = ad L ab = cd 29. cÉPziK wbYÆGq (f1 + f2) ‰i gvb KZ?
M 8 N 32 M abc = d N bcd = a K 4 L 6
1 20. tanA = 1 nGj, cosA ‰i gvb KZ?
10.  1 + 3  ...... avivwUi 8g c` KZ? M 8 N 10
3 1 1 30. DcvîmgƒGni gaÅK ˆKvbwU?
K 27 3 L 27 K L
2 2 K 26.2 L 26.5
M 27 N 27 3 M 2 N 2 M 31.0 N 36.5

1 L 2 M 3 K 4 N 5 L 6 N 7 K 8 N 9 M 10 L 11 N 12 K 13 K 14 N 15 L
Dîi

16 K 17 K 18 M 19 K 20 K 21 M 22 M 23 L 24 K 25 K 26 N 27 K 28 M 29 L 30 L
34 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
379 AvIqvi ˆjwW Ae dvwZgv MvjÆm nvB Õ•zj, KzwgÍÏv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
a b 2 3 
1. 
 = = nGj, a : c ‰i gvb KZ?
b c 3
11. tan = 4 nGj, sec2 = KZ? 20. 3.2 ‰i mvaviY f™²vsk wbGPi ˆKvbwU?
K 2:3 L 3:4 9 16 25 9 1 2 5 7
K L M N K 3 L3 M3 N 3
16 25 16 25 3 9 9 9
M 4:9 N 9:4
2. wbGPi ˆKvbwU x2  11x  12 ivwkwUi ‰KwU 12. 0.000045 msLÅvwUGZ mvaviY jGMi cƒYÆK 21. AebwZ ˆKvGYi gvb KZ wWMÉx nGj, LuywUi
KZ? Š`NÆÅ Qvqvi Š`GNÆÅi 3 àY nGe?
Drcv`K?
K 30 L 45 M 60 N 90
K x  12 L x4  
M x3 N x1
K 5 L4 M4 N 5 22. 
 a, b, c  R; a > b > 0 ‰es c < 0 nGj
13. 
 wbGPi ˆKvbwU mwVK?
DóxcGKi AvGjvK 3 I 4 bs cÉGk²i Dîi `vI: A
K ac = bc L ac > bc
A 50
D E M ac < bc N ab < bc
8 cm 64
23. `yBwU msLÅvi AbycvZ 5 : 6 ‰es ZvG`i
B C j.mv.à 150 nGj, M.mv.à KZ?
B F C
K 5 L6 M 11 N 30
10 cm
wPGò DE | | BC nGj, ADE ‰i gvb KZ? 24. 
 eMÆGÞGòi ‰K evüi cwigvc x ‰KK nGj,
3. 
 ABF ‰i cwimxgv KZ? K 50 L 64
K 40 cm L 22.43 cm
Dnvi cwimxgv I KGYÆi Š`GNÆÅi AbycvZ KZ?
M 66 N 114
M 20 cm N 18.43 cm K 2 2:4 L 2 2:3
1
4. 
 AFC ‰i ˆÞòdj KZ? 14.  1 + 3  ------ avivwUi 8g c` KZ? M 2 2:2 N 2 2:1
3
K 20 cm2 L 40 cm2 25. ABC mgGKvYx wòfzR nGe, hw` ‰i
K 27 3 L 27
M 60 cm2 N 80 cm2 evüàGjvi cwigvY nq@
M 27 N 27 3
5. y = 2x + 1 dvskGbi@ i. 5, 12, 13 ‰KK
15. BsGiwR S eGYÆi NƒYÆb ˆKvY KZ? ii. 6, 8, 10 ‰KK
i. ˆjLwPGòi ‰KwU we±`y (1, 3)
K 90 L 180
ii. ˆjLwPò ‰KwU mijGiLv iii. 14, 16, 20 ‰KK
M 270 N 360
iii. ˆjLwPò ‰KwU e†î wbGPi ˆKvbwU mwVK?
16. hw` A = {a, b, c}, B = {b, c, d} nq ZGe,
wbGPi ˆKvbwU mwVK? K i I ii L i I iii
A\B wbGPi ˆKvbwU?
K i I ii L i I iii M ii I iii N i, ii I iii
K {a} L {d}
M ii I iii N i, ii I iii
26.  wPGò, O ˆK±`Ê wewkÓ¡ S
M {a, b, c, d} N {b, c}
6. ˆKvbwU wewœQ®² PjK? e†Gî PQRS A¯¦wjÆwLZ O R
17. 

nGqGQ| SPQ = KZ? P 80 T
K Zvcgvòv L cvwLi msLÅv A
Q
M eqm N DœPZv C K 80 L 90 M 180 N 360
DóxcGKi AvGjvGK (7 I 8) bs cÉGk²i Dîi `vI: 60 10 wg. 27. 6x  y = 5 ‰es 5x  2y = 2 nGj, x + y = KZ?
B
B K 2 L3 M4 N 5
S D
2x  1
C
60
A wPGò AB ‰i Š`NÆÅ KZ? 28. ( 5)x+1 =  5
3
nGj,
O
K 20.0 wg. L 21.55 wg.  
M 24.14 wg. N 30 wg. x-‰i gvb KZ?
O ˆK±`ÊwewkÓ¡ e†Gî AC = 12 ˆm.wg. 1 5
× × K L M1 N 5
7. AB PvGci Š`NÆÅ KZ? 18. x = 0.4 ‰es y = 0.8 nGj@ 7 7
K 40.84 cm L 12.57 cm i.
×
x + y = 1.3 ii. xy =
32 DóxcGKi AvGjvK (29 I 30) bs cÉGk²i Dîi
M 6.28 cm N 3.14 cm 81
`vI: A
x
8. e†îKjv AOB ‰i ˆÞòdj KZ? iii. = 0.5
2 2
y 5 ˆm.wg.
K 150.80 cm L 75.40 cm
M 40.84 cm2 N 18.85 cm2
wbGPi ˆKvbwU mwVK? C B
12 ˆm.wg.
7.2x+1  13.2x K i I ii L i I iii
9. 

2x
= KZ? 29. cosC ‰i gvb KZ?
M ii I iii N i, ii I iii
K 2 L 1 M 1 N 2 5 12 13 13
4y + 1 1 K L M N
10. hw` a + b = 5 ‰es a  b = 3 nq, ZGe 19. f(y) = 4y  1 nGj f   2  ‰i gvb KZ? 13 13 12 15
  30. cotA + tanC = KZ?
a2 + b2 = KZ? 1 1
K 1 L M N 1 5 3 181 169
K 2 L4 M8 N 64 3 3 K L M N
6 2 65 60

1 M 2 K 3 L 4 K 5 K 6 L 7 M 8 N 9 M 10 L 11 M 12 K 13 M 14 L 15 L
Dîi

16 K 17 N 18 N 19 M 20 L 21 M 22 M 23 K 24 N 25 K 26 K 27 L 28 N 29 L 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 35
380 beve dqRyG®²mv miKvwi evwjKv DœP we`Åvjq, KzwgÍÏv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. m : n ‰i w«¼fvwRZ AbycvZ ˆKvbwU? 12. 
 a > 0 ‰es a  1 nGj@ 20. 

6 D C
K n 2 : m2 L n:m i. loga1 = 0 E
M m2 : n2 N m: n ii. logaa = 1
2. A = {1, 3, 5} I B = {2, 4, 6} nq, ZGe iii. log a0 = 1 10
A I B ‰i Dcv`vbàGjvi gGaÅ x ≥ y  1 wbGPi ˆKvbwU mwVK?
mÁ·KÆ weGePbvq mwVK wiGjkb ˆKvbwU? K i I ii L i I iii
A B

[ˆhLvGb, x  A, y  B]
M ii I iii N i, ii I iii ABCD ‰KwU eMÆ nGj, AC = ?
K {}
L {(1, 2), (3, 4), (5, 6)} 13. 0.0024 ‰i mvaviY jGMi cƒYÆK KZ? K 5 2 L 6 2
M {(2, 4), (5, 2), (5, 4)} K 2 L 3 M 8 2 N 10 2
N {(3, 2), (3, 4), (5, 2), (5, 4)}   21. wZb cvLv wewkÓ¡ ‰KwU dÅvGbi NƒYÆb ˆKvb
.. M 2 N 3
3. 5.132 ‰i mvgvbÅ f™²vsk cÉKvwkZ i…c 2 KZ wWMÉx?
1 1
ˆKvbwU? 14. x + x  = 4 nGj, x3 + 3 = ?
x K 90 L 80
51 53
  M 60 N 120
K L K 0 L 2
90 90 22. 
 9y2 + 36 ‰i mvG^ KZ ˆhvM KiGj
512 5081 M 5 N 8
M
99
N
990
ˆhvMdj cƒYÆeMÆ nGe?
1
4. 
 (180  A) ‰i mÁ·ƒiK ˆKvY KZ? 15. 
 x(3x  2) = nGj@
3 K 54y L 27y
M 18y N 36y
K A L 180 + A 1
i. 9x2 + =6 1 + sin2
M 90 + A N 180 9x2 23. 
  = 45 nGj, =?
1  sin2
wbGPi ZG^Åi AvGjvGK (5 I 6) bs cÉGk²i Dîi `vI: 1 2
ii. (3x + ) = 8 1
ˆKvGbv we`ÅvjGqi wbeÆvPbx cixÞvi 40 Rb 3x K 1 L
3
wkÞv^Æxi MwYZ welGq mGeÆvœP bÁ¼i 96 ‰es 1
iii. 3x + = 2 1
3x M N 3
meÆwbÁ² bÁ¼i 55| 3
5. DcvGîi cwimi KZ? wbGPi ˆKvbwU mwVK?
24. ABC mgevü wòfzGRi AB = 2 ˆm.wg.
K 40 L 41 K i I ii L i I iii
M 42 N 43
‰es AD  BC nGj, AD = KZ?
M ii I iii N i, ii I iii 4
6. ˆkÉwY eÅeavb 5 aGi ˆkÉwY msLÅv KZ? K L 3
K 7 L 8 16. 
 3 + m + n + 81 àGYvîi avivfzÚ nGj, 3
M 9 N 10 n =? 2 1
M N
2
7. 
 ABC ‰ C = 90 ‰es B = 2A K 27 L 9 3
nGj, wbGPi ˆKvbwU mwVK? M 6 N 3 25. ABC ‰i AB I AC evüi gaÅwe±`y
K AC = 2 AB L AB = 2 BC wbGPi ZG^Åi AvGjvGK (17 I 18) bs cÉGk²i h^vKÌGg P I Q nGj,
M BC = 2 AC N AC = BC ABC : APQ = KZ?
Dîi `vI:
8. (x) = x5 + 2x  a, (1) = 0 nGj, a = ? K 1:2 L 4:1
K 7 L 3 6% nvi gybvdvq 15000 UvKv 3 eQGii RbÅ
M 1:4 N 2:1
M 3 N 7 wewbGqvM Kiv nGjv| 26. cÉ^g 100wU Õ¼vfvweK msLÅvi mgwÓ¡ KZ?
9.  wbGPi ˆKvbwU Agƒj`?
 17. mij gybvdv KZ UvKv? K 1999 L 5500
3 9 K 2900 L 2700 M 5050 N 1000
K L
8 16
M 1800 N 450 27. e†Gî A¯¦wjÆwLZ mvgv¯¦wiK ‰KwU@
5
M
243
N 7 18. PKÌe†w«¬ gybvdv KZ UvKv? K eMÆGÞò L iÁ¼m
32
K 1862.24 M AvqZGÞò N UÇvwcwRqvg
10. wbGPi ˆKvb ˆRvov mnGgŒwjK? L
K 3, 18 L 6, 18
2865.24 28. 
 `yBwU abvñK msLÅvi eGMÆi A¯¦i 3 ‰es
M 3, 19 N 4, 10 M 3865.24 àYdj 2; ‰G`i eGMÆi mgwÓ¡ KZ?
11. 3x  4y = 14 ‰es 6x  8y = 22 N 4865.24 K 5 L 3
5x2 2x+1
mgxKiGYi ˆRvUwU@   3
19. 
11
=  3
5
mgxKiGYi M 2 N 1
i. mgém     29. ‰KwU e†Gîi cÉwZmvgÅ ˆiLv KqwU?
ii. AwbfÆikxj mgvavb KZ? K 1 L 4
iii. mgvavb ˆbB 3 1 MAmsLÅ N 9
K L
wbGPi ˆKvbwU mwVK? 5 5 30. 
 ˆKw±`Êq cÉeYZvi cwigvc KqwU?
K i I ii L i I iii M 
1
N 7
K 3 L 4
M ii I iii N i, ii I iii 7 M 5 N 6

1 N 2 L 3 N 4 K 5 M 6 M 7 L 8 M 9 K 10 M 11 M 12 K 13 N 14 L 15 K
Dîi

16 K 17 L 18 L 19 N 20 M 21 N 22 N 23 N 24 L 25 L 26 M 27 M 28 K 29 M 30 K
36 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
381 KPzqv miKvwi cvBjU DœP we`Åvjq, Puv`cyi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
. . 10. 
 3 ˆm.wg., 4 ˆm.wg. ‰es 4.5 ˆm.wg. 19. avivwUi 1g 7wU cG`i mgwÓ¡ KZ?
1. 
 0.3  0.6 = KZ?
.
evüwewkÓ¡ ‰KwU wòfzGRi cwie†Gîi eÅvmvaÆ K
364
L
1093
243 729
K 1 L 0.2 wbGPi ˆKvbwU? 3280 6560
.. . K 5.88 L 5.75 M N
M 0.18 N 0.18 2187 6561
M 2.30 N 3.29
2. x = {a, b, c}, y = {b} ‰es z = x\y nGj, 20. H AÞiwUi NƒYÆb cÉwZmvgÅ ˆKvY KZ?
11. cot(  60) = 3 nGj cos = KZ? K 60 L 90
P(z) ‰i Dcv`vb KqwU? 1
K 0 L M 180 N 360
K 1 wU L 2 wU 2
21. ˆKvGbv eMÆGÞGòi ˆÞòdj Zvi KGYÆi Dci
M 3 wU N 4 wU 3
M 1 N
2 AwâZ eMÆGÞGòi KZàY?
3. 
 x2 + y2 = 9 ‰es xy = 3 nGj@
K AGaÆK L mgvb
i. (x  y)2 = 3 wbGPi ZG^Åi AvGjvGK (12 I 13) bs cÉGk²i
Dîi `vI: M ˆ`oàY N 2 àY
ii. (x + y)2 = 15
iii. x2 + y2 + x2y2 = 18 P 22. 
 e†Gîi ˆK±`ÊÕ© ˆKvY 144 ‰es eÅvmvaÆ
wbGPi ˆKvbwU mwVK? 1 10 ˆm.wg. nGj e†îPvGci Š`NÆÅ KZ ˆm.wg.?
tan =
3 K 15 L 12
K i I ii L i I iii 
M N M 8 N 4
M ii I iii N i, ii I iii
12. PN ‰i gvb wbGPi ˆKvbwU? 23. ‰KwU eGMÆi evüi Š`NÆÅ wZbàY KiGj Dnvi
4. 0.000432 ‰i cƒYÆK KZ? K 2 L 2 ˆÞòdj KZàY e†w«¬ cvGe?

K 4 L 4 M 3 N 4 K 3 àY L 4 àY
 13. cosec ‰i gvb wbGPi ˆKvbwU? M 8 àY N 9 àY
M 3 N 3
1 1
5. logx16 = 2 nGj, x = KZ? K L 24. ‰KwU NbGKi KGYÆi Š`NÆÅ 6 3 wgUvi nGj
4 2
K 2 L 4 ‰i AvqZb@
M 3 N 2
M 4 N 16 K 36 NbwgUvi L 144 NbwgUvi
14. 
 12 wgUvi `xNÆ ‰KwU gB ˆ`Iqvj ˆ^GK
6. mgxKiYwUi mgvavb ˆmU M 216 NbwgUvi N 512 NbwgUvi
3x  6 + 5 = 2 6 3 wgUvi `ƒGi f„wgi mvG^  ˆKvY Drc®²
ˆKvbwU? 25. cixÞvi bÁ¼i I RbmsLÅv ˆKvb aiGbi
KGi ˆ`IqvGji Qv` Õ·kÆ KGi|  ‰i gvb
K  L {5}
PjK?
KZ?
M {3} N {5} K wewœQ®² PjK L AwewœQ®² PjK
K 30 L 45
7. M evÕ¦e PjK N AwebÅÕ¦ PjK
A M 60 N 90
26.  5 ˆ^GK 5 chƯ¦ cƒYÆ msLÅvàGjvi gaÅK
40 15. `yBwU msLÅvi AbycvZ 3 : 2| ‰G`i j.mv.à.
KZ?
D 42 nGj msLÅv `yBwUi M.mv.à. KZ?
B C K 5 L 1
1 K 16 L 7
ABC ‰i AB = AC nGj ACD = KZ M 0 N 5
2 M 14 N 21
27. 10% nvi gybvdvq 6000 UvKvi 3
eQGii
wWMÉx? 16. 25% jvGf KÌqgƒjÅ I weKÌq gƒGjÅi AbycvZ
PKÌe†w«¬ gybvdv I mij gybvdvi cv^ÆKÅ
K 50 L 55 KZ?
KZ?
M 90 N 110 K 1:4 L 4:3
K 186 L 1800
8. 
 ‰KwU wbw`ÆÓ¡ wòfzR AuvKGZ cÉGqvRb@ M 5:4 N 4:5
M 1986 N 6000
i. wZb evüi Š`NÆÅ
17. 
 6x  y = 5 ‰es 5x  2y = 2 nGj 1 1
28. 
 p + = 2 nGj p13 + 13 = KZ?
x + y = KZ? p p
ii. wZb ˆKvGYi cwigvc
K 2 L 3 K 14 L 10
iii. `ywU ˆKvY I ‰G`i msj™² evü M 11 N 2
M 4 N 5
wbGPi ˆKvbwU mwVK? wbGPi ZG^Åi AvGjvGK (18 I 19) bs cÉGk²i 29. (x + y,  1) = (3, x  y) nGj (x, y) ‰i
K i I ii L i I iii Dîi `vI: gvb KZ?
M ii I iii N i, ii I iii 1 1 K (2, 1) L (1, 2)
1 + + + ... ... ... M (1, 2) N (2, 1)
9. mgGKvYx wòfzGRi ˆÞGò cwiGK±`Ê wòfzGRi 3 9
x+3 2x1
ˆKv^vq AewÕ©Z? 18. avivwUi 7 Zg c` KZ? 30. ( 5) = ( 5) nGj x = KZ?
1 1 1
K Afů¦Gi L ewnfÆvGM K L K L 1
729 243 7
M AwZfzGRi Dci N jGÁ¼i Dci 1 5
M N 3 M N 4
81 3

1 L 2 N 3 N 4 K 5 M 6 K 7 L 8 L 9 M 10 M 11 K 12 L 13 N 14 K 15 L
Dîi

16 N 17 L 18 K 19 L 20 M 21 K 22 M 23 M 24 M 25 K 26 M 27 K 28 N 29 L 30 N
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 37
382 AvZvZzKÆ miKvwi gGWj nvB Õ•zj, ˆdbx

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 0.37̇  0.5̇ = ? 12. 
 ABC ‰i B I C ‰i mgw«¼LíK 21. secA ‰i gvb
2
nGj, tanA ‰i gvb wbGPi
K 0.0068 L 0.068 O we±`yGZ wgwjZ nGqGQ ‰es BAC = 40 3
M 0.68 N 6.8
nGj BOC ‰i gvb wbGPi ˆKvbwU? ˆKvbwU?
2.  (x  y, 6) = (3, 2x + y) nGj, (x, y) ‰i

K
1
L
1
M1 N 3
K 130 L 110 M 70 N 50 2
gvb KZ? 3
wbGPi ZG^Åi AvGjvGK (13 I 14) bs cÉGk²i
K (0, 3) L (3, 0) 22. cos sec2  1 ‰i mijgvb wbGPi
M (0,  3) N ( 3, 0) Dîi `vI: ˆKvbwU?
wbGPi ZG^Åi AvGjvGK (3 I 4) bs cÉGk²i Dîi A K cot L sin M cos N tan
`vI: 23. 8 wg. jÁ¼v ‰KwU gB f„wgi mvG^ 45 ˆKvY
60
A = { 1, 1, 2, 3}, B = {x : x2  2x  3 = 0} E
D F Drc®² KGi ˆ`IqvGji Qv` Õ·kÆ KGi|
C = {x : x2  6x + 8 = 0}
ˆ`IqvGji DœPZv KZ wg. nGe?
3.  C ‰i Dcv`vbàGjv nGjv@ B C
K 3 L2 2 M3 2N 4 2
K  2,  4 L  2, 4 wPGò DE | | BC ‰es BD | | CF
M 2,  4 N 2, 4 24. 8cm I 6cm eÅvGmi `yBwU e†î ciÕ·iGK
13. CEF = 50 nGj, BDE ‰i cwigvY KZ?
4.  A  B = KZ? ewntÕ·kÆ KiGj ZvG`i ˆK±`Ê«¼Gqi `ƒiZ½
K 50 L 60
K {1, 2} L {1, 3} wbGPi ˆKvbwU?
M 110 N 120
M { 1, 3} N { 1, 2} K 4cm L 6cm M 7cm N 8cm
1 14. AB = AC nGj, ABC + EFC = KZ? 25. ‰KwU mgevü wòfzGRi NƒYÆb cÉwZmgZvi
5. x + x = 5 nGj@ K 100 L 110 M 120 N 130
gvòv KZ?
i. x2  5x + 1 = 0 15. AwRf ˆiLv AâGb y-AÞ eivei ˆKvbwUGK K 0 L1 M2 N 3
1
ii. x3 + 3 = 25
x aiv nq? 26.
5
tan = nGj, sin = KZ?
12
1 2 K KÌgGhvwRZ MYmsLÅv L ˆkÉwYi gaÅgvb
iii.  x   = 21 K
5
L
12
M
13
N
13
 x M ˆkÉwYi MYmsLÅv N ˆkÉwYi DœPmxgv 13 13 12 5
wbGPi ˆKvbwU mwVK? 16. 
 13 ˆm.wg. DœPZv wewkÓ¡ ‰KwU ˆejGbi DcGii ZG^Åi AvGjvGK (27 I 28) bs cÉGk²i
K i I ii L i I iii f„wgi eÅvm 12 ˆm.wg. nGj ‰i@ Dîi `vI:
M ii I iii N i, ii I iii i. f„wgi ˆÞòdj 113.10 eMÆ ˆm.wg. wPGò 'O' ˆK±`ÊwewkÓ¡ O
6. 25% jvGf ‰KwU `ËeÅ weKÌq KiGj ii. AvqZb 1470.27 Nb ˆm.wg. e†Gî OA = 10 ˆm.wg.
weKÌqgƒjÅ I KÌqgƒGjÅi AbycvZ nGe@ iii. mgMÉ c†GÓ¤i ˆÞòdj 490.09 eMÆ ˆm.wg. OP = 6 ˆm.wg.| A P B

K 5:4 L 4:5
wbGPi ˆKvbwU mwVK? 27. AB ‰i Š`NÆÅ wbGPi ˆKvbwU?
M 4:3 N 3:4
K 6 ˆm.wg. L 8 ˆm.wg.
2 K i I ii L i I iii
7. 
 x2 + 2 = 3x nGj,  x  2  ‰i gvb M 10 ˆm.wg. N 16 ˆm.wg.
 x M ii I iii N i, ii I iii
28. e†îwUi cwimxgv wbGPi ˆKvbwU?
KZ? wbGPi ZG^Åi AvGjvGK (17 I 18) bs cÉGk²i
K 31.42 ˆm.wg. L 62.83 ˆm.wg.
K 1 L5 M 13 N 17 Dîi `vI: M 125.66 ˆm.wg. N 314.16 ˆm.wg.
10 Rb wkÞv^Æxi AvB.wm.wU welGqi bÁ¼i ˆ`Iqv 29. 
8. ( 3 7)2x  1 = ( 7)x + 1 nGj, x ‰i gvb KZ? 
1 5 nGjv: 36, 35, 43, 50, 40, 43, 38, 33, 45, 46 A
K L M1 N 5
7 7 17. 
 DcvGîi cwimi wbGPi ˆKvbwU?
9. log93 + log 33 ‰i gvb KZ? K 20 L 18 M 17 N 10 B C
1 5 18. 
 DcvGîi gaÅK wbGPi ˆKvbwU? D E
K L1 M N 4
5 2
K 40.5 L 41.5 M 42.5 N 43.5 wPGò BC | | DE
10. 
 jMvwi`Ggi ˆÞGò@
19. ‰KwU mgevü wòfzGRi ˆÞòdj 36 3 eMÆ wPGòi ˆcÉwÞGZ wbGPi ˆKvbwU mwVK?
i. 10 ˆK mvaviY jMvwi`Ggi wfwî aiv nq K AB : BD = AD : DE
wg. nGj, ‰i cwimxgv wbGPi ˆKvbwU?
ii. 25.345 ‰i jGMi cƒYÆK 1 L BC : DE = AD : AE
iii. logaM + logaN = loga(M + N) K 4 3 eMÆ wg. L 12 3 eMÆ wg. M AD : BD = AE : EC
wbGPi ˆKvbwU mwVK? M 36 wg. N 48 wg. N AD : DE = AE : CD
K i I ii L i I iii 20. 4 wg. evüwewkÓ¡ ‰KwU mylg lofzGRi 30. ABC ‰i AB I AC evüi gaÅwe±`y
M ii I iii N i, ii I iii ˆÞòdj KZ eMÆ wg.? h^vKÌGg M I N nGj, ABC : AMN
x5
11. (x  5) = x nGj, x ‰i gvb wbGPi ˆKvbwU? K 24 3 L 12 3 KZ?
K 4:1 L 2:1 M 1:4N 1:2
K 2, 5 L5 M2 N 1, 5 M 6 3 N 4 3

1 M 2 L 3 N 4 M 5 L 6 K 7 K 8 N 9 M 10 K 11 N 12 L 13 M 14 M 15 K
Dîi

16 K 17 L 18 L 19 M 20 K 21 L 22 L 23 N 24 M 25 N 26 K 27 N 28 L 29 M 30 K
38 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
383 we‰‰d kvnxb KGjR, PëMÉvg

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
. . AB || CF, DF || BC nGj,  EFC = KZ? 22. e†Gîi NƒYÆb cÉwZmgZvi gvòv KZ?
1. 0.4 + 0.3 = KZ?
. . . . . . K 90 L 70 M 50 N 60 K 1 L 2
K 0.7 L 0.1 M 0.102 N0.148 13. ˆKvGbv `G´£i Qvqvi Š`NÆÅ Zvi Š`GNÆÅi KZ M 3 N Amxg
2. P = {x  ô : 2 < x < 6 ‰es x ‰KwU àY nGj D®²wZ ˆKvY 30 nGe? 23. n evü «¼viv MwVZ mylg eüfzGRi cÉGZÅKwU
cƒYÆeMÆ msLÅv} ˆmUwU ZvwjKv c«¬wZGZ 1 1 ˆKvGYi cwigvc KZ?
K L
cÉKvwkZ i…c wbGPi ˆKvbwU? 3 2 180(n  2) 180(n + 2)
K {2, 3} L {2, 3, 4, 5, 6} K L
M 2 N 3 n n
M {4} N  14. 
  = 30 nGj@ 90(n  2) 90(n+2)
M N
3. 4x  3 + 5 = 2 ‰es mgvavb ˆmU ˆKvbwU? 3
n n
K {} L {0} M {3} N {3} i. 4 sin  =
cos 24. ‰KwU mgw«¼evü wòfzGRi cwimxgv
4.  {a, b, c, d} ‰i KqwU DcGmU AvGQ hvi
 ii. tan 2 = 3 cot 2 16 ˆm.wg.| ‰i f„wg 6 ˆm.wg. nGj DœPZv
cÉGZÅKwUi wZbwU KGi Dcv`vb AvGQ? iii. tan 2 = 2 sin2 KZ ˆm.wg.?
K PviwU L wZbwU wbGPi ˆKvbwU mwVK? K 12 L 8
M `yBwU N ‰KwU K i I ii L i I iii M 6 N 4
5. hw` a + b = 5 ‰es a  b = 3 nq, ZGe M ii I iii N i, ii I iii 25. ‰KwU BGUi Š`NÆÅ, cÉÕ© I DœPZv h^vKÌGg
a2 + b2 = KZ? wbGPi ZG^Åi AvGjvGK (15 I 16) bs cÉGk²i 8 ˆm.wg., 5 ˆm.wg. I 4 ˆm.wg. nGj,
K 2 L4 M8 N 64 Dîi `vI: A AvqZb KZ Nb ˆm.wg.?
6. 35  2y  y2 ‰i Drcv`K ˆKvbwU? K 120 L 160 M 320 N 460
K 5+y L y6 26. wPGò, O ˆK±`ÊwewkÓ¡ C
AC = 8 wgUvi
M 7+y N 7y ABC e†îvKvi gvGVi
ACB = 60
7.  log366 + log 6 6 = KZ?
 B C
mxgvbv ˆNuGl 2 ˆm.wg.
O

K
1
L1 M2
1
N 5 15. 
 BC ‰i Š`NÆÅ KZ wgUvi? PIov ‰KwU ivÕ¦v AvGQ| A B
6 2 4
K L 4 OA = 6 ˆm.wg.| ivÕ¦vwUi
3 3
8. ( 5) x+1 = ( 5)2x1 nGj x ‰i gvb KZ? ˆÞòdj KZ eMÆ ˆm.wg.?
M 4 2 N 4 3
1 5 K 87.96 L 113.09
K 1 L5 M N 16. 
 AB ‰i Š`NÆÅ KZ wgUvi?
7 7 M 201.06 N 210.06
9. wòfzR AuvKGZ cÉGqvRb@ 4
K L 4 27. ‰KwU UÇvwcwRqvg AvK‡wZi ˆjvnvi cvGZi
3
i. wZbwU evü mgv¯¦ivj evü«¼Gqi Š`NÆÅ h^vKÌGg 3 ˆm.wg.
ii. `yBwU evü ‰es ZvG`i A¯¦fzÆÚ ˆKvY M 4 2 N 4 3
I 1 ˆm.wg. ‰es ZvG`i jÁ¼ `ƒiZ½ 2 ˆm.wg.|
iii. `ywU ˆKvY ‰es ZvG`i msj™² evü 17. p, q, r KÌwgK mgvbycvZx nGj,
p2 + q2 cvGZi ˆÞòdj KZ eMÆ ˆm.wg.?
wbGPi ˆKvbwU mwVK? = KZ? K 1 L2 M3 N 4
q2 + r2
K i I ii L i I iii r p q q 28.  KÌgGhvwRZ MYmsLÅv cÉGqvRb@
M ii I iii N i, ii I iii K L M N
p r p r i. cÉPziK wbYÆGq
wbGPi ZG^Åi AvGjvGK (10 I 11) bs cÉGk²i 18. `yB AâwewkÓ¡ msLÅvi A⫼Gqi A¯¦i 3| ii. gaÅK wbYÆGq
Dîi `vI: P
msLÅvwUi A⫼q Õ©vb wewbgq KiGj ˆh iii. AwRf ˆiLv wbYÆGq
3x msLÅv cvIqv hvq Zv gƒj msLÅvi w«¼àY wbGPi ˆKvbwU mwVK?
O
AGcÞv 2 ˆewk| msLÅvwU KZ? K i I ii L i I iii
wPGò, O e†Gîi ˆK±`Ê| 2x+60 K 25 L 36 M 41 N 63
Q S M ii I iii N i, ii I iii
10. 
 QRS = KZ? R 19. 5 + 11 + 17 + ... ... ... + 59 avivwUi wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
K 30 L 45 M 60 N 135 c`msLÅv KZ? Dîi `vI:
11. 
 PSR = 90 nGj, K 27 L 29 M 10 N 18
ˆkÉwYeÅeavb 21-30 31-40 41-50 51-60
QRS + PQR = KZ? 20. cÉ^g wòkwU Õ¼vfvweK msLÅvi mgwÓ¡ KZ?
K 100 L 135 M 180 N 225 K 435 L 235 M 465 N 265 MY msLÅv 4 12 8 10
12. A 21. p + q + r + s + ... ... ... àGYvîi avivfzÚ 29.  cÉ`î DcvGîi gaÅK ˆkÉwYi Ea»Æmxgv
60 nGj wbGPi ˆKvbwU? KZ?
D F
E 50 p s K 40 L 41 M 51 N 50
K pq=sr L =
q r 30. 
 DcGivÚ ZG^Åi cÉPziK KZ?
q s K 27.07 L 37.67
B M = N p+q=r+s
C p r M 47.67 N 57.67

1 K 2 M 3 K 4 K 5 L 6 M 7 M 8 L 9 N 10 N 11 N 12 L 13 N 14 K 15 L
Dîi

16 N 17 L 18 K 19 M 20 M 21 M 22 N 23 K 24 N 25 L 26 K 27 N 28 M 29 N 30 L
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 39
384 evsjvG`k gwnjv mwgwZ evwjKv DœP we`Åvjq I KGjR, PëMÉvg

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 0.2̇  0.04̇ ‰i gvb KZ? 13. wPGò, ABC mgevü wòfzR nGj, 21. 
 hw` ‰KwU wgbvGii DœPZv 20 3 wgUvi
K 0.5 L 0.05 M 5 N 6 cÉe†«¬ x = KZ? A ‰es Qvqvi Š`NÆÅ 20 wgUvi nq, ZGe kxGlÆi
2. 
 3% nvi gybvdvq 10,000 UvKv 3 eQGii K 60 D®²wZ ˆKvY KZ?
RbÅ wewbGqvM Kiv nGj PKÌe†w«¬ gybvdv KZ L 120 O
K 30 L 45 M 60 N 90
x
UvKv? M 180
B C 22. 
 `yBwU msLÅvi AbycvZ 5 : 7 ‰es ‰G`i
K 927 L 927.27 N 240 M.mv.à 4 nGj, msLÅv `yBwUi j.mv.à KZ?
M 790 N 790.37 14. wbw`ÆÓ¡ ‰KwU PZzfyRÆ AuvKv mÁ¿e hw` ˆ`Iqv K 120 L 140 M 160 N 180
wbGPi ZG^Åi AvGjvGK (3 I 4) bs cÉGk²i Dîi ^vGK@ wbGPi ZG^Åi AvGjvGK (23 I 24) bs cÉGk²i
`vI: i. PviwU evü I ‰KwU ˆKvY Dîi `vI:
A = {1, 2, 3} ‰es B = {2, 3, 4} ii. wZbwU evü I ‰G`i A¯¦fÆyÚ `yBwU ˆKvY log2 + log4 + log8 + .......
3. A ˆmGUi cÉK‡Z DcGmGUi msLÅv KZ? 23. 
 avivwUi mvaviY A¯¦i KZ?
iii. `yBwU evü I wZbwU ˆKvY
K 8 L4 M6 N 7 K 2log2 L 3log2
wbGPi ˆKvbwU mwVK? M log2 N log3
4. A – B = KZ?
K {1} L {2} M {3} N {4} K i I ii L i I iii 24. 
 avivwUi 7g c` ˆKvbwU?
5. logx400 = 4 nGj, wfwî ˆKvbwU? M ii I iii N i, ii I iii K log112 L log120
D
K 5 L2 5 M5 2N 2 15. 
 C M log125 N log128

wbGPi ZG^Åi AvGjvGK (6 I 7) bs cÉGk²i Dîi

3x + 5
3x + 15
25. ‰KwU NbGKi KYÆ 4 3 wgUvi nGj, ‰i avi
`vI: A
KZ wgUvi?
B
a2 – 5a – 1 = 0 K 4 L5 M2 3N 3 3
ABCD mvgv¯¦wiGKi x ‰i gvb KZ?
1
2
26. 
 ‰KwU wmwj´£vGii DœPZv 8 ˆm.wg. ‰es
6. a+ 
 ‰i gvb KZ? K 20 L 22 M 22.6 N 26.7
 a f„wgi eÅvmvaÆ 4 ˆm.wg. nGj@
K 29 L 25 M 21 N 20 wPGòi AvGjvGK Q
i. mgMÉZGji ˆÞòdj 301.59 eMÆ ˆm.wg.
1 (16 I 17) bs cÉGk²i ii. eKÌZGji ˆÞòdj 201.06 eMÆ ˆm.wg.
7. 
 a2 + 2 ‰i gvb KZ?
a
Dîi `vI: O
iii. AvqZb 100.53 Nb ˆm.wg.
K 19 L 21 M 25 N 27
8. 
 ( 3)2x + 1 = 27 nGj, x ‰i gvb KZ? 100 wbGPi ˆKvbwU mwVK?
A B
2 5 K i I ii L i I iii
K L M9 N 12 P
5 2 M ii I iii N i, ii I iii
9. `yBwU msLÅvi cv^ÆKÅ 4 ‰es ˆQvU msLÅvwUi 16. APB PvGci A¯¦MÆZ ˆKvY 100 nGj, AQB 27. ‰KwU eGMÆi A¯¦e†GÆ îi eÅvmvaÆ 3 ˆm.wg.
eMÆ eo msLÅvwUi w«¼àGYi mgvb| eo PvGci A¯¦MÆZ ˆKvY KZ nGe? nGj, eGMÆi evüi Š`NÆÅ KZ ˆm.wg.?
msLÅvwU KZ? K 120 L 130 K 4 L5 M6 N 8
K 4 L5 M6 N 8 M 260 N 360 ˆkÉwYeÅvwµ¦ 11-20 21-30 31-40 41-50
10. mgGKvYx wòfzGRi mƒßGKvY«¼Gqi A¯¦i 8 17. wPGò APB PvcwU ˆKvb aiGbi Pvc? MYmsLÅv 5 15 10 20
nGj ‰i Þz`Z
Ë g ˆKvYwUi gvb KZ? K DcPvc L AwaPvc DcGii ZG^Åi AvGjvGK (28-29)bs cÉGk²i
K 36 L 38 M 40 N 41 M e†îPvc N Pvc Dîi `vI:
wPGòi AvGjvGK (11 I 12) bs cÉGk²i Dîi `vI:
18. cot( – 30) =
1
nGj, sin = KZ? 28. gaÅK ˆkÉwYi wbÁ²mxgv KZ?
wPGò AB = AC A 3 K 21 L 31 M 40 N 41
‰es BO I CO 60
K 0 L1 M 3 N
1
29. DcvGîi cÉPziK KZ?
h^vKÌGg B I C O
2
K 33.30 L 30.03 M 41.33 N44.33
B C 3
‰i mgw«¼L´£K| 19. tan = 4 nGj, cos2 ‰i gvb KZ? 30. AvqZGjL Aâb KiGZ cÉGqvRb@
11. BOC ‰i gvb KZ? 16 25 9 16 i. x AÞ eivei AwewœQ®² ˆkÉwYeÅvwµ¦
K 120 L 100 K L M N
9 16 16 25 ii. y AÞ eivei MYmsLÅv
M 80 N 60 20. 
 mgevü wòfzGRi evüi Š`NÆÅ 8 ˆm.wg. iii. ˆkÉwYi gaÅgvb
12. OBC ‰i gvb KZ? nGj, ‰i DœPZv KZ ˆm.wg.? wbGPi ˆKvbwU mwVK?
K 30 L 60
K 2 3 L 4 3 K i I ii L i I iii
M 120 N 180
M 16 3 N 18 3 M ii I iii N i, ii I iii
1 M 2 L 3 N 4 K 5 L 6 K 7 N 8 L 9 N 10 N 11 K 12 K 13 N 14 N 15 N
Dîi

16 M 17 K 18 L 19 N 20 L 21 M 22 L 23 M 24 N 25 K 26 K 27 M 28 L 29 N 30 K
40 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
385 PëMÉvg KÅv´ŸbGg´Ÿ ˆevWÆ Av¯¦t DœP we`Åvjq, PëMÉvg

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 0.2˙  1.1˙ 2˙  0.08˙ 1˙ = KZ? wbGPi ˆKvbwU mwVK? 21. eGMÆi KGYÆi Š`NÆÅ KZ?
K 0.02139 L 0.02039 K i I ii L ii I iii K 70.028 L 70.208
M 0.20239 N 0.02239 M ii N iii M 70.280 N 70.820
2.  q weGRvo
 Õ¼vfvweK msLÅv nGj wbGPi wbGPi ZG^Åi AvGjvGK (13 I 14) bs cÉGk²i 22. CD ‰i gvb KZ?
ˆKvbwU ˆRvo msLÅv? Dîi `vI: K 49.571 L 94.517
K 4q + 1 L 2q  1 M 49.517 N 94.371
ax  cy = 0
M q2  2 N q2 + 1 cx  ay = c2  a2
23. 
 e†Gîi cwiwa = mgevü wòfzGRi cwimxgv
3.
x y x y
  +  1 = 1 +  nGj, (x, y) = KZ?
 13. 
 DóxcK nGZ mgxKiY«¼q@ nGj ‰G`i ˆÞòdGji AbycvZ KZ? eÅvmvaÆ
 2 3   3 2 = r, evüi Š`NÆÅ = a
K mgém, wbfÆikxj, ‰KwU gvò mgvavb AvGQ
K 56  56 5 6
L   L mgém, AwbfÆikxj, 1wU gvò mgvavb AvGQ
K 2 3: L 3:
   6 5 M 3 3: N : 3
6 6 6 5 M Amgém, wbfÆikxj, 1wU gvò mgvavb AvGQ
M    N   24.  NbGKi ‰KwU avi 2cm nGj@

5 5   5 6 N Amgém, AwbfÆikxj, AmsLÅ mgvavb AvGQ
i. mgMÉ c†Ó¤ZGji ˆÞòdj 12 cm2
4. B = {x : x, 4 ‰i àYbxqK} C = {x : x 14.  DóxcGKi mgvavb (x, y) = KZ?

ii. NbGKi AvqZb 2 2 cm3
ˆRvo ˆgŒwjK msLÅv} nGj, B\C wbGPi K (a, b) L (b, c) M (c, a) N (a, c)
iii. NbGKi KYÆ 6 cm
ˆKvbwUi mgvb? 15. ZGji gvòv KqwU?
wbGPi ˆKvbwU mwVK?
K {1, 2, 4} L {1, 4} K 1 L2 M3 N kƒYÅ
K i I ii L i I iii
M {1, 2} N {2} 16. BDwKÑGWi Õ¼xKvhÆ@
5.  144x2 + 36y2 ‰i mvG^ KZ ˆhvM
 KiGj M ii I iii N i, ii I iii
i. ˆiLvi cÉv¯¦ we±`y AvGQ
ˆhvMdj cƒYÆeMÆ nGe? 25. wbGPi ˆKvbwU ˆÞGò mgGKvYx wòfzR Aâb
ii. ZGji cÉv¯¦ nGjv ˆiLv
K 144x L 414xy Kiv mÁ¿e bq?
iii. we±`y nGjv kƒbÅ gvòvi mî½v K 3cm, 4cm, 5cm L 6cm, 8cm, 10cm
M 144xy N 441xy
1 8p3  10p + 14 wbGPi ˆKvbwU mwVK? M 5cm, 7cm, 9cm N 5cm, 12cm, 13cm
6. P +
P
= 2 nGj 6p3 + 4p + 2
‰i gvb K i I ii L i I iii 26. ABC PQ || BC nGj wbGPi ˆKvbwU mwVK?
KZ? M ii I iii N i, ii I iii K AP : PB = AQ : QC
1 3 17. PQR ‰i ˆÞGò ˆKvbwU mwVK? L AB : PQ = AC : PQ
K L1 M N 2
2 2
K PQ + QR < PR L PQ + PR < QR M AB : AC = PQ : BC
7. cot ‰i cƒYÆi…c nGjv@ M PQ  PR > QR N PQ  PR < QR
N PQ : BC = BP : BQ
K cotengant L cotengent 27. 
 wPò nGZ@
M cotangent N cotanjent
wbGPi ZG^Åi AvGjvGK (18-20) bs cÉGk²i Dîi A
8.  mƒßGKvY nGj sin ‰i ˆié@ `vI:
B
i.  1 < sin  1 2cm
ii.  1  sin < 1 60 C D E
iii.  1  sin  1
wbGPi ˆKvbwU mwVK? B C

K i L ii M iii N i I iii
D i. ABC I ADE ciÕ·i m`†k
30 AD CE
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i 30
ii. =
BD AE
Dîi `vI: A E
ABC BC2
wPGò RS = 20 wgUvi iii. =
ADE DE2
P 18. 
 wPò nGZ AC ‰i Š`NÆÅ KZ? wbGPi ˆKvbwU mwVK?
1
K 2 6 L 6 M N 3 6 K i I ii L i I iii
6
M ii I iii N i, ii I iii
R
30 60
Q 19. 
 ABC ‰i ˆÞòdj KZ?
wbGPi ZG^Åi AvGjvGK (28-30) bs cÉGk²i Dîi
S K 1.73 L 2.73 M 1.37 N 2.37
9. 
 PQ ‰i Š`NÆÅ KZ? `vI:
20. 
 DE ‰i Š`NÆÅ KZ?
K 10 3 L 15 3 ˆKvb ‰KwU AçGji 10 w`Gbi Zvcgvòv 10,
2 2 3 2 4 2 6 2
M 20 3 N 25 3 K L M N 9, 8, 6, 11, 12, 7, 13, 14, 5
4 4 4 4
10.  (PR + RQ + PQ) ‰i Š`NÆÅ KZ wgUvi?
28. DcvGîi Mo Zvcgvòv ˆKvbwU?
 wPò nGZ (21 I 22) bs cÉGk²i Dîi `vI: K 8 L 8.5
K 81.64 L 81.47 M 81.74 N81.96 A B
M 9.5 N 9
11. H AÞiwUi NƒYÆb ˆKvY KZ?
K 30 L 60 M 90 N 180
r 29. DcGii msLÅv mƒPK DcvGîi cÉPziK ˆKvbwU?
O K 120 L 50
12. logaa = 1 ‰i ˆÞGò ˆKvb kZÆwU mwVK?
i. a > 0, a = 1 M 14 N cÉPziK ˆbB
ii. a < 0, a  1 D C 30. DcvîmgƒGni gaÅK ˆKvbwU?
iii. a > 0, a  1 e†Gîi cwiwa 220 wgUvi ‰es ABCD ‰KwU eMÆ K 9.5 L 9 M 8.5 N 8
1 2L 3 4N 5 M 6 L 7
L M 8M M 9 K 10 N 11 N 12 N 13 L 14 M 15 L
Dîi

16 M 17 N 18 L 19 K 20 L 21 K 22 M 23 M 24 N 25 M 26 K 27 L 28 M 29 N 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 41
386 AvMÉvev` miKvix KGjvbx DœP we`Åvjq, PëMÉvg

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 wbGPi ˆKvbwU Agƒj` msLÅv? 12. mgZGj `yBwU iwkÄi cÉv¯¦ we±`y ‰KB nGj wbGPi ˆKvbwU mwVK?
5 75 32 15 wK ŠZwi nq? K i I ii L i I iii
K L M N
4 27 8 2 K ˆKvY L ˆiLvsk M ii I iii N i, ii I iii
. . M iwkÄ N we±`y 3
2. 0.3  0.6 = KZ? 20. 
 sin  = nGj cot = ?
13. `yBwU cƒiK ˆKvGYi AbycvZ 3 : 7 nGj eo 2
. .. .
K 1 L 0.2 M 0.18 N 0.18 ˆKvY ˆKvbwU? K
1
L
1
M
3
N 3
3. A = {2, 3, 7, 9} nGj, A ‰i cÉK‡Z K 27 L 70 3 2 2
M 63 N 124 21. 0    90 ‰i RbÅ sin  ‰i meÆwbÁ²
DcGmU KqwU? 14. 

K 7 L4 M 15 N 16 gvb KZ?
1 x K 1 L0 M1 N 
4. 
x+ =2
x
nGj@ 22. cosA = sinA nGj 2sinA.cosA = KZ?
105 y
1 1 1
2
i. x  2 = 2 5
ii. x + 5 = 2 K 0 L M1 N 2
x x 2
wPGò, y = 2x nGj x = ?
1
iii. x7  7 = 0 K 25 L 30
23. 
 mgGKvYx wòfzGRi 30 ˆKvY AâGbi
x ˆÞGò@
M 35 N 45
wbGPi ˆKvbwU mwVK? i. f„wg > jÁ¼
15. R
K i I ii L i I iii ii. jÁ¼ = f„wg
M ii I iii N i, ii I iii O
iii. f„wg < AwZfzR
5. a + b = 7, a  b = 1 nGj, wbGPi ˆKvbwU mwVK?
a3  b3 + a2 + b2 = KZ? P Q K i I ii L i I iii
K 62 L 49 M 48 N 53 wPGò, P I Q ‰i mgw«¼L´£K«¼q O M ii I iii N i, ii I iii
6. (x) = x3 + 2x2 + x  3 ˆK (x + 1) we±`yGZ wgwjZ nGqGQ ‰es R = 40, 24. 
 x : y = 2 : 3 ‰es 2 : x = 1 : 2 nGj
«¼viv fvM KiGj fvMGkl KZ nGe? POQ = KZ? y = KZ?
K 70 L 110 1 3
K 0 L1 M 2 N 3 K L M6 N 8
M 130 N 140 3 2
7. ‰KwU eBGqi weKÌqgƒjÅ 30 UvKv hv eBwU 16. wbGÁ² ˆm.wg. ‰KGK wZbwU ˆiLvsGki Š`NÆÅ 25. (2, 3) we±`ywU wbGPi ˆKvb mgxKiGYi ˆjL
ŠZwii eÅGqi 60%| eBwUi cÉK‡Z gƒjÅ ˆ`Iqv AvGQ| ˆKvb ˆÞGò wòfzR AuvKv wPGòi Dci AewÕ©Z?
KZ? mÁ¿e? K xy=1 L 2x + y = 7
K 50 L 48 M 20 N 18 K 2, 3, 5 L 6, 7, 8 M x + 3y = 5 N 2x + y = 6
wbGPi ZG^Åi AvGjvGK (8 I 9) bs cÉGk²i Dîi M 4, 5, 10 N 7, 5, 2
26. 25 + 21 + 17 + ... ... ...  19 avivwUi c`
`vI: 17. ‰KwU iÁ¼m AuvKGZ nGj KqwU Dcvî
cÉGqvRb? msLÅv KZ?
3x = a, 9x = b K 13 L 12 M 11 N 10
8.  a = b nGj, x = KZ? K 2 wU L 3 wU M 4 wU N 5 wU 27. Z ‰i NƒYÆb cÉwZmgZvi gvòv KZ?
wbGPi ZG^Åi AvGjvGK (18 I 19) bs cÉGk²i K 1 L2 M3 N 4
K 2 L 1 M0 N 1
Dîi `vI: S
9. 
 wbGPi ˆKvb mÁ·KÆwU mwVK? 28. 
 ‰KwU wòfzGRi `yBwU evüi Š`NÆÅ 6 ˆm.wg. I
a O 7 ˆm.wg. ‰es evü«¼Gqi A¯¦fzÆÚ ˆKvY 60
K ab2 = 1 L =1
b2 nGj, wòfzGRi ˆÞòdj KZ?
b2
M
a
=1 N a 2b = 1 P R Q
K 10.50 eMÆ ˆm.wg.
10. 0.000567 ‰i jGMi cƒYÆK wbGPi ˆKvbwU? wPGò PQS e†Gîi ˆK±`Ê O, OR = 3 ˆm.wg. L 14.85 eMÆ ˆm.wg.
   ‰es PR = 4 ˆm.wg. M 18.19 eMÆ ˆm.wg.
K 7 L6 M5 N 4
18. OP ‰i Š`NÆÅ KZ? N 36.37 eMÆ ˆm.wg.
11. mgxKiY@
2x  3 + 5 = 2
ˆm.wg. ˆm.wg.
K 3 L 4 wbGPi ZG^Åi AvGjvGK (29 I 30) bs cÉGk²i
i. ‰KwU ‰K PjK wewkÓ¡ mgxKiY M 5 ˆm.wg. N 7 ˆm.wg. Dîi `vI:
ii. ‰i mgvavb ˆmU s = { } 19. wPòvbymvGi@ 46, 45, 33, 38, 43, 40, 50, 43, 35, 36
iii. ‰i mgvavb x = 6 i. PQ ‰i mgw«¼L´£K OR 29. DcvGîi cwimi KZ?
wbGPi ˆKvbwU mwVK? ii. PQ PvGci Dci `´£vqgvb K 10 L 16 M 17 N 18
K i I ii L i I iii e†îÕ© PSQ 30. DÚ DcvGîi gaÅK KZ?
M ii I iii N i, ii I iii iii. PQ < OP + OQ K 40.5 L 41.5 M 42.5 N 43.5

1 K 2 L 3 M 4 M 5 K 6 N 7 K 8 M 9 N 10 N 11 K 12 K 13 M 14 K 15 L
Dîi

16 L 17 K 18 M 19 N 20 K 21 L 22 M 23 L 24 M 25 L 26 L 27 L 28 M 29 N 30 L
42 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
387 evsjvG`k ˆbŒevwnbx Õ•zj I KGjR, PëMÉvg

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 2.02̇ ‰i `kwgK f™²vsk ˆKvbwU? wbGPi ˆKvbwU mwVK? 22. 
x 2
 hw` = nq, ZGe
6x + y
‰i gvb KZ?
182 200 K i I ii L i I iii y 3 3x + 2y
K L 4 14
9 9
M ii I iii N i, ii I iii K
5
L
15
182 20
M
90
N
90 13.  wPGò, O e†Gîi ˆK±`Ê A
5 20
nGj, x ‰i gvb KZ? M N
2. (2x + y, 3) = (6, x  y) nGj, (x, y) ‰i gvb KZ? 2x + 10 4 13
K (0, 3) L (3, 0) O 23. 5x + 2y = 7 ‰es 10x + 4y = 14 ‰B
M (0, 3) N (3, 0) B C
x + 110
mgxKiY ˆRvUwU
2n + 1
3. f(n) =
2n  1
nGj, f(2) = KZ? K 18 L 26 i. mgém
K 3 L 5 M 30 N 90 ii. ‰i ‰KwU gvò mgvavb AvGQ
3 3 1 iii. ciÕ·i wbfÆikxj
M 
5
N
5 14. cos2 = 2 nGj, sin22 =?
wbGPi ˆKvbwU mwVK?
4.  hw` f(x) = x3 + kx2  4x  12 nq,
 1 1
K L K i I ii L i I iii
ZvnGj k ‰i ˆKvb gvGbi RbÅ f(3) = 0 4 3
M ii I iii N i, ii I iii
nGe| M
3
N 3
K 3 L 2 4 24. wPGò BC | | DE A

3 15.  1 + tanx = 2 nGj, cotx ‰i gvb

 KZ? nGj, BC ˆiLvi 4 ˆm.wg.
M N 3
2 K 21 L 1 2 Š`NÆÅ KZ? D E
3 ˆm.wg. 6 ˆm.wg.
5. nGj, n ‰i gvb KZ?
n
32 = 2 M 2 N 2+1
K 3 L 4 16. 31 + 29 + 27 + 25 + ........ avivi B C
M 5 N 6 i. mvaviY A¯¦i  2 K 10.5 ˆm.wg. L 11.5 ˆm.wg.
wbGPi ZG^Åi AvGjvGK (6 I 7) bs cÉGk²i Dîi ii. 12 Zg c` 9 M 12.0 ˆm.wg. N 13.0 ˆm.wg.
`vI: 25. 
 cvGki wPòwUi cÉwZmvgÅ
iii. cÉ^g 5 cG`i mgwÓ¡ 145
p2  3p + 1 = 0 ˆhLvGb p > 0 ˆiLvi msLÅv KZ?
1 wbGPi ˆKvbwU mwVK?
6. p2 + p2 ‰i gvb ˆKvbwU? K i I ii L i I iii
K 4 L 6 M ii I iii N i, ii I iii K 1 L 2
M 7 N 9 1
1 17. 
 ,  1, 3 ..... AbyKÌgwUi PZz^Æc` M 3 N 4
7. p3  3 = KZ?
p
3
26. ‰KwU mgevü wòfzGRi ˆÞòdj 6 3
KZ?
K 2 5 L 8 5
1
eMÆwgUvi nGj ‰i cwimxgv KZ?
M 10 5 N 20 5 K 3 L  K 4.89 wgUvi L 14.69 wgUvi
3
3
M 19.59 wgUvi N 72.00 wgUvi

1 3  1
8.  x ‰i mij gvb wbGPi ˆKvbwU? M N 3
 a  3 27. 2x  3 + 4 = 3 mgxKiYwUi mgvavb ˆmU
x a 3
18. iÁ¼Gmi ‰KwU KYÆ 16 ˆm.wg. ‰es ˆÞòdj ˆKvbwU?
K L
a3 x 96 eMÆ ˆm.wg. nGj, Aci KGYÆi Š`NÆÅ KZ K {2} L {4}
x x3 ˆm.wg.? M {2} N 
M 3 N
a a3 K 6 L 10 wbGPi ZG^Åi AvGjvGK (28 I 29) bs cÉGk²i
9. 0.00000538 ‰i mvaviY jGMi cƒYÆK KZ? M 12 N 18 Dîi `vI: A 30 D
K 6̄ L 5̄ 19. ‰KwU mylg AÓ¡fzGRi cÉwZwU kxlÆGKvGYi 20 wgUvi
M 5 N 6 cwigvY KZ?
10. KqwU Õ¼Z¯¨ Dcvî Rvbv ^vKGj ‰KwU K 45 L 60
B C
wbw`ÆÓ¡ PZzfÆR
z AuvKv mÁ¿e? M 120 N 135
K 4 L 5
wbGPi ZG^Åi AvGjvGK (20 I 21) bs cÉGk²i 28. AB = KZ wgUvi?
M 6 N 7 K 5 3 L 10
11. wòfzGRi wZbwU evüi Š`NÆÅ ˆm.wg. ‰KGK Dîi `vI:
M 10 3 N 20
ˆ`Iqv nGjv| wbGPi ˆKvb ˆÞGò wòfzR ˆkÉwY eÅeavb 30-39 40-49 50-59 60-69
29. BC ‰i Š`NÆÅ KZ wgUvi?
Aâb Kiv hvq? MYmsLÅv 13 21 24 2
K 15 L 17.3
K 5, 6, 18 L 6, 7, 19 20. mviwY ˆ^GK gaÅK wbYÆGqi RbÅ FC ‰i gvb M 20.3 N 30
M 7, 8, 17 N 9, 6, 13 wbGPi ˆKvbwU? 1
12.  e†Gîi ˆÞGò K 60 L 58 30. 
 sin = nGj, tan ‰i gvb KZ?
2
i. AaÆe†îÕ© ˆKvY ‰K mijGKvY M 34 N 13 1 3
ii. ˆh ˆKvGbv RÅv ‰i jÁ¼w«¼LíK ˆK±`ÊMvgx 21. DcvGîi cÉPziK ˆKvbwU? K L
3 2
iii. e†Gîi mgvb mgvb RÅv ‰i gaÅwe±`yàGjv K 48.8 L 51.2
M 3 N 2
mge†î M 57.8 N 60.2

1 M 2 L 3 N 4 K 5 M 6 M 7 L 8 L 9 K 10 L 11 N 12 M 13 M 14 M 15 N
Dîi

16 K 17 N 18 M 19 N 20 N 21 L 22 M 23 L 24 K 25 M 26 L 27 N 28 L 29 L 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 43
388 ˆecRv cvewjK Õ•zj I KGjR, PëMÉvg

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 2cos  3 = 0 nGj,  = KZ? 10. 1  1 + 1  1 + ..... avivwUi 1g (2n + 1) wbGPi ˆKvbwU mwVK?
[ˆhLvGb 0 <  < 90] msLÅK cG`i ˆhvMdj KZ? ˆhLvGb n  ô. K i I ii L i I iii
K 30 L 45 K 0 L 1 M ii I iii N i, ii I iii
M n N 2n + 1 1
M 60 N 90 21. x = 5 + 2 nGj x ‰i gvb KZ?
a 11. 'H' eGYÆi NƒYÆb ˆKvY KZ?
2. hw` cosec = b nq, ZGe tan = ? K 360 L 180 K 1 L 5 2
M 90 N 0 1
2
a +b 2 2
a b 2
M N 2 5
K L 2 5
b b 12. 
 mgevü wòfzGRi evüi Š`NÆÅ 8cm ‰i
b b DœPZv KZ ˆm.wg.? 22. p2  p  2 ‰i Drcv`GK weGkÏwlZ i…c ˆKvbwU?
M N K (p  1) (p  2) L (p  1) (p + 2)
a2  b 2 a + b2
2
K 2 3 L 5 3
M (p + 1) (p  2) N (p + 1) (p + 2)
3. 
 cosec(A  60) = 2 nGj, cotA = KZ? M 4 3 N 3 3
23. 
 ( 3)
2x + 4
= 27 nGj, x ‰i gvb KZ?
K 3 L AmsævwqZ 13. ‰KwU mge†îf„wgK ˆejGbi DœPZv 2cm
1 3
M 1 N 0 ‰es f„wgi eÅvmvaÆ 1cm nGj ˆejGbi K 1 L0 M N
2 2
4. 
 ‰KwU wgbvGii DœPZv 5 3 wgUvi ‰es AvqZb KZ cm3? 24. 
1
 logx =  2 nGj, wfwî KZ?
36
Qvqvi Š`NÆÅ 5 wgUvi nGj D®²wZ ˆKvY KZ? K 4 L 3
K 2 L6 M2 N 6
K 90 L 60 M 45 N 30 M 2 N 
14. wbGPi ˆKvbwU avc wePzÅwZ? 25. y2 + 7 y = 0 mgxKiGYi mgvavb ˆmU
5. 
 wPGò PQ ‰i Š`NÆÅ KZ wgUvi?
P xi + a xi  a wbGPi ˆKvbwU?
K ui = L ui =
h h K {0, 7} L {0,  7}
K 40 xi xi
R M ui =  a N ui = + a M { 7} N {(0, 7)}
L 74.64 h h
30
20 3 wbGPi ZG^Åi AvGjvGK (26 I 27) bs cÉGk²i
M 34.64 wgUvi
wbGPi ZG^Åi AvGjvGK (15 I 16) bs cÉGk²i
Dîi `vI:
N 60 S Q Dîi `vI:
ˆKvGbv e†Gîi ˆK±`Ê ˆ^GK ˆKvGbv RÅv-‰i Ici
6. b, a, c KÌwgK mgvbycvZx nGj, wbGPi ˆKvbwU ˆkÉwYeÅvwµ¦ 30-35 36-41 42-47 48-53
AwâZ jGÁ¼i Š`NÆÅ AaÆ-RÅv AGcÞv 2 ˆm.wg.
mwVK? MYmsLÅv 8 11 14 7
Kg| e†Gîi eÅvm 20 ˆm.wg.|
K a2 = bc L b2 = ac 15. DcvGîi cÉPziK ˆkÉwY ˆKvbwU? 26. 
 RÅv ‰i Š`NÆÅ KZ ˆm.wg.?
2
M c = ab N a=b+c K 30  35 L 36  41 K 10 L 12
7. a1x + b1y = c1, a2x + b2y = c2 mgxKiY M 42  47 N 48  53 M 16 N 20
ˆRvUwU ˆKvb kGZÆ ciÕ·i wbfÆikxj nGe? 16. gaÅK wbYÆGqi ˆÞGò, fm = KZ? 27. 
 jGÁ¼i Š`NÆÅ KZ ˆm.wg.?
a1 b 1 a1 b 1 c1 K 7 L 8 K 6 L8
K  L =  M 11 N 14
a2 b 2 a2 b 2 c 2 M 10 N 16
a1 b 1 c1 a1 b 1 c1 ×× 28. 
 wPGò, ABE ‰i gvb KZ?
M = = N   17. 0.45 ‰i mvgvbÅ f™²vsk wbGPi ˆKvbwU?
a2 b 2 c2 a2 b 2 c2
5 41 41 45 K 60 A
8. 
 2  4 + 8  16 + .... avivwUi 7-Zg c` K L M N
11 99 90 90 L 90
ˆKvbwU? 18. wbGPi ˆKvbwU Agƒj` msLÅv? M 120 E
B C 60
P

K 128 L  128 M 64 N  64 8 3 64 32 N 150 Q

K L M N
9. cÉ^g n msLÅK Õ¼vfvweK msLÅvi eGMÆi 32 2 16 8
2x + 1 1 29. `yBwU e†î ciÕ·iGK ewntÕ·kÆ KiGj,
mgwÓ¡- 19. hw` f(x) = 2x  1 nq, ZGe f 2  ‰i gvb
  ‰G`i ˆK±`Ê«¼q I Õ·kÆ we±`y Kxi…c nq?
n2(n + 1)2
K
4 KZ? K wZhÆK L wecixZ
n(n + 1)(2n + 1) K 4 L2 M0 N 2 M mgGiL N mgv¯¦ivj
L
2 20. {(2, 1), (1, 0), (0, 1)} A®¼qwUi 30. 
 mgGKvYx wòfzGRi ˆÞGò cwie†Gîi ˆK±`Ê

M
n(n + 1)
i. ˆWvGgb {2, 1, 0} ˆKv^vq AewÕ©Z?
2
ii. ˆié {1, 0, 1} K wòfzGRi Afů¦Gi L AwZfzGRi Dci
n(n + 1)(2n + 1)
N M wòfzGRi ewnfÆvGM N f„wgGZ
6 iii. ‰wU ‰KwU dvskb

1 K 2 M 3 N 4 L 5 L 6 K 7 M 8 K 9 N 10 L 11 L 12 M 13 M 14 L 15 M
Dîi

16 N 17 K 18 L 19 M 20 N 21 L 22 M 23 K 24 N 25 L 26 M 27 K 28 N 29 M 30 L
44 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
389 wPUvMvs BDwiqv dvwUÆjvBRvi Õ•zj I KGjR, PëMÉvg

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 3x  6 + 5 = 2 mgxKiYwUi mgvavb 11. log9 3 ‰i gvb KZ? 21. 1 ˆ^GK 19 chƯ¦ ˆgŒwjK msLÅvàGjvi Mo
ˆmU ˆKvbwU? K 2 L
1 KZ?
K  L {5} 2 K 9.63 L 9.5
M {3} N {5} 1 M 8.67 N 8.23
M 3 N
3
2.  ‰KwU
 wgkÉGb Pv I wPwbi AbycvZ 7 : 3 22. ˆKvGbv e†Gîi DcPvGci Dci `ívqgvb
nGj, wgkÉGY PvGqi cwigvY kZKiv KZ? 12. 
 tan(  45) = 3 nGj,  ‰i gvb KZ?
ˆKvY
K 30% L 40% K 15 L 60
K mħGKvY L mgGKvY
M 50% N 70% M 75 N 105
M cƒiK ˆKvY N Õ©ƒj ˆKvY
3. A = {a, b, c, d, e} nGj, A ‰i cÉK‡Z 13. mgGKvYx wòfzGRi ˆÞGò cwiGK±`Ê wòfzGRi
1 1 1
DcGmU KqwU? ˆKv^vq AewÕ©Z? 23. 
 1, , , , ... ... ... AbyKÌGgi mvaviY
3 7 15
K 8 L 15 K Afů¦Gi L ewnÆfvGM
M 31 N 64 c` ˆKvbwU?
M AwZfzGRi Dci N jGÁ¼i Dci 1 1
4. ‰KwU wòfzGRi wZbwU evüi Š`NÆÅ ˆ`Iqv K L
2n + 1
AvGQ| wbGPi ˆKvb ˆÞGò mgGKvYx wòfzR 14. 
 `yBwU msLÅvi AbycvZ 4 : 5 ‰es ‰G`i 2n  1
j.mv.à 120 nGj, e†nîg msLÅvwU KZ? 1 1
MVb Kiv mÁ¿e? M n N
2 n
K 2, 3, 4 L 3, 5, 7 K 20 L 24
M 30 N 120 24. wbGPi ˆKvbwU AwewœQ®² PjK?
M 5, 12, 13 N 10, 15, 20
15.  u2 =  2, x2 = 12 ‰es h = 4 nGj, K RbmsLÅv L eqm
5. R 
AbywgZ Mo, a = KZ? M R¯Ãmvj N wkÞv^Æx msLÅv
O
K 20 L 10 25. 
 2x  y = 8 ‰es x  2y = 4 nGj,
P Q M 6 N 4 x + y = KZ?
 wPGò P I Q ‰i mgw«¼LíK«¼q
 16. D
K 0 L 4
M 8 N 12
O we±`yGZ wgwjZ nGqGQ ‰es R = 40 O
26.  ‰KwU
 e†Gîi eÅvm 26 ˆm.wg. nGj, ‰i
nGj, POQ = KZ?
K 70 L 110 B
cwiwa KZ ˆm.wg.?
A C
K 530.9 L 81.68
M 130 N 140
ABD e†Gîi ˆK±`Ê O nGj M 40.84 N 13
6. 
 A = {2, 3, 4} ‰es B = {1, 3} nGj
i. A  B = {3} i. C, AB ‰i gaÅwe±`y 27. A

ii. A\B = {2, 4} ii. OAC = OBC

iii. (A  B) ‰i Dcv`vb msLÅv 5. iii. OAC + OBC = ‰K mgGKvY P Q
wbGPi ˆKvbwU mwVK? wbGPi ˆKvbwU mwVK? O
O
B
K i I ii L i I iii K i I ii L i I iii
C

M ii I iii N i, ii I iii
M ii I iii N i, ii I iii
wPGò BC || PQ nGj
7. 1.4̇3̇ ‰i mvaviY f™²vsk ˆKvbwU? i. BOC I POQ m`†k
142 143
17. 
 ˆKvGbv `Gíi Qvqvi Š`NÆÅ Zvi Š`GNÆÅi
K L ii. AP : BP = AQ : CQ
99 99 KZàY nGj, D®²wZ ˆKvY 30 nGe?
142 143 1 1 iii. BO : OQ = CO : OP
M N K L
100 100 3 2 wbGPi ˆKvbwU mwVK?
8. iÁ¼Gmi cÉwZmvgÅ ˆiLv KqwU? M 2 N 3 K i I ii L i I iii
K 1wU L 2wU 18. 
 0.0000975 ‰i mvaviY jGMi cƒYÆK M ii I iii N i, ii I iii
M 3wU N 4wU KZ? 28. wbGPi ˆKvbwUGK kƒbÅgvòvi mîv eGj MYÅ
9. mgvbycvGZi D`vniY wbGPi ˆKvbwU? Kiv nq?
K 4̄ L 5̄
K 3:4=4:3 L 2 : 3 = 6 : 10 K ˆiLv L Zj
M 6̄ N 5
M 3:2=9:6 N 2 : 5 = 4 : 25 M ˆKvY N we±`y
10. 5x  2y = 13 ‰es 2x + 3y = 9 19. 
 ‰KwU mgevü wòfzGRi ˆÞòdj 25 3 eMÆ
29. 
 'T' eYÆwUi ˆgvU KZwU cÉwZmvgÅ ˆiLv
mgxKiY«¼q wgUvi nGj, wòfzRwUi evüi Š`NÆÅ KZ wgUvi?
K 5 L 10
AvGQ?
i. mgém K kƒbÅ L 1wU
M 50 N 100
ii. ciÕ·i AwbfÆikxj
20. hw` f(x) = x3 + ax2  6x  9 nq, ZGe M 3wU N AmsLÅ
iii. (x, y) = (3, 1)
wbGPi ˆKvbwU mwVK? a ‰i ˆKvb gvGbi RbÅ f(3) = 0 nGe? 30. e†Gî A¯¦wjÆwLZ mvgv¯¦wiK ‰KwU
K i I ii L i I iii K 6 L 2 K UÇvwcwRqvg L iÁ¼m
M ii I iii N i, ii I iii M 2 N 4 M eMÆ N AvqZ

1 K 2 N 3 M 4 M 5 L 6 K 7 K 8 L 9 M 10 N 11 L 12 N 13 M 14 M 15 K
Dîi

16 K 17 N 18 L 19 L 20 L 21 K 22 N 23 K 24 L 25 L 26 L 27 N 28 N 29 L 30 N
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 45
390 ev±`ievb miKvwi DœP we`Åvjq, ev±`ievb

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1
1. sin(A  B) = 2 ‰es B = 30 nGj, A ‰i wbGPi ˆKvbwU mwVK? 21. RÅv AB ‰i Š`NÆÅ KZ?
K i I ii L i I iii K 7 L 7 M2 7N 5
gvb KZ?
M ii I iii N i, ii I iii 22. AOC = 55.5 nGj, OBC = KZ?
K 0 L 30 M 45 N 60
K 34.5 L 55.5
2. ˆKvGbv mgevü wòfzGRi ‰K evüi Š`NÆÅ 3 12. 32 = 2 nGj, n ‰i gvb KZ?
n

M 65.5 N 95.5
ˆm.wg. nGj, ‰i ˆÞòdj KZ? K 3 L4 M5 N 6
3 3 3 9 9 3 13. 4 x+1
=2 nGj, x = KZ? 23. ˆKvb e†Gîi AwaPvGc A¯¦wjÆwLZ ˆKvY Kxi…c?
K L M N
4 4 4 4 1 3 7 K mħGKvY L mgGKvY
K 1 L M N
3. ‰KwU NbGKi AvqZb 24 3 nGj ‰i KGYÆi 2 2 2 M Õ©ƒjGKvY N cÉe†«¬GKvY
Š`NÆÅ KZ ˆm.wg.? 14. 4sinA = 3 nGj, tanA = KZ? 24. PQR mgevü wòfzGRi cwiGK±`Ê C nGj,
K 6 L2 3 M4 N 3 7 3 4 3 QCR = KZ?
K L M N
4. ‰KwU mge†îf„wgK ˆejGbi DœPZv 2 wgUvi 3 7 3 4
K 45 L 60 M 90 N 120
‰es f„wgi eÅvmvaÆ 1 wgUvi nGj 15. sin cosec2  1 = KZ?
25. ‰KwU iÁ¼m AvuuKv hvGe hw` ˆ`Iqv ^vGK
i. eKÌZGji ˆÞòdj 3 K sin L cos
i. 1wU evüi Š`NÆÅ
ii. AvqZb 2 M sin.cos N sec
wbGPi ZG^Åi AvGjvGK (16 I 17) bs cÉGk²i ii.1wU evü I 1wU KYÆ
iii. fƒwgi ˆÞòdj 
iii. 1wU evü I 1wU ˆKvY
wbGPi ˆKvbwU mwVK? Dîi `vI:
wbGPi ˆKvbwU mwVK?
K i I ii L i I iii cosecA + cotA =
1

M ii I iii N i, ii I iii
2 K i I ii L i I iii
16. cosecA  cotA = KZ? M ii I iii N i, ii I iii
wbGPi ZG^Åi AvGjvGK (5 I 6) bs cÉGk²i Dîi
1 1
`vI: K
2
L2 M
3
N 4 wbGPi ZG^Åi AvGjvGK (26 I 27) bs cÉGk²i
cÉvµ¦ bÁ¼i 51-60 61-70 71-80 81-90 91-100 17. secA = KZ? Dîi `vI:
MYmsLÅv 8 12 15 7 8 5 5 1 1 tan(2A  45) = 1 = 3sinB
K L M N
5. gaÅK ˆkÉwYi cƒGeÆi ˆkÉwYi KÌgGhvwRZ 3 3 3 2 26. A ‰i gvb KZ?
MYmsLÅv ˆKvbwU? 18. 1 + 3 + 5 + .......... avivwUi 1g n msLÅK K 30 L 45 M 60 N 90
K 20 L 30 M 42 N 50 cG`i ˆhvMdj ˆKvbwU? 27. cos B = KZ?
2

6. cÉPziK wbYÆGqi ˆÞGò f1 + f2 = KZ? n(n + 1)  n(n + 1)2 10 8 10 2 2

K L   K L M N
K 11 L8 M5 N 3 2  2  9 9 3 3
7. cixÞvq cÉvµ¦ GPA ˆKvb aiGbi PjK? M
n2
N n2
28. an+1.an = KZ?
2
K wewœQ®² L AwewœQ®² 1
+1 n+
1

19. 2 + p + q + r + 162 àGYvîi avivi 4^Æ c` KZ? K a L a1 M an N a n

M AwebÅÕ¦ N webÅÕ¦
K 18 L 27 M 54 N 81 29.
8. wbGPi ˆKvbwU wbYÆGqi RbÅ KÌgGhvwRZ P
1
MYmsLÅv mviwY cÉGqvRb? 20. ,  1, 7 ,........ mvaviY AbycvZ KZ?
7 O 60 M
K MvwYwZK Mo L eÅewa 1 1
K L 
M cÉPziK N gaÅK 7 7 Q

9. A = {2, 3} nGj, P(A) ‰i Dcv`vb msLÅv KZ? M 7 N  7

K 2 L 3 wbGPi ZG^Åi AvGjvGK (21 I 22) bs cÉGk²i
M 4 N 6 O ˆK±`ÊwewkÓ¡ e†Gî PM I QM `yBwU Õ·kÆK
Dîi `vI:
10. P(x) = x  4x + 3 nGj, P(2) = KZ?
3 ‰es PMQ = 60 nGj, POQ = KZ?
K 0 L2 M3 N 4 O K 300 L 270
11. A = {0, 2}, B = {1, 0, 1} nGj M 120 N 90
A C B
i. B\A ‰i cÉK‡Z DcGmU msLÅv 3 30. log55 5 ‰i gvb KZ?
ii. A  B = {0} 2 3 5 3
iii. A  B = {2} wPGò O e†Gîi ˆK±`Ê OA = 4 ˆm.wg., OC = 3 ˆm.wg. K
3
L
2
M
2
N
2

1 N 2 L 3 K 4 M 5 K 6 K 7 L 8 N 9 M 10 M 11 N 12 M 13 L 14 L 15 L
Dîi

16 L 17 L 18 N 19 M 20 N 21 M 22 K 23 K 24 N 25 M 26 L 27 L 28 K 29 M 30 L
46 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
391 ev±`ievb KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, ev±`ievb

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 B = {2, 4, 6, 7} ˆmGUi cÉK‡Z DcGmU 11. cos215 + cos275 ‰i gvb KZ? wbGPi ZG^Åi AvGjvGK (20 I 21) bs cÉGk²i
msLÅv KZ? K 1 L0 M 0.5 N 1 Dîi `vI: S R
K 7 L8 M 15 N 16 wbGPi ZG^Åi AvGjvGK (12 I 13) bs cÉGk²i (3y + 8)
3
2. f(x) ‰i gvòv abvñK nGj ˆKvb kGZÆ f(x) Dîi `vI: A
2y + 7
ˆK (ax + b) «¼viv fvM KiGj fvMGkl  P 4 Q
wPGò PQRS ‰KwU mvgv¯¦wiK|
f
 b
 a 
‰i mgvb nGe?  20. y ‰i gvb KZ?
K a0 L a=0
B C K 15 L 30 M 33 N 41
M a>0 N a<0 21. PQRS ‰i cwimxgv KZ?
wPGò, ABC = 90 ‰es sin = x
K 7 L8 M 12 N 14
wbGPi ZG^Åi AvGjvGK (3 I 4) bs cÉGk²i Dîi 12. 
 sin ‰i gvb KZ?
22. mgGKvYx wòfzGRi cwie†Gîi ˆK±`Ê ˆKv^vq
`vI: 1  x2
K L 1  x2 AewÕ©Z?
x = 7 + 4 3 ‰KwU exRMvwYwZK mgxKiY| x
1 1 K wòfzGRi Afů¦Gi L wòfzGRi ewnfÆvGM
1
3. x2 + x2 ‰i gvb KZ? M 2
N
x M jGÁ¼i Dci N AwZfzGRi Dci
1x
K 190 L 194 M 198 N 200 13. 
 tan ‰i gvb KZ?
23. ˆKvb wòfzGR KqwU ewne†Æî AvuKv hvq?
4. x
1
‰i gvb KZ? K 1wU L 2wU M 3wU N 4wU
x 1  x2
x K L
x 24. 
 ˆKvb e†Gî A¯¦wjÆwLZ mvgv¯¦wiK ‰KwU?
1  x2
K 2 3 L4 M 8 3 N 14 K eMÆ L iÁ¼m
1 1
5. 
 0.000435 msLÅvwUi mvaviY jGMi cƒYÆK M
x
N M AvqZ N UÇvwcwRqvg
1  x2
KZ? 25.
A
14. ‰KwU UvIqvGii Š`NÆÅ I ‰i Qvqvi Š`GNÆÅi
K 4̄ L 3̄ M3 N 4
AbycvZ 3 : 3 nGj, D®²wZ ˆKvY KZ?
3m + n
6. nm
= 9 nGj, m : n ‰i gvb KZ? K 60 L 45 M 30 N 15 B
O
D

K 1:5 L 5:1 15. ‰KwU mylg cçfzGRi NƒYÆb ˆKvY KZ? C

M 2:3 N 3:2 K 36 L 45 M 60 N 72 wPGò, O ˆK±`ÊwewkÓ¡ e†Gî BAD = 85
wbGPi ZG^Åi AvGjvGK (7 I 8) bs cÉGk²i Dîi 16. ˆKvb eMÆGÞGòi KGYÆi Š`NÆÅ I evüi nGj, BCD ‰i gvb KZ?
K 100 L 95 M 90 N 85
`vI: Š`GNÆÅi AbycvZ KZ?
26. ˆKvb wòfzGRi f„wg jGÁ¼i w«¼àY nGj f„wgi Dci
‰KwU cÉK‡Z f™²vsGki ni I jGei mgwÓ¡ K 2:1 L 1: 2
AwâZ eMÆ jGÁ¼i Dci AwâZ eGMÆi KZàY?
13 ‰es àbdj 42| M 3:1 N 1: 3
K 2 L3 M4 N 8
7.   f™²vskwUi gvb KZ? wbGPi ZG^Åi AvGjvGK (17 I 18) bs cÉGk²i 27. ABC ‰i A = ‰K mgGKvY ‰es BE I
K
5
L
6
M
4
N
7 Dîi `vI: CF gaÅgv nGj
8 7 9 6
A D i. BE2 = AB2 + AE2
8. 
 f™²vskwUi je I ni ˆ^GK KZ weGqvM
ii. CF2 = AC2 + AF2
12 cm

1
KiGj f™²vskwUi gvb 2 nGe? iii. 4(BE2 + CF2) = 6BC2
wbGPi ˆKvbwU mwVK?
K 2 L3 M4 N 5
B E 5 cm
C K i I ii L i I iii
9. f + g + h + k + l + ........ mgv¯¦i avivfzÚ nGj
M ii I iii N i, ii I iii
g+f 17. ABCD UÇvwcwRqvGgi ˆÞòdj KZ eMÆ ˆm.wg.?
i. h=
2 28. 
 MYmsLÅv mviwY ŠZwi KiGZ cÉ^Gg
K 72 L 102
h+l
M 174 N 204
ˆKvbwU cÉGqvRb nq?
ii. k =
2 K ˆkÉwYmsLÅv L ˆkÉwYeÅvwµ¦
f+h 18. ABED eMÆ I CDE wòfzGRi cwimxgvi
iii. g = M cwimi N MYmsLÅv
2 AbycvZ KZ?
29. 1 ˆ^GK 22 chƯ¦ 3 «¼viv wefvRÅ msLÅvàGjvi
wbGPi ˆKvbwU mwVK? K 3:4 L 5:7 M 5:8N 8:5
19. 
 kyaygvò `ywU evüi Š`NÆÅ ˆ`Iqv ^vKGj gaÅK KZ?
K i I ii L i I iii
K 12 L 15 M 18 N 21
M ii I iii N i, ii I iii wbGPi ˆKGbwU AuvKv mÁ¿e?
30. 
 avc wePzÅwZ ui ‰i mgvb wbGPi ˆKvbwU?
10. 
 3 sin  cos = 0 nGj,  ‰i gvb KZ? K iÁ¼m L AvqZGÞò
xi  a xi + a xi a  xi
M mvgv¯¦wiK N UÇvwcwRqvg K L M N
K 30 L 45 M 60 N 90 h h h h
1 M 2 K 3 L 4 K 5 K 6 M 7 L 8 N 9 M 10 K 11 N 12 L 13 K 14 K 15 N
Dîi

16 K 17 M 18 N 19 L 20 M 21 N 22 N 23 M 24 M 25 L 26 M 27 K 28 M 29 K 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 47
392 KÝevRvi miKvwi evwjKv DœP we`Åvjq, KÝevRvi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. wbGPi ˆKvb mÁ·KÆwU mwVK? 10. avivwUi n-Zg c` KZ? 18. OAB = 45 nGj, AOB = KZ?
K NQZR K n+3 L n3 K 75 L 80
L NZQR M 3n N n+1 M 90 N 95
M NRZQ 11. 1g 7 wU cG`i mgwÓ¡ KZ? 19. BAC = KZ?
N ZNQR K 7 L 7 K 60 L 70
2. 
 wbGPi ˆKvbwU gƒj` msLÅv? M 35 N 35
M 80 N 90
K 2 L e 20. 
 OM = 4 ˆm.wg. nGj, AC = KZ ˆm.wg.?
wbGPi ZG^Åi AvGjvGK (12 I 13) bs cÉGk²i
4 K 3 L 4
M
3
N 1.7325... Dîi `vI: M 6 N 8
A
3. duvcv ˆmU-‰i Power Set-‰i Dcv`vb 21. wòfzGRi wZbevüi Dci AwâZ eMÆGÞòmgƒn
KqwU? KqwU mgGKvY Drc®² KGi?
K 0 L 1 K 8 L 9
M 2 N 4 B D C M 12 N 16
4. AGf`-‰i ˆÞGò@ ABC mgevü : AD  BC, AB = 6 ˆm.wg. 22. sin = cos nGj,  = KZ?
i. mKj exRMwYZxq mƒò AGf` 12. 
 BD ‰i Š`NÆÅ KZ ˆm.wg.? K 30 L 45
M 60 N 90
ii. mKj AGf`B mgxKiY K 3 L 3
23. ‰KwU LuywUi Qvqvi Š`NÆÅ 3 3 wg., mƒGhÆi
iii. mKj mgxKiYB AGf` M 6 N 2 3
D®²wZ ˆKvY 60| LuywUwUi DœPZv KZ wg.?
wbGPi ˆKvbwU mwVK? 13. 
 AD-‰i gvb KZ?
K 4.5 L 9
K i I ii L i I iii K 27 L 2 3 M 12 N 5.4
M ii I iii N i, ii I iii M 3 N 5 2 24. 
 wòGKvYwgwZK mÁ·GKÆi ˆÞGò@
5. 
 0.000345 msLÅvwUi mvaviY jMvwi`Ggi 14. e†Gîi eÅvmvaÆ r nGj@ i. cosec2 = 1  cot2
cƒYÆK KZ? i. eÅvm 2r ii. sec2  tan2 = 1
  iii. cos2 = 1  sin2
K 5 L 4 ii. ˆÞòdj r2
wbGPi ˆKvbwU mwVK?
M 3 N 4 
iii. e†îKjvi ˆÞòdj 360  r2 K i I ii L ii I iii
6. 2x  5 + 3 = 2 ‰i mgvavb ˆmU ˆKvbwU? M i I iii N i, ii I iii
K {3} L {3} wbGPi ˆKvbwU mwVK?
25. ‰KwU lofzGRi ˆgvU ˆKvGYi cwigvY KZ?
M {3} N  K i I ii L i I iii
K 4 mgGKvY L 6 mgGKvY
7. p, q, r KÌwgK mgvbycvwZ nGj@ M ii I iii N i, ii I iii
M 8 mgGKvY N 10 mgGKvY
p q 15. e†Gîi A¯¦wjÆwLZ mvg¯¦wiK ‰KwU@
i. =
q r 26. ˆejb I ˆKvYGKi AvqZGbi AbycvZ KZ?
2 K eMÆ L iÁ¼m K 3:2 L 2:3
ii. q = pr
p r M AvqZ N UÇvwcwRqvg M 1:3 N 3:1
iii. =
r q 16. iÁ¼Gmi cÉwZmvgÅ ˆiLvi msLÅv KqwU? 27. 
 ‰KwU NbGKi mÁ·ƒYÆ c†GÓ¤i ˆÞòdj
wbGPi ˆKvbwU mwVK? K 0 L 1 216 eMÆ.wg. nGj AvqZb KZ?
K i I ii L ii I iii M 2 N 4 K 6 6 Nb wg. L 36 N. wg.
M i I iii N i, ii I iii wbGPi ZG^Åi AvGjvGK (17-20) bs cÉGk²i Dîi M 144 Nb wg. N 216 Nb wg.
8. 
 ‰KwU eGMÆi KGYÆi Š`NÆÅ I cwimxgvi `vI: 28. DcvGî eÅen†Z msLÅvmgƒnGK Kx eGj?
AbycvZ KZ? C K NUbv L Z^Åvw`
K 1:2 2 L 1: 2 O M WvUv N PjK
M 1:2 N 2:1 29. 
 ˆK±`Êxq cÉeYZvi wbfÆiGhvMÅ cwigvY
A M B
9. ˆKvb wòfzGRi KqwU ewnte†î AuvKv hvq? ˆKvbwU?
K 1 L 2 K Mo L gaÅK
M 3 N 4 wPGò, M, AB-‰i gaÅwe±`y OA = 5 ˆm.wg.
M cÉPziK N meàGjv
wbGPi ZG^Åi AvGjvGK (10 I 11) bs cÉGk²i 17. OM = 4 ˆm.wg. nGj, AB = KZ ˆm.wg.? 30. meGPGq ˆQvU ˆgŒwjK msLÅv ˆKvbwU?
Dîi `vI: K 5 L 6
K 1 L 2
‰KwU mgv¯¦i avivi 1g c` 2, mvaviY A¯¦i 1 M 7 N 8
M 3 N 5

1 L 2 M 3 L 4 K 5 L 6 N 7 K 8 K 9 M 10 M 11 L 12 K 13 K 14 N 15 M
Dîi

16 M 17 L 18 M 19 N 20 N 21 M 22 L 23 L 24 L 25 M 26 N 27 N 28 N 29 N 30 L
48 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
393 LvMovQwo KÅv´ŸbGg´Ÿ cvewjK Õ•zj I KGjR, LvMovQwo

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 Mo wbYÆGqi ˆÞGò AbywgZ Mo, a = KZ 11. 
 ‰KwU mgGKvYx wòfzGRi mƒßGKvY«¼Gqi 21. 0.000575 mvaviY jGMi cƒYÆK KZ?
nGe? ˆhLvGb, u5 =  5, x5 = 17, h = 5 cv^ÆKÅ 12 nGj, mƒßGKvY«¼Gqi gGaÅ e†nîg K 3̄ L 4̄
K 40 L 42 ˆKvGYi gvb KZ wWwMÉ? M 2̄ N 5̄
M 50 N 52 K 49 L 50 22. (5y  1) + 9 = 3 ‰i mgvavb ˆmU ˆKvbwU?
2. 16 ˆ^GK 50 chƯ¦ ˆgŒwjK msLÅvàGjvi M 51 N 52 K {} L {6}
gaÅK wbGPi ˆKvbwU? 12. S eGYÆi NƒYÆb ˆKvY KZ wWwMÉ nGe? M {6} N {0}
K 23 L 29 K 180 L 190 23. 
 ‰KwU eGMÆi evüi Š`NÆÅ wZbàY KiGj
M 31 N 37 M 260 N 60 ‰i ˆÞòdj KZ àY e†w«¬ cvGe?
1
3. 
 hw` cos(x + 45) = nq, ZGe x ‰i 13. `yBwU e†Gî mGeÆvœP KqwU mvaviY Õ·kÆK K 9 L 8
2
AuvKv mÁ¿e? M 6 N 15
gvb KZ nGe?
K 45 L 30 K 3wU L 4wU 24. x + 3y  1 = 0 I 2x + 6y  2 = 0
M 25 N 15 M 6wU N 10wU mgxKiY«¼q
4. cotx sec2x  1 =? 14. e†îÕ© mvgv¯¦wiKGK wK ejv nq? i. ciÕ·i wbfÆikxj
K 1 L cosx K UÇvwcwRqvg L eMÆ ii. Amgém
M secx N tanx M AvqZ N iÁ¼m iii. mgvavb AvGQ ‰KwU
5. 
 ‰KwU MvGQi Š`NÆÅ I MvQwUi Qvqvi Š`GNÆÅi 15. ˆKvGbv eGMÆ A¯¦e†Æî AvuKGZ KqwU avc wbGPi ˆKvbwU mwVK?
AbycvZ 1 : 3 nGj, D®²wZ ˆKvY KZ? AbymiY KiGZ nq? K i L ii
K 30 L 45 K 6 L 4 M iii N i, ii I iii
M 60 N 90 M 2 N 3 25. 
 4 + m + n + 32 MyGYvîi avivfzÚ nGj,
6. 
 mgGKvYx wòfzGRi 60 ˆKvY AâGbi 16. a I b `yBwU cƒYÆmsLÅv nGj a2 + b2 ‰i mvG^ m2 + n2 ‰i gvb wbGPi ˆKvbwU?
ˆÞGò wbGPi ˆKvbwU ˆhvM KiGj ˆhvMdj ‰KwU K 520 L 620
i. fƒwg < jÁ¼ ii. fƒwg > jÁ¼ cƒYÆeMÆ msLÅv nGe? M 820 N 320

iii. fƒwg < AwZfzR K 8ab L  2ab 26.  2 + 2  2 + 2  2 + .... avivwUi 2(n + 1)
wbGPi ˆKvbwU mwVK?
M 4ab N ab cG`i mgwÓ¡ KZ?
K 0 L 1
17. f(m) = m2 + 2m  7 nGj, f   3  = KZ?
1
K i I ii L i I iii
M 2 N 3
M ii I iii N i, ii I iii
K
49
L
60 27. cixÞvq cÉvµ¦ wRwc‰ ˆKvb aiGbi PjK?
7. ‰KwU mylg lofzGRi ‰KwU kxlÆGKvGYi gvb 9 9
K wewœQ®² L webÅÕ¦
KZ? 58 68
M
9
N 
9 M AwewœQ®² N AwebÅÕ¦
K 180 L 135
M 120 N 90 18. a3 + 3 3 ‰i ‰KwU Drcv`K? 28. sin = 2 nGj, cot  2    =?
3 
8. wPGò e†îwUi
2 2
K a + 3a + 3 L a  3a  3
1
OA = 6 ˆm.wg. nGj-
A M a+ 3 N a 3 K 3 L
O 30 3
i. cwiwa = 20 ˆm.wg.
B 19. 
 ABC ‰ AB = BC = CA = 6 nGj, M 1 N 0
ii. ˆÞòdj = 36 eMÆ ˆm.wg. gaÅgv AD = KZ ˆm.wg.? 29. 
 PQR ‰ Q = 90 ‰es P = 30
iii. AB PvGci Š`NÆÅ =  ˆm.wg. K 3 3 L 2 3 nGj, wbGPi ˆKvb mÁ·KÆwU mwVK?
wbGPi ˆKvbwU mwVK? M 6 2 N
3 K PR = 2QR L PR = 3QR
2 PQ
K i L ii I i M PQ = N PQ = 4QR
1
M iii N ii I iii 20. 
 x + = 5 nGj
x
3

9. ˆKvb eGMÆi A¯¦e†ÆGîi eÅvm 10 ˆm.wg. nGj, 1

30. U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3, 4},
i. x2  = 5 21 P = A nGj
eGMÆi ˆÞòdj KZ nGe? x2
1 i. P ‰i Dcv`vb 8wU
K 20 L 10 ii. x3 + 3 = 110
x
M 25 N 100 ii. A  P ‰i Dcv`vb 8wU
iii. x2 + 1 = 5x
10. 
 ˆKvGbv eMÆGÞò Zvi KGYÆi Dci AwâZ iii. A  P ‰KwU dvskb
wbGPi ˆKvbwU mwVK?
eMÆGÞGòi KZàY? K i L ii wbGPi ˆKvbwU mwVK?
K i L ii
K w«¼àY L ˆ`oàY M ii I iii N i, ii I iii
M i I ii N i, ii I iii
M mgvb N AGaÆK

1 L 2 M 3 N 4 K 5 K 6 L 7 M 8 N 9 N 10 N 11 M 12 K 13 L 14 M 15 N
Dîi

16 L 17 N 18 M 19 K 20 N 21 L 22 K 23 L 24 K 25 N 26 K 27 M 28 K 29 K 30 L
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 49
394 Rvjvjvev` KÅv´ŸbGg´Ÿ cvewjK Õ•zj ‰´£ KGjR, wmGjU

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. cot(  30) =
1
nGj sin = KZ? 12. 
 5 5 ‰i 5 wfwîK jM KZ? 20. cot  cosec = 3 nGj
4
3 1 5 3 5
K L M N cosec + cot = KZ?
1 3 2 2 2 3
K L0 M1 N
2 2 13. A = {1, 3, 5}, B = {2, 3, 5} ‰i ˆÞGò@ 3 4 3 4
K  L M N
2. x + 2y = 10; 2x + 4y = 18 mij 4 3 4 3
i. A  B = {3, 5}
mnmgxKiY `yBwU@ 21. y = 2x + 1 ‰KwU mgxKiY
ii. P(A  B) ‰i Dcv`vb msLÅv 16
K mvgémÅ L AmsLÅ mgvavb i. (1, 3) we±`ywU mgxKiYwUi ˆjLwPGòi
iii. A\B = {1, 5}
M mgvavb ‰KwU N mgvavb ˆbB wbGPi ˆKvbwU mwVK? IcGi AewÕ©Z
3. 
 5 + 8 + 11 + 14 + ... ... ... avivwUi ˆKvb ii. mgxKiYwUi ˆjLwPò ‰KwU mijGiLv
K i I ii L i I iii
c` 383? iii. mgxKiYwUi ˆjLwPò ‰KwU e†î
M ii I iii N i, ii I iii
K 127 L 129 wbGPi ZG^Åi AvGjvGK (14 I 15) bs cÉGk²i wbGPi ˆKvbwU mwVK?
M 130 N 132 K i I ii L i I iii
Dîi `vI:
4. 5 ‰i NƒYÆb ˆKvY KZ wWwMÉ? 1 M ii I iii N i, ii I iii
K 90 L 180 x+ =2 2
x 22. wbGPi ˆKvbwU Agƒj` msLÅv?
M 270 N 360 1
14. x2 + x2 = KZ? . 36
5. 12 ˆm.wg. DœPZv wewkÓ¡ ‰KwU mge†îf„wgK K 2.5 L
121
wmwj´£vGii eÅvmvaÆ 4 ˆm.wg. nGj ‰i K 4 L6 M8 N 10 3 64 7
eKÌZGji ˆÞòdj KZ eMÆ ˆm.wg.? 1 M N
15. x + 3 = KZ?
3
x
125 7
K 96 L 128 . ..
M 192 N 384
K 22 2 L 16 2 23. 
 0.28  42.18 = KZ?
M 14 2 N 10 2 .. .
6. ‰KwU MvQ I ‰i Qvqvi AbycvZ 3 : 3 nGj K 12.185 L 12.15
16. 
 ‰KwU e†Gîi ˆK±`ÊÕ© ˆKvY x + 80 ‰es
mƒGhÆi D®²wZ ˆKvY KZ wWMÉx? .. ..
e†îÕ© ˆKvY x + 10 nGj x ‰i gvb KZ? M 12.85 N 22.185
K 60 L 45
3m+n
M 30 N 15 K 50 L 60 24. hw` nm = 9 nq ZGe m : n = KZ?
wbGPi ZG^Åi AvGjvGK (7 I 8) bs cÉGk²i Dîi M 70 N 80
K 2:3 L 3:2 M 5:1N 1:5
`vI: wbGPi ZG^Åi AvGjvGK (17 I 18) bs cÉGk²i
wbGPi ZG^Åi AvGjvGK (25 I 26) bs cÉGk²i
ˆkÉwYeÅvwµ¦ 20-29 30-39 40-49 50-59 Dîi `vI:
Dîi `vI:
MYmsLÅv 5 6 7 2 B
`yB AâwewkÓ¡ ‰KwU msLÅvi `kK Õ©vbxq
7.   mviwYi ZG^Åi gaÅK ˆkÉwYi gaÅgvb C
O
A
Aâ ‰KK Õ©vbxq AGâi w«¼àY ‰es ‰KK
KZ? Õ©vbxq Aâ x|
K 34.5 L 38
25. 
 msLÅvwU KZ?
M 38.33 N 43.33 O ˆK±`Ê wewkÓ¡ e†Gî AC = 12 ˆm.wg. ‰es K 2x L 3x M 12x N 21x
8. 
 mviwYi ZG^Åi cÉPziK KZ? BOA = 60 26. 
 A⫼q Õ©vb wewbgq KiGj msLÅvwU
K 46.67 L 41.67 17. 
 AB PvGci Š`NÆÅ KZ ˆm.wg.?
M 38.33 N 37.5
KZ nGe?
K 40.84 L 12.57 K 3x L 4x M 12x N 21x
9. ‰KwU e†Gîi eÅvm 24 ˆm.wg. nGj cwiwa M 6.28 N 3.14
27. 0.000337 ‰i mvaviY jGMi cƒYÆK KZ?
KZ? 18. 
 AOB e†îKjvi ˆÞòdj KZ?  
K 15.07 L 37.7 K 4 L3 M3 N 4
K 150.8 L 75.4
M 75.4 N 150.77 28. 3  3 + 3  3 + ... ... ... ‰i 19-Zg c` KZ?
M 40.84 N 18.85
10. 
 log625  2log5 = KZ?
19. b`xi ZxGi ˆKvGbv Õ©vGbi Aci cÉvG¯¦i 90 K 3 L 3 M0 N 30
K log600 L log125 29. 2sin2y + 2cos2y = KZ?
M log25 N log5
wgUvi jÁ¼v MvGQi D®²wZ ˆKvY 60 nGj
K 2 L4 M0 N 2
. b`xi cÉÕ© KZ wgUvi?
11. 0.25 mgvb wbGPi ˆKvbwU? 30. 
2
 y  = 2a nGj 2
6a
‰i gvb KZ?
K 90 3 L 30 3 y y  2ay  1
5 25 23 50
K L M N M 120 3 N 60 3 K 3a L 2a M 3a N 6a
99 90 90 99

1 M 2 N 3 K 4 L 5 K 6 K 7 K 8 L 9 M 10 M 11 M 12 M 13 K 14 L 15 N
Dîi

16 L 17 M 18 N 19 L 20 K 21 K 22 N 23 K 24 K 25 N 26 M 27 K 28 K 29 K 30 N
50 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
395 Imgvbx ˆgwWGKj DœP we`Åvjq, wmGjU

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. ˆKvbwU Õ¼vfvweK msLÅv? 12. 4x3 + 3x2  2x + 1 = 0 mgxKiGYi NvZ 20. ABC ‰-
K 1 L 2 KZ? i. cosA = sinC
5 5
M  N 3 K 4 L 3 ii. cosA + secA =
2
2
M 2 N 1 1
2. 
 wbGPi ˆKvbwU Agƒj` msLÅv?
13. 75 ˆKvGYi ŠiwLK mÁ·ƒiK ˆKvGYi cwigvY
iii. tanC =
3
64 32
K L KZ? wbGPi ˆKvbwU mwVK?
36 64
K 15 L 105 K i I ii L i I iii
3
M
81
N
8 M 195 N 265 M ii I iii N i, ii I iii
625 3
27 14. 
 mgGKvYx wòfzGRi mƒßGKvY«¼Gqi cv^ÆKÅ 21. wbGPi Z^ÅàGjv jÞÅ Ki-
wbGPi ZG^Åi AvGjvGK (3 I 4) bs cÉGk²i Dîi 8 nGj, e†nîg ˆKvGYi gvb KZ wWwMÉ? i. 30 ˆKvY AâGbi ˆÞGò f„wg  jÁ¼ nGe
`vI: K 41 L 42 ii. 45 ˆKvY AâGbi ˆÞGò f„wg = jÁ¼ nGe
U = {1, 2, 3, 4, 5, 6} M 45 N 49 iii. 60 ˆKvY AâGbi ˆÞGò f„wg  jÁ¼ nGe
A = {1, 3, 5}, B = {2, 4, 6}
15. iÁ¼Gmi- wbGPi ˆKvbwU mwVK?
3. {A  B} ‰i gvb wbGPi ˆKvbwU? K i I ii L i I iii
i. mw®²wnZ evüàGjv ciÕ·i mgvb
K {} L {2, 4, 6} M ii I iii N i, ii I iii
M {1, 3, 5} N {1, 2, 3, 4, 5, 6} ii. mw®²wnZ ˆKvYàGjv ciÕ·i mÁ·ƒiK
22. ˆKvbwU mgvbycvZ?
4. A\B ‰i gvb wbGPi ˆKvbwU? iii. KYÆ«¼q ciÕ·iGK mgGKvGY mgw«¼Lw´£Z K 1:2=3:4 L 2 : 5 = 6 : 15
K {1, 2, 3, 4, 5, 6} L {2, 4, 6} KGi M 4:6=9:4 N 10 : 5 = 5 : 10
M {1, 3, 5} N {} 23. (2, 3) we±`ywU ˆKvb mgxKiGYi Dci
wbGPi ˆKvbwU mwVK?
5. a + b = 16 ‰es ab = 1 nGj, (a  b)2 = AewÕ©Z?
K i I ii L i I iii
KZ? K xy=1 L 2x + y = 7
M ii I iii N i, ii I iii
K 12 L 14 M x + 3y = 5 N 2x + 2y = 6
M 22 N 24 16. 
 5 ˆm.wg. eÅvmvaÆwewkÓ¡ e†Gîi ˆK±`Ê ˆ^GK
24.  16  8  0 + ...... avivwUi mvaviY A¯¦i
6. 
 x4  5x2 + 1 = 0 nGj, x + ‰i gvb
1 ˆKvGbv RÅv ‰i Dci AwâZ jGÁ¼i Š`NÆÅ 3 KZ?
x
ˆm.wg. nGj, e†Gîi H RÅv ‰i Š`NÆÅ KZ? K 8 L 8
KZ? 1
K 16 ˆm.wg. L 8 ˆm.wg. M 2 N
K 7 L 3 2
M 4 ˆm.wg. N 2 ˆm.wg.
M 7 N 3 25. 
 12 + 22 + 32 + ....... + 502 = KZ?
1 1 17. AwaPvGc A¯¦wjÆwLZ ˆKvY- K 1275 L 42925
7.  a + = 5 nGj, a + 3 ‰i gvb

a
3
a
KZ?
K cƒiK ˆKvY L mƒßGKvY M 1625625 N 54587
K 21 L 23 26.  eGMÆi cÉwZmvgÅ
 ˆiLv KqwU?
M Õ©„jGKvY N mgGKvY
M 110 N 140
K 4wU L 3wU
4n  1 18. 
 `yBwU e†î ciÕ·iGK ˆQ` KiGj ZvG`i
8. 
 n ‰i gvb wbGPi ˆKvbwU? M 2wU N 1wU
2 +1 gGaÅ mGeÆvœP KqwU mvaviY Õ·kÆK Aâb
n
K 2 +1 L 2n  1 27. 
 ‰KwU mgevü wòfzGRi evüi Š`NÆÅ 6
Kiv mÁ¿e?
M 2 n+1
N 2n1 ˆm.wg. nGj ‰i ˆÞòdj KZ eMÆ ˆm.wg.?
K 1wU L 2wU K 3 3 L 4 3
3
9. m=3 nGj, m = KZ? M 3wU N 4wU M 9 3 N 18 3
K 3
3
L 3 wbGPi ZG^Åi AvGjvGK (19 I 20) bs cÉGk²i 28. ‰KwU eGMÆi ‰K evüi Š`NÆÅ a ‰KK nGj,
M 3 N 27 Dîi `vI: ‰i ‰KwU KGYÆi Š`NÆÅ KZ ‰KK?
10. 
 log164 ‰i gvb KZ? A K 4a L 2a
1 1 M 3a N 2a
K L
16 8 2
29. MYmsLÅv mviwY cÉÕ§Z KiGZ nGj cÉ^Gg
1 1
M
4
N
2
ˆKvbwU cÉGqvRb?
K ˆkÉwY msLÅv L ˆkÉwY eÅeavb
11. 
 0.000567 ‰i mvaviY jGMi cƒYÆK B 3 C
M cwimi N MYmsLÅv
wbGPi ˆKvbwU?

19. A ‰i gvb ˆKvbwU? 30. 
 ˆKvbwU wewœQ®² PjK?
K 7 L 6 K 90 L 60 K Zvcgvòv L cvwLi msLÅv
 
M 5 N 4 M 45 N 30 M eqm N DœPZv

1 N 2 L 3 N 4 M 5 K 6 M 7 M 8 L 9 N 10 N 11 N 12 L 13 L 14 N 15 N
Dîi

16 L 17 L 18 L 19 L 20 N 21 N 22 L 23 K 24 L 25 L 26 K 27 M 28 N 29 M 30 L
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 51
396 nweMé miKvwi DœP we`Åvjq, nweMé

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 2a4 + 16a ivwkwUi Drcv`K wbGPi ˆKvbwU? 11. 4sinA = 3 nGj, tanA = KZ? wbGPi ˆKvbwU mwVK?
K 2a2 L 2+a 1 3 3 7 K i I ii L i I iii
M a2 N a2 + 2a + 4 K L M N
5 5 7 3 M ii I iii N i, ii I iii
2. {x  Ñ : 4 < x  5} ‰i ZvwjKv i…c 12. 
 n msLÅK Õ¼vfvweK msLÅvi NGbi mgwÓ¡ 23. e†îÕ© mvgv¯¦wiK ‰KwU
ˆKvbwU? 225| H msLÅv àGjvi mgwÓ¡ KZ? K iÁ¼m L eMÆ
K 0 L {4, 5} M {4} N {5} K 13 L 15 M 20 N 125 M AvqZ N UÇvwcwRqvg
3. {a + b, 2) = {4, a  b) nGj (a, b) ‰i gvb 13. 
 logx324 = 4 nGj, x ‰i gvb KZ?
24. wPGòi AvGjvGK tanP + cotR ‰i gvb KZ?
KZ? K 81 L2 3 M3 2N 8 2 3
K (1, 3) L (3, 1) M (4, 2) N (2, 4) K P
14. 3 3 ‰i 3 wfwîK jM KZ? 2
4. hw` f(x) = 3x2 + 4kx nq ZGe k ‰i ˆKvb 3 1 1 7
K L M N 32 L 4 5
gvGbi RbÅ f(2) = 0 nGe? 2 2 3 5
3 3 15. ax = ay nGj, hLb x = y nGe ˆmGÞGò 8
K 3 L M N 3 M
2 2 3 Q 3 R
i. a > 0 ii. a < 0
5.   ‰KwU avivi cÉ^g c` 2 ‰es mvaviY 5
iii. a  1 N
3
A¯¦i 3 nGj wbGPi ˆKvbwU mwVK?
25. tan( + 30) = 3 nGj,  ‰i gvb KZ?
i. avivwU 2 + 5 + 8 + ............... K i I ii L i I iii
K 30 L 45 M 60 N 90
ii. avivwUi 6Ó¤ c` 17 M ii I iii N i, ii I iii
wbGPi ZG^Åi AvGjvGK (26 I 27) bs cÉGk²i
iii. cÉ^g 10wU cG`i mgwÓ¡ 155 wbGPi ZG^Åi AvGjvGK (16 I 17) bs cÉGk²i Dîi `vI:
wbGPi ˆKvbwU mwVK? Dîi `vI: M N
m+1
K i I ii L i I iii 3m+1 (3m1)
a=
(3m)m +1
‰es b = 9m1
M ii I iii N i, ii I iii F E
wbGPi ZG^Åi AvGjvGK (6 I 7) bs cÉGk²i Dîi 16. 
 a = 3 nGj m = KZ? A D
`vI: K 0 L 1 B C
A D M 1 N 2
3y + 8
wPGòi ABCDEF ‰KwU mylg lofzR|
17.  b = 81 nGj m = KZ?

2y + 7 CD = 4 ˆm.wg. ‰es EN = FM = 13 ˆm.wg.|
B C K 3, 3 L 3,  1
26. 
 mÁ·ƒYÆ ˆÞGòi cwimxgv KZ?
M 3, 1 N 3, 5
ABCD ‰KwU mvgv¯¦wiK hvi mw®²wnZ evü«¼Gqi K 46 ˆm.wg. L 50 ˆm.wg.
18.  a  b = 27 nGj m ‰i
 gvb ˆKvbwU?
Š`NÆÅ 3 ˆm.wg. I 4 ˆm.wg. K 0 L 1 M 54 ˆm.wg. N 58 ˆm.wg.
6.   y ‰i gvb KZ? 1 27. 
 ABCDEF ˆÞGòi ˆÞòdj KZ?
K 3 L 5 M 2 N 
2 K 44 3 eMÆ ˆm.wg. L 50 eMÆ ˆm.wg.
M 30 N 33 19. AwewœQ®² PjGKi D`vniY ˆKvbwU? 24
7. 
 ABCD ˆÞGòi AaÆ cwimxgv KZ? M 24 3 eMÆ ˆm.wg. N eMÆ ˆm.wg.
K Zvcgvòv L RbmsLÅv 3
K 7 ˆm.wg. L 14 ˆm.wg. M QvòxmsLÅv N eÕ§i msLÅv 28. ‰KwU NbGKi ‰K avi 2 3 ˆm.wg. nGj,
M 28 ˆm.wg. N 56 ˆm.wg. wbGPi ZG^Åi AvGjvGK (20 I 21) bs cÉGk²i KGYÆi ˆ`NÆÅ KZ?
8. hw` ‰KwU e†î ‰KwU wòfzGRi wZbkxlÆMvgx Dîi `vI: K 6 ˆm.wg. L 6 3 ˆm.wg.
nq ZGe e†îwUGK eGj ‰KwU ˆejGbi DœPZv 8 ˆm.wg. ‰es f„wgi eÅvm M 3 3 ˆm.wg. N 24 3 ˆm.wg.
K A¯¦e†Æî L A¯¦wjÆwLZ e†î 12 ˆm.wg.| 29.
1
 1 + 2 .... avivwU àGYvîi aviv nGj
M cwie†î N ewnte†î 20. ˆejbwUi eKÌc†GÓ¤i ˆÞòdj KZ eMÆ 2
9. wPGòi e†Gî ˆm.wg.? i. avivwUi mvaviY AbycvZ =  2
i. PS = 12
O
K 37.70 L 50.27 ii. avivwUi mµ¦g c` = 4 2
10 1
ii. OQ = 4 M 301.59 N 603.19 iii. cÉ^g wZbwU cG`i mgwÓ¡ =
6
iii. e†îwUi cwiwa 20 P Q S 21. ˆejbwUi AvqZb KZ Nb ˆm.wg.? 2 2
wbGPi ˆKvbwU mwVK? K 904.78 L 3015.91 wbGPi ˆKvbwU mwVK?
K i I ii L i I iii M 3619.12 N 150.81 K i I ii L i I iii
1 M ii I iii N i, ii I iii
M ii I iii N i, ii I iii 22. x2 + 2 = 7 nGj
x
1 30. ÷ay cwimxgv ˆ`Iqv ^vKGj ˆKvb cÉKvGii
10. = KZ? 1
x  1x 
2

1 + tan2 i. x + = 3
x
ii.
 
=4 wòfzR AuvKv mÁ¿e?
K cos L sec 1 K mgw«¼evü L mgevü
iii. x4 + = 47
M cosec N cot x4 M mgGKvYx N welgevü

1 N 2 N 3 L 4 M 5 N 6 N 7 K 8 M 9 L 10 K 11 M 12 L 13 M 14 K 15 L
Dîi

16 K 17 L 18 N 19 K 20 M 21 K 22 L 23 M 24 K 25 K 26 L 27 M 28 K 29 K 30 L
52 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
397 kvGqÕ¦vMé Bmjvwg ‰KvGWwg ‰´£ nvB Õ•zj, nweMé

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 2x  5 + 3 = 2 ‰i mwVK mgvavb ˆmU wbGPi ˆKvbwU mwVK? 19. 
 `ywU msLÅvi AbycvZ 3 : 4 ‰G`i M.mv.à.
ˆKvbwU? K i I ii L i I iii 3 nGj, msLÅv `ywUi j.mv.à. KZ?
K {3} L {} M ii I iii N i, ii I iii K 4 L 9
M { 3} N {3} M 12 N 36
11. 
 x2  2x + 1 = 0 nGj
2.  log2 2 64 ‰i
 gvb ˆKvbwU? 1 1
20. 8 ‰i NƒYÆb ˆKvY KZ wWMÉx?
i. x+ =2 ii. x =0 K 90 L 120
K 2 L 3 x x
M 4 N 8 M 180 N 360
1
iii. x2 + 2 = 4 21. 3x  5y = 7; 6x  10y = 15 ‰B mgxKiY
3. 0.555̇ ‰i mvgvbÅ f™²vsGki i…c ˆKvbwU? x
5 11 11 50 wbGPi ˆKvbwU mwVK? ˆRvUwU
K L M N
9 10 9 99 K i I ii L i I iii i. Amgém
4. D = {2, a} nGj, P(D) wbGPi ˆKvbwU? M ii I iii N i, ii I iii ii. ‰KwU gvò mgvavb AvGQ
K {2}, {a} L {2, a} iii. ciÕ·i AwbfÆikxj
wbGPi ZG^Åi AvGjvGK (12 I 13) bs cÉGk²i
M {{2}, {a}, {2, a}, } wbGPi ˆKvbwU mwVK?
N {{2}, {a}, {2, a}} Dîi `vI:
P K i I ii L i I iii
5. 23x + 2 = 16 nGj, x ‰i gvb KZ?
M ii I iii N i, ii I iii
6 6
K 2 L 4 O
2 4
22. 
 12 + 24 + 28 + ... ... ... + 768
M N Q R àGYvîi avivwUGZ KZwU c` AvGQ?
3 3 6
wbGPi ZG^Åi AvGjvGK (6 I 7) bs cÉGk²i Dîi K 5 L 6
PQR ‰ Q ‰es R ‰i mgw«¼LíK«¼q M 7 N 8
`vI: ciÕ·i O we±`yGZ wgwjZ nGqGQ| 23. ‰KwU mgevü wòfzGRi ‰K evüi Š`NÆÅ
ˆkÉwYeÅvwµ¦ 1519 2024 2529 3034 12. 
 OQR ‰i gvb KZ? 3 ˆm.wg. nGj, wòfzRwUi ˆÞòdj KZ eMÆ
MYmsLÅv 2 8 10 6 K 30 L 45 ˆm.wg.?
6.   cÉPziK wbYÆGq f1 + f2 ‰i gvb ˆKvbwU? M 60 N 90 3 3 3 9 9 3
K 4 L 6 K L M N
13.  QOR ‰i
 gvb KZ? 4 4 4 4
M 8 N 10 K 30 L 60 24. ‰KwU PvKvi eÅvm 8 ˆm.wg. nGj ‰i
7.  Dcvî mgƒGni
 gaÅK ˆKvbwU? M 120 N 180 ˆÞòdj KZ?
K 26.2 L 26.5
14. cosec sec2  1 = ? K 8 cm2 L 16 cm2
M 31 N 36.5
K sec L cos M 32 cm2 N 64 cm2
8. A = 15 nGj,
M sin N tan 25. ‰KwU wòfzGRi wZbwU ˆKvGYi AbycvZ
i. tan 3A = 2sin 3A
15. sin3A = cos3A nGj, tan4A = ? 3 : 4 : 5 nGj Þz`ËZg ˆKvGYi cwigvY KZ?
1
ii. cot4A = 1 K 15 L 45
3 K L 3
3 M 60 N 75
iii. sin 4A = cos2A
wbGPi ˆKvbwU mwVK? 3 26. 5 ‰i Pvi `kwgK Õ©vb chƯ¦ Avm®² gvb
M 1 N
K i I ii L i I iii
2 ˆKvbwU?
M ii I iii N i, ii I iii
wbGPi ZG^Åi AvGjvGK (16 I 17) bs cÉGk²i K 2.2269 L 2.2263
Dîi `vI: M 2.2361 N 2.2260
9. D
27. f(y) = y5 + 6y  5 nGj, f(1) ‰i gvb
A
BAC = 2x KZ?
O
4
ACB = x K 0 L 2
C M 12 N 14
A B C

16. ACB ˆKvGYi gvb KZ? 28. KqwU evüi Š`NÆÅ ˆ`qv ^vKGj ‰KwU eMÆ
B AOC = 88
K 15 L 30
AuvKv hvq?
ABC = x K 1 L 2
 x ‰i gvb KZ?
 M 45 N 60
M 3 N 4
K 44 L 88 17. BC ‰i gvb KZ?
29. ( )
16
M 136 N 272 K 2 2 L 2 3 5 = KZ?
1 K 5 L 5
10.  a + = 2 ‰es a > 0 nGj,
 M 4 2 N 4 3
a M 25 N 125
18. ‰KwU `ËeÅ 20% jvGf weKÌq Kiv nGj,
2 1
i. a + 2 = 2
1
ii. a3  3 = 0 30. 33x  2 = 81 nGj, x-‰i gvb KZ?
a a weKÌqgƒjÅ ‰es KÌqgƒGjÅi AbycvZ ˆKvbwU? K 2 L 4
1 K 5:4 L 6:5 2 4
iii. a4 + 4 = 4 M N
a M 5:6 N 4:5 3 3
1 L 2 M 3 K 4 M 5 M 6 L 7 L 8 N 9 M 10 K 11 K 12 K 13 M 14 K 15 L
Dîi

16 L 17 N 18 L 19 N 20 M 21 L 22 M 23 L 24 L 25 L 26 M 27 M 28 K 29 M 30 K
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 53
398 eWÆvi MvWÆ cvewjK nvB Õ•zj, kÉxgãj, ˆgŒjfxevRvi

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
.
1. 0.69 ‰i mvgvbÅ f™²vsk wbGPi ˆKvbwU? 13. 0.0336 ‰i ŠeævwbK i…c wbGPi ˆKvbwU? 22. 
 A

7 69 23 7 K 3.36  102 L 33.6104 90wg.

K L M N 30
11 100 30 10 M 36102 N 0.336102 B C

2. 
 ‰KwU cvLvi AaÆNƒYÆGbi cwigvc KZ wWMÉx? 14. (x) = z + 5z  3 nGj, (1) ‰i gvb
4 2
wPGò, BC ‰i Š`NÆÅ KZ?
K 90 L 180 M 270 N 360 KZ? K 51.96 wg. L 103.92 wg.
3. 4 + a + b + 32 + ... ... ... avivwUi mvaviY K 3 L 4 M 155.88 wg. N 180.43 wg.
AbycvZ KZ? M 7 N 9 23. e†Gîi ˆÞGò@
K 1 L2 M3 N 4 15. y2 = 9y ‰i mgvavb ˆmU wbGPi ˆKvbwU? i. ˆhGKvb mijGiLv ‰KwU e†îGK `yBGqi
4. ‰KwU `ËeÅ 20% ÞwZGZ weKÌq Kiv nGjv, K {0, 3} L {0, 3} AwaK we±`yGZ ˆQ` KiGZ cvGi bv
KÌqgƒjÅ I weKÌqgƒGjÅi AbycvZ ˆKvbwU? M {0, 9} N {0, 9} ii. e†Gîi mgvb mgvb RÅv ˆK±`Ê nGZ
K 4:5 L 5:4 M 5:6N 6:5
16. wòGKvYwgwZK mÁ·GKÆi ˆÞGò@ mg`ƒieZÆx
5
5. tan = nGj, cot2 ‰i gvb KZ?
2 i. sin(90  ) = sin iii. e†Gîi eÅvmB e†nîg RÅv
29 25 4 4 ii. sec2  tan2 = 1 wbGPi ˆKvbwU mwVK?
K L M N iii. sin2 + cos2 = 1
4 4 25 29
K i I ii L i I iii
6. p + q = 5, p  q = 3 nGj, p2 + q2 ‰i gvb wbGPi ˆKvbwU mwVK? M ii I iii N i, ii I iii
KZ? K i I ii L i I iii 24. 

P
K 34 L 19 M 17 N 8 M ii I iii N i, ii I iii
7. 
 evÕ¦e msLÅvi ˆÞGò@
17. 
 60 1 ˆm.wg.
i. 81 ‰KwU weGRvo msLÅv A
R
Q
ii. 21 ‰KwU weGRvo msLÅv 70
iii. 0.21 ‰KwU cƒYÆmsLÅv 60 wPGò, QR = KZ ˆm.wg.?
wbGPi ˆKvbwU mwVK? B
C
D
K 1 L 2 M 3 N 9
K i I ii L i I iii 25. 1 ˆm.wg. aviwewkÓ¡ ‰KwU NbGKi AvqZb
M ii I iii N i, ii I iii wPGò, ACD ‰i gvb KZ? KZ Nb ˆm.wg.?
8. C = {y : y  ô ‰es 5  y  10} ZvwjKv K 50 L 60 K 1 L3 M6 N 9
c«¬wZGZ wbGPi ˆKvbwU? M 70 N 130 26. ‰KwU e†Gîi eÅvm 12 ˆm.wg. nGj, ‰i
K {5, 6, 7, 8, 9} L {6, 7, 8, 9} 18. log5125 ‰i gvb wbGPi ˆKvbwU? cwimxgv KZ?
M {5, 6, 7, 8, 9, 10} N {6, 7, 8, 9, 10} K 3 L 5 K 37.70 ˆm.wg. L 75.40 ˆm.wg.
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i M 6 N 8 M 113.10 ˆm.wg. N 452.39 ˆm.wg.
Dîi `vI: 19. 5x + 3y = 4 I 2x + 7y = 9 mgxKiY 27. wbGPi ˆKvbwU mylg eüfzR?
x=3+2 2 ˆRvUwU@ K wòfzR L eMÆGÞò
1
9. 
 x + ‰i gvb KZ? i. mãwZcƒYÆ M cçfzR N lofzR
x
ii. AmsLÅ mgvavb AvGQ wbGPi ZG^Åi AvGjvGK (27 I 28) bs cÉGk²i
K 6 L4 2 M2 2N 0
1 iii. ciÕ·i AwbfÆikxj Dîi `vI:
10.  x2 + 2 ‰i gvb KZ?

x 10 Rb evjGKi IRb (ˆKwRGZ):
wbGPi ˆKvbwU mwVK?
K 28 L 30 M 32 N 34 46, 45, 33, 38, 43, 40, 50, 43, 35, 36
K i I ii L i I iii 28. DÚ DcvGîi cwimi KZ?
11. (21 + 31)1 ‰i gvb KZ?
1 2 5 6 M ii I iii N i, ii I iii K 10 L 16 M 17 N 18
K L M N
6 3 6 5 20. 3 : 8 = y : 32 nGj, y ‰i gvb KZ? 29. DÚ DcvGîi gaÅK KZ?
12. 
 hw` p + q = r nq ZGe@ K 3 L 12 K 40.5 L 41.5 M 42.5 N 43.5
i. p3 + q3 = r3  3pqr M 24 N 48 30. 

P
ii. (p q)2 = r2  4pq 21. 
 2x + 3y = 2 mgxKiGY x = 2 nGj cÉvµ¦ 5
4
iii. {(p + q)2}2 = r2
we±`ywU ˆKvb PZzfÆvGM? Q R
wbGPi ˆKvbwU mwVK?
K 1g L 2q
K i I ii L i I iii wPGò, QR = KZ ‰KK?
M ii I iii N i, ii I iii M 3q N 4^Æ
K 3 L9 M 16 N 20

1 N 2 L 3 L 4 L 5 M 6 M 7 K 8 M 9 K 10 N 11 N 12 K 13 K 14 K 15 N
Dîi

16 M 17 N 18 K 19 L 20 L 21 L 22 K 23 N 24 M 25 K 26 K 27 L 28 N 29 L 30 K
54 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
399 cUzqvLvjx KvGjÙGiU Õ•zj AÅv´£ KGjR, cUzqvLvjx

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. a, b, c, d ‰KwU mgv¯¦i avivi c` nGj 11. 
 3sinA  2cosA = 0, cosecA.cosA = ? 22. cos2B = KZ?
wbGPi ˆKvbwU mwVK? 3 1
K
10
L
8
K L 9 9
a+c c+d 2 2
K a= L b=
2 2 3 10 2 2
b+c b+d M N 1 M N
2 3 3
M a= N c=
2 2 12.  ABC ˆZ AB = AC = 25 cm ‰es
 23. 
 ABC mgevü wòfzGRi cwiGK±`Ê O nGj
2. 4  4 + 4  4 + 4  4 ... ... ... avivwUi BC = 30 cm nGj, ABC ‰i ˆÞòdj BOC = ?
(2n + 2) msLÅK cG`i mgwÓ¡ KZ? KZ cm2? K 30 L 60
K 4 L 4 K 250 L 300 M 90 N 120
M 0 N 8 M 340 N 360 24. ˆKvGbv ˆkÉwYi DœPmxgv 65 ‰es gaÅgvb
3. hw` sec = 2 nq, ZvnGj tan = ? 13.  3 ‰i 3 3 wfwîK jM KZ?
 62.5 nGj H ˆkÉwYi wbÁ²mxgv KZ?
1 4 3
K L 3 K L K 60 L 65
2 3 2 M 70 N 75
1 3 2
M 1 N
3
M
4
N
3
25. AwRfGiLv Aâb KiGZ `iKvi@
4. ‰KwU eGMÆi ‰KwU evü 3 àY e†w«¬ Kiv 14.  9x2 + 16y2 ‰i mvG^ KZ ˆhvM KiGj
 i. x AÞ eivei ˆkÉYx DœPmxgv
nGj ‰i ˆÞòdj KZ àY e†w«¬ cvGe? ‰wU ‰KwU eMÆ nGe? ii. y AÞ eivei KÌgGhvwRZ MYmsLÅv
K 3 L 6 K 6xy L 12xy iii. AwewœQ®² ˆkÉwYmxgv
M 9 N 15 M 24xy N 144xy wbGPi ˆKvbwU mwVK?
5.  hLb log10x = 3,
 ZLb x ‰i gvb KZ? 15. 4, 0, 6, 12, 8, 4, 9 ‰es 15 DcvîàGjvi K i I ii L i I iii
K 0.11 L 0.10 gaÅK KZ? M ii I iii N i, ii I iii
M 0.01 N 0.001 K 7 L 8
6. mgevü wòfzGRi ˆÞòdj 25 3 eMÆ ˆm.wg. M 9 N 12
wbGPi ZG^Åi AvGjvGK (26 I 27) bs cÉGk²i
nGj ‰i evüi Š`NÆÅ KZ ˆm.wg.? 16. mgvb mgvb f„wgi Ici AewÕ©Z ˆhGKvGbv Dîi `vI:
K 10 L 8 `yBwU wòfzGRi wkitGKvY«¼q mÁ·ƒiK nGj, ˆkÉwY 11-20 21-30 31-40 41-50
M 6 N 4 ‰G`i cwie†î«¼q Kxi…c nGe? MYmsLÅv 4 18 22 16
7. 5 2 ˆm.wg. eÅvmvaÆ wewkÓ¡ ‰KwU e†Gî K mgvb L Amgvb 26. DcvGîi gaÅK ˆkÉwY ˆKvbwU?
‰KwU eMÆ A¯¦wjÆwLZ nGj, eMÆwUi evüi M ciÕ·iGœQ`x N Õ·kÆK K 11-20 L 21-30
Š`NÆÅ KZ ˆm.wg.? 17. 
 12 + 24 + 48 + ... ... ... 768| avivwUi M 31-40 N 41-50
K 10 L 8 27. gaÅK wbYÆGqi RbÅ ˆÞGò fm = ?
M 6 N 4 mgwÓ¡ KZ?
K 1524 L 2545 K 40 L 22
8.  ‰KwU wòfzGRi ˆKvYàGjvi
 AbycvZ M 2054 N 2124 M 20 N 18
1 : 1 : 2 nGj, wòfzRwU@ 28. ˆKvGbv e†Gîi AwaPvGc A¯¦wjÆwLZ ˆKvY
18. 2  2 + 2  2 + ... ... ... avivwUi (2n + 2)
i. mgw«¼evü wòfzR Kxi…c?
msLÅK cG`i mgwÓ¡ KZ?
ii. mgGKvYx wòfzR K 0 L 1 K mƒßGKvY L mgGKvY
iii. mgw«¼evü mgGKvYx M 2 N 3 M Õ©ƒjGKvY N cÉe†«¬ ˆKvY
wbGPi ˆKvbwU mwVK? 19. mylg cçfzGRi ‰KwU kxlÆGKvY KZ wWwMÉ? 29. mgGKvYx wòfzGRi cwie†Gîi ˆKG±`Êi AeÕ©vb
K i I ii L i I iii K 108 L 120
ˆKv^vq?
M ii I iii N i, ii I iii M 180 N 360
K wòfzGRi evwnGi
wbGPi ZG^Åi AvGjvGK (9 I 10) bs cÉGk²i 20. e†Gîi ˆÞGò@
Dîi `vI: L e†Gîi ˆKG±`Ê
A i. AaÆe†îÕ© ˆKvY ‰K mij ˆKvY
ii. ˆh ˆKvGbv RÅv ‰i jÁ¼w«¼L´£K ˆK±`ÊMvgx M AwZfzGRi gaÅwe±`yGZ
O iii. e†Gîi mgvb mgvb RÅv ‰i gaÅwe±`yàGjv mge†î N f„wgi Dci
wbGPi ˆKvbwU mwVK? 30. `yBwU exRMvwYwZK ivwk x I y ‰i àYdj
xy = 0 nGj@
B C
K i I ii L i I iii
E
BAC = 60 M ii I iii N i, ii I iii i. x = 0 A^ev y = 0
9. BOC = ? wbGPi ZG^Åi AvGjvGK (21 I 22) bs cÉGk²i ii. x = 0 ‰es y  0
K 110 L 120 Dîi `vI: iii. x  0 ‰es y = 0
M 150 N 160 tan(2A  45) = 1 = 3 sinB wbGPi ˆKvbwU mwVK?
10. ABE + ACE = ? 21. A ‰i gvb KZ? K i I ii L i I iii
K mgGKvY L mÁ·ƒiK ˆKvY K 30 L 45
M ii I iii N i, ii I iii
M cƒiK ˆKvY N mijGKvY M 60 N 90
1 N 2 M 3 M 4 N 5 N 6 K 7 K 8 N 9 L 10 N 11 M 12 L 13 N 14 M 15 K
Dîi

16 M 17 K 18 K 19 K 20 M 21 L 22 L 23 N 24 K 25 K 26 M 27 L 28 K 29 M 30 N
kxlÆÕ©vbxq Õ•zGji cÉk²cò: wbeÆvPwb cixÞv 2024 55
400 eiàbv miKvwi evwjKv DœP we`Åvjq, eiàbv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
1. 
 2x + y = 1 I x =  4 mgxKiY«¼Gqi 4
12. x2  x2 = 0 mgxKiYwUi@ wbGPi ZG^Åi AvGjvGK (22 I 23) bs cÉGk²i
mgvavb we±`y ˆKvb PZzfÆvGM AewÕ©Z? Dîi `vI: A

K cÉ^g L w«¼Zxq i.PjGKi mGeÆvœP NvZ 4

M Z‡Zxq N PZz^Æ `yBwU gƒj ( 2 ,  2 )
ii.
O

2. 35  2x  x2 ‰i Drcv`GK weGkÏwlZ i…c aË‚eK c` 4

iii. B D C

wbGPi ˆKvbwU? wbGPi ˆKvbwU mwVK?

O ˆK±`ÊwewkÓ¡ e†Gî ABC e†îÕ© wòfzR|
K (7  x) (5 + x) L (7  x) (5  x) K i I ii L i I iii
BC = 8 ˆm.wg. I OD = 3 ˆm.wg.
M (7 + x)(5  x) N (7 + x)(5 + x) M ii I iii N i, ii I iii
22. ABC e†Gîi ˆÞòdj KZ?
3. 256 + 128 + 64 + ... ... ... avivwUi KZZg 13. e†îÕ© mvgv¯¦wiK ‰KwU@
K 78.54 eMÆ ˆm.wg.
1 K iÁ¼m L eMÆ
c` 4 ? L 31.42 eMÆ ˆm.wg.
M AvqZ N UÇvwcwRqvg
K 9 L 10 M 11 N 12 M 50.27 eMÆ ˆm.wg.
14. 
 P
4. p, q, r KÌwgK mgvbycvZx nGj, N 201.06 eMÆ ˆm.wg.
p2 + q2 23. OBC = 30 nGj BAC = KZ?
= KZ? E F
q2 + r2 K 30 L 60 M 90 N 120
K
r
L
p
M
q
N
q
Q R 24. A = 30 nGj@
p r p r D
3
5.  x  2y = 5 I 2x  4y = 10

wPGò, PE = EQ, PF = FR ‰es PD,
i. sin2A =
2
ii. cot2A = 3
i. mgxKiY ˆRvU mgém
QPR ‰i mgw«¼L´£K nGj@ iii. cos 3A = 0
ii. mgxKiY ˆRvU ciÕ·i wbfÆikxj i. EF || QR wbGPi ˆKvbwU mwVK?
iii. mgxKiY ˆRvUwUi AmsLÅ mgvavb AvGQ ii. QR = 3 EF K i I ii L i I iii
wbGPi ˆKvbwU mwVK? iii.
QD QE
= M ii I iii N i, ii I iii
DR RF
K i I ii L i I iii 25. 
 ‰KwU MvGQi DœPZv I MvQwUi Qvqvi
M ii I iii N i, ii I iii
wbGPi ˆKvbwU mwVK?
Š`GNÆÅi AbycvZ 3 : 3 nGj D®²wZ ˆKvY
K i I ii L i I iii
6. 3x1 = 27 nGj, x ‰i gvb KZ? KZ?
K 4 L6 M9 N 7 M ii I iii N i, ii I iii
K 90 L 60 M 45 N 30
7. log5p2 =  2 nGj, p ‰i gvb KZ? 15. iÁ¼Gmi cÉwZmvgÅ ˆiLvi msLÅv KqwU?
K 0 L1 M2 N 4 wbGPi ZG^Åi AvGjvGK (26 I 27) bs cÉGk²i
1
K 5 L
5
M  2 N  10 16. ˆKvGbv e†Gîi AwaPvGc A¯¦wjÆwLZ ˆKvY@ Dîi `vI:
8. p2 + q2 = 3 ‰es pq = 3 nGj, (p + q) ‰i K mƒßGKvY L Õ©ƒjGKvY ˆkÉwY eÅeavb 21-30 31-40 41-50 51-60
gvb KZ? M mgGKvY N cÉe†«¬ ˆKvY MYmsLÅv 15 25 35 25
K 3 L9 M 11 N 15 17. 2 sin (  30) = 3 nGj,  ‰i gvb KZ? 26. gaÅK ˆkÉwYi cƒeÆeZÆx ˆkÉwYi KÌgGhvwRZ
9. 5 + 7 + 9 + 11 + ... ... ... K 0 L 30 M 60 N 90 MYmsLÅv (Fc) KZ?
i. ‰KwU mgv¯¦i aviv 18. ABC-‰ AB = BC = AC = 8 ˆm.wg. K 25 L 40 M 70 N 100
ii. avivwUi 23 Zg c` 49 nGj, gaÅgv CD ‰i mgvb Š`NÆÅ wewkÓ¡ 27. cÉ`î DcvGîi cÉPziK KZ?
iii. avivwUi cÉ^g AvUwU cG`i mgwÓ¡ 96 mgevü wòfzGRi ˆÞòdj KZ eMÆ ˆm.wg.? K 35 L 41 M 46 N 51
wbGPi ˆKvbwU mwVK? K 4 3 L 8 3 M 12 3 N 16 3 28. ‰KwU mylg lofzGRi KqwU cÉwZmvgÅ ˆiLv
K i I ii L i I iii 19. 
 AvGQ?
Q
M ii I iii N i, ii I iii K 2wU L 3wU M 4wU N 6wU
5 ˆm.wg.
10. 
 B = {x  ô : 2  x  4} nGj@ P
29. D
13 ˆm.wg.
O
i. B ˆmGUi Dcv`vbàGjvi gGaÅ ˆgŒwjK
P Q
msLÅv 2wU R E F
ii. P(B) ‰i Dcv`vb msLÅv 8wU
iii. B ˆmGUi Dcv`vbàGjvi gGaÅ 2 «¼viv O ˆK±`Ê wewkÓ¡ e†Gîi PQ I PR `yBwU DEF-‰ PQ || EF nGj, wbGPi ˆKvbwU
wefvRÅ msLÅv 1wU Õ·kÆK POQ ‰i ˆÞòdj KZ eMÆ ˆm.wg.? mwVK?
K 65 L 60
wbGPi ˆKvbwU mwVK? K DP : PE = DF : QF
M 32.5 N 30 L DE : PE = DF : QF
K i I ii L i I iii
20. mylg cçfzGRi ‰KwU kxlÆ ˆKvY KZ? M DE : DF = PQ : EF
M ii I iii N i, ii I iii K 106 L 108 M 110 N 120 N PQ : EF = PE : QF
.. .
11. 42.18  0.28 = KZ? 21. iÁ¼Gmi KYÆ«¼q 20 ˆm.wg. I 30 ˆm.wg. nGj 30. 
 y2 + (a + b)y + ab = 0 mgxKiGYi
.. .. ˆÞòdj KZ? mgvavb ˆmU wbGPi ˆKvbwU?
K 0.132 L 11.810
K 50 eMÆ ˆm.wg. L 100 eMÆ ˆm.wg. K {a, b} L {a, b}
.. ..
M 12.185 N 13.285 M 300 eMÆ ˆm.wg. N 1000 eMÆ ˆm.wg. M {a, b} N {a, b}

1 L 2 M 3 M 4 L 5 N 6 K 7 L 8 K 9 N 10 K 11 M 12 K 13 M 14 L 15 M
Dîi

16 K 17 N 18 M 19 N 20 L 21 M 22 K 23 L 24 L 25 L 26 L 27 M 28 N 29 L 30 N
56 cvGéix ‰m‰mwm ˆUÕ¡ ˆccvim ˆgBW BwR: cÉk²cò  MwYZ
401 ˆfvjv miKvwi evwjKv DœP we`Åvjq, ˆfvjv

MwYZ welq ˆKvW : 1 0 9
mgq– 30 wgwbU eüwbeÆvPwb AfxÞv cƒYÆgvb– 30
[`ËÓ¡eÅ: mieivnK‡Z eüwbeÆvPwb AfxÞvi DîicGò cÉGk²i KÌwgK bÁ¼Gii wecixGZ cÉ`î eYÆmÁ¼wjZ e†îmgƒn nGZ mwVK/ mGeÆvrK‡Ó¡ DîGii e†îwU () ej cGq´Ÿ Kjg «¼viv
mÁ·ƒYÆ fivU Ki| cÉwZwU cÉGk²i gvb 1|]
.. . . 11. tan(B  A) ‰i gvb KZ? 21. ABC I DEF ‰i A = D, AB = 5 cm,
1. 
 8.243 ˆ^GK 5.24673 weGqvM KiGj
K
1
L1 M 3 N 
1 DE = 3 cm, AC = 4 cm, DF = 2cm nGj
weGqvMdj KZ? 3 3 wòfzR«¼Gqi ˆÞòdGji AbycvZ KZ?
. . . . 12. mƒGhÆi D®²wZ ˆKvY KZ wWwMÉ nGj ‰KwU
K 2.99669761 L 2.9612761 K 20 : 6 L 16 : 8
1 M 5:4 N 10 : 12
. . . . MvGQi Qvqvi Š`NÆÅ DœPZvi àY nGe?
M 2.99669760 N 2.92142760 3
22. H ‰i NyYÆb ˆKvY KZ?
1 K 90 L 45 M 60 N 30
2. ˆKvGbv ˆmGUi DcGmGUi msLÅv 24n nGj K 30 L 90 M 360 N 180
13. hw` ˆKvGbv LuywUi Qvqvi Š`NÆÅ ÷bÅ nq ZGe 23. mylg cçfzGRi NƒYÆb gvòv KZ?
ˆmUwUi Dcv`vb msLÅv KZ? LuywUi D®²wZ ˆKvY KZ?
4n n
K 4 L5 M6 N 1
K 4n L2 M 4n N 16 K 90 L 180 M 0 N 45 24. Q
3.
3
x+
3
‰es x  y = 3 nGj
y=5
3 3 14. 5, 10, 8 ‰i PZz^Æ mgvbycvZx ˆKvbwU?
K 12 L 13 M 16 N 14
x + y ‰i gvb KZ?
15. 
 100 I 64 ‰i w«¼fvwRZ AbycvZ KZ?
K 50 L 63 M5 N 65
K 50 : 32 L 10 : 8 P R
4. 
 0.00836 ‰i jGMi AskK KZ? M 10 : 6.4 N 200 : 128
D

K 0.0778 L 0.9222 16. x  y  4 = 0 ‰es 3x  3y  10 = 0 wPGò, PQ = QR, QD = 4 cm ‰es PQ = PR


M 2 N 0.5723 mgxKiY«¼Gqi KqwU mgvavb we`Ågvb? nGj PQ ‰i gvb KZ?
ax bx K 1wU L AmsLÅ K 4.6 cm L 6 cm
5.  = a2  b2 nGj x ‰i gvb KZ?
b a M 4 cm N 4.25 cm
M 2wU N bvB
K 0 L ab M1 N  ab 5 3
17. 4x + 3y =  12 ‰es 2x = 5 nGj 25. 
 ‰KwU mgevü wòfzGRi DœPZv
6. 2
(x, y) = KZ?
A wgUvi nGj, ˆÞòdj KZ?
K 52  22  L 25  10 75 5 3 25 3 5
 3   K
4
L
4
M
4
N
2
22  5
B C M (5, 2) N  3 2 26.
  D
1.5 cm
C
wbGPi ˆKvbwU mwVK? wbGPi ZG^Åi AvGjvGK (18 I 19) bs cÉGk²i
3 cm
K AB < AC  BC L AB > AC  BC Dîi `vI: 2

M AC < BC  AB N AC  AB > BC ‰KwU mgv¯¦i avivi 1g c` 2 ‰es mvaviY

A 4 cm B
7. 
 UÇvwcwRqvg AuvKGZ DcvGîi cÉGqvRb@ A¯¦i 3|
K 3wU L 4wU 18. 
 avivi n Zg c` KZ? wPòwUi ˆÞòdj KZ?
M 5wU N 2wU K 3n + 1 L 3n  1 K 4 cm2 L 4.6 cm2
n(3n + 1) n(3n  1) M 5.5 cm2 N 2.25 cm2
8. M
2
N
2
O 27. 
 P = 30 nGj 1  cos2P ‰i gvb KZ?
C
19. 
 mvaviY A¯¦iGK mvaviY AbycvZ aiGj
1 1 3 1
A
D
B 5g c` KZ? K
2
L
2
M
2
N
2
K 81 L 243
wPGò AB = 24 cm, OC = 5 cm nGj CD M 160 N 162
28. avc wePzÅwZ ui = ˆKvbwU?
‰i gvb KZ? 20. 
 A
K
xi  a
L
a  xi
h h
K 8 L6 M 13 N 12 F
E xi + a xi
9. ˆKvGbv wòfzGRi KqwU ewne†Æî AuvKv hvq? M
h
N
h
+a
K 1wU L 2wU M 3wU N 4wU 29. ˆKvGbv MYmsLÅv wbGekGbi cÉPziK = 72.5,
B C
wbGPi ZG^Åi AvGjvGK (10 I 11) bs cÉGk²i D L = 70, f1 = f2 = 3 nGj ˆkÉwY eÅeavb KZ?
Dîi `vI: wPGò AE = CE, EF = 7cm nGj AC ‰i K 10 L5 M 12 N 15
1 gvb KZ? 30. 3, 3, 3, 4, 5, 8 I x ˆhGKvGbv 7wU msLÅv
cot(A + B) = , 3 sec(A  B) = 2
3
K 49 cm L
7
cm
nGj, msLÅvàGjvi Mo cÉPziGKi wZbàY nq
10. A ‰i gvb KZ? 6
ZGe x ‰i gvb KZ?
6
K 30 L 90 M 60 N 45 M 42 cm N cm K 27 L 63 M 37 N 33
7
1 M 2 K 3 N 4 L 5 L 6 L 7 L 8 K 9 M 10 N 11 N 12 M 13 K 14 M 15 L
Dîi

16 N 17 K 18 L 19 N 20 M 21 K 22 N 23 L 24 K 25 M 26 M 27 N 28 K 29 L 30 M