q.GG).� (e,.Gce) ciIDJGJ'3 - 2019

10 - CJo�dm �ma> I

( m0 �B®f!�a>)

A 6C>:>0f5: 10 x 25 · - 250

B 6C>J0W : 05 x 150 - 750

. .,,

1. -� � -� ��- .s�g nez+

·t.,.� ·i(ir-l), =·rl@f) �c:) �omm.

J'crl

n = 1 e.5�!:!X>, L.H.S. = 2 x 1-1 = 1 ro:> R.H.S. = 12 = 1. 0

:. g.636cJc.:i n = 1 e:i�ro:> e:lt5).l5 @eJ.

�ro,® p E z+ ©CD25.> g£36ec.:, 11 = p e:l��):) e:ll:5)1) c.:,z8 Ctll:i)(302S:>C.:, tllOIDZSJ.

p

625.>® L (2r - 1') = p2. 's' r-I �

p+l

�;sS L (2r - 1)r- I

=±(2r-1)+(2(p+l)-I) 0 r•I

= p2 + (2p + 1)

2 =(p+l). 0

t!J m8zrl, n = p, e:i�o.1:> g.636cc.:i �mn m® n = p + l o.-;!:lx> � 9£18cc.:i coz,,u 0/J .. :tr= 1, r.o�roJ g.636e;c:i

e:lt:ll:l:l @el <glo.lZ5) ©025)e)J q1_1::D. 6@ &,le,:, m&m qro;grom @ewb@c.'.l @li:32:5:1' 8c.'.l�@ 11 E z· 'C:l�<.})J g.636c;c.'J

e.5.t:lllS ©!), 0

2� �,-® 010 �my=l4x, ... 3:J ·�· ,��rZlx:J ea g�� � �ts.t �m�

d� � ·c)d.· �.-,, ]'2x-.3f:+lxl<J; ·�� �· x la ,B� ® �a, gmc,trl �z,). .

.

y .. 4x-3

y=2x+3

1 ·J

@@@ gdt5)J6®"'mto 01!�� ccl@0c�

4x :- 3 = 3 -2x => X = 1 (v - 4x + 3 = 3 + 2x => X = 0

I 4x '-3 I < 3 -2 Ix I

:. l4x-31+12xl<3

¢> 0 < X < 1 &t),

-:=> O<x<l

I 2x -3 I + lxl < 3 -::>,O<x<2. 0

X

. ·. · I 2x - 3 I + I xi < 3 qe.:i@:)2:5)t:i)Je)C, madz:i:> %:l)0m 8c.,� X qroom&ro �ez:i)co

{x : 0 < X < 2} @El. 0

I '", .:·· ,·.' •

Cii'E>mm E)®'-'t:sf

<yto.tD om: gd.tD:>d e:i�ro:> 0 + 0. x 63 qmc.:i2S) e:i�to:> e-Elmcl �®c.:icl

12x-31 +Ix!< 3

:x s O:

6eJc) I 2x - 3 I + I XI < 3

0 <x s 1

ae:io I 2x - 3 I + I x I < 3

(iii) qvd'6:iE) X > l_

6�0 12x-31 + lxl < 3

2

¢>-2x+3-x<3

<:;> 3x > 0

¢;> x>O

<*-2x+3+x<3 ¢;> X > 0

<* 2x-3+x<3

¢;> 3x < 6¢> X < 2

6m8ID, �®® q0d'6:>®�� qo®:imtil:>0 t:1ladtil i:i>dm x $ qm,.n,1 1 < x < 2 ©�.

q0d'6:> 3 ® £:\E}i:� ®ci�® e:i@l:l)E)

@)mi® q0d6:> 2 cl 25,E)l� eJe:l�®

e:)$%:i)E)

@ 0

.). q;.?ttidf;, ��� • .Arg(;,t..,.z, .... zt)= ... ¥ ,to�!$) z�ab:,iio., �m �- -��- o=sr.,.:.oac.;

� •� �� q�trltn. . I

• /

a�-��- Atg(1"�2,..'Zl:)=·-, ��·-�· ,./i.;·+1f&3qe® q���-;

0M

p -i

u"z+11 = 1 i(z-i)I ==lz -i 1=1 z +ii

= lz + i I

==lz-C-i)I 0 t!J 25)825), Ii z+ 11 63 q0@ Cf<DG:l PM ee). 0 <'il,25:i, PM = l . sin 1! = 1

04 {2. 5

2+2i

0

'·

7 7 7-r

. {x3

+ �2 ) L7

(x3)' (-\-) =

C ,-o r X 0 7

� 7

Sr-14

C X

,-o

6 5r- l4 = 6 r = 4. 0X <=>

6 C= 35 0: • X 63 Qo(Q-@lnz:;,c.:, =

4

iyrotil 9ao6-@lncc) x, 0(5� d'0:1c.:icl'%5.l O'tc.:izsi £)�® e:i�e0:i Sr - 14 = 0 5o 9.t,Jc::i. 0

r E_ Z+

&-r5� e@c.:i Bf15c emoe01t:i1. 0

�· . .... .

5.-_ ]' .J± .. _;'2-l .. .L:�� :Jc--P-, JID. . . -- -- ---- - - :::: - �- <IN::).:::,;x;.,,:iJ;Q.J, -�� sfo{j[{x-a» 2;,r ,. - - ·

lim x-+-3

-Vx-2 -1

sin (n(x - 3)) =

=

=

=

lim x-+-3

Jim x-3�0

lirn x-3�0

1

0-'

1

2rc

.../x-2 -1

sin (n(x - 3))

x-3

sin (n(x - 3))

1

sin (n(x - 3)) n(x - 3)

1

:rt 2

0

(�+ 1)

(-Vx-2+ l) 0

lim 1

0x-+3 5. (�+ 1)

_L : _L n: 2

r 0

.. .<,�,.

6. y s;Jjl£ , ,.; "'o, p I mo , .;.,O'flla � <jde.;,; t>,,, � x-.• � •!:!ii. � .Zli �··�� . .eoo'\'J eS)f!. '.!Yl� � ·oos•·�·p��. i{#'.+liJ4J � ��. ·

y

0

I '

I - \ I I \ \·

I I \ ,

I I t I

I I I I

I I t I

I I I I

I I I I

I 1 I I

I t l I

I I I I

I I I I

\ I I I \ I I I I I I I \ - I ,_,

=

=

=

=

1 X

-x2

-:-1- dx + J )

��B.·

.,,. ,_....;--------------=---------------------:, 1. C�t€R�-��·ai2 �y,=:;2,Qt·�--��.-�@m�<oi:e:�·�a¥0�.­

Q� (ar·,2:ai) ;�-� �-� -� � y +.fx:;iqt+ a� ®@d ��eamoo�

,C� ·$i:,) ,,,=,{4.a�4a)��-, fJ �� � �. �· ��· Q ::(4T�,2aT) ��: �-� �- T.·=--300 ��. .

d.x = 2at, dy = 2a dt dt

dt ·-

y - 2at = -t (x - at2) e-e).

1 = 2a.-2at

= 1

t

y + tx = 2at + ar 0 (@®� t = 0 e::i�ro:> 0GoCQ @!).)

Pa (4a,4a) on C � t =2.

0

P 63� lfe'3e®ai e-�iSJ:>0 : y + 2x = 4a + 8a = 12a 0 2

e)C,:} C 63� (aT , 2a1) w� 0� &iehrl

2aT + 2aT 2= l2a. 0

2

<=;, T + T - 6 = 0 ¢;, (T- 2)(T + 3) .= 0

<::. T = 2 @a:ii T = -3.. "\

T=-3 0

Si (1·coo !z-�- ·ege:E>•� -� + 1 -� 4 ioo 4x + 3y. :.:lo �m: -�� e� �e � c.ois CD��:·

_ P� {J ��' �» :�z;:s,. ,lt � ®ll> �-:qt�� ¢t:,J'C� ��� � li��,n.B cf>@: .g.S ��� 1 m{ � �. � P $i � (J .&l ��= emx.otrlt».

/2

: 4x + 3y = 10

/1 ®m ®7j)1@ edll!lJ5'5d

(t, 4 - t) qaz;s,:i6e-c� (3elc., co1t:i>; ©®63 t ER. G)-1 ),

14,1

+ 3 (4 - t1 ) - 10 I -= I \.

!·

lt1 + 21 = s 0 t1 = -7 G'Wi t1 = 3 0

(-7, 11) 00� (3, 1) (3'E), 0 + 0

)\_ .. '"'k, j ... ·> . ,:X.-: ·j 'i,. :·

I I

9 .. Aali(""·'Zt�) �� S.':$'.�+ f.;.�+·6y-1�:*i(t�� �-8� oo, c,o�.

$::; 0�. em �A· �j;c®)�� �.�• Q)��� ��.

S = 0 @ @d2rl�C.'.l C @d2rl�C.'.l (2, -3) @eJ. 0

S= 0 63 R qcrx:,../4 +9 +12 = ../zs = ��. 0

I. ·;-

.. ' . �.

A= (-7, 9) S = 0 0arn!5)c.'.lo I q:ie:)2rl�!5)@ A G�©Jllc.'.l/\ 10

. �: ;: . ·. '. • .l • .. ·•· •• ( •

CP : PA = 5: 10

= 1:2 ·-0 ... ·,.

p "" ( 2 X 2 +/ (-7) ,

6m@ P= (-1, 1) 0

.-.. • . .' �·, .. :;

I .

- 13 -

cos 0 =

=

=

0 =

cos2 0

• 2 0

0- -sm -2 2

0• 2 0

tan2 .!!... cos2- -sm - 1-2 2 2

0 • 2 0 0 cos2 - + Sill - 1 + tan2 -

2 2 2

I-l2 I + t

E 1-/

c.:,1,8 cnz3�. ��O 2 1 + t

==> F c1 +l) = 2(1- l)

(2+ [3)/ = 2-F

2 (2-V3) t

{2+/3)

==> t lf = tan- =12

0 ;,, (2n + 1) lf t'..'.l�roJ

0

0

2-/3 0 I • . ' :\.:

( ·: _tan x > 0) , 12 .

.. ..

11. (a) pER to:i �<ps; 1 c,tf3 m�. p2,il + 2x+ p = 0 chd6i�. 1 �C.'lcl � 00 <tdclt;>mm.a too Pa)� <.t®® �t:nd-4G� �c 0tS ©63�. a ci,:, /J �r::l)® ZSX>�l-1) oo �zrlm. p q-cser6d a + fJ 'P:> af3 8c.,:i <i� l 1

_ P

2

(a -'- 1) • {/J-1) - p2+p+ 2

�� �tdE)clz.n. . .

a fJ a_ 1 wo p _ 1 �e E>m Vt5md e,®�6-f.!"iir.:i (p2 + p + 2) x2-2 (p + 1 )x + p = 0 ®@trl �� eats) oo.cl,Clt®® @� �tn® Q2:n f),m @D.me,emt:>trlt».

( b) c i:o:, d �22l � .i:oo.cl� coa@;o:i e<jz:s,.cl c.,1S � /(x) =; :,.3 + 2x2 - dx + cd "'1,8 'i w�. (x-c) cmm/(x) @ coo{.)f5)01.m ��. (x - d) ®@trl f(.x) ea>� � �c.') cd @e)rn � qirn. c ro:i d t3 qcnc.,� �:ic.,2:rlm.C .w:l d 63 "'®® q<D�� �to:>, (.t+ 2)2 ®@m /(X) � &) �cg� c,co:i�mm.

(a) p2 x2 + 2x + p = 0 63 1 @ec.,zs$ c.,1.8 8%5)�.

x = I, p2 +2+p = Oei®°'· 0 25)§%5) p > 0 � l + 2 + p > 0, @ieJm e@c., 8� �c., @25):>tl:>ttsl, 0

2 2 :. p X + 2x + p = 0 @ 1 @ec.,zs$ @25:l:>@e:l.

® = 4(1-i)

> 0 (·: 0 <p s 1) 0 :. a ro3 /J @�tsi@ %5)3clcle32Sl e�. 0

a+ /J 2

ro:> a/J =

1 0+ 0 - --2

p p

'cim,

1 1 (a - 1) (/J - 1)

=

1 (a/J - (a+/J) + 1)

_1 + � +1p p

p2 +p+2

0

0

�ja;· =======================================

a /J + a-1 /3-1

= a(/3 - l) + /J(a-1)(a-1) (/J- 1)

= 2a{J -(a+ fJ) (a-1) (/3- 1) 0

p2+p+2 0

a

a -1

x2 _ 2(p+ 1)2 p+p+2

= 2(p + 1)

p2

· 2(p+ 1)= 2 p + p+ 2

=

p22 p+p+2

0 a{J /J

/3-1 (a-1) (/J- I)

=

=

p 1

p p+p+2

p p +p+2 0

X + pp2+p+2

0 @EJ.@

(p2+p+2)i-2(p+l)x+p= 0 0·

-- 00:>

(a-1)

a

(a-1) + _fl_

(/3 -1)2(p+ 1)

p + p+ 2> 0, (·: p > 0 ), -•: . 0: -,

a --.

(a-1) fl_=

(/J -1)

p 2 p + p + 2

> 0,i

(·: p > 0 )<

t!J m8ID ©®® �e @�t,)@ w� @E,. 0 ------------------------------- ·----------------------------------------

..... , .. -=====================================

(b) f{x) = x3 + 2i - dx + cd

(x- c) e,:,wt:i)c.,cl 0l1e3zs5 f (c) = 0 e�. 0 => c

3 + 2c2

- de + cd = 0 0 => c2 (c + 2) :::; 0

=> C = -2 ( ·: C ;t 0) 0

/{x) c.,�m (x - d) @@zs5 ®@� �D ecdiac eel_ @i�zs5

j{d) = ed. 0 =;, i + 2tf- cf+cd =cd 0 => i + cl- =O

=> J (d + 1) = 0

=> d = -1 ( ·: d � 0) 0 C = -2 a,:, d = -1.

/(x) = / + 2i + X + 2.

6e3c) /{x) a (x + 2)2 Q(x) + (A.r + B); e-®8 Q(x) �:,�c., 1 � @� c,c:;e5£1.

c>Ql1e3�. x3

+ 2x2

+ x + 2 E (x +2)2 Q(x) +Ax+ Be!). 0

qEl2:,°)G25)C!l §13e-®�

3x2 + 4x + 1 = (� + 2)2 Q 1 (x) + 2Q(x) (x + 2) + A ��. 0 m10tD X = -2 q:ie<a;i®Ozrl

12-8 + 1 =A eiea>, 0A =5oo:iB =10

6 2')8�. e<JJ'a11c., 5x + 10. 0 ------------------------------------------�---------------�-------------

---------------------------------------------Ii··,·-

X - 2x2 + 4x + 4 i + 2i + X + 2

x3 + 4x2 + 4x

-2i-3x + 2-2x

2 - 8x - 8

5x + 10. @

x3 + 2.x2 + ·x + 2 - {.i2: ·+4x + 4) (x -2) + (5x + 10)

I

qE)<3;1:15 ®Cdr1Jc.:i 5x + l O eel. . ® ---------------------------·---------------------------------------------

--·======================:;;;;;;===================-t ----

. 18

12. (�) P1ro:>P2�8g(i,E)gd {.A,B,C,D,E,1,2,3,4} too {F, G, H , l, J, 5,_6, 7,8} ®Szrl�l23e@�re·ez:l>�z:!) c.:i18 co.63§. p 1 u P

,. .m IDll3 �@2:l) <!'C)�m qt;lOz 3 am �>:> CJ'Dtnd e:locl:.00ol5> 3 Btrl ed, (ft)(::.t)

6 £2:rl o®� �6ot;c.:i.d e11�®c) qDcimt> tfIIS>. e,eots, �d �d q�6:i@D � ei�c., to1B C,�� �t)mtJ!.. ' §6e<; · cD�zn ��dm:(i) qEx..,D 6 ® p

l zrl o®� ® <!'m10:> wm ei@@,

(ii) qooD 3 r:n P1 m � P2 m q�m.cl qDa:u 3 't ��&d:i m� eieai.

Z+ 1 · · 1 (h) rE eo'ieo3 U = --.,....,---c---- t0:> V: = __,...___,,,____,,.,.. c:18 mm�.r r(r + l)(r + 3)(r + 4) r r(r + l)(r + 2)

rEZ+ c:)"'5); v,-V:.+2

= 6UT

00 �dt.)�:;,,.'

� {, 5 .(2n + 5) d �. n E z eo'1to� LJ Ur = 144 - 6(n + l)(n + 2)(n + 3)(n + 4) �D ��mw.r=l

n

z+ · �w _ 5 (4n+5) nE e,�:o:i L, r - 144 - 24(n+l)(n+2)(2n+1)(2n+3) OO � 2:S)(5zrlm.

· r:.:l00

c5 �. :}: W,. qo6�Q ©g.8iio tfOO:>O' 00 �-� � �z:!)ZJc:i @cx.,n-5tn. r=l

(a) P1 = {A, B, C, D, E, 1, 2, 3, 4} 1m P2 = {F, G, H, /, J, 5, 6, 7, 8} .

(i).

5 4 qclce6 cn�m = C . C 3 3 ·

d 2518zrl, qEJ<.:Jt) 6 @' P, @(!)25) 01�'-' ro1B �6 O<i ro�m = 5CJ

. 4C3

' 6 ! 0

� 28800 © 0 ·- - - - - - - - - - - - - - -�---- - .... -·-·--- ·- .. ""'!:6. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

(ii) G-z:n5Q i:f.ll £ <P'E>z:ncd' q:>t:i:io

P, � pl 25)

qclllllO �o@:i:llotll qclllllO �o@:l5Jots)

' 5 - - 3

s 2 1 1 2

s

1' 2 i l

4

- 3 3 -

§0 O(i (.0-Eiu�

C· 3 C3

• 6 , = 288004

C· 2

C· l

4

C· l

C· 2

s

s

s

C · l

C · 6 ! = 864000 2

C · C · 6 ! = 8640002 l

C · ,C · 6 ! = 28800 ] 3·

.

@ @

@

@

ee)ZSJd �d' o� a:,6:zs, = 28800 + 864000 + 864000 .+ 28800

. ·. �

= 1785600

.@ ------------------------------------------------------------------------

l (b) U

r

= -----­

r(r+l) (r+3)(r+4)

V, - V,q = r(r + l) (r +2)

= I · r.EZ+ ·

r(r+I) (r+2) '

(r +2) (r + 3)(r + 4)

(r+3)(r+4) -r(r+l) r(r+l)(r+2) (r+3)(r+4).

6 (r + 2) r(r+l)(r+2) (r+3)(r+4)

0

= 6U, 0 ------------------•-•••r-••--·----·----------·---------------�--

. --- ----

6uD. .

r = 1; 6 u, .. = V1 - f,{',

@r =2; 6 V2 = V2 ;- Tf.;, , I ,'

r =3; 6 U3 = y;-Vi,

' r =4; 6 u4 == ,r:-r6,

•

... ..

' r = .n-3; 6 Un-3 = Vn.; - ,r:. I

,'

r = n-2; 6 Un-2 = vn-2 - �; @' ' '

n- 1; 6 Un-I w Vn +I r = = "�-1 -

' r = n; 6 U,.

=

f/- vn +2

- 20 -

n

@6 Lu, = Vi + V2 - Vn+I - Vn+2 r = I

I 1 1

0=

6+

24 (n-t-l)(n+2) (n+3) (n +2)(n +3) (n + 4)

5 2n + 5 24 (n+1)(n+2) (n+3) (n+4)

n 5 0 LU,

2n+ 5

0=

144 -

r = I 6(n+J)(n+2) (n+3) (n+4)

------------------------------- . -------------------------- -----

n

.;> L w, -r= I

n

L w,r = I

limn-..oo

n

: L (U2,-1 + U2)r = I

2n

'':. 1 U, Is'_·r=1 \V

5 4n+ 5 =

144 6(2n + l) (2n +2) (2n + 3) (2n + 4)

5 4n + 5 =

144 -

24(n+1)(n+2)•(2n+ I) (2n+3) 0

n 1· ( 5 ___ 4_n_+_5 _____ )� W, = n!!oo -144 - 24 (n +1) (n +2) (2n + 1) (2n + 3) r = l

lirn 4n + 5

144 n�oo 24(n+l)(n+2) (2n+ l) (2n+3)

0

13.(a) A= (: �I �

I ) , B = ( : � ! ) .,, C = ( !I b-:, ) '"'il ABT = C0m o� � m�= ..,S

ID.53�; e®@ a, b E IR��. a= 2 in:i b. = l sm @omDrom. lJ)[) � c-1 �� 00 002"0znm.

P = ½ (C-21) c:i1El m253�. P-1 (3w:i -i�:i. 2P(Q+3I) = P - I Dm o� Q m�:100 atc:o:i��:;,;,. $®@ I c:,m <D� 2 E>m a°ttitD m:2X>QQ cit� •.

· (b) z,zl 'z2EG:�il3 ro� •

. (i) Re.z�lzl, ioo

( .. ) ... ••. i0 I z, / I Zi I u z1 ... e�a:Y.> ,--=- = -

. . z2 Jz2 I Q)E) cfb�filID.

Z1 + Zz 'I- 0 O"c�:l R_c(·� .Z-1 ) s Jz1 I @€) � Z:iXJ2'5m.·-_._ '-i +zz lz1 +z-zl

z1 + z2 ;t O �'lt0:i Re I� z1 ) + Re ( Zz ) = 1 oo e:im.ri:1025)0 rsio,

� "'I + Zz Z1 + Li

ZpZzE(�� lz1 +z2I :S IZil+tzzl S)E) <M,�wm.

_ .. {�) w = ½(1-JJ;) �t8 "'�-(��t'\,

(a)

l+w �ts) r(cos8+isin0)q�:>d�m 9� Z'.1'��; e@a, r(> 0) '11:l o(-; <8< ;) om��

z:D@ -� &m e!>.

't �Doe)� 9<it�QQ �.tS>�C-'tl), ( 1 + ro )10 + (1 + wi o = 243 @f) ,.�cnlillmm;

ABf. = (�

ABT= C

0 -1

¢>

1)(2 l) (2a ...:3 a:4)0 ; ·-: -1

0

(2:/. a:l ( b

a -1 -2

) b + I

@

.;:> 2a - 3 = b, a - 4 = -2 c:ii:o a = b + I. @

G)

c = ( 1 -2) -1 2

1 1 -2 ,-1 2 = O 0

:. c-1 @zn:>0£3. 0

p, q, r, s E R 0w 06� oi0BC!) S�'-'·

( !1 -; J (; � l - ( � n 0 . ... ...

� p - 2r = 1, -p + 2r = 0, q - 2s = 0 z:o:> �q + 2s = 1

:. c-1

@Z5):>0el£3. 0 -------------�-------------- ---------------------------------------�---

P - ; (C- 21)- ; {( !1 -; ) - ( � �)} - � ( =: -i) 0

=Y· -2c�i(� -n - P -n,® , ... ,, ..... .

V ,_.:·.

2P (Q + 31) = P - I

(;'\ .. , ¢> 2 (Q + 31) = I - p-' \:_.I ·, ... 2(Q+3[) - (: n 0.i

=Q - 1 (: t)- 31

'.'-

----------------------------------------------------------------------. -

(i) z = X + iy, X, y ER c:l18 (!)��-

. Rez = x :s; .../ x2 + y2 =lzl ©

[cos ce, - 02) + i since, - 8:z)]

l

- - - - - - - - - - - - - - ... - - - - - - - -·- .. - - - - - - - - - - - - - - - - ·- - - - - - - - - - - - - - - - - - - - - - -·- - - - - - - -

0

+

) �

f (i) @©Z5S

= 1

= 1

. ( Z1 ) ( Z2 ) l Re z + z + Re z + z

=

I 2 I 2

0 (ii)@0Z5S

0

0

--================---==================== 10 - �a, a>&= - 1 c�� �ee �) 2n0 �= 1 q.C1>01.e.Cc..os> - 20191 qDe:>m -�r:n en� e>e iaqif.l lfla>. _ 24 _

(c) co = J_ (1 - .ff i)2

=

121 I+ 1 22 I

I 21 + 22 I

( ·: I z1 + z

2 I > 0 )

1 + uJ = -If [ � + i (-_; )] - r ( cos 8 + i sin 8),

(i) © ®@m

0

0

. <; §DJ0� ge@c.:iQ @@� (1 + w)10 = (./3)1° [ cos (108) +{sin (108)] 0

l + w = l + co = ../3" (cos 0 - i sin 0) =v'T [cos (-0) + i sin (-0)]

::::, (1 + w)10 ::;; (v13)1° [ (cos (-108) + i sin (-lO{i)l 0 (_m,o

.'. (1 + W)lO + (1 +w)10 = v � / X 2 COS (108) 0

= 35 X 2 X -2

= 243. 0

. 2 3 9(x -4x-1) 1�;{�) xi,1 �«xi J (x) = 3 �s mv3@. ,,:,:,_ f. �. (x-3)

(a)

x;t3 �f(x)�V� f'(x) QIDV) j'(x)=- 9(x+3)(.x/S)®tom�c@z,,0000Zrle)trn:i).. �-� .

.

r.do�@. y-�� 111::1 '°!Or® �.i, .:;de)littf. y = f (x)@ seo� �-��� q�m�. , ,, 18(x2 -33) · ·--., ..t ¢ 3 � / ( X) ::;: 5 W ( c,tm. y ;::; f ( X) 63 gc&ooo'� m&.iesO'>zn e•l:lt>e X -@��z:D

(x-3) · · .

(b) @� c.1to0<:>W Ott)� c@tl) r:es e),dfil:Jl:D:)� <Btnt=jlif2rlm�ai::n q:>1:s1�2rl � �1.m �w08. 0Qlc:)@@@qie, �ID 30 CIIl l!if 'l <:,film �aOO�d (i� qd'c.,o�@ qdc:> 13®� 13'���,; @D. a�d qoo remCl{@ (I)�.

�c:,\!'® 06®::ie:> V cm3 CDww O:< r<30 ��to:> .

'

V::::: '7srr?..J900- r2 ®&Jtrl 0"1� �@tn @e) ©C.,cle)zmn.�ep-.@ o6®:iE> co5® Elm � r 81 qcw, �c:i.trltn. x i- 3 o�ro::i; j(x) = 9 (x2-4x -1)

(x - 3)3

f'(x) = 9 [ 1 3 (2.x - 4) _ 3 (x2 - 4x - 1) ]· 'zo' (x - 3)

(x _ 3 )4 � = 9 [ 2x2 - IOx + 12 - 3(x2 - 4x - 1)]

(x - 3)4.

= -9 (x+3) (x-5)

(x - 3)4

for X / 3 ©

616d cloe5®aa:fm®ol: limf(x) = 0 :. y = 0. ©x�±oo

}�3-f(x) = oo eoJ · }�3+ f(x) = - oo. Socl c.&0�1:to;i:fm§@ : x = 3. © a:>tdi® ez:»1111:l:l to�j1

(x) = 0. ¢:>. X =; -3 tl:JJ X = 5. ©

IO - � � - I (� � .. �) mf.) �Ille I q.00:u:o.(c�) - 20191 qt>eim -�m qt�� a� qtm. - 26•'

·/ f (x) 53 Gti1.@lii

:,·

:,.-: :,. . .

,'·, .' · · f(x) is

-oo <x <-3

(-)

0 \

-3 <x< 3

(+)

0 /

3<x<5

(+)

0 /

eo1.01.® er:rl'�:l5 ®�ts>r:rl' q1.%5): (-3, - �) d6:1.i>o qEJ®o7" (s,

© I i

y . ' I .

I

., .

I

( 5, �)

x = 3

5 <x <oo

(-)

0 \

X

-------- ·---------------------------------------------------------------

X -:j::. 3 O�tl:>l;

!'1

(x) = 18 (x- -y33 )(x + v'TI)

(x - 3)5

/11(x) =.O <=> x = ± v33. 0

-oo<x<-./33 -v'33<x<3

J\x) 53 Cti1-Elii (-) (+)

q0ts)el5:):i0c:, c:,B qe:itS)e cf} q0%5)e

:. 25.>63 E>e5rom etsf�:i5 �'ct:i)cl q-iz:n. :

3<x< .Jii

(-)

111cJ q015:le

X = .:... .fjj to:> X = '\f'33 c5@ 25.>ti)E)e5%5)25.l X- Q)�e)Jot:i) @E). . 0

,133 <x < oo ® (+)

cf) q0%5)e

(b)

. ' \ :

·\ h l \ : � \ / � . .

\ / \/

0 < r < 30 e:ili'cro:> ;

h = -v' 900 - r2 ©

1 1 '

V =

3 n (2r)2 x 2h -

3 nr2 h @@m 0�� eiG'Q). ·©

=; nrh

= ; Jt r2 v' 900 - r2 . ©

0 < r < 30 e:i��x:i

dV =37 Jt [zr v'900 - r2 + r2 (-2r)

] . ©dr 2 v'900- r1

7 = -Jt3

[ 2 r (900 - r2) - r1] v'900- r1

= 7

Jtr (600 - r)

v'900 - r2

av

dr = 0 � r=IO-v6

©·

(': r>O) ©

0 < r < 10./6 C'.l�cr,:, dV > 0 ro:> r > Io./6 e:i�ro:>dr

av <O dr

© ·�. ,\ ': � .. ·-.... : -· .. \

-----------------------------------------------------�------------------

--.·======;;;;;;;;;;==============================

4

15. (a) Os O :s: � �roo .x=2sin2 0+3 r,�a;ic., ts:K>� f �� =; dx q({)CJ'-"U>, ·3

(a)

(b) tav.t� to'J<D too�� f (x - l)!x _ 2) dx '9'C>:>Qfflm.

. f 1 t> 2 � J(t) = (x _ l)(x _ 2) dx crS en�.

t>2�w::i/(l):=ln(t-2)-ln(t-l)+ln2 oo � t)X1wm.

�.t::>:x'JW e)a;i� lr�2".)a, to;i5�. Jin (x-k) dx �C,O�ID; G'®@ "al� � 8:.,tl)""m

��. J f(t)dt����. b b

(c) aw:i b.i3c:l't5'J 0m J j(x)dx= f f(a+b-x)d.q;-@C!I w:i��.a a

d � f COS2

X d.x � qwa, �m. 1+?

-n

Jt 0 :S 0 :S - O�.:Ol :

4

x = 2 sin2 0 + 3 � d.x = 4 sin0 cos0 dB © x = 3 ¢> 2sin2 8 = 0 ¢> fJ = 0 © x = 4 � 2sin2 8 = 1 <» sinO = ...!_ <» 8 = 2!.. © J2 · 4

�elo . J � x - 3 d.x3

5-x

:n:

= 4

f / 2sin2 B , 4 sinOcosO dfJ o 'V 2-2sin2 0 2!.

= :J 4 sin20 dB © 2!.

= 2 :J (1 - cos 28) dB © 1t

· 1 4 ©

©

= 2 (8-2

sin28) 10

5

=;-1© . El ------------------------------------------------------------------------

(b) X ;,,! 1 , 2 ��roJ1 A .(x - 1) + B

(x-2)

A

= (x-l)(x -2)

.

x1:

¢;> 1 = A (x - 2) + B (x - 1)

A+ B =0 0 x!J: -2A-'B = 1 0= -1 roJ B = 1 0

6e;c) f 1 . dx(x- l)(x-2)

f -1 = (x-1) dx

J 1 + __ dx(x- 2)

= ln j x -2 j - In j x -1 j + C , @®8 C c.'.l� qeo@t.:i) 253c.:Jt,)c.:JB.0 0 0

------------------------------------------------------------------------

fi.t) - I' 1 . dx(x- l)(x-2)

3

= ln (t - 2) - In (t - 1) + ln 2 for t; 2. �

f ln (x - k ) dx = x ln (x - k) -f _x_

d.x 0(x- k)

= x In (x -k) -f 1 dx - f-k - dx 0 (x -k) = x ln (x -k) - x - k In (x - k) + C 0 = (x -k) In (x -k) -x + C, G-@B Cc.,� q@®m 253c.:i%:5'.lc:iB.

f f(t) dt = Jin (t- 2) dt -f In (t- 1) dt + J In 2 dt 0 = (t - 2) In (t -2) - t - [ (t -1) In (t - 1) -t] + t ln2 + D

= (t - 2)ln (ii - 2) -(t - 1) In (t - 1) t t ln2 + D, e®63 D c:i� q63®to 253c.:J!.5.lc.'.lW . . , 0 , ·0*----------------------------------------------------------------------

- 30 -

b b

(c) Jj(x) dx = I (a+ b - x) dx a@c::) ro::i��®'<.:lzrla a

lt lt

. f cos2x dx =

f cos2(-x) dx 0 1 +e" 1 + e-• -n -lt

lt

0 f e" cos2x dx= 1 + e" -lt

lt

2J cos2 x d� = 1 + e"

lt 71

-lt 1+ e-x l+ e"

f cos2 x dx +f e" cos2 x dx

-x -n

= f (I + e") cos2

x dx(1 + e")

-lt

� J cos'xdx ©

= 1 -l{f (1 + cos 2x)dx 0

. J cos2 x-lt 1 + e"

= � 0

0

10 • �d'11 � - I(� �Cl>fl �oo&>) mt)�= I q.900.c.(c.�l • 20191 q� �1:1> tft� �· � tfta>. - 31 -. .

16. 12t-5y-7 = 0 ·u,o y= 1 mde �e �m edoll<-' �m A '3 @�m � ��t1>.

lo� ct®® ��m cot�m � �6' eo� attS en�. I �e � Q�c., �:n.

P � l !!>m .;) ewe� i::>18 a,.15,§. P � @«6tt:>:iot11 (3.t + 1, 2A. + 1) "Cc., B!)o '°1B 00 �trlm; 0®63lER�

. B::(6,0)c.,t8 a>�.BtooP�� e'5�!0Qt1l qmm � � f:>�&3 cwll�<!I S+lU=O� 6&!!OlB 00 ��;�Ss.r·+y2-7x-y+6coo Um-3x-2y+l8�.

S=O� AB��f$)=·� �B f:i�lo �� 00 � 1.1>0mii, •

. , u = o a1� t c> otQ)t), B �oa,:, c.,m moo �� c� Qlf> �:m2".

' SCl§ A. E IR C'i�:I S + J..U = o :o®�Q c:>@0) f)ad,m ®.m � <c B E>8trl e6:I� � c; qf)e �:.o&cc63 @�� <B'Oa�.

12x-5y-7=0ro:>y=l=>x=l, y=l

:. A a (1 � 1)

e:l®0e��%S)e)e e,®%5'>6-t!!ic.,

12x - Sy - 7 = ;1: (y - 1) ® 13 1 .

®

=> 12x - Sy -7 = 13 (y - 1) or 12x - Sy - 7 = -13 (y - 1)

=> 2x - 3y + I = 0 or 3x + 2y - 5 = 0 0 + 0

y = 1 ro:i 2x - 3y + 1 = 0 qz:n6 ez:;,:i�c.:i 8� 8 25)@

l : 2x - 3y + 1 = 0. 0

< 1 ee). 0

l @m � (x, y) cdllllJ5c:l e,�t:0:i (x- 1) (y- 1) __ = _. _ = i. (c:>18 m�§.)3 . 2 0

� X = 3A + 1, y = 2A + 1. 0 ------------------------------------------------------------------------

:. P l!i (3i.+ 1, 2A+ 1), :\ER.

(jiz:rl B = (6, 0) i:l:l:i P ;:;; (3:\ + 1, 2A + 1)

(x - 6) (x - (3:\ + 1)) + (y - 0) (y -(2i. + 1)) = 0 @@z:rl @�� e1e@. @�zn®(x2+y2 -7x-y+6)+A(-3x-2y+18)_ =0 0 e@e.5 S + AU = 0, q:iii:sx16@c:>z:rl @0. @®@ S = x2 + -y2- 7x - y + 6 crn U 2 -3x - 2y + 18 @0.

0 0 0 ------------------------------------------------------------------------

S = 0 odts) A = 0 o q�610 @0. � P = (1, 1) ;e A. © :. S = 0 c:l� AB El�t!il®roc:id � E>at:nt::rx::) @�. 0

B C�ll!D'.le5 3x + 2y + µ = 0 @z:n &ie3m 18 + µ = 0 =:,, µ = -18

:. q0aa:i5 �®ts>d-®iic 3x + 2y - 18 = 0 @e).

e'.:12:5)@ U = -3x - 2y + 18 = 0.

0

----------------------------------------------------------

--------------

@

u '°" -3x - 2y + 18 = 0

IDl / • 2x - 3y + 1 = 0

� X:4 ZO:J y=3

:.· c ... (4,3). 0

S = O i:o:> S + AU =0 gci®& et>.

� 2(-; (3�+7))(-; )+2(-; (2A+l))(- �)

� 131. = 26

� A= 2.

0

0 0

= 6 + 18A+ 6

0

-'---====================;;;;;;;;;================

10 - Qo\3=1� C3l&z= - I <�<a �elb ��) � ��OK., I q.�.co.(C,,1J0E.) - 20191 �f)Q� Ci'lG�m Ill� tDe 8� lfllll,

17. (a) sin.A, cos A, sjnB rm cosB qi.-al!K5m sin(.A+B) 8® �:i. sin(A-B) e;,"icoo �Or.� �(3)2'X:lJ e&. m�t». 2sinA�B= sin(A+B)+sin(A�B) ll):i

2cosA sin B = sin (A +·B)�� (A :..e)

@e) � ts1d:t.rlt».

"� O < (} < 10�ro:> 2sin30cos20=sin70�mm.

(b) ABC�%:!) BD;;DClO:IAD=BC�m � DCtMJ:tiwAC®t:t! � <t(t:D. BAG� a coo ACB-:::. {)�ls CD�. ass @0� co�coo �m �r.., �0<"Dm, 2sinacosp = sin(a + 2P)OO 0-eroDmm.

a : fj = 3 : 2, �. <r)W� (a) 63 qDamn s&e'-' -eoo���tt\ a � � 00 cJtO��.

(a) sin (A + B) � . sjn A cos B + cos A sin B . ---- (D

�w sin {A - B} "" :tin (A- ·H-B}} © = sin A co� t� B) + cos A sin ( - Bi

:. sin (A -B) = sin A cos B - cos A sin B ------- @

0

0 ----------------------------------------·----·--------------------------

CD + @ � sin (A + B) + sin (A -B) = 2 siri A cos B, ©

CD - @ � sin (A + B) - sin (A - B) = 2 cos A sin B. 0 0 ------------------------------------------------------------------------

2sin38cos20 = sin78,

� sin 58 + sin8 = sin78 @: � sin 70-sin50-sin8 = 0

<;:> sin (60 +8) - sin (68 - 8) -: sin 8 = 0

� 2 cos 68 sin 8 -sin 0 = 0

<=> sin 0 (2 cos 6 fJ - 1) = 0

<;:> cos 68 = .!.. since O < 8 < :rt , sin 8 > 0 2 ·2

0

0

=:> 68 = 2mt ± ; ; nEZ .. 0 + 0 = mt :rt EZ± - ; n

=:> 8 =3 18

2!.. Sn: ?:rt ( .. 0 < 8 < �) 018 ' 18' 18 ' . . 2

------------------------------------------------------------------------

(b) A

B

�a?P' mb1c:i ®C!l�@®m :

ABD @@%:ilo-E!iic:, e::,�ro:i

=:>

BD .

I\

smBAD

BD sin a

ED sin a

=

=

.AD .

I\

smABD @

sin (rt - (a + 2/J))

AD sin (a + 2/J)

BDC @@%:ilO�C.'.l c:l�WJ

CD =

I\

sin DEC

=> CD =

sin fJ

BC

sin BDC

BC sin 2/J

@

C

0

(2)

I\ I\

CBD = /J, ADB = 2/J,

I\

w:i ABD = rt - ( a + 2/J)

(1)

0

. . BD = DC and AD = BC, (1) zrl �:i (2) �.

sin a -- =

sin fJ sin (a + 2/J)

sin 2/J 0

::;, 2 sin a cos /J = sin (a+ 2/J). ©

a : /J = 3 : 2, �®

2 . . 2a . ?a If\ f;\5 sm a cos

3 = sm

3 @c;.,. \V

::;, 2 sin 3 ( � ) cos 2 ( �) = sin 7 ( � ) ©

=> a= n 15n: 21n: -,-, 6 18 18 ©

:.a=�·©

(c) 2 tan-1x + tan-1 (x + 1) = I!.2

650 2a + fJ = ; . © ¢> 2a = I!. - /3

2

� tan 2a = tan (; - /3) © ¢::,

2tana cot/J © 1- tan2a =

� 2x 1 ©=

1 -x2 x+l

� 2x = 1-x ( ·: XI'!± 1)

¢> X = 1 ©-.

�

2 tan-1 ( ! ) �- tan-1 (; ) = ; .

1

0 3

0 ---

---------------------------------------------------------------------