1 .. - -.. - -,.· .. ··I . . 0,- : ·:: c, - .a:l:...11.(a) a ERcotS
27
1 . . -� - .. -�-,.· .. ··I . �. �-��:�·:: -��.::
Transcript of 1 .. - -.. - -,.· .. ··I . . 0,- : ·:: c, - .a:l:...11.(a) a ERcotS
1 . . -�- .. -�-,.· .. ··I
. �. �0,-��:�·:: c,-��.a:l::
10x25
05 x 150
.....-
250
= 750
= 1000/10
100
n . . .
1� C!D.Cil8a> � � �::,!!31.:il@c.:>m, s� n E .z+c,<li� L r(3r - 1) = n2 (1Pi- 1) 00 ��c:i wdmm.
n = I �0, c>. e.,i . = L r(3r - 1) = 2 eo:>r=I
�-ei .=12 (1+1)=2. �Z)Eazrl n = I Ve:> s��cc.o Zil;:i:, 0E>. 0
r=l
<!>2nt® k ez+ @<nZ), n = k e.,�ao� �253e:lec.e i:.,!S):is coiS ce26)daZ)c;.; 2S)d§.
�Z)e t·r(3r-1) = k2 (k+ l}@!>. 0 k+I k
Z:r(3r -1) = �r(3r -1) +(k + 1)[3(k + )-1r=I�
-�
r=I
= k2 (k + 1) + (k ·-r-'1)(3k + 2) = (k'+ I)[k2 + 3k ·t 2]=(k+I)1 (k+2) � 0=(k+l)
2 [(k+l)+1J
2. ge4tna8'= �= -�� � qtrl qQOSz,:$ <MO:t, Ix ... 1 j > 3x + 7 qc:o®:ii»al:iE> eoqo:i� x@ Sc.,�tl);�tSIDi:n �©com <9'Q:>G!)fflttl. . . . .
1
@D�� :Wel"'21if
I Ix+ 11 > 3x + 7 q�®:>tn�:iD Ix+ 11- (3x + 7) > 0 {,OWZ)� �G� (;'�.
{- 2x-6�©0 y = -4x-8
,x�-1.�5
· x�-1-� <ii':-'
'
�-\:� .
___ ,, ___ l
· 1
----...X
G) I. 0 2
3. �IJI ® q:,a,ffl!) e,c)e(:)11)�
(i} Arg(z+l)=f .:
(ii) Arg(z-1)��
�ts) z C,�41ij eo.;a):i;o ®@isf' &it� .=a'� e@� ��� cc.'E>c � �m� ·qi�. �erj
0el�m OM:,,Q ®@trl �� m>d� c�-.�._ � �:ic.,mt>l.
Arg{z-J):-: � . � . V\'J ·.,
, s:r 1r 1r ·
0.s···· 1,�
,J',$
. ACB=---=- @�.
6 3 2 . . . . tr AB= 2 @z.l'�, o�� BC= 2cos- = I "'"··
. . ·3 . :r Ji D tr l ,..
·@w:1 z+ =cos-.: -+zsm-�iZ'! CD= sin-::-· to� B =cos-= - ��. 3 2 3 2
. --�[ 1 .In-: . .-. 3 . 3 ·
. 1 Ji � • 7 =--+-·-; \ I .. -.. - . . \. / 2 2 .....___ ....
.et>Z)2S.f W®�� · . tv ZJJ.}1) ( + J �)AC "" BC il � .. o..:. 8�@!}82" : )' = -': }'(;t.c_ l) c ,,,, y � $(; + I) �. 0
-
l·.25. L.__
3
j s,
'Z ---+-1 1 ../3. 0 .. " - 2 2 .
x"-2 ii �0�616>� 120 @O � qt� Rl't�iri,
G)2 "Cn-l +3 "Cn-l = 120©� 9�'-'· ' ...._,....,.--·····-·
r-0--..,n! · n!
�Z'J� 2 +3 = 120. (n-2)!2! (n-l)!l!
·<=> n(n-])+3n = 120 �
'("' \:)
<=> n2 + 2n-120 = 0 <=> (n + 12)(n- IO)= 0 <=> 0 n = 10 · · (': n ·E i+)
. tan 2 2x ·1. tan�2x 1+.Jl+xl t)·�
hrn . } = 1m l }x+---== x�O x(t -.J1"+x · x"'° X\} - .J1 + X ) + .Jt + Xv
_ r sin 2 2x (1 + ft+x) r.;" .. -)� cos2 2x (.:..x2
) \J . (sin2x)2
( -4 }( �) = hm -. -. 2
l.+vl+x . .-.o 2x cos 2x
l
4
),",
������_;_;::,-,i.:;..-:;;-----.�����2�--x
12�� I·_____ ___;.__
7. x=·t+·, t, y�)-e-1 ®&1��;: -�.C c.,zS «)�;,.� ... t �·,�:�.d '
' ' '
t qiaGtd!d t ��. t=- ll:1.2-f) �e:,-e>: c ®m -fJ emfe.z,��, -�G)·-�� d��
5x-3y-� = 0 00 _<te�:r,ft,),
I -f dx I -I x=e +e �-=e -e
dt
y=e' -e-r � dy =e' +e-1
j ' dt
C � t = ln2 c) if�61e �•'lSc.o '°tS <I>��- c>�� C = ( 2+}2-f) @�:
�e c�(H) �-. �(HJ ;!��;i 0�o.;,;, .. ,.,d..,., y � f ; f ( x -%) ..e. 0 elm!> 5x- 3 y-8 ; O <>!>.
AB �ijlle �®�o� (1 + ,1.)x- 2(1-;L)y- 2(1-2) = o e-!).
t)@� y=O !,€) x= 2(l-J) u,:, x=O 8c:i =-1 ctll.. (l+l)
:. �OAB iS D mocc., = - x 1 x = 4 5 t 12(1-J)
0· 2 l+.:l. .. ¢::> 1(1 - A. )I = 4
I+ ,t ' .
y
/
19. �0.3)��(y-cpac.,d�m)O�.U):>i" r+,1-8x+4y-5=0·��@eea��I ��:, � <i. !>a� �-a,�.
. . . .
.
���+-�---·����-+ x
6
qe:>a;,::o e>.amtsi�d ���o,(,lio"' (x -,t) 2 + (y __: 3J 2 == ,1.2 eeeo (96w:, tot�"'· . . . .
e)Z)® x1 + y1 -2AX�6y+9=0.2 2
·
@®c., x + y -8x + 4j-5 = 0 Dazs:ttl)c.,c.) 9C�� �!lz-!,2(-4)(-it)+2(2)(-3)�-5+9 @!)�
¢:;} 8...l. -:- 12 = 4 <=> 2 = 2 (;\ . ·0
�Z)Szrl 'fD�:i, e>azs:tzs,@'4 eo�wd-Goc., x2 +. y:2 . ...: 4.x - 6 y + 9 = 0 @e'.>.
0
H . (it . , . Itana=-1 w, ·-. <a<27Z'=:>a=-:=)cosa= r;; w, sma=- r;::· · 2 · 4 . · v2 · -.J2
sin /3 = � t<>3 7t < P < 1t => cos /3 "f-= -Jt ..:.. sin 2 /3 = - � .· � l)
v5 2 · · · . ....;5 � ti''IJ)
(lz!, .cos(a + P) = cos a cos P- sin a sin P 0 \\"
2 . I =--+-
O Jw QD·
7
. . 11.(a) a ERcotS <tf<x)=3.x3+Sr+ax-l cotS ii. ll'�j. (3x-1) CMSlmj(x)@ eoowimc.:,,m 00 ' �m.
a 6\qroc.,�. . .
J(x) c.'.12rlm (3x-l)(x+k)2 q:i� 9� �o�m; "®@ k � 6l�r.,B.41wm s�t!)� Jx-1 =� b eoo c �"'1:» � b(x+ l)+c � 8��. J(x) Q.'!ffl.S'.l (x+ 1)3 � $Q)' & Ne� @(00�:m;,).
(b) a,b,cElta»ac;t0QtSa:iaift��r..,.a.t2+bx+c=Oe:,h��..SS@c>�.e®®. �8mdG� le a rs»/J etiS ·� .!=j '°la Cc _G:i�.ac(.t+l)2:b2A. ·@e) ec,�.
p,q, rER m, pr-¢0 �tS G>� •. 2:DE>� px'-+qx+r=O ce®�Cilcl @e y ln:I 6 QtS " µ=iei@ � «>�. l. =µ � ,\ = ! E)mm acq2, = prb2® d c®4182.Sf 00 ec�mm.
. kil- 3x + 2 = o W3 Sr+ 6A:x + I =O. e,ho'dE'le le· m ® q��f.> !).a,, QJe,> · qim; �kER@!j.ke q<D'° �-
I + 5 + 3a -- 9 = 0 :. a= I. 0
f�25 _J ------------ - ---- · - ----------- - --------· ---------------------· - -----------�------ �.:::\f;.\=fj.;0\---- ----
/(x) = 3x3
+5x2
+x-1 =(3x-l)(x2 + 2x+ 1) 0 <!--- . i�htf, ,:):��\._,,��·;.�,#· 1..'J(].
, � \_,,., ·,trf''\ ( = (3x-l)(x+ It\ 5 J
<;--.... __ . (f"}'.'v , _ _,, . r:;;l
G,&i)i;:i k .::: I �eS�v qDe,v���i;oe-·ai�-1 ;]. 0 ·-.. 1_::__··-___ _J_J
,"'{ I r;i. . . \.::..,.) ./ �.n:y<v t·n:·'('.;., :Y)Lf:3 � -----------------------------�------------------- ---------------------------------------------------. 3x-1=3(x+J)-40
<S>®<,0, b = 3 w:i c = -4 eot1�f.) cfeJ(gl3 t:p!fi".la6e-co� @E,.
/(,x)=[3(x+1)-4](.:t'+l)2 =3(x+1)3 -4(x+1)2 0
:. - . @o:fUal =-4(X+ 1)2
. 0 ------- ------------------------------·-----------------------------------.---------------------(b)-'oe��tt, ·ax2 + b:x + c = 0 lID �wwcew �eoo ©��.
�l}p, X'=0�:i(5'<©�c.oz-f, c=:Oe1.��. G a<;:#- Q �tt)2rl, e®Q tc.,oO:,'c'-'�·
·. -��:i,� �2 + bx+ c � 0 f3ie�� C,Z)a@E:>� 0. . . --------·-----· ----------------------------�-------------------------------------------------
8
b c 5)$�, a+� 11'3 aP_= a.
G �
(a )1
b2 fs\ 2 -+1 ( . )2 -, . 2 . \.:...) _(..i+l)
= fJ a+fJ =£=!?.__ ·:.ac(i+l)2 =bi).
,t 0 ; . 0 ap0 � . 5 QC
- --.-----. ---�----�------·-----�·�----------�-----------·--------------------. ---------------------
; ac(.l + I )2 b2 A 2 ( )2 2 ( ) · · (µ )2 = -:i- ail>Zl', acq µ A+ 1 = p rb ..i µ + I pr +I q µ
acq2 = prb2
� µ(_i + lf =A.(µ+ 1)1 10 ·¢> .i µ + 2A,u+ µ := .f.µ2 + 2J.µ + 1
, Jo'1)i" ¢)_ J.,µ (,t.,.. µ)-(.1 �µ) = 0 �. (l .- µ)(J.µ-1) = O:<t.." 10)
.\, �' ¢> .i = µ- ..9rJ = - · · s \ 6� µ . -----------------. -------·-- ·-. ·----··--·-. '·.-----. ---. ----- 1 -------- - -------------------------------
�<' �21!)® ffiC,:,�'°� E)�(!J,�' <==> ,i = µ. �� 2 = -. . µ
:.· acq2 = prb2 6"' 9�01
ac,q2 = prb2
CD ��, 2k(6k)2 =8x-9 . ·&i�� :. k = 1
3 ..
k :;:: J �� a1\1"'·
0
9
12. (a) �e,e Wc:.'ll'm toot6i � ��E>� �:,@ e>m qmcf, @::DO @6:>.tnctq��. �c�t @e.i�zrltoo� ��zrlMl�m �Sid���1m{!)d �.il� �c@S. 0®® @�m �ts! mo@:16m,Sm ��m sm =>®flt)d. �� en•& �l> � 41m.
(i) � 6d @� ·@E>:s,Z32rl �� ai@m qi� � Wil �.(ii) eatAG � ® � �m �.·ezsl 6d @�td @fil,s,£3� ��� @t@:rl ttt� �
��.(iii) � �t1,6hrl � �d � @£.>�@���St@� 'i � ooe:id �mi75m �mt�
�� f!>I@ @� si@trl � qii;� � 9t\l m®,<t®® �ssc:i t.,'t�Q anml -,E)� (1:>t9:Kl' <D.flltn �t,oQ!ffl'1.),
(b) rez+ �too CJ = ,2 - 7 - 5 "'le G)�. · '
r(r + l)(r + 4)(r + S) n = 0, l, 2, 3 �too,,. " � =te�e®.zrl, rEZ+ e,� r2-r-5 = A(r2-l)(r+S)-Br2(r+4)
e1m c� A m:J B &.m t)f.)&l) @e> @e�mm.
rEZ+ toCiU)il U,=/(r)-/(r+l} f>m o�/(r)�.
nEZ+
m�:, iu, = - (n+l;n+S) SEl���.00 ,.,,1
}:u,qm�m � � f)a, al� �cx>m ec�. �@ �:i,Q �m .. r=l
Q)
c!Sdd. 23V, �m�.
Jr:=<3
------.,...--, ,,��,:;" ---. t;0 0 er--- 'ti? £:<S�' ,,
(a) (i) �!lam�@.,�., ='C,x'C,x'C, = 05)3 = 3375.SJ) .0 0. -'i)tn-�1-1'{/$) �
- JO • --1?
,>
(ii) qf.)a.>)S !)® �&2S> =6C�)c2 x2 C:<:::90 0j 1s--]
--------------------------------------------------------------------------------------------
.--·----------------- . -J--�-----------·---------------------------------------·· ---------
(b) U = r2
.-r-5
' r(r + l)(r + 4)(r + 5)
·r2 -·r-5 =A(r2 -l)(r+5)-Br2 (r+4)
=(A-B)r3 +(5A-4B)r2 .,..Ar-SA 0
10
@��'td@!S'.i t,5o�-@!r,?Jil e.:\oe.52.rl��c:8 26:>6�.
r3 : 0, ='A·-B - .-............................... (i)
l.:==5A-4B 1 ................................ (ii) 0. 40.
7(ir; i,, . ..._qr2 :
ri:-:ls-- I
-1 = -A ......................... � ... (iii) . l �i"iJ
r0 : · -5 =-SA .............................. (iv) Q �. ;:;�Ji"> . ..s-:H:) :J- :_'.,'--....·• "'3 '11'),t9' ( .-�) 6' 't >
�� {i) W:> (iii) � A = l t:O:;i B = L """' - \:j . . �®® qco<-0:s!liJccoir/ (ii)� (iv) <t ��?il <S-0. �®z3e.,:, , lftzil �c)<ro,%l):;ic) :s)at!ti) tl)0Z) e8� W:> B=l.
0 '
----- ---------------------·----------------------.------------------------------------------
·r E z+ Mti�)
U = r2 -r-5 = (r2 ""'."1Xr+5)-r2 (r+4) 0 ' · r(r + l)(r + 4)(r + 5) r(r: l)(r + 4 (f + 5)
r·-1. rr7\ 't'tzi'' Ur = r(r + 4) - (r + l)(r+ 5) \__5j
= f(r)- f(r+ I)� 0 e-®£ f(r)
= r-.l fs' r(r+4) \:._}
-- ·�©8 .'
r:::: I: ,. :::: 2:
r=n-1:
r =n:
:. ru, = /(1)- f(n+ I)r=l 0.
n = (n+I)(n+S). ·o
---------------�-----------·----------- --··------ ·------------------ ------------------------. lim :tu
r = I� -n =0 c=»r.!)z-1 fu
r qs3�:,B @!': 0 ·-,,, ·-(n+1Xn+5) l'� \.:...J
""' .,,.,aj!) 0 " . 0 . 0 ---- --------.-----. ----------------.-. -------------��-·-••·----·---- . -,-------M--------••
11
13. (a) a, bER <0t8 4 A = u n "" B = u :)"ts 4 dfl. AT A = B !>tn o8!f a "'' ball q,,x.,,,i�z:rl�; · ©®ta AT ®@m A io»�®'c.,€3 �"® «�·
C = ( 7 5 )" wo x_ = ( u ) OtS <Dzi@; \HI� uER @!>, CX = )..BX mis � (3)��; �®@ 5 3 u+ I ). E IR �. 11. a, qax.:i c.0:, u Ea qmc.:, 0e00t:)m:m. A @ 1:r®® t)!CDQ �a,-:i C- .AB z,;e:;e%,'l �. � � �Bite 00 (!)QfflE)�.
(b) zEC _ Q,8 mzJ�.(i) jt-zl2 =1-2Rez+lzl2
00 �
(ii) z:# 1 e.:,'1t� Re(_J._) = I -Re; 00 @t.'l�dm.\1 -· z 11- zl Re(1 � z) = t t)m0m J.tl = 1 ti):) z.� .t ® � �.i@-e!i1m oo � =o'�.· · I 1 \ 1 % •
· . · S �. Rel\_l -,·::: -2 � -·c- < Arg z < -;- c.,m �o«>Zlt!'l:l �� ® ��J!l z ��w �ahi�m · -z 3 . ., co®� �co c.:ttS ��. s@ "'9�-GD C:>oQ)� �oeu� =dUI e='ltl!l �C!>trlm eaw�zs, ti!�z:f��z� S� @e ZS'l® � Rez+ imz= i-�. z = cos(�)-isio({;) 00 c90tl'ID�z:i.
!\ - (-, , ' 1·11' (0,,. .::_ . -I ' , ) �, .., . elZl 1:}!2"6 01S>8z-J', A '.A = : _ 1 , l 10 )\. I a + I '-..____/
.P�-t .dr.d
-R,......_,(2 l )_(b 1) '(L-•, •� " - ,..-, i a7. + 1)- l 1 i)
<:...> ·2 = l; �i:, a:i. + l :;: 1 · 0 ¢:}2=b tb3 a=Oo
1-
1
---- ·-----_(7 .-,\----.------ 11 - ·-----
.-------------------------. _____________________________ ..: _____ _
C;::; ·-Jto3X=(-Jcvit}z-$ · 5- 3 u+ l
ex== (l2u + 5) r:"\_58u+3 '\_:_} . w3 BX=(? �J(' u ) = (3u + lJ
I i u + l 2u + I e:iz:if.a?rl', CX = JJ3X <=> i2u+ 5 = A(3u + !) 1 �:1 8u+ 3 = 2(2u + i) :. 12u + �5 = .:l(3u+ 1) 0 ·08u + J il(2u + 1) i!>®�eoo 24u 2 +22u+5=24u2 +17u+3 �5u=-2
u=-i0 16 ( 4 ) ���;-.5+3=A �5 +1 :.l=-1 0 ,-....
--------� -- .�--------·-------------------�-------------------------------------------------------12
(f)96
�
c- ;tB = (7· sJ+(2 J) = (9· 6') r;\ h 5 .3 I 1 6 4) \..:_).
6t-'
= O i5>i�� C - AB$ g�@e:l®c.o <:J':znac,c)�.4
e��lll®"� �,zl, c-:W=G �F(� :)=(! :) G)
(: !) (; !)=(� �) eeuZ>@. 0 <::> 9 p + 6r = 1 . . . . . . . . . . . . . . . . . . . (i) .
�
I
9q + 6s = O ..................... (�i) )· · 6 p + 4r = 0 ..................... (iii) ,.
I �1 -•.-•--•�----------------��----------·-•-••-�--------�•••r•-•--�--------- .�--------·-----------·
-------- - --- - -- ------- --- --- -----·· .... -· - ---------- ·-- ---- - ·. ----- ------- --- -- -.---- ---- -- --------1 1 (1 .-" 1 =
,. ') / 1 1 1 . , - .r., l - z l JI Zj -:f:. , t:eic�3 ·- = �--X . = ----;, · 1 -z O - z) �� · j1 -- z) ·
:. Re-= I-Rei= r-Rez · �-('"";;\ 1-: z i,q f 1 - zJ2 It - �· 0 \.:J _ _,
-�---------------------�- .-------------------------------"-------------------------------
@'��ti!' '!!l®�atf z = x + iy l!'er.:i ���- <§'@� x� y e 9l 00�
I (i) cl}-:, 1-z = 1-x-!)• Q Q 15
:. r,-4' ={1-x)'. + y1 = l --2x + ;:1 + y2 = 1 ·-2Rez+jzj'0 �
f------·---···----·---- .----. ______ .. _ f -- ----. ---·---------. -- ·--·-- _------------·-------- .-----·
{ .. ) · 1 l . 1 \1-x)+,y (1-x)+iy (0-·· n. z.:t= • -- =
0x - )- = 5 . · e,�w:> 1-z 1-x7"iy 1-x +iy (1-x)2 + y
2 _
5 . 1?;:._1__ 1-x _ 1-Re.?f'0 - ··n"'1.:..z-(t-x)2+y2 -K�
.0 � . .-�- ------------------------------------. ------�--------·-----------------------�� . -·-----------
13
1 I 1-Rez z -t:. I eo�ao:i, Re- = - <::> 2 =
1-z 2 jI-zl 2
<=> 2(1-Rez)= 1-2-Rez+kf 0 112
�-- . <:=>z =1
�lzj=I 0 -
;
\"))
@
� ---------------, -------------------------------------- ___ - _____________________________ : __ _
. I . n- l OP= 1 W3 00 = r;; sm _;_ =-, :a)�:)- ...;2 4 2· 0
0
0
- . - --- . - - -*---------------------·-· --------.---------------------------.--���-,-.�-��-�--,-.������-
.@�Z)cl we�� ,l
Z ES:::> Z = cos8+ isin O; "'.'." 7r < B < 7r r (;. \ 3 '�
ez +Iinz= � q cosB +sinB = � :::::) cos2 B+ 2sin BcosB +sin2 B =_!_ ,v2 ...;2 · . 2
-------i
14
�sin2B=-_!__ · 0.2 \,.:_)
J! 1! .
1!0 -- < () < - ' o>i�Z'! () == --3 3' · 12
· · 8x 4.(a) x� -1 c�w:> /(x) = (x.+ l)(x2 + �) Q'[ (DiS�..
2 ..
X¢ -1 co� f'(x) = 8(1 - x)(2x + 3� + 3) @€> eotrle)mzl. . (x+t)l(x2·;t- /)2 . .
wi6{(t ���a) to:> �� .��l®m y = /(x) �· e� CiG eoawmd q'i1.fi,). y = f(x) €3 sd::i,ao� mo!J�id (x.� 1)(x2 + 3) = 16.t e:i®�o'� �® m&f!)el) �ts:fm.
(b) qdQ &,c)d r � ¥d qt1w @<D�e �?;,, ·�® � ® e,@,:n c�
�d h � ea9 etd,:n � ia8id�� �=ad ��m "��ad) @Co) e,�zrll:> 80'� �t:06 :e,09mfm �� (Ot�c:i 8�
. . · . ,· .· �-��- r.,oewm � �c o6®:,c) 36n m3·@D. h =
3�Si) �tdfl�.
te):6 �� � �® t38me.:JO':,i:1to(fcia�co �zoo � @aio�e>·d.{8c.,d 300 = 't q� ·(l(I)�&D' ba� �too � �c)O'co�r:Jz,8CJd 1000 tsf 't 0�. · �®. ·too9�1.11 Ddtn() C:l'f�®.e) ��cs:i,·
,�.a �� (03) � l)cti® diS� C comm O < r < 3 °'Cceooc = sooJr (4r2 + 2,?)®@irl ��3 �.it,) @e �ts1�.
C qD® E>� � r � �c:, �mtt,..
8x
(a) f(x) = (x+ ()(x2 +J) ; x * -1 @ee,o,��.
x -:t:.-1, e,'t�
/'Cx)= Cx+1)(x2 +3).s-sx[cx2 :r3)+Cx+l).zx] .. (};\ (x+1)2(.x2 +3)2 �
8(x2 + 3)-16x2 {.x + 1)
. (x+I)2(x2 +3)2 .
. = 8[3�x2 -2x3l ew, . (x+J)2(x 2 +3)r . .::._;
_ 8(1- x)(2x2 + 3.tf 3) ·
- (x+1)2 (x2 +3}2
,,-----,, ......... .
h
15
2 8(2x2 +3x+3) Bco� x e.,<cwa 2x + 3x+ 3 > 0 Q)1_©Z3" S'°'� x ;t:. l e.,�oo:i > 0 eD. (x+ 1)2 (x2 +3 2
-oo<x<-1
I /'(x)£e�� (+) .
! f(x) Ot��e:,.
-l<x<l
(+) f--,.,, f(x) Dt�0!),)
� /,/ l) .)-···/ ·-...,,,-
�e)� , lim f(x) = +oo w:i lim f (x) == -oo ef).. . X-+-1- . l-->-l•
¢@-it3� x = - l _ �20® B6M �C3t5@m,/2tl�S>co @!,. 0 iS3o� �e�@�lzf�a: liq., f(x) = O @!>. :_ y = o �25)@ id6� �:;:.Galz:f�SC:l.@El. 0
y
I
� l<x<oo
(-) .i?,
I (x) �<S-!'. \I
�-' \ 5 f .,__.
-----------------�--------------------�----------------------------------------------·----�----
16
( ) , 1 8x x+ l (x- +3) == 16x<::::> - = . ···
2 (x+l)(x" +:')
---------- . ---- - -------------------- -------------------------------------.---------· ··-----�----�-
�1�, h > 0 => r < 3. e;o>tfJ� . 0'< r < 3 �c.,· Stil"'· 0 $€):), e.'S4rt03 (02.S) !e) �"'�® � . �
C = 300 x2nrh + J 000 x 4m- 2 0 �2oo+fo��,4,')+wJ (D� 800,r{4,' + 2:} ; 0 ::: 3.
-- - -- - -· - - - - - - - - ... - ---- - - - - -- - - ... - - ..... --- --·-·- - ·- - - .... - - .. - - -· · - - - .. -·. -· - .. ·- ·- - .
17
J 3x+ 2 d ---�...i
lS,(a) 2 2 5 x ��m. x + x+ . e"
(b) ecmea.1 � � ��m f cos(lnx)dx = - �(cilt + I) &e eo�mm.t
. a a ·
(c) f /(x) dx = f j(a - x) ch � S�zs!zs,; � a � 6.ses,�. 0 0
If --r. p(x) = (x-n) (1.r + 3iJ c..1t6 � I = J •;:: dx Q{S � «1 ••
0
"""" !ltileec - I = I«:.: dx "" "" -· 0
,r 2
1 e:1era,;, � �G).Q) q�e � to� I = J f _ p{x)
dx @f> � i=>�m.
t!J � l = in .Ill{!) gle) ���-0
(a) j 3x+2 dxx2 +2x+5
= f 3(x+ 1)- I dx �x2 +2x+5 · �
=If -2x+2 dx-f . t dx (;;'\2 x2 + 2x + 5 (x + 1)2 + 4 �
- l_ in( x' + 2x + 5) -!. tru1 t� + 11 + C , @®,ii C "",!l "'31!>$ ol""'"'s\. � 2 ® _ 26 2 J®
_ [x' +2x+5>0a.> "'ll"""'..r"'.] . ..
) ,._!-fJ�;�;':�� . ·,/;, <� {b) I= �cos(lnxJdr. :) ,/v,·· . �--:.�-' { . ��� -�
v t.\D. r·F ·r e• dx @k, � sir' == Jcos(lnx)-· dx 05 · \ 1
. dx . . (9 . -�·�.>
'""
. �
-� . ,\) .,, e' .· . t K' . \ =xcos(l_nx)J.;. + Jxsin(lnx)-dx · ,�--1� . ;) '"' ·. x v
:i·--, _ err
;-,., • '.
�H
dx 6t?n7:\ . • �1<�)'! = e" cos(ln e�)- cos(ln 1) + J sin(lnl.X) dx dx 6 �\��----� �l
® · = eir ?)-; n: -cosO�-x� x;,:� -j xcos(ln x)_!_clx N'-' .. · . x ...__;_,,,/ ---.,.--... • . _.. 1
18
�------ .. , ______ . "" /
""' dx \ .... -· .-- . '·
( c) u = a - x c.eiS "'��. �c:IE> x = a - u w:. - = -J => dx = -du <S-©. x = a �i;)
G. . . �
u=O �-x=O �� u=a 1t 0l). ------.:..-�··_..,
..-,
,· --
f f(x)dx=f f(a-u)(-du) = Jt(a-u)du = ]t(a-x)dx � · ·0
· �s --·--------------------------------------------------------------------"----------------------�-
.·._I= 1 7si�2 x+cos2 x dx- 17�dx �.
2 0 p(x) 2 0 p(x) · \.::.J
I { re .} I=--- _In--ln2n 611· 2 .
=�In 2 =-1 ln-[ 7Z'J 6,r· . 2ir 6,r l4J
19
16. 11
wo Li. Qf[} �� 2x+ y= S !lo:> x+ 2y=4 ®@trl �� e@m e,d'e � c.,1S a>:rl�. 11 too '2 q:rno
9� Gt!:006C3 tan-I(!) �f) etoZ'IDl, e®® 0�,Secl to��� eo�CM!IDQ �zrm.
/1 eoo 12 la �· c=fe;i:,c., A mrS � R = { ( x.y): x + 2y s 4 t00 2x + y iit 5} '-'i.S � IDz3�. A ed•»ed
@-eltc),o� @'CO:>COJ. R � xy- � � mxSmm.
11
w::i .l.i. � � ® ��CD m41®m R ��� Sa,om 4� J's. :I � s � =�Q r+f-14x+sy+oo=o@O �•-
� o:i,:,a,,.e,� es�� tti3&1@� A C-ll� Sos��� �ca� dotsm 6�
co@m� x-y = 10 @!> (:IO�zft$.).
A ea:.oa, � 11 w:i 12
�®m s s; -c.1� et:ll•lS 'i @deciJ CO:,) � eo®Ddeu� �zr�:
----------. ------·--�----------------------------------------------
j2x+ y-5j _ lx+2y,rAI �
Js - Fsv
i.e. 2x+ y-5 =±(x+2y-4)@
- x + y + 1 = 0 or 3x + 3 y - 9 = 0
----�-�---�-- ----·----·-----.---�----------�-------------------------------------�-----------------
20
2x + y = 5 ?DJ x + 2y = 4 W®«b®D ©��@®�, X = 2 t:O:> y = 1 eie-S. :. A$(2, I ). ®
··Y
2x+yL
. . --------- .-------�------------------·-----------------------------------------------------
S lS 021i:t��c., x + y - 3 = 0 ®m-883�"' 9Zi1"'· e:iVc:>. Sa, e-��S"' (2+t,1-t)_q:,�d(:)� 8!)a, w1im.
Slil 11oro . ..[fa,�. 2(2+1)+(1-t)�s =..fs. 0 {H,
l' .= ... t=,+�® � C
0
�{7,-4)�<3-to! (-3,6); <!'�Di� �Zl'flll:l!li:;, R �e <3-3'�885€)8. 0
CS) sh.,� '6-elo"'
(�:� 1)+:+(y+4)2 ·=5 ··-:·.1·
.i�<.:...14x+49 +y� +:8y.+Y6-;� s
·.x�:+y? ·�-i4x+8J,+:6.0'·:;.;.0 \�___________________________ . ___ : ------·------------------------------------ ·-----------
21
Cj'f)Z) 153' !1®"'2:rl' '
c "(t',3- t') 0 · S 63 qoca .Js �t�Z'l,IZt' +(3-t' -5) r; G)o
r::: = -v 5v5Ir' -21 = s
t'=7 ort'=-30
c � (7, - 4) .,,o,1 (-J, 6); .. ��,ii e..r"""' R zte .. ,,,.s.ici s. 8 · S £ eo®wo-eico:(x-7)'+(y+4)' =5 G x
1 -14x+49+ y2 +8y+16=5 · . .r' + y' -14.r+By+ 60 =0G
Xo = 2,Yo = 1,g =-7,/ = 4,c = 60 =®<D XoX+ YoY+ g(x+�o) + f(y+ Yo)+c = 0®a3z.f 2x+ y - ?(x + 2) + 4(y + 1) + 60 = 0 "e1.e-�.
1.e. -5x+5y = -5.0 G05 x-y=10------------------. . ------ .. _...,. ______________________ �------------------- ....
�0©21 Oad?S>w-(.'.Js3 co®�,f�ca, · j�x2
+ y2 -14x+8y+60+2(x ..;. y-10)=0 �
i'pZS)lOc..50 8�ai Wt2S)·A s {2, I)@®® DaZi'?i><:-' ®15> �t,.�\:4 +l-2�+.8+66+ l(x- y-10) = 0
1ftt=0 ®· .·
J;·-��3 <fDS)i'�hf�(O.; x2 + v2 -9x + 3y + 10 = 0
22
7r Jr f I - tanx 17.(a) -2
< x < 2
eo�a:io (x) =1 + t
anZ xG'lts <n��. /(x) cotn:s"J Acos(2t + a)+ B cpz:»oOQC) g,.m:irsi
zsidv!2:n; e®@A(>O), B �o a (o <a< f) z3'1Ec., � 9�-&m �.
d �. /(x) = 2 +4..fi. C,tl) (0/!lts,�C.:, &�mm.
f(x} �W:> ·�z:, C(i §e 9��mc.:, �o CD�/ihrl /(x) = 2 \ t �15.fm 2 tan2x + 4ktanx- k2 = 0
�JdQc) B�c rl'.>l� @f> @azro�; e®2 k�2-../2. �.
tan�= ""6-../3+./2-2 @E) � �mm.
?:i)E} 'i -1 < x < ; U)�w:, y = 2f (x) 63 �:>d'�0 ti� e.,c;>a,m� q��m:
(b) es� qotSl�0QtS"t, ��d �a):> mt!datim �(g �z:f.w.
ABC c=� @�� '°IS ID�. �, (Fots>�. a:b:c= 1: ..il: µ @D � (t(m; �i, � eo:i µ. �
�QQ) e-f), µ2(sin2A+szn2B+sin2C)=4l. sin3C 6>� ee,�::r.fui.
, 1-tanx 1r. tr(a) f�x)= 2
;--<x<-.l+tan x 2 2
= cos2 x(t -sinx
)= = cos 2 x --sin xcos x (s'\ cosx \....:__) ® = I+c
;s2x _ sin:x ®
= .! { cos 2x - sin 2x}+ .! 2 2
I r;:{1 I. } 1=
2 ·v2 .fi.cos2x- ./ism2x +
2 ·l·f TC · - . 1r . · .} 1= ..Jilcos4-cosLx-sm 4 sm2.x + 2
= }ico{2x+;)+i®
I 1 1r(�-� A=-. B=- a=- 05 ,JSY 0 ____ .J}' '-�cT-� .fP!
,f{J ---· - .------------------ ---·--·-- . _________________ !_�---·. .. . . [?) .
f(x) = 2 +Ji
.-l
(2 1r). I 2+..fi. -cos x+- +-.=--
..fi. ·, 4 d-®. 4 · · • 05.
( . :Jr) 1 1l cos 2x+- =-=cos. 4 2 3
2x+�=2mr±n
;nel ·Gos .. 4 3 . ,· · 1 .
. .
r �n ·
� CfV> U U·· JJ\,:.,,,.,"p. 23
1C '/( 2x=2mr±---3 4 1l . 7,r 2x = 2nv: + - @�! 2x = 2mi --12 )2
fr . 71t
Gs x = mr+-@eMx=mr--. 24 24 .
x = ;4, - :: (·: -; < x < ; )G
1-tanx 2+.fi.l+tan2 ; = -4-4 - 4 tan x = 2 + Ji+ (2 + ii) tan 2
x
4-�+"2)-4tanx= (2+ "2)tan'x ............ (i) G {i)x(2-.J2) ::::>
(2-Ji )2 -4{2-Ji)tan x = (:'-2 - (Ji)2)tan2 x 2tan 2 x+4{2-fi.)tanx-(2-(.fi))2 ::::0 �J
'-E) r.:-1 2tan 2 x+4ktanx-k2 =0, e®t8, k =(2- '.;.1
, 05 ·��-----�-�--�-·--------------· -�- ·------------------ -�-------·------��---��-----··--------
84H �I 6k' + 8k' - 2k± Ji,ktanx= -- ·-· --. 4 2 Gtan!!__ = -(2 - .fi.) + ./6 (2 -..fi. ); (·: tan!!__> o) {o024 2 24 �Vo
- 16-..fi+"2-2G0
--�--------------�--M·-----------------·------------�----·------
y = 2/{x)
= cos 2x+- +l·--<x<-Ii ( . 1{) 1C 1( 4 ' 2 2
----------·----�------------·
24
/--.... \�) �31'� Grek)j:}3£bG'!>3"J
• .
.. ,_._� .. �---.. ,.,/S .: .. ,,�---\'\\,,.l.. .... "1\'-,·� - -....:......_� ,.,_..,,i[
(b) n • __, sinA sinB sinC ®5 ="'v!' se�c.e=: -- = -- = --. · a b c ·
. .
� 0) sin 2A + sin 2B + sin 2C = 2sin{A + B}cos(A-B} + 2sin C cos C
= 2sinCcos(A-B)-2sinCcos(A+B) G == 2sin c[c9s(A- B)- cos(A + B)j
(��� = 4sin C sin A sin B \.:.J
4. CsinA sinB b ® =· sm --.--.a 05 a b
. = 4sin C sinC
. sin C .o!J
c c
� . .05 => c.2 {sin 2A + sin 2B + sin 2C) = 4ab sin 3 C \Q (
. " ,--....._ µ2a 2 (sin2A+sin2B+sin2C)=4a;tasin 3 C t:a:b:c=.:l :;L:p) ( 05)\ /.... _
µ2
(sin 2.A + sin 2B + sin 2C) = 4..:i sin 3
C
25
