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1ff?
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=~-~~
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[ )'If ♦'dlt Jfl~Clli•1.,_, 6f'ttil CtJl'lt (+) 'l!lllt c•IC•BI ♦'151t Jit(Jft Clli•I.,_,
6rltii HUtl•I (-) ] I

>111,qlt{ lllft'f .s,:qJ 'tcjfft = a

a
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2
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gi6'6I = h : He,cagonal pris~
I
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= Q-11,0 + (t)~ = Q-11,0 + ~t = ttrt

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2980 dlt "'iliifl-, 4f 111°'11 ♦ifO♦if\ ~ ~ 'Q I
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filt'fo{ I
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► 1'11f.l Attil•I ♦ft'I 'ol•lc-t4 ~'iill
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~ I '1111'{, ~000 ~ fii"f ~ ~ l!AI 800 'ffl 9111 ~800 Jttlrfii"f ~ I
t.w ~ooo 9Rl9Rl )oo ~ ~)oo, ~~oo, ~~OQ•'Jl'llfffl ~on! 1
Leap Years 1800 • 2400
1804 1904 2004 2104 2204 2304
1808 1908 2008 2108 2208 2308
181.2 1912 2012 2112 2212 2312
1811 1911 2011 2111 2211 2311
1820 1920 2020 2120 2220 2320
182A 192A 202A 212' 222A 232A ~ooo ~ ~o~91W
1828 1928 2028 2128 2228 2328 ~~ ~ 111'91 ~ ,
1832 1932 2032 2132 2232 2332
1831 1931 2038 2131 22341 2331
1840 19'0 2040 21"41 22AO 2340 11t-1 f illflf ~ .
1844 1944 2044 2144 2244 2344 C\ At! 1Bl'f flifl ~
1848 1948 2048 2148 2248 2348
1852 1952 2052 2152 2252 2352 llfif,
1856 19541 2056 2156 2256 2356 {(A + B + C) :I: ) or~ or~
1860 1960 2060 2160 2260 2360 0'\
1884 1914 2084 2184 2294 2384 ..... • o 1.11 ~ -
1888 11188 2088 2188 2288 2388
1872 1972 2072 2172 2272 2372 1.11, '61(Cfl f.i ,dt w,.:q 1'51
1871 1978 2078 2171 2278 2371 ~ I
1880 1980 2080 2180 2280 2380
1884 1984 2084 2184 2284 2384
1888 1988 2088 2188 2288 2388
1892 1992 2092 2192 2292 2392
18N 1998 2098 2198 2298 2391
2000 2400

I 'Mt11:· Rony NII Abdi

 ~11C'IFil ~ ldllt• 111$ ~ "ll!l'~hl !lfirr 1 ~ (~ooo ~ ~oi-i.) I
1lfif' ~ooo )lit•HI 'l/5fft'f 151.,it. )1-oo ~ )2-2-lD ~ c.t-1 ◄(ClHI A"if.
♦fCl!l 61""1 l!l(C'I '9it1if '\tall ~ ◄il!ji~ ) """1 ♦ft'! ~ I
91t1i C\ frftl ~ ♦ft\! ~~ I Q!Sltlt"lci ~ C41◄1t◄ ~ ~ ~ A"if I
151.,it.,
~ = {(A + B + C) + ) } / C\
ClflR, ) il> l!>1T ~ ) iSi 11{1 IHI ~ <ffif?
A= :.1,-, =
B ~ct C =)
I!!~.
{(A + B + C} + ) } / C\
= {(lDlr +~ct+)}+ )}/C\
= )~ct/C\
~ ♦11tii Q51 9tc"lct ~ illl♦t◄ 1 15/j.,.._, ~ 1( •l~lt I

1511◄ 11, ~o:.~ ~1 9ff ~)oo ~t♦ ~)1-lD . , . 'J.caHI 'fft'III ◄ Ni~ ;, f.4ttll'I
♦11Cl!l ~ I ~ = {(A + B + C) - ) } / C\
ClflR, ~ ) 00 ~ l!, eil1411HI ~ ~?
A=o~ a~ o C=~
I.fl~.
{(A+ B+ C)- )}/ C\
={{O +':O,+ 1-) - )}/C\ =lr/C\
~ ♦11t'I '81'1t"lct ;, illl♦t◄ I 15141"-, ~ "lf-1◄ 111 I

IA + B + C) + 'I
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1Mtll:· Remy NII Absh

II
I""1 ~ m ff
~~=
"l~clBI = ), tMclFII = ~. (.Jliilclit = "• :;qs14111 = 8,
!.lf = ~ , 1( •i~lt = -i,, Sf= o
~ = (A + B + C) + '\ [ '51'11 <1511111 ~ ~1•1C•ltlil llT ~ t'ffTI '41<15Ccl ~~ ~t~

f.!of-t qig(<q c.~ RS~] I

A = 31tilft;'II C"ltlilt lil. Niftt~
8 = l!JIT~ c:1f Aif "'11 +llt<lil C)lfiiil 91W dt-1161~"5 ~ ♦1$1 fil,f bff Nlf I
C = ~ ~ ~11 +llfAil C'IM-i .,,_-el ◄\Ct ~wiir-n

Ex:-
~o)-i, '11Ci1'51 ~ <:'ft~ RS 'cll1I F.:tiil?

~"""'·
A • )-!, (~o)-i, 'li'lltt C"l(.flHI 'IJ.\ fQi(ift'J
8 = 8( ~000, ~008, ~o)~)
~00lr,
( ~o)ll, 'lit'l<J '1_t(~till1il'9f ~ f\C'li ~till il'IIC\6 ~~I ~o)ll,
C◄S'e 'ft1 'lit<lofl)I
C = ") + ~i- +"~ + "°
+ ") + "o + ") + ") + ~= ~82- (~o)ll, 'lltil'li
)-!, ~~~~lff~ ~ I CcM'liil ~~ ~WI' '\ ~~~ '-'lr
di"""'~ lA) I
~1'-t, (A+ B + C)+ <\

11!51'1t"llil ,.:,
~ '5t-11t fi.t'l'i 0,111<011

I 'Mtlt· R.ony NII Abdi

1~~, ~ ~ f f I
~~=
"1~◄ 111 = ), 11fq:q111 = ~. 01111◄111 = "· .... ,;::;111 = 8,
l1f = t ' 1' •1~◄111 = ~. - =0
'Pl= (A + B + C) + <\ [ ~t <li11I-!I "ffl '6I•1t• lt'I 1fl "f~~ lljl<JiC<I '<3~ ~t.~

f.ttif"I <li-!IC<I ,"f1t f.j NO!) I

A= Jtlii1fth1 c-ttct'li lJ.'t Nilit'G
8 = 1511'1f.l c:1f ~ C'l'II +llC◄-, CJt~-1 9f(s ~-1~1~1.'6 C1f ~fii91 brf ~ I
C= c:11 fifit c,q ♦11C<l-l CJJtflt-1 91W~ ◄t.C11 '11$ W fir-JI

Ex:-
~O ) ~ >iiC"l-!I t 01tffl f.j ~ lw,I?
Ill~,
A • )~ (~o )~ Ji l'llt11 C"lf.1111 'If.\ fqifitt)
8 = 8 ( ~000, ~008 , ~OOlr, ~O)~
( ~O)~ )jjtt;t 1J.tcf~t'ltfil'-r lJmr fwlt CJl'dt'II i:11itle f{c-1 I ~O)~

~ lf1i'I 'IIC<l-10 I
C= ") + ~~ +") + "° + ") + "° + ") + ") + t = ~8~ '11Cil11
(~O)\':,
) ~ l l l t . f ~ ~ ~ ~ ~I CclliHI~ ~~~~ C\ ~~~ l:)b-
lll~ffl~) I
Ill'(;{, {A + B + C) + <\

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{ '6j{j,iJ) \!ll ◄~'11 9IR ♦t'Slil ,~ 11M 8i UMiif ( '61!1~ Ill~
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)11' ~ -
~ 1"it1111 ~ 'ft'ffl ll'I' 151l"iiifl11 ~ ~ W I

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a) 'Iii, 111-.lt "J\~ t
b) Ii' X ~ "' )O

c) )t + t = )t
d) ) t X ¢0 = C\to
e) -r1f, Cii'Q~ ( '6j{jic) 9fR ~ ~ C\tO + )CN,lr = ~t)lr
f) ,\lt~lr - )~i.) = t~C\ (lli'V 'fTlf = ),.,.))

4-IIQ.,I : -
t~C\
t ll'I' A 'l~Ji ~tti.t'lil I
~C\ ~ cl1Pf I

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HCifi.,4SJff?
~ cqi51C<1 <mi,
~ = vo firoli ~ = 80
151141'dl♦Bi ◄ i•Utof'll C1N.4 If = vo x 80 = ~00 ~ fiielt
)11 ◄ i'llt-,t ◄l~tt'II bl'IIM♦ 111S ~.
'dliltil, 'fllf'I'~ ◄l'llt-,t~ = vo + ~ + ~ = v8 fil'Cit
'fllf'l'lc. ◄l"lit.,t ~ - so+~+~ • 88 Mlt
1llf"1c. ◄1•11c.,11 C1N.<-1 =vs x ss .. ~ '◄'f '1ittt
'llRil,i C'Niqiij = ~ - ~oo = 8~ '◄'f '1iDlt 1 (Ans.)
~1 ◄1•11c-,1 fti«t'li tsi'liM♦ 11'Rlt tffii,
'5liltil, •4'1tir c11•11t-,'!l~,fi=vo - ~ - ~ = % liiGi'li
naf-◄roi ◄i•llt-,t ~=so - ~ - ~ =~ fil'Clt
nlr -◄rof ◄1'1 1C'1'll (.liiiijjQ S: .. ,~ x ~ = ~C\~ ~fil'Cl'li
tNl'II Cifi.,4 SJ= ~oo- ~C\~ =8~ ~fll'Cit 1 (Ans.)

"lltc'C♦F-t♦,
~ X = vo Fil'Ci'!I ~ Y = So 11NBi R&it a = ~ fil'Clt
'IiNii -.1 +ilt',fll C..ai'M = ~a (X + Y t ~a)
)ll'1.«a1 •en~
(.liiiijjcjArj: ~a
◄ 1~'11,
(X + Y +~a)= ~ x ~ ( vo + 80 + ~ x ~
-= 8 x ( )~O + 8) .. 8 x )~8 • 8~ ~ fil'Ci'li (Ans.)
~11(.'li'tiilfM~+illJ~ffl
C""'i4', = ~a (X + Y - ~a) =~ x ~ ( iro + 80 - ~ x ~
= 8 K ( )~O • 8) • 8 x ))~ -= 8~ ~Fil'Ci'li 1 (Ans.)

'ft,IIQf;• Rony NII Alwh

I~!) 1 111:c:il -tt 911C~<SI C'iNJ-is'I 'ftiJltRi I
~ 151141'6C'iftiiHI -11 <li'lif.ifil ?lffi Xfil'Clt di~~ Yi 151141'6C'Mt ell
<ti•ilt-111 <ti~t oHIMC4 <IT ~'6Ct oitMC4 a fil'Cit S'Glf 1l'Rlf <IT~ 151ittl 1
1l'Rlfci1 +ii(lfil c..ai, I ff?

<l!RJI cf1 "11C~~ C'iN!~ = ~a (X + Y ± ~a)

nltcll +iT'f <ti~~. + ~a~ ~'- C'INlqAII = ~a (X + Y + ~a)

nltcll +iT'f k¼!Cil ~. - ~a~"' 151~'- C'INlq,61 = ~a (X + Y - ~•)

I •-~~
•
.....
•
~Y=Bo~a

?lf(r X =vo fin

~

-
• •
-
~y =
sofif:

·-~-. ?lffl X = b'O Nl

mtcll ~ <tlfQ:"51
 Ex:-

~cl51t '51141'6(.'ijQf~J mltJBI N"Gct I ~ifii 512 ~~I'll~.

91Al~sil ff fii61:a l

~1ft.{, ?siC'fJf q'ff"1 ~ Gitt, I

-i;z,ect"' AOltff 1"ct = 2

~ = ✓(:llliiiiifii X q\&Gct

= ✓512 X 2 =32

(m
= ✓ z =16

:. 91ffi~si1 = 2 ( ~ • ~ )
= 2 ( 32+ 16)
= 96 fiiGlf (Ans.)

1 Page 34 °' 34 1 lJUPtt:- Rony NII Alcash

 ~ce..A,..,
l!l'IIIIC-l, ~ = 1u Fi$t11

151111'111 ~ .

(.'iNjcjlfl =( J~ - ~ ) ~ X

=J15 2 - 102 X 10

=J225 - 100 X 10

: ✓ 125 X 10

= 5"'5° X 10
= so"'5 ~1fiiGB1 (Ans.)

I 'l) 'dllQ'dC 1'CilHI 91Ri~ail iill1'a flt I

1

~'1C~fs1 <II !!C''SI 4S&Oct ~ "il♦Ciil,

~:- C'T'U =v'C'ljiiljql61 X ~Cf

~:- ~ = ✓C'il'JliM
~'1

:. "1B~ill = ~ ( ~ +~)

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Ex:- ~'1111t 151141'6C'lft:iJBI ~ 10 f.®111 -u ♦cfil ~ 1:, fs$111 ~ ,
151I41«'1Ptii1'11 (.11iiJiq111 • '¥11' I

'i ji:ff'11'1 M41Cil '!lliil<lli c,il\SIC<il emf~.

111'tljlti1, ~ =10 f@BI

~. '51l 4116C1RilHI ?t;'(J = a ~ = b = 10 fiiGl<ll ~.

C'iji(ijqiiij = ab
♦t'f'!l?t;'(J= ✓a 2 + b 2 = l!J fil'GBI

:. ✓a 2 + b 2 =15

⇒ ✓a 2 + 102 =15

⇒ a + 100 = 152 ~ 40il)

2

⇒ a2 "' 225 - 100

⇒ a2= 125

⇒ a = ✓5 X 5z

⇒ a=SJs

:. (.11iiJiq111 = ab =sJs x 10
= so Js ~ fifflPa (Ans.)

Page32ol 34 ,nffi,. Rony Nll Akasll

 <s1fclc~~ ~ dJq5~<J$
'Sll~lil,
s r
q~l'-C•Fli
<
C~"'1 = - z
- 13.612 X 12
=13.60 x 6 = 81.66 ~ ~.fit. (Ans)
Ex:- 2

44 filtl<ll c11i:»1 Ml-fl 1c~<ll 1114~ ,'11'411T'(;11f ,..,,, ,s ~ ~"-9flf

4'AC'I ~'.t"Af C;iNi'M""'~?
l!i'otllt;i,

11111tti{, r = 22 fir. 8=75~ ,~i'-c-1'11 ,~qi., =?

Glla!<lll ~ .
8 .
1~1'-c-1'11 C1i'Jl'PI = 360 x 1t r z

7S - z
= -360 X 3.1416·><- 22

- 7S x 1520.5344
360
= 316.nB fiii;B1 (Ans.)

Ieo, 10••-•,rwiw I
I~ ~-e ~ ~ll'I '414C'I c.ilcf'I .cRt I
,pr:- C."'NI~ =( J~ ~ - !fl_~ ) X !fl_

I ~ ~ -e ~ csrelll iill4C'I C...-l♦'I .'4'At I

"i_i1I:- C'lfii1! qii'I = ( J~~ - ~~ )x ~

lftml':- Rony NU Akasll

1 Pap31o, 34 1
 ------ A

AB = s = 1C~'II 6f1'f
r = 1C~'II ,qJl'11(
0::-1-----fB
AOB = 1~1(C81'111,,_1Nl......,.il'_.i1
L AOB=9

<J) 1c~'II ~ e <1Jl:>il( i<ll♦Cil ~ 6lt.11f?i;lfJ.<J<lltlltV I

~:- s = nr8
ieo [ s =~ -~~
blC"f"II' '-"'~l ]

~, 1cfi <1J1'11'r e "6191 ~m~1«111~1~c.11'II C'INiijlolf •<Rtlll'V 1

s r
~:- ~~~ C'JliJlcJi81 =2

~) 100'11 ~ ~ <1Jl'11( i<ll♦c-, 1~1(C"1'11 C-.al¥61 ~<Rllfnr I

~:- ~f~C"l'II c-.aiaii61 = 3: 0 x n r 2
Ex:-1
!JI~ 1C~11 ·,qJl'11( 12 C'f.~. ~'iq''- 1C~'li ~~'6141 C♦Ca:\!f 65 ~ C4lil
,~"ffl'c!Stf I 1~1>1C9RI ~ -e 1~1(C"1'11 C'INlil'i1 ~ cR ?
~lltlltil, r = 12 C'f.~. 9 = 65 f\if3;f s =?
1511¥1'11 I. - , -,W'6T9f,

S
= 180r 8 = 3.1416180
1t >< t2x 65 =2450.448 =13 61-Q,.
180 • 1.•i.l~.
( Ans.)

l Paae30ol 34 I
~JI' f.Bl $1:- '1 11:iFftCf ISICci Cllleft<i ~ ffl,
89
2
Ex:- 2 ( 89) = ? 89
801
'ffi, a = 8 b =9
712x
b2 = 9 2 = 81, 1 ~ '(IC-6 tlll♦C<il = 8
7921
2ab = 2 x 8 x 9 = 144 + (b ~ tllt\bjl 8 l = 1S2,
2

2 ~ ~C-6 '4l<ISC<I 15
2 2
a = 8 = 64 + ( 2ab ~'II tllf.\!>'11 1S )= 79
'5l(Cl1, a2 lab b2 = 7921
2 2
a 2ab b

2ab =2 x 8 x 9 =144 =81-

= 64 :. 144+8 = 1S2 •'

' ,, ...
,.-.... \

a2 2ab b2 (.illC♦ ~1ft~'.~'Cl'fl 111"11"11f-t

,q>t IC~ 89 ~ = 7921 ~1" 'I ICci I
(R::H3• ~tffitCf f.t~Cif 151111111<1Sll61tC01G11 ~ CcN5IC<I ~,n.~~
"♦tf ~'\bf~~~ I ♦Fil-I l!ltilC<I ISll11111 91Jlll>t ♦C111' -lt '♦t'A
\Sf\1!5.l<I I N-91C1111 FHIJI ~ 111111'1 ♦'IIC'I >tliql11•1 M41CJ111 ~~~
cilf\'11 f l t ~ I 911111:»f ~11~9ffl M<S'II ♦'IICct .!1-'.b"f ) I

Pap29of 34 1fUT~:- Rony NII Absh

)'Sf Milli:- '11~~"3, Cill4c◄q ~ ~~'lfJff )1f~t."'1ff 'tiCCil

( )1f~'--.fn + )) •-1~ !ft~ IPtf:◄ql ~~ "'IIC-1 'SIIIIIWS ~ I

1!11'""' ~ll~IIIJT'SI 'IICCil ~'V~-.fn~~ )'Sf ¥Cil'SI 'tit◄
eic~ Mtirfl.~~'-~~ "'lt'e11t ~lc◄q I

2
Ex:- 1 (35) =?

~ll ~IIIJt = 5

~t.◄JIID-st )'Sf~t.◄Jl-st '11'14 ()'11'~~ + )) .-.t' = 3 x (3 + 1) = 3 x 4 = 12

~ll~lilJT'SI 'IICCil ~'1"1\-.fn~ =·5 x 5 = 25

) 1f ¥t•HI 'I ICCil ~lJ lPf - fiRi'r ~ "'I1"1I~ 'SI IIllf.11 9ftl = 1225
:. (35) 2 = 1225 (Ans.)

35
35
175
105x
1225

Pap28of 34 ,n~,- Rony NQ Akasll

 Ex:-2
360 l.fl'f ISiet♦ ♦41fe t
2!360
2 !180

2~
31§_
3l!L
5
:. (2)3 + 1 =(2)4, (3)2 + 1 = (3)3 I.fl~ (5)1 + 1 =52
-019'4 =lj_6C4'SI (9ft'6llt'SI) "-1'l'i1 =4 x 3 x Z =24 (Ans.I

~) ~ :- -Sli!!O = '5jejq5 X -Sl'flq., + ,SpfC"'~

Ex:-
~~~ ~ c♦l-1 Afifl SR.~~~ ♦'S!Ci1 1Sl'1'M ~ ~ e1,w-1t1 ~1:>
'4l♦C<il 1 [ 'Slit♦ 'dl"1t. ~~1Sl't<R'l~]
lfl"IIIC-1,
ISl\siJ = ~l\loo '5l'1""1 = ~
ISl\si<IS = ? [ llt~1Sl't<R'l~]
'di Ia1111 'lflf.{,
CSIIO = '5l1t4 x '51•1 "'" + '5l•1t.-t 'I
⇒ ~~\l,O =ISiet♦ x '-'I:> + ~I:)
⇒ ISiet♦ = (~~~o - ~'-') + ~
= ~~'-'C\ + ~ = b-~ (Ans.)

I Pace27of 34 I 11\ffl:• llonY NII Alcuh

Ex:- 2
4 3 2
x - 4x + 5x +ax - 10 "' x - 3 ~'18M +!IC'I ISl•tC-l'i f l ~ ?

-mi, '61114 = o, '51"1t., X- 3 = 0 :. X = 3

~~ x ~1111ti1 tellf.le16 :citstca 1'f1t
4 3 2
x -4x + Sx +8x- 10
= (3) 4 - 4(3)3 + 5(3)2 + 8(3) - 10
= 81- 108 + 45 + 24 - 10
= 150-111
=32

:. <sl'>tt.•lli 32 (Ans.)

I!:~~-•••~ I
~ A~~=- ~"'f& '1.6<.4?1 (9fte~~ '511611'11'511611'11~ ~ ~
~ ~i1 41:gca, -.:,iet4i ~~'41JT ~ ~ I
Ex:-1
72 ~111Slet4 411' ?
2lzL
2~
21is
3l.L.
3
:. ( 2 )3 + 1 =(2 )4 ~~~ (l )z + 1 = (l)3
\5let◄5 = 't6C+sl ("'ft'e1ITTI) v-t.-t = 4 x 3 = 12 (Ans.)

hce26of 34 lf\ffl:. Rony NII Absll

1-.., llil11Acua:,t,11.21◄1iWw I
f.m'sr:- ~ Cl1'<3'ffl ~letCi:15 ~ ~ X ~~ ~ <11~'!1 i:15'!1"5 ~ I '15111fi'-,
~ \SletJC<l5 eti \Sleti:15 Jf~~ ~ 'l5t'Ff '<l5"!IT ~ ~ \Sleti:15
"1,'41JlftCii15 "BJ~ X ~~~ ~'<15"!1T I ~~9m ~~ X~'ii ill-iC<IS
~'5let)C'5<1P1Cil ~ '5l'fC"l'4 "'fl'61ffcllC<I C'lft"l.ffl~ I
Ex:-
(X + 3) (X - 3) '°' X
2
- 6 ~ ~ 4'!1C'I tSl"iC-114 ~~?

'lilqi'!l"f f.UICil ~'6IC<I ~ .

2
(X + 3) (X - 3) = X - 9
:. X2 -6) X2 -9 ( 1
X2 -6
-3
:. 'Sl"tit.-1'4 - 3 (Ans.)

~4111 Mil 11,

'diet♦ = o, 15j"1~, X2 - 6 = o :. X = ./6
I!!~ X ~~ tiletlt'6 <IP1Cil -Pftt

(X+ 3) (X-3)
=x2-9
= (./6 )z - 9
=6-9
=-3
:. el•tc-t-i - 3 (Ans.)

I Page 25 of 34 I 'Rffl:• Rony NII Akash

I
Ex:-1

lSfil', X • ./3 + -./5 Q, ~ X +: 3 = ~ ?

3

X + :3 = ( 2 )( ~ 1119') 3 ( 2 X ~ 1119' )
3 3
-

= ( 2 X -./5) J - 3(2X -./5)

=8x5./s - 6-./5
= 40-./5 - 6 -./5

= 34 -Is (Ans.)
Ex:- 2

~. X = Y3 + ..fi Q ,~ Jt'- :3 = ~?
3 3
X - : 3 = ( 2 X ~1119') + 3 ( 2 X -1119')

= ( 2 X ./z) 3 + 3 ( 2 X v'2)

=8x2x./z +6./z

= 16 ..fi + 6 ..fi.
= 22 ..fi. (Ans.)

'"'110t- Rony NI Alcash

,-,e24of 34