@ .(got))) e}c)JG) <i)t6)�a>@l�OJ

11.@tJJ.�. (c.eee) t>m�,� - 2014

10x25 250.·

= 1000./ 10

= 100

1. £Md M:sf ®a, � 0 ezsfc�a,= Sc) u �ID(i>c:ltd S'1e, e,®ID !!_ ���c.:,l.Sf e.,:,�ts) ,m:>f:liiuf, Gt;) a fl � . . . 4 . O Be> 2a &s� soBtd. f;d �- Bd� �zsf �tso. �o�!>!Sf �d,� oa�m gt:SfMO at011) et�. ·u > 2.Ji'a m8, lfoGAD m.,dc,c.,o �gm c.,m Se �2Sl�.

0 2a

O Sc> A 1�£.b c.>(3!Slc., t-'�ttl:>: s = ut + :.!:at2 @��@�?.l'l . 2

_. 1a=uco{:)i O · 2J2a

t=--

U. 0t h=usin(:Jt- �gt 2v. 1 . 8a2 4ga2=2a�-g.-.- =2a--. · 2 u2 · u2

4· .2 )> 2a-:-�:ga (·: u} > 4ga · ·4ga

=·a. =;i, h > a o,,, '1"'6 c®'!I ... <!•<el> &.rillco<l 91Dg,d <OS. 0

··: .• ....

2. dm,� �kg�'. �c.,m,f, mt�� � �-:-� C,i) -�c;)c.,, e e,e� �e&.,zd��:-�ecr_:� l� :q-(tc����·-��<J<J e� _�;<.�� f.)�� � seal)����.Ra)3�R�.o.;a,� ��:·PkW dc)=!uf.·� �tz,J f):,to� Vins�l-���t)etnf..id�-�� �N>��J::f�� �e�:-�� }(.R+J��) @D ecirle)zrit1l.

-�f

I I T

:1 M .... M •F

•• 2R R

F=mc ·� -- -�

·----+(i) {'.01$.) (ii) m�

t,,f@,;J<> "'"'°" F- 3R = 2.}rq- - - - - (i) 0 0s)eOco �""' T - 2R = .141· - - - - - -(ii) 0

. 2(T-2R)=F-3R.

01 T=-(R+F) . 5 2

= J_(.R + WO.OP) N.2 · V

3. d�t.rlo� m � P qo�1..)f 9®D &;:1eti �� @tl), u <2�}��· l3'J�; ��"'� �e.,o, @�Q" e®'�-r,,dc, �� €/emro ev. a� eo®<o 6lt� ete:.,O P q�, c;&J �::))© �a.fOeEJ ttt.il) ��mwc., ® 0!)6,� � Q "9� C>®© rede <see, mi0em q.c,o, Q q� �a� @z»s.b-.J IDtl:'ee:io:i c:,8!. <n?,g@ � e e,�wo -�8 �<t?� e (0 < e < 1) ee. Q q� ®a> �z! inB���c.; i(l+efz,m, ���-

..

·-,��

.. ��� u'Jtg® ���!)

!.. = A(mv) �4�1.il<Ae> � mv+mw '-- mu=O O 'l+W=U

v-w=eu0v=-1..(l+e)u . 2 .

c,,�D.t� �® e,'tw:>, . �

L�6\mihQ:t�{l5;lr.;, ��-.l-,-=:mev-�(-'-·mv) V

=(t.t�Jmv_ < . ;

\ -�· J, .... _/

I

.0

2

4. �eo:>!'z:» �a., a t0a g.e:,��:> �C,:>o!$C'J. 4mg � «stwtd'1! ���e.{C, .enzd'�tS> �ZSl @!:9st)CJ=fqc>e O e-�e coie> roeo:>. qiB � ���z:» �d -� m g qo�t:i)C) eo®@mQ �o(tt.tn, 0 63 �t:>ez,,o� So qo�e> �c;ncl00 c.,G)0d ��� to8'� e1�. (.';)�B C$ot&e;63 t)ee:>�®G:, acs,��. o� s� f>2S'l �� � �� ce6® �<.0 @��Q'l.

0 P.E.=0 --- -�

a I 4mg( )2 O+,-·- b-a -mgb =0

'2 <'¥

2(b-a]" = ab

. z .,

2b + 2a� - 5ab = 0

(b·-2a)(2b-a)= 0

5. . eSd-t� <fo�zrl, i + ZJ «J-J 31 + Jj dt) 0 q!Je te� q� 8�� A a13 B ewsl)

� 8� "'il»Z:O Q'[@ �- C a,t3 OABC 6'®:.m�:.i©� Dm � � �s:i:se.3 "IS <t a>53§.- . .. . .

OC=21+J 6'f> ·eo�

AOC=BQtS ��. BA·OC �ieiBCl®td cos9�·1 &e) ��.

y

f.--�5.·.; v

B 3i+3j .

·-

x

3

6. @o WfJ 6zm:,z:r,::id t":etS) (!)o(D:fec.:>� dtei@cl �m@e)7S) o� Bd'eoc) a<&�:f41i>�tsi�{f:lZ).e:i) " cc, meQ= ®tn 53efe)e€> �1�@-m, (!,•il)�@Q e,£lc>.m® (3d¥:6C5 �A c) �:> qomm ts>e� B e2:SJ•:oc.:,ZS)c) e:i®@� 'b6� e1e} e:>'l.�lc!� q�m�s:u:,.w�:cgc:>t.:i> q:>Q�d<:,,c.,�. AB m2:>:f�f) J!ikJ= f) c:ie:>.Sts:1 �c.) @©'Jee., e\®:,z�x16

e.®�emm:ie,t:) :tn0$. -co!!fl!M!li> c..J:s>:J-eiuc:, !!. @e; eo�:>. z:t)���i=tE:> qai::D�

@f'.5:>ctJtrlts). 2 '

C0�¥6 @�J�GO = OCA, fs"\

� ao uto:i OCA=-

2 5

@UZlW ?!)6C�2b'.I'

· C� Wasina=T(a+acosa)

T = Wsina·· =

_ 1 + cos a )'

......___.,

a Wtan-

2

7. A Jg!) 1l o� n ��, q��� Q.1c'J 0�1$1:sf G!ltS G>.il§. ESOi� qom,m�� P((A UB)f"!(A'UB'))= P(Al+.P(B)�ZP(AnB) @f> ��2rl�.

' ,

P((Au B)ri(.A'uB')). = P((A �S,)n(A n B )'J = P(A u B)-P((A u B)n (A riB)) 8 - .· (I) ' ' [·: P(x nY')=P(X)-P(X nY)]

0= P(Au BJf P(;4nB) = P(A)+ P(B)� P(A nB)-P(A nB)::. P(A)-�1>(_�)'--2P(A n .B) s [�

8. ®ee,:s,, s®o�ec;� C)®om � c:5� @Q)�e 6 r.z$ 't· E� � 4 rsf <c � �- @@&e �z.,:st, �dZilD om>@t��. sBt!&.Jo�c.:i§� @tn:i6El, tl')Q$b'.)� � ®c::f�trJ <gc)toc) (D� Ct�- ��.ii �GO 86 Om>l'd@€) � qitn 00, tlltrlOrz3 0@dca, Oq} �'lSJd e.35<:1® e,@t!O:,e,� @�c.,3.ftn.

�m,-;!f

R:�.W:$3$

5

9. �de-em cnnm, ®Ql3�:i,a,.mo �a, �iz$7 � 9 ee,. ��e � ® &.a,Q 11 �- �e'�Sc.,oo 03' �� <!I'� cttS �ome� �<1®m, D{�m® �O'=f•tfzic., � q9.m® �ZHEc.i �u!'21'.>.

x+y+31'=75 =>x+y=4

10. c,�.t,} ��� �� 100 m, �l):1%;)) D:ixrdSt;,d SQ:z,mz,c., 31.8 � .. • I s • 15 l 15 • 25 I 25 • 35 I 35-45 f 45 _ � l_ 16 . . x _ . 30 · _ y _ 20. =·

·x � y � q� �. �� Wiidoa, � r:n6mm. z35�•-eio 100 to �o <¥co�� 10 x 16+ 20x + 30 x 30 + 40 y + 50 x 20 =- 31.S ;-< WO 2x+4y = 318-206 = 112

x+ 2j,'= 56 ......... : ..... : .... ( i) 66+x+y = 100 8

.x +y::= 34 .......... i: ....... .. ( ii) ··x�12c=�i:;;;22······.�··

. ·

.

(?\ tO

v �"

6

--------------- -----------,-----·-··-----.11.(a) &)'eo0a(o<a<j'.}1·���z,1 �mo, ·�e 9®1:) �t:L) � 0 er.:n�CI'> pf(;Y;) Qqo<Jl @�tn�.tn@o

tftlS>.· 0 tOO� � ce,5® � �E) �� C� qma P qo�OO IA �Qd e�@ e@r;r; ep:s,o, � �tt'):>i:Ji'frl ®, Q (fo(QC) 25'iaft)� 6e) §(? we� e(�. (JO\Js �::\, � Z:OCc.:> &f.6 ettio� ·cW co=dozx., ��m. P a» Q e €:i&;n eoc;;w:, �-� s�m:>oc:>e <cs eo6w� �mi ® cfzc,c.o.i:D

(a

�,:mm. . . . .

GI-®® 9� to:>!'j�,:ri. p �o�f) 0 �-ll� �'lf).t1) Ct�m C!>®:)ca,a,oe,.eif � Q qo'@€i O SO2,1- . .

gs{iia �d'fntrl �E)t,) 00 itt,m&,rlz,,.

(b) =eg � ·� =& IDUlmf u �6 �i@trl �@a.A ID:i B �� ��'dz:nffl 6.f!lf �2:i'I

. <itJ�!lf)������Ai}�u�aeo���c.:1.86®s®�A� q:>d®� Z"S>d, dee.co eo:x&dd. E> qt>e ��� e�� 2u �-&m��aii$i,-18e�ttf •

. B � �:> �; � u=jul �- �� «»trloe,-B o�m (io� �o A 0e>m ��!!:1 oi®e�m � de�� 6qVc�-�2u�®��m�. A@a8�!}�.J!DG' �� <i B �o A <cmf>:i oa.r:nc.:, �rn!) ic 9�(5) ��� <ce WO!lOttt �?.:\) ® �OC.,1.:l) q�m. d � A � B -c;i:dc>.> �'° � 't B Sc) A � t>8= e� <t. -� t�cl��f.> .. !>� 9�� ��- Ai eoo Bl � -�1:S) ® 8�mf =t� az;i @!.) ec,�mm; (l'(i)@sin8 = ;sma �- .B Se> A �� e@�. a,=J zmoec.,. A So B ¢c.b e@�c) � � (t>@i,$ k (1 < k < 3)

��tllf �s. c:os8=!(z�Dcosa _ Q)E) ee��-sin6 � eos9 �3 � (f�m gm><D� t0&� ooso== {k-l} 1 QJ.E:J <c ""°�i,1�.

' I v

J c.,

7f ./ � r�

;;-.· , .. /.

-- _/_· _ -···--. jJ --

. I.J

----------·---·-----·------------·*·---------------------------- -. • . u

,,.,....-----, tan e = g Stn a == r I,,. .. _. � . 7;2 0_.,.... s,i'\.�-" .t.,

;.T= S 2u ·

0· . gsina . ·

p l:fc><iE> �=e o �o� l:.'te��is) �� Q e o e� sd = -1soi 1S)e ,@25}1-G>e-'4 t)�wce� 1· ·2u2

8 =-xTx2u= 5 2�) gsina �

·---------- - - - ------• - --- -- --------·- ---- ---- ------------ ----- -- --- - -- - ------ ----- 7 ----

(b)

. [(Boy,E)=V(Boy,W)+V(W,E)

. . . .

B

------------·---M----�----- - --- · -·- --------· ------- .R--------------�------------. -----------.

. . . . -

g

( )2 1 . 2 I k+I 2 -sm a+ - -·- cos a = 1

4 4 k-1 .

. . 3(k _;_ 1)2

� cos2 a = --'---'--4k :.cosa

= k-l {3 82 Vt

.

.

12.(a) �<nB���ABC��M�·�eseqa:�'-'m> �dtaffJ �mrtd,Q ll>o'U>:, "� ·Bad· wdd=!Jml�d't.O.e!oQ z:r,dS. AB ��@:,e) �as qr.om f3E.e!iietD§t 'co6®Q)�t!> '1i�aD=J D� qco ABC = a, ACB � ; ro:> B<; = a �-.� &sd �· ®m BC qc.,d � (ft&> ti�� (tttn. =� m �.q�!)zdAB-�� ®co A czllf•�63 E!dt�m mah �Elcr.ooe:>GK4 Be f3� a,6'11.3 et�. qoaae> B -=--------.::::����--!z:e� Wtd' '-'ZS> emm, ��C!l'Cd .tdf)d-4!1iQ mgsina _cos a ... <.----

a---.-

. · · · - M + mS1n2a @El� �a,c)=� e qo� � �m=.

<a,irl'. a.;;; ... ; (00 M = s; ..:tcS �. qaLiD .�«.'ii'-' a:>td c.,in e&�m � �Gtc.d <:>f>a>�

.J�f @E) �mm.(b) � a eot0 �<., 0 �· s� tgSc> f)a.a,b� � $:$d

me.r.,m co!)•d' <,t�· �(3). 3;a.a'·e!):, 0c�· C,tS>tda q!5m�1i

� c')mf <&m>�m.- OP add E> q1Sc> tt,@C., t;e q��. ·�!.QQ ,n fJt,) P �o�.� qii:b

'. ·ot� .� �tS

�. :#� �� � � � .� .� e:::&w� q1@ �6)!) l

e<f)a Sgdtlsf. � 'i c.,8i)1 CF'9'!Sd· -� :cf�co 2m � Q q�E)d. �d:, �. � is>�E) ,nBe �.� � _p ��.ee�.�mea;��s���e(o·< fl< :3;J���2'f ·op.rs>t� ·�i l)a P (JO'i� �

v_c.1� v1:�··.l�a(28�sin9),!��:�,n cwm St).'"�·P q�. ®m �� tjtS �ms6\@aiof)�t5S. ·

(a>

ii,.•·.

Q

9

g(M, E) = ---'> F-�

d g(m;M) = f ----7 .,s a,,!l@. 0e>1'0 g(m,E) =

70 F

. f F = ma : @�=>'i�·

. e,(f &)153� ��to:,

m t::J�w:,/

___ .., O=MF+m(F-fcosa) --------- {i) © mg sin a =m(f-.Fcosa) :, ________ .,.. (ii)8

(i)=> f = (m+M)F

mcosa

( .. ) . · (m+M)F F 11 => gsma = - cosa

mcosa

mgsinacosa = (M +m-mcos2 a)F · :.F= mgsina_cosa �·

M +msm 2 a \...:::_,,}

f = (M +m) mgcosasinamcosa (M +msin 2 a)

(M+m)gsin� Q.M +msin. 2 a \...:::_,,)

------ .�- ·--, �--4------ .-.--------� ·--------------------·--------------------------------·. .

� . 5,n . g ·. 7g· ·0 a = - t0:> M =-. - 0�.,. !,D F = .;._. eo:> f = r;:; • 5

4 · · 2 11 6 · 6.y2 · ·

..J2a=I., 7g xT2 fi' · 2 6..J2 . ·. \...:__,)

T. : :J24. a. C'\.. s

· ... ··

· = 7f '\ZY

/ 1 ., ¥ s=ut+-at1.ec.,:i.-Q.·.. 2 ·. 't�

.. - ··.·_ .. :-� :� ·:--- -.- ... --.. - . ..,._ '· - - - ------ .

_,·.; .. ··- . ---· .. :; _· .-· -:· - - -.. ·; __ ._ ----------------- -- ----- -- - ------ - --: - - -

10

12.(b)

:

• •

i T i (lf"

, •

a

1 v? 1 ( )v? . 1 ) · �

2m - +2

2m - +mgasmfJ-2mg\aB+h =-2mgh \.:..)

3 v ? 2. o · e-: � = ag -gasm 2

·V2 =�ga(20-sinB) 0 . 3

R

.... vt ......... "'.""' P.�O

40

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

. , . mV2

mgsmB -:- R:::=-­a

.,ii&-'[:. . . ..... . :;:::_· »: t5sfri B 2:.,40]' ",. .. 3 · .. ;-:

·,· '· � �-·:. ;, .< :::cc,:

11

8 c

I

13. �� �© 4a aoo Stl)l:I� � 8mg � 63@=' °'l:tt>Icl'� gm:ooe&a ¢:m=f, it,$ �s emr#)o· o i;c:,e � Ml! Bot& t) � tnm· �a, rri t;j P q� � �s � �i�:i �. P q.,(ie>

O c> Sod. f> �gm � A o=�c:::sr.:s, t:o� e> ,rta. OA = �a @€> 0otxfE>wtf}.

4i{m. � m �mQc., ® =63� � Q·qo� Pe> 80l<!t!)� qi�� e@� �c:S f;,}os• �� A@ �e� @a � � l!'D6S. teoam q:o'@� � �4llo(.') X = - Jx � 00�e>�;

· @®6i x <-'� OB=3a !>2t1 � 0 � �� (!>� � 8 c,de:c� � � qo� �mo&�.

t:e"'9::l1:S> qo�� S@:> �i,> � S 0-:):)c., C �S «>5°3�. OC � � A Be C � t>ewc., � �e=t:oq� imsl� ZS,:,CQ " @e:JQm& � q..� C ta (¥t%» �' Q qoqtf:1 � <l>f)uf �?}} et0$. � i%Dis:) P �et>c>� o�·e>Bm ��� j= -�y @e ��; e-@63 y '-'� A emla� BE> Pq<,�9!>�ma,�.� �=.J�ro0 y = acoot:81 i' psm.a,t c:pzs.,ooecd' �� ��� �®1n a,/J&13 w &� jtJG>G.'W1 �mm. i

ddal c� D � �Zl)Cl;'I � 'p qoea!) ��� �"' ?f J'¥ @!) ee��!',}. e®e D � oo=4a e>m ef.3 Oe> � £> 'ftoim � � .0 . .p C9f>m aani !)z'} !Se Pq .. � �ct��.

A mg

0

"®�8">""" """" j·j T, = mg 8 � · g ls)

as>()� · To = .mg ( 4a - OA) 4a ··

. 8 : . '

8;. ;:(4a _:_ OA)=mg

"';.s . 78a-2(0A) =a=> OA = __!!. ' . 2

}.! l

,r c

p �"'ll" "�"'• l F = m� f'::\,-J;+mg = my v

88mg( a} ..

--- y+- +mg=m 1

4a 2;

i:ge,iila®-0 -· 2g

c·> '}' = -.....--y ......... · • . .. I

a . 0 . -- ------ ----- _Q_ ._ -------- -- ___ ._ ----------- ------�---- --- ----- - ------ - ---- ------ --- --- - -----

u· 2..,,

_J__

y = acosa,t + /3s1n (J)f ••• :......... (ii)

C j-.::; -aOJsin mt+ {Jaicosoi.......... .. . (iii) 6

.Y=-m'(<rcosar+Psinmt) 8 =-{iJ'y 8

I I I__ :ts _ _.!

�-----------��------------- ·----· ---.-------�---�--- .---------------�--��---------------------

y=-a@Sm<IJI =-a -·-=- -- 5 . , . . · J¥2g Jj ·filga 8 · 1

· a 2 2

13

14.(a) ABCD "'23 DC= lAJ � e,� � �Sc:>$d Q"tl:a ��- me> ii AB= p a'.l3 AD= q c.,zS � - IJ . . .

G>�. BE= 3

BC E:>m � .BC®£2>E��f?S�.AEs»BD� � �:z,c., � F@@vJ

BF'== .u.o���Ja.(O <A.� 1) �� AE = �p+ �g Sf>a» AF= (1".'" A)p+ MMa®��ffl-

' .t!J amd. ,i &3 �CDQ �&ffl:D.

(b) ABCD� c,i� ,cs, 8eid a� e� c:ttS CD83j. � �= 4, 6/i ,8,JO,Xl:03Y� @e .� AD, CD,AC,SD,AB a» CB�. qiaf&l6·4'l)� �� �� �edS. � OE � � d � ¢'0� c,!); ed 08>:>E '-''B �fjZlf.AC el» CD� Sc;);i, �:i., ·�� x a» Y6' 4:p�=J �. m� � 6.!00tti 4Kac.)��; e®Q K = 2-Ji et!).Fe.,\!} OAFD� � c� 1' etife� cotS ��· � @c � � !>� �--

AD '" '( qomm Fe--� � 4i E>t,), &e e;a �-@o 0�=eedAJJCD� � �� 9c>d6K4� Q}C ���c� 11\0'V} er�. me �s«:d eoeged� t= ctt1a):,f.) <9e�.

. . :� � . .

(a)

A""""""::;;;.;_����� ..... �0������--B

p

------------��-·--------------------·---�--------------· ----------�----�------------� ------

' ' ' ------�---------------------------------------�-- . . ---�---------- .------- ------�----14

(b)

. 2 �A,=-

7

F r.@

� DE>:> ����G'.>t��ez-1� Xx a -Yx a +6./2 x a -4x a= 0· 2, . 2 2 . 2

Y;;, 1,/i �6,/2-4 = I3ifi:C4 N O 0 �ege..tm<A =4-Y+ifi+s.Jz O . 8

� 4- 13.J2 + 4 + 9f2 . == 4(2 -.J2) = 4K N .. t�ee K = 2 -Ji @O . � --------------- .--------------------- .-------------:--------------------------- -------------

. -:-+ Q = O . . t P4-� = 40

� 4KxJ1a= Px J2

22a_·. fs\ . . . v

):]) N "" BK

J 0 . . . . �·-f ��{����4!(]=.:.�.tt» .: 4zj��ci1@e� =8K Nf.-- - ------ - -------.---------------- ---�-------------- ·------------------

15

:�t B r 4'-�H_ . 4K h

�

\o 4Kh=6Ka

114.(<J) -� ""' w - ll < AB• AD• JJi - BC� DC� l ll � AJJ.BC. CD.,. DA ---,

� � ABCDdo.fl a>t� �- I!'�. ���sec�� D6 tQ't.Cl).

I � 2l ia l"3tWtdQ q� Z;D� A:�· C .m0 ae&tdD -4- f.W$. 6;i8 !:Oladc A�

. �:i�� C;i;�' ,\ik'"4 i�-� 6 �� �� lt�°S«; !'Ji(S.--t··)i).� f.>-e.,�.

I

(b)

AB w� BC eolito:i CD e.,�w:i . �

.8CW3CD a,�� t T-·2wl-2Y=0 0.{s) r _ ,.,,.., , wl.( Ii_�) _ wl f0 _,_ �/, 31 _ wl f1;. :i.. .fj) M �O .\.,_ J • - ,. '" ' v .j � I - l" ' v J. 1 - �. t" • I ._. ./ __ ..,, 4 4 . u. .....__.....

.• (b). P

.¥ •• •• ,·········0·,, ······.--········ ,----·QN·····--.-K ------- •··· .. :- .· .. :-.·· C . D

120.:V

-

D\, P x 4/ = J 10 :dl + 6<fx l

lOS 70.J)

17

.

. .

to. "°"' a roo e,aai�� e$�=&, "'� �="� �o' ��� � e,13 e) .. Of!ll:)6 �&,c,&3 =<,c,o .es@:,mmd" §1 � On2'1,ec4 So aoosa ,dam�� �-6\zrl t11ts; !JO et�2" 6mts:imie9cl �� �sc:, OC_ � ®G:>;i, et1i"•21� SSlElm 00 tfi}m>O�,� eomE>mm; e@l� C c,,n �E.>3 �� m1� �dm��. e>® er oaeilaz.,, mmatf)C) ® :.,In.Cl) qoc:, a sin:a g} � t!f�� e>am� 2;!)'{6ai2S3 tilli:x:, �oolS.l lit>.1m=>�c4

q� f>ar:dzm=o miebe> c;� eeas a,�� eoocl� �:, qz.m. ee>e> eD:ld�CK:>@ �O't� @!Sf��.OC ®.m O BD ( 1 + cosa - cos2

t )acosa ,�m &\;)� @f) �ffl�t1).1 + 2cosa - cos a ·.

a=; a,iS'i ll>o�SO"Wa11S'iC3>��.,<ribU>:tad·f

� •

�-

A {l S6'h:d &s,,moo AB�� l)E)z:sf eee,. O,.A l!>'J B � 1'= _

O __

�"' l)2S) �. dtot!ld ta=f�m �8'm UX>d2',Md G>tQ� �ae., eeee e!'J:S>d �ffld � cnm. =��m@ �<1lde>

,_. .. c:--� ..

�='�� e&W® �"-lzrlm.

iJ •

�. $� B �trl �le�� q-<8 �· 86 c.,a qm Sd= e.,&)l?i) tan-1( �) ,.�c.,mf_�®= e.\®�� "d<!'CS. 3h = 4a Q)E) �.�mm.

= 2mr2 og sin 85(}

.t:-,,··C)

. -

oao 'l?otdc> <3'as:tzts"'0 6d x ,az.@ CJ)��:

= (2nasin8 )aoBog

r, 8 _ J 2m:z2u gsinB · acosBdO

�6� x = __ a ___ -.,... ______ _

· 12nu'c:rgsinOdO 8 ()

[ cos 20]'.fz

= . . . ·.,a=......,. .

· · . a;:; a= -a cos a--�- :· 1 (i+cos2a) J cos2

a

1 ... 0· .. t-2cosB]Z'J. 4 cos,z 2 cosa 2 • ··-·!: : -. .. ,-...

0.. . .

------- - - ... -------·- -- ------·- - ·-· ----... - .. -......... ·---· ··---- __ .... --------- ...... -· - ---- --- ___ .,. ___ --- ---�Od�()

r=- :o-:::,i.\

' I \ : /\�-·--1--.. ; , ___ . t ---··

'(.') · . ._____/

,-1- :-=i

I \ I \ :

\,:;":·�,?��,'..... �

@6 + OS€l ,o

og · 2Jra' cos a 8 I (:-a cos a 2

agmi sin 2 a 8 . acosa8 . ognir'(2cos�+sin' a)Q. -

y

-�

...,M....i �4.ro ...i,,,s" oc ®" o so ]¢,&t oeos; ..e,i O. 1

. C\ · u gna' (2 cos a +sin a )Y = u g2nn' cos ax i" a cos a + a g1111' sin 2 ax a cos a �

-:. acosa(cosa+sin 2 a). ( l+cosa-cos2 a) ·o· .. y = - . - = a cos a · s r:;-i · {2cosa+sina) 1 +2cosa-cos2 a . · . � . . -----------------------�---------�-------------------------------------------------------------

7l a=- �ei 3 ( ·1 . l

J1+--- ' . .

-. 24a· 5a·u· ... y= � ;;:- 5

·l 2 14 · 1+1--

E>e!�O C)}o

� ·W

-\�

·w ·-

sw(). 4 .

AB aa rio

5a -. 14

. . '

OC �tiaa.e-0 @D

GO

0 G)

() .b G) 0 a+-

2

y -x

19

o?'-i

---------------�----·------------------------. -- ··-�-- ·-----------------------�--------------

8a+9b= 2da

4a�3b ® _-.:·.--\;._

20

t?.(a) A to:sB � 06.1t� � P<.B)> Ofi,s) 8(a �id cqS QJ •• B � .(ft� A e <,�.» � i P(AfB) qtsd �a P<A> • PCB>P(AIB) ... P<B'>P(AIB') so �• • ..e o <, (B) < a � .,c1 lf smd "e

·���.dee.!.D:,c othG)•· eclt>o dlgal�d 8" Zif 8$8 e)m ffr»d 20'6 • otm1� �.�· �....,= 51c.l DINI 'fctg 8 � MF8 q.ctOJ.e. (ea:,.� E)d) qt:,d32'6'DHJ - � �- �� �- �e. &,a •. Nf&J � ��.cs. .... � .. �1§ � �J:t&.,a;,;sj 409', »>ecd .,,.,. f> �l:S) e&s-8 4.ec,.,.a. (� � <pad 45* iNd dS SGe-& �a Ce.� .t,. ·� et4&:,�-,� Od � --.e .� �. IDm a.a. � �- edi �Se�.(i) ff)IID6. ����a. (tA:,.eee} » �· � 89.

(ii) t*IS • q0� &96f!P8 c,� <=-eec> Q w � ��

(iii) 89 .. .,.. • t ttt• eo, � � � � •·

(iv)�·� at>�� e& c,e � � G'llWI'!!...,. 21$.f;l')8) � "- 6NnS ..,� �.

{'1). �, .. � ..... x.} oa � � eo»IO'i*" • � Sl•tlld J mo� otS 4P�.

ff

o> �=l Ix1-:ri 1$�,.,

(ii) a "'3 fJ � 6- f;O(li ��.I (q + fl'p • Ja; + n(a:i ,{,, Pl'' S& ��.l'l)�

lal

lo t',2, .... • 1t� y1 �.orx, + /J ota�. y = a'Z+/Jaf>.,.,�,91=' (i)a,(li)� G; �� r�� � � �; eN 1 tm a; � � {Y1111i,-••,1it} � �O:is-�l.1et � ��

cWmcb Dl'om� t 4'1'Cf� c:,Q, «d eclfj!JC eo-.-4S'lfll «).. CIM- � eQ»mBCI so.cw·�..,._ 1St1oaam:6td�.�--qia.�� a.s6S����f0ao,«a.id�._qoceaxoo,t1mca �.. -... ·

4•dd-CIII c&> N td <)1141& t) .m • .-m or.-� e&ilPm' eta �- iDD � tcna.m.r. 20 ao eotde)-..

· ·

0 c-05 .··1------------------------------�·�------�·------------------------�-----------------"�--�.

21

,,,-----.... �) <T --

------- ___ ... ----- -·- ... --·- - -· .. ·----. -"'

P(A)= P((An B)u(AnB')) [·.·(AnJ!)u(AnB')::;; An(BuB')= An.Q= A] ""'\= P(An B)+ P(A n B') 0[·.· An B & An B' q@2S)JSaZ))s E>�©-c.,2;:f 6>\fh1/im:,6 )

6>1B?.rl] \_J �

/

_ P(B)P(AnB)+ P(B')P(AnB') 0 . .. 0 ",it;' , rs l-P(B} P( B') . �v_:i) �-:.(?rt•)) Jo-v". .J .!, t.. /

= P(B)P(AfB)+P(B')P(A/B'J °},,:.;'{\ ii f�l ----------------------------------------------·----------------------------------------------

. Jt.1 - 98® F - <DT.Wt"a OL- :.. �e.,a® ate�® OIL AL - ce8® 9�9w® AIL

G -c���lh5� P' • �) � o nl ·-) _, 2\M = V.6 .t V' = u .. P(OL)=0.57 P(AL)=0.32 P(G)=0.11

. P(OLIF)=0.4 P(ALjF)=0.45. (i) P(F nOl)

;. P(F)P(OljF) 0

. 8 =0.2x0.4=0.08 �

------�:.----"-P

(OL

)�.P

(Oln. M)+ P(OLnF)� .· . - .

P(MnOL);,o.-s1-0.08=0.49 fl-0 � -�-�--------------------------------.----------------·------------- -- ---(iii) P(GIM)=? . . �

P(G) = P(lvi )P(GiAi) + P(F)J>(GfF) . 0 0.11 =<i.8xP{a/M)+o.2x(l-0.4-0.45) 0

'--v---' � (GIM) = 0.11-0.03 = 0.08 =l�� 1 :0

P\}_ · 0.8 0.8 10 · ' fl..

--- --

----

-__ .,. ;_ -

------- ._ -

----

---- -- ---- ·-� ----

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

-----

-------

---

-·-----

---.- -

( ._1.,,)- P(FnG') _ P(F)P(a;jFf-(iv) pr G - P(G') · - ?- P(G) �

o· h 0.2�{o:4o'� 0.45)

r;c..o,t1 -----�------------------------·--------�-�---·---------- .. ---------------- - ---

22

I ,, 17. (b) x =-Z:x,

n i=J

2 'v< ->2

O"x = -.,::,;_ X.1

- X f1 i=I

(i) 2 I �( 2 2 � -2)

O"_, = - L..i x, - x,x + x n ,,,,1 p I " I ,, l ii

=-I;x,2-2x-Lx, +-.x2 z:1 n ;:1 n 1=1 n ,=1

1 � 2 2-2 1 -2 0= - £..., X; - X + -x X n 5 n 1:1 n

I 11 2 -2 =-Ix, -x .

n l=I -----------------------------------------------� · ----------------------------------------

II It

(ii) :Z::(ax, +f])2

= L(a2x,2 +2af] x, +/J2

)

'"'' hi

n n

I:(ax, +/3)2

= L[a(x, -x)+ai+PYt=I

. 1-1

. 2 JI �

na = '°'x - ·-nx2

X "-'. I j,:J

� t.�'(x, -x)' +2(ox+ p)a(x, -x)+ (ar � p)'} 12-0 n . n n

·

= a22)x: -x)2 + 2a(c.i + PfL (x, -x)+ �(�+PY -� � �

=a2na/ +2a(<ii + pf tx, -tx)+(ax+ p)2n.rt'"T0I

I �ax+fJ-nn

. - \ ,=I ,=,I , \::..J

= na'u./ + 2a(ax + PXnx -nx)+ n(a< + P)' 0 · 1 ..... --

15__,I

_ -...,v2....,.; 2 ..L-,,,{,,...;:;;:·, /3\2 v - ,...,.. . vx , u\........, J • I

-�_+fl, 0 _ � - . ,, _ --.,. - - --- _g - - ------- --------- --- - -------- _ _. ____ - ----- - ---------- - -----� --

·: Y; = ax; + /3, by (ii)"\""" 2 ., 2 -2,t....Y; = na-ax + nyl=I

I� 2 .-2 :z 2:. -L,;Y; - y = a CY:,

:. ;;;;, a,' =a'a,'. 8 . . 0-------------------------------- .--------------------------------------------------------------

=> 68 = 60a + f3 ----------------- (ii)

(i) i>n:> (ii) ::::::> 15a = 18

. a=-.

6 /3 = 50 - 45 x- = - 4 . 5 .

j; 12� 'J. 15 6 "G·rJ = a a ::::;> = -CY

)' . .r 5 .x

r, = m => y, � m.

�6 =>-x.-4�m5 1

6 ·:::::;> - m - 4 � m: 5

(,'\ \:._.)

}O