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TRANSCRIPT
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. . ··-i
. I
n = l �0, �. ot. = })2r+ l) = 3 l!i>l
,,
· t;.. Ol. ;;:;;t(l+ 2)=,'�f. · '·:.··.-. . .. -·-· ·.
. , ' ' ,,, I ,
OOcN®k eJeD 8,Ql@Q�f�,--n·::-:·k eo(.eD>. �-q,ffi Qffi� Qf:W OO.COtS>@d6JC:O �-
�f ±(2r+l)=k(k+Z). G) -r•l ·-
-hi- , 1r ···
�en 2)2r +I)= L (2r + i) + [2(k + 1)-t- l] :=-; i<:tk + 2) + 2k + 3r=l r�t
. .
:. � ·Cf�6.\ Q�@C:O n = kD roa,;, eu@, tiw· n = k + 1 O e;, c,t.00 �- n = 1 �i!i:>) _ � ·qr;B Q� ro� 00 @t:00>-��> qim:_.<S)�ci>Cf�eo �e,' � � weo ��- n �Ci)J® �S�©Q roaie, ee' .. _
U
2-x
· 3x-,:.1 ·•
. .. � . ! ·. I" I . •·· . -••···--·,1,_<..._i ____ .... _ L ___ , _,·_x_..::_2 __ +---7-___ <_x __ -J
,,{+.} ( I·) H
-,.,..--1---- .... ..;. ,,·. �---------.-..-,·•(..:) (I) (+)
··-·-•······ . ----"'l'---------i---------1
(-) (I) 2-x
3x..:..l '---�-'--�--'----------- c-) I.
0� n e1/! � (fOOh> �ut, 01@ t,� n e � 6- ;::: _e&3� �iJ Ot.,a, e88 �e
n � �ai �. �. � �6 �d M> �m()8fln.
. L
---··-2x --2 ,, (' -. )Jl-·1
- y,•-! - _J 0
0
(D a e JR <.:tt,� iii�. ( x + 7}"'
� ¢� �@ � t.>8� ��� o.:ui ¥· ..0. a &\ lfUl<J �M,
20 = I 20C,a' x'l.O· 4r
r=O
_._ r = 5.
20! 5 969 <:---:> --·-a =-
l5!5! 2 5 l' <=> a =-. 32 ®
0
<=>a=-
®
( .. \ \ "))
25
( ( ) . :-: I • A \ 1 srn •····
:._. lr'n 1 1' [. 2 ! (' r----. 1·····-···-.. -··) ·- 1- '\ / / l ..;� 7 T
j
,,..,o 4 ! r r' I · " 1 + x · + -v 1 ·- r
l l·�- l I. .
·, -,• )
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.·. J ln(x +-i�-;··-�;.:·1·)dx "" xln(x .f.J;;-;"()-f >f";i -t-C •.i11:!iL�'. i • .::�-f'.>1 tyEi>i�ru l:Pi .. lt:5><.'.l&\
0
· @ (l.l) oa.J;l<IQ� x + 2y + a aa O � a� em{;';� (},b) o•»� �� M,., « re, bf. · a en.'\ b B qq:it.ffl't ��-
A (3.1) '
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\A 'r.· \ ' '
l :x+2y+a=0
, B( ¾,b)
14 :. a+ b = ··-- .................. : ............... (1)
. 5
AB .L !-@�, t � -� ] x ( -½) � -1 0--3
. 5
24 ,', b,...... I ::l ���
19 0b=--
5
(l)⇒ci=l 0
'ff{hffiW"I ��c§V ®
-{ 'v e_ Yr' CJ v (J_g
a.· � -a @ 1
· · @ x = 2 ·cos0. y .;.. sin 0; ®�m �1ll oa>� e� · C �� 11>��; e�B e {!)ta �t1:�61��- C o�.;.i. .
o::: f o q�qo o.z:s!w�@ �· �. �t,�(M), · C ei�� �em e • a Q ·q�o c•�i �.
. I . . . .
�. 2sina;,_8cosa+·3.../2 =O�.o <K)�u-M. ·
.. x::::: 2cos0,y= sin0 · y
'•\ :
8 :;;: !!_. D q�to @tsw�. P eri®,4
p =( ✓2,}i) cfx· · . dy . . dy -cos0 - = ---2sm0,-.::: cos0 ⇒ - = -d0 d0 d 2sin0 .. \,G)J:i&{L� .
. I sin a-- r;; = 2(2cosa--J2) · v2 ·· 2sin(x --8cosa+ 4✓
2
- ✓2 = O
0
© q&t · 1 =! Q �- a,�,� xtyi:iO o,6c, �JD�{).�. C Dil!rlmc.6!:St,. x2 + l + 4,Y + 3. 0 !lam� eo� Q
- �me;.; r:oo8. C @ <,imti"l�c,d &61��-•tl'I �:rkr>.
·_q�a tDoo �adcn6cl etro'�co x + y == O@bQ))!!) ®cu 86300 �,�rn �0 1/�al (I, ·&i <!>(ll6 ffi�i., CDl!ll, f®Bl I eo 91.
_etmerl'½�(t, - t) �� (,fOCOl � e:>aclt:Wcl cn®t1l6eJ:Jctl,.
-0 i 2 . .., ,,.. . 2 .2 1 0 ,r X + y -· Ltx + LfY + [ ··- • =
2(0(2) = (2t 2 --1) + 3 0
i.e. 2t 2 -4t+2=0; (t-1)2 =0.
:. t = 1 ®ooe,C} =(1, -0
/
�¼, cos 20 = 1 - 2 sin 2 0 = l - I = 2. 9 9
' 9 tan20 = sin 20 = (·4✓
2
J(P.) = 4✓2 .cos20 9 7 7
0
0
E)�,(-1.,-JJ o/1� M,-zo'1'V>Cq
-�- r-·�" ®
f-1- ,f'�c..,'L.\ ®
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® (a) f(:c):.ax.3 + fr.x2 -1Jx+6 �G, <D��� � a,b ElR Gi>. (x-l) �irl'ts::, f(x) t3 oowr;ix.,d � � tflJ J(x) c.,� (x-4) m �!, 00 C"{e6l-o) �� -6�. a too· b Ve 4'�� �m. /(x} lo qo!S)cl &:.'>t5 ·o::iwt.i) ��isi' �tl'fm.
(b) a ioo fl d� ;? +bx:+ c === 0 �i:i6� @c.··c.6tl3 .:;. r cs:,:, 8 (.)ll,) xi + mx -t· n r.-, 0 �r;;)O<.m01:J·� "'te
l
<; a:i�; e@S b, c, Nl, n e IR �-
' (i) b l:03 C q-z_9m5!:51 (a - {J)2 »�JJ(J,J, C, �- m m:i n r.tc�6t!1 (r -8)2 8c.,:i <;�Ou.fr.;-,.
a+y=/3-t-8 :'S)� b2 ··4<"=m2 -411 �D q•� r:;,o'/Shs-,,
(ii) (a:-•Y)(a .. -8)(/l·-rX/J-o)=(c·-,,)2 +(v-m)(bn-cm} wD "°�t:rl .....
x� +bx+,-':.: 0 i\'D x1 + mx + 11"' 0 Q°'��.&'JCO isr:,:i[i §e;t-:>z:.i <l'<i:J{C!id
(c-11)2 ==(m-b)(bn-cm} ® m� e,@�� oo �� �mtS'l .
. -''1 +10x·1 k=O· cm .x1 +kx+l0.:.-0 ��CV Gb�f; �od <ftt:il; �� k (.)l{)t:t.1��1'31 inc.:>f.)J<.)8. k &\ �m �omm.
-----------------------�· ---�--------------···��---------··-------- --------··---�----------.------
(:,"·) (b) (i) , ..... _ _,,/ a � j3 <,'.J?:;?") x:• + hx t· c ·"" 0 EJticoej ��25)06i�cl S\(". c:v,0::.-
r
a 1 /f · h ;:;i:: a/J , c· 0�).
�����®�· (r-0) 2 == rti2 .-· 4n 0
��--�----------�--------·--------------��-���--
G) 0
-�>" - (ii) 0�;,· '\.e,'(a-y)(a-8)(/J-r)(/3�0)=
. [a2 -a(r-to)+rS][/31 -P(r�ro] s
=[a 2 +ma+.n][/3 2 +m/J+n] \..:)
� a 2 fl� + af}(a + ,B)m + n(a 2 + /32 ) + a,Bm 2 +-mn(a + /J) + 11'
== c2 -hem +n(b2 -2c) +cm 2 -mnb +n2 0
= (c1 -2cn+ n 2)+ (b 2n-bcm-mnb +cm2
)
=(c�n) 2 +(b-m)(bn-cm) 0
x� + I Ox+ k = 0 ro:> x 2 + kx + .10 = 0 .,,$._ZS) �®2:llol!li, DcO 0e,:i� %}c�?.Si t)[J-6J�@m' Zl'J�
• ( C \ 6®�Z6f (c-n)" -.::::(m-b)(bn-cm) \..:..)
" oi:.k · ...--.. 0<:11'.fl b = �O, c = k,�e,w n ==IO ef'>.(:)
(k - 10) 2 = (k. - lOXl 00 - k 3)
¢::;> (k-lO)fk -:- 10-.. IQO+ k 2 J::; 0
¢:> (k-lO)(k 2 -1 k--110]=0
<=> (k-- lO)(k-lO)(k +!I)= 0 0 .
(,.'1 e:i@a3� £:::, 10 ew:! k.=-11 V'''1"
- . . . ,., . , .. .. � ·- - - - - - - - - - - - - -- - - - - - - - . - - - - - - - - .... - - - - - - - .. - - ... - - - -· - - .. . .. . - - - - - - - - - - - ---- - - - - .. - - - - - - -
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. -··- - -- - . --- --· ___ .... ---·-- - --·-•------ .. - -------- ---- --- ,..,.. ______ -- �- ---- - ----- - ------- --------- --- --___ . __ . __ --
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S9z.n" 6 G-<;�:ic)® tl'eisilS'J:>2S):J0 .:,(!ie.,c., 80:, €):)� 5� IDt2ili 6ill)025)ZS)c) tp[)z,,et <'.tJlIDOO (J)-On2:5) = 5! � . . (
(b).
'-1, "'· ('3"'1-t'.)) '2 - 6-r--1 ) 2..
(ti-() 12. ( ]�t � l 1
.,, _!..-, _ _ _J_ C ® n-1-1J 'l-- O'lt-i) 1
/J"A -fi_) C,·�
3(6r + I) A B - . =---+---;
·z+r E. (3r -1) 2 (3r + 2)
2 (3r --1) 2 (3, + 2) 2
<=> 3(6r +!)=(A+ B)9r 2 + (12A-6B)r + ( 4A + B);
--�---------------------------�----------- . ----------------
. 1 � f I j(r)=(3r-l)
2 �lSCD01§. 66c) (r+l} =
(3r + 2}" .'. U, = f(r)-· f(r + l)
r = I BD
r=2 Bo 0
0
1 lim
? = 0 IHOO (3n + 2)-
--------------------------------------------------------·---··· ....
II{' II l · 10-f, 0,� -- :::·---<. 5,·-" 4 On+ 2l
<::> 1)n +- 2f > i 1/' ':n,1 n <7: ;z,•:•
<.:::; 3n + 2 > l O 1 W3 n � ?: .,.
c� Jn > 998 tfl'J n c ?i. •
�;⇒ , ... , :> .) 3 2 .::-:. :,:_{')J n I..; • • '.2'. ·•·
:·· 'n s z.ub€- �;)c@(,'.l .:ti.s);,:. ,�· 333. (i) IN
@(a)
;:�co. QrQ ·:;:: ( l . .--1
AP= PD
·1 #II II I 7 Pi F ,-;--qpg
r·1s-··:
i ............ _,_,.i
17 � C rtKF FI ·1;
l__ io ... l
,········· ··•1
i 15 i ! ........... ..1
-:. ' . ,.r-.
.... /-\i_. ··0::_·.·::···_ ... ' 5 .. . · . . . · .
. . .
a+ b 2
12 = {2
c + d 2
.,,fj_ - .fi
-c +d 8
./2 = ./2
u+b=2
-a+ b = -8
c+d=2
-c + ct = 8
-----
C = --J
d = 5
O - O � o -� .. - - O •• •• - - 4 •• 00 HO .. oW •• " 4 - - •• • 0 O � o, 00 .......... ,.. ......... - ... - - - - - - - - - - - - - - - - ... - - - - ... - - - - - - - •
:fi1:) lz\ = Jx' + Y2 0 "" f. = x - i! G) - .
--- ---- [ 10 ]
. t ,,-....,_
!z\2 = x2
+ y2 = (x + iy)(x-iy� zi l:_)
· 0
___ ,.. ____ ,, ____ _ -----·----------------
lz-3il >!1+3iz[
·¢:::}. jz - 3il 2 > i1 + 3izli 0� lzl2 - 6 Im z + 9 > 9lzl2 - 6 Im z + ·t
<=> 8(l.zl2
- l) < o
lzl2 <1 0
.lz!<1 G) !z - 3il >\1 + 3izJ ¢::;, hi< 1
.,_ -
·® {a)
-
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(x:1 �-1)2
; x=t. I
0 .,
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nws
0 - ·• - - - - ,.. 0 -- - .. _.., - - - ,. - - - ... - - - ... - - •- •· .. - - - - - - - - .. - - - -- - - - - ••• - - -• •• • • •• ••.
- - - - - -- - - - - - ,<O - - - - - - - ... - ... - 0• ,o - .. H - - - - - - - - - - •
(� / (x) =c. 0 <;=:> x =.c O G-l'S):1 r·1 = -·2
X X < -2.1 -�
f'(x)_ � GesJ� H
X = 0 � /(0) = 0
@(b)
I .
-"➔+a) 50 .f(x) -� 0 X ➔ -oo 5c) j(x) ➔ 0
X ➔ 1·· [le) /(x) ➔ -00
x -➔ ! + 50 f(x) ➔ +oo
"-•
0
I
-2·
:.
-�· ..
<x<O
(+)
10
6-toe.J tr86i. ©©D?Sf1.insf a8@&.:.,; P = l&x + 4y
@©ud0tsi Eib©6c;�; _800 = 4xy + 8x2
(2) m, y cc
200 ... 2x 0
X
O<x<I X> I
(-) (-) ---
0
0 < X < 10
0 - 800 0IO -- --- -= 0 5
_x
2
<=> x2
= 80
¢.:> X = 4✓5
LEJ
0 < X < 10
-------------------------------------------------�---------------------
4✓5 10
X ---_ -- --\
0<x<4✓5
----------------------------------------------------------------------
·-------·------------------------------------------------------------@
(b)
X1 + 3x + 4 Xi+ JX + 4
(x2 •-1)(.x + 1)2 (x- l)(x + I)� G)
x;,· + 3x-+ 4"" A (x + 1)3 + B (x + 1)2 (x - 1) + C(x - 1)(� + 1)
+ D(x - 1)
x :,;:; l ; 1 + 3 + 4 = 8A => A = 1x ''" -· l; 1 -- 3 + 4 = -2D ==> D = -1
1 = 3A - B + iB + C .=> C = -1
,_' !c
i 3 4 .•x + x+ · 1 l I I·•·· � ·· :.{t2 - l)(x� l) 2
= (x-1! - (x+ I) - (x+ 1)1 - (x+ 1)3 �
f x1 + 3x + 4 . f 1 J 1 f
1 f
1
0· · · -;c-(b: = ,--.:._--dx - ---'fr - - --dx- ---dv: 5· (x2 -l)(x+l) 2 ·(x-l) {x+l) (.x+l)2 (x+l)3
:::: lnjx-lj-ln(x+lj+-. -1-- +- 12 +Cc@· · · (x+ l) 2{x+ 1) 20,. ·· ·
• • I t ., ; - . . ; . .
--�--*----------------------- ------------- ------------------------.- . ·._
f .. ' ' •. ,;,,,,, ,· )
()
0 0
--- --•-----h ••--•• - •• •• - � --� -- ----- •---------�-----------•---• •-- • • • • • ••
OP= OC+CP = ✓2 + l, �ts:> P == (OP"os n:, OP�in n) t;,2.,� t;"1ro, . 4 · 4
'·(M)
·I so
- --
- - - - - --
... ,# - - - - - - - -- - - � --
.. - - ... -
• -
• -
- - - - - - - - .. - - - - - - •• - - - - - - - - - - - - - - - - - - - --
•• - - - - ... ·- - - - • -- -
- - - .. - - .. .. - - - - - - - •
,--· ----
I.) ,._ •✓· ') \ - / h .J... 1,- \ / '\ ,- '
-, \' ',.., \:.) ii · ·-----'--- 5 -Ji
CC'·=:l.+d G "'1 CC''....:(! +df
• 2 1 I,-
('J,+ ✓2
)-(h+k) ·1
·'·-� (h -1) +(k-1) -"' l l-;':•·--··r:::-·--·-
: L �L J
r;'\°1> J/-2hk+k 2 +4✓2(h+k)=-8(✓2 +1) \.J
G
· ------------------------------------ ··---- - - --------------------------------
2 (a+/1) (<t-ii\ 2 . (<r+/J+:>.y"'l _ , (tt-t/J)= COS - COS -1 - COS _ COS -.-. -
2 2 1 \ /. I \ /, .
(a+{J) [ (a·•P.) (·rr+/1+2Y)] ,� = 2 cos 2 cos ·--;- - cos ---.;--·· C.:.)
(a+i3\ (a+v) 1 {J+y) 0 = 2 cos 2 J 2 sin 2� sin �-2 s
= 4cos�(a + {J)sin;(a + y)sin½CP + y)
-- ------------------------ ------------------ --------------------------�-
b)
::::: (1- cosx) + v3sln.x + 2(1 + wsx) 0o) �
= ✓3 sinx + cos x + 3
f../3 1 l r\:.::: 2 l-- sin x + - cos x + 3 t s_ )
2 · 2 J �/ �-2 rl !!: • • -re ]+3( 55)" .. :;:::; cos6smx + sm6cosx '--
/.--�
= 2 sin (x + �) + 3 (__:__)b =-- 3,
0 ---------------------------------------- ----------------------------------
-1 � sin ( x + �) :;:; 1 0-2:;; 2 sin ( x __ +;) s; 7. 0
-2 + 3 :s 2 sin ( x + !:) + 3 5 2 + 3. 6,
1_� f(x) 5 5 0 ------------------------------------- -----------------------------------
/(x)
ff
3
.
.
GJ
A
I'
C
c.,Slrl �ta� ���e-e2S!.
sin A = � = :!2� = k �ts St»@. 0 p-fq P P·-t/ \..:_)
''0.0 ==> sinA - 2sin8 + sinC = k(p + q) ··- 2kp + k(p - q) = O G)
------------------- -�----- - - - ------------------------------�-�-------- ·
Sin A + sin C :::; 2 sin B
2 sin 11;c cos A;c = 2sin(1r-A + C) 0 + 0
. A+c. 1:-c . (.A+ C) o srn-cos-= sm . .;, Z 2
· ·
• A+C A-C\ · . ll+C A+C sm (-7) cos (2-.J = 2sm (-l-) cos ( 2) ( .. (A-C) · (A+C) 0 cos 2 ::::: 2 cos ·
2-
( A+C tr • (A+C) . ) 0·0 < - < - :. sm -- > 0 , I ' 2 2 . 2 · .. "- . . 30�-- C - - - - •. - - - • - - - - • - •. - - - •• • -.- ••• - - - ••• - • - - - • - • - • - ••••••••.•• - •••• • - - •• - • - - - .:
·-· ··--·-····•·J••··-·-··"'---·