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e�"9 . �e® ooo:>oa,

� � ��dtme<It -�� =<.ii�. e,z::M.d �6& .. � � oif)ide&» qf>c.i� � CJ!�"� tK� fflE:>. e®� �� � ecea.

. t,.�>.m.<e.@os> !la,)(5)(.5 • 2015

10 · � (!l.tc»Q

�<@CCDl®

• II oq,p

A (Dx)c) 10 x 25 = 250

05 X 150 = 7.50

�- ,. 1000 / 10

• 100

l, -� �� m cm _2m ta.-4 tW. 8.�.� �-� �d� e®o r.;� c�

·ra����=�������;t

:��Ot�-=-� -� �. &1 · t!>td lMtJ·· �(< I) ,diii i:>CiDQ . .!) <n.6l t,e) ��. eJd q<>�v e� v1 = �fx Mttf �� e�2n @e> eemV•. �- 8'G'>l �ffl �� � Ocf� ���:>,;'!·; . . . . .· . �

2v dv == 2gdx 3

c

L3· 2 (2 )x=>.:..... · mv = mg-mg 2

t eJ�Gll@c.:!ZS"l (1Uz:i)CZ'lG'WZJ

2v!�� 2[(!:)

trl'D66c,� = dv = g dt 3

1'· ;;:�'!��,i-�:!,71::'�:.=;::.=�:,:;::::: O � ij_����:�� ei�_q;;;Ji�� ���-���;�.ttt,�r:hS#�· ca� .. �®D q� A �e;�.¢ i,>����,iu1*) e�ol'J;· Q Ba ;.fl;t �-1;:t' ��,U :lJ.c�i!t��� � �i3� .. �- iti�@. (i) A .6>. :;· qc� · ��� fii' �9 il-(�iJ . h ==·:�, rJtJ -�:·we�. . .

h

0

' '

h'

' ' •I

• '' ''.:9•·

..... ��-:.Jjz-,-���

\ v2 �u2 +2as

v2 =2·gcos!!..·t f0. ·3 \J

v2 ;_,_ gl; v = ./ii

-

2 s.=;.ut+-at .,.,. ::·2''

./3z-lsin 7f =t/iisin.! Q ·.. 3 3 �

Jj J Ji .-/·-· -. =-t

2 fiL 2

. G ,�JIG ! h'�IJgr,fI+-g·!..=z2 vg 2 g

:. h =I+ I cos 1i = 31

03

0 3. ��·��-���·��m�:J!�.P:_. . .._�

�-·�:�;��;Q.��·�x.it�f�� ... ··•��"*1·���:

10<.'';ictJ .... 'iP,.,jl:',,,;at,,m�Jr.anJs:���:··· .. -��� .. �:•!Oie� -�, .e .... �-�·.,. .. � . .,.·.,.¥,.·�- . .. . . D'S��·�- . .... . ,eiw.'6\1Sffl).,�ffl,.4J�;�7'!,.�,�:P-I·�· .... ��,���-�-�·�:t:Sz;3:,q�. (�+i}.:=2 as·�·

-+u

p OQ

. ..:.-+V �W

P00Q : C'J®:msh �odd�c6 :

. �?Hf -r)C:--f' . mu-mv+mw ............ (1)

r._\...JJ'�'\.t";J./ u=v+w f0

v if)"?' J / \:.:_) � )( ��i;)� 9e)J:>(DJ5j Z)(6e)C., :

(:\ eu;;: w-v ........... ... (2) 0

� ===:=:t=:;:��;;:=.w'�;;;. ��-=as�-��-��' .. ��::���-(}8�tft. � S ·81}_ C98fl· �or�i��·�:��/,.; ::��

$iJfa:_J?1$�.,�\1,;�nmn ..

;• u --+ .; .

y ·--a =0

' I � FR� . .

5 dil� =HkW@.(nrl' � (:

_,l:n lOOOH (:;\ --(IC) ·/"'(q) . � 925)�¥-ei;; @e� F = u N�0 / . �v p

__.. F=ma · .'d � -tP. F=ma.�·r--::::> <b

lOOOH ,,.\ # �

. Q)· · · 3 --- -R = 0 ov

,..tr;,.· F'-R-I OO<Rfgsma 5 � .. u . 5

IOOOH ·r-:\ 2000H - --lOOOH : ·· .-.R =.. N� ....... �- .. .. . �. ±IOOOM"gsma

lt,-l]f, ,>r. .u

v '? = #: :¥:"i.in� ._,;s.., 'v ""·..,,,. e.��--:-.' .. _'"". r-

.> ·

.,v-

.,� :\!··�.:-·. '.;'.,:··:21:f.."'f. :--. . ",m e:! . •.c::;-1.25 . · · :, :lf +Afgus�tx .· .. . . -�

. � .. . ' :

.... ' . -

0A=2i+ µj_

OB=µl:__Ai._

�µE lit+ e,�� OA·OB=Aµ-,ia =OG

=> AOB=!!. - - -

OC=OA+AC=;if+µf_+AC

A(2,µ)

=Ai+ µj_ +t {(µt -,,tt)-C,,ii + µL) }_· --,ea------r-----.. =.!.(A+ µ)i+!(µ-..t.)j f02 2 -\..:_J

Oc'::{ __:. 2 cos;;·= J:j.f/r(i + µ)

OC· I= 2sin 1t = 1 =_!.(µ---t - .6 2

�§;:}-�e-G'�� · � ·6rm� ,i£o8<:i-@z-r ::;::>

i�J°3-I� µ-;;c:Ji+I

� �

0

6. t:!t�·.e��-(,�'�\�;"��):fci�iid�.$m�'.qe•.�•�•�E.i-Q·aea-�a�,��-..-a.·.-��-�:�_:a.··•,'*-'•;J·•=i•t��--�,� -a.aJ!\.l:l..·.-�!J'. .rs-�..--··-. .�.;;;,.,;,.....Ja.Qea,c1·'�·-·

.. --� ,- . · . . ;'l'·fc :-,-r��:�v.- --��--W�-� :JI . ."·�

_-µ:��t� -�,:--��-

s B

R

A

1. A;,a·· ··:c··-·· s.�� .. �c::i=-��-... ��--�a-�P(�·usu:C>.• . _) u:i.o: .. �We- ... �-.: " .. , .... ··. . . . '. .. .· --j!;f'.�f.. .

·�:I!) ,t"(A' P(B)ti),;·F(C)�--�---� . , .. • "-_,..'#,'fl. __ • .• , . ,. ""'"'�_.....·_-�

. -P.tAf:;; .i·j .Pt/J}:�ij;.:� .!(�V.�9,e) :fj �,��-�!-.ta,,� �9).�v-��

""°' e4 '°"'""l:l"": P(A) = ! , P(.1,1)�f and P(AvBv�:=!

A,B .,, C ...,_,,,.,,, S<fl.l "7.1:l , jl(�<' (!,) 1/ctt)P(AuBvC)=P(A)+P(B)+P(c)�P(A}P(B)-,P(s}P(c)-P(c).P(A)+P(A)·P(B)·P(C)

-=-+-+PC)---- -+- ·P(C)+-.·P(C) s 3 . l I ( 1 (1 1) 1 0 4 4 2 . 8 2 4 · 8

_!. = P(C{l +!- 3]. = P(C)·[�J fs\ 8 1 8.4 8_\...J :. P(Ch=i 0)

a�!:fE�!:�:::� ��e @e@ Tw l:fZSJ©Oz! 2 25f ©{?� �£� Elm q25ld 5 'd. o::,c:}!)8 25)� ·Wi�k@D.·

D= @ecf� e:1�?5>-C,®, U(=D') = o::iO�� ?@C �'-zS �@D2

u

·:rf2

.o5zsfll!:> �'c26lzsl'. .. o@-€1ow e:ii.§:5e-® e,i!ll'ri::iC,2il:iE> = P(D1D

2) == ; l

. �

08?.lS'f@3 �25125i'· o®-€1iiw e.,l�<=""® w®i:o:,c)t>):,e}- = P(D1U

2D

2)+ P(U,D,D2 ) = J_ +"....!._=.!:__ts\

----- . . 21 21 21 '-..J

9. �1\oq'j't;t!::;.�,;:-�<l!lPt.,._��m ... :�'-. �(0-,.,,,�

,.��:i;·J,;@jj�;#.3��:�-��- �� ��c) 'f: ae eed'.l:cf=-

ep@6:ltD-6> 8g©DGcl w2n 9�-€lo eo�SIJ.:>:> Wz» : l 2 . 4

®wl!lZ1:15CO =y =:i, 1+2+4+x+y+ll+l3::::7y

� 6y-x=3l G x'=4c.,iS Sm�.: 6y-4=31

6y = 35. y G:l�ts,::, Wis) 9b'-€1ii C,e,SJ©zsf G'ID:>@7?.S).

x y 11 13

. ·ei:>2n?S! W®"'ti:r ,. '·-.·�· 4 .. 5 X: 5 ll �.�� .35 < 6y<42 . �g}(b �� @�ey: .. · )'.' = 6 @ID:f ··� x ··s' (s,

�e.,�® ; x = ; y = 6 = µl 7

. . .

5�e25):1e:. ·S2 .;;,;(f:ix1 -µ)2, ' . "', fa,:l.

= $k-5)' +('-4)' +(-2)' +(-1)' +o+s' + 7' 1 G

Number x

Frequet1cy

. f( 120 . = 7 25�.16+4+1+�5+.49)�7

1 2 3 4

I a 9 r 11

Lf =: 50=a+y+35 ==> a+y:;15 0

5 6

8 7

?5)f}·4 ,Ji>@J:JZ)JX.1 =·3.66 �50x3.66=1�3=1,a+2·9+3·y+4·11+5·8+6·7

Q =a+3y+l44 =,:, a+3y = 39 0

je,�<:t®zn: r==.J.2 �w �=3 . G JZl)'-0 = 3 . ��ef¢� = 4.

�· . . ft!-0 'i .·

tf:{a)·-,, :Dl;g.�� <H�--�:-�--�,:h_.� �.:8.6.. -�rif-�:�72�eu;_

;;;�;;;;;;;r��t:*r�::

.. • . ...

. !b)· ��-��:({:��-:� A��,��{c�,�,'/���-.:ca,�,, .. , P.�� ae .. =.t.�'3d.:�'{�:.pe�:'£jdi9�::� ·-4t.l;I);·;:��-�·"� ,� .. :� $ -�& ��\��:��:�:'@U,f.$,--S-&�:8tn_,�:'-8

1�·Bj·�eSet�.

::r�:r:�:1�,:::t:t�t1��

(a)

. -�'� e--�,-� ·p.t"J�":,&'.:r S�·�-��o.e-(''B·· #I 8 -�&e-

=t,.;�� '' > > ' I i . .

·[2/ . U/4g

p

. :u

6 ,'2 Q

ii) Q' qog;ic)

iii)

v1

2 � (·1.)

2

-·2g(;g� ' = 0; © Q �o�u, �:i, ge!,a,co2Szl S�<S�m ellO=> G-BS.

v � t.(J--�1, 858®£> ·bwil,®c) G)ti) ZIS)QZ) 2macc.;r2 0�,a c@'Zf!e-zs! o = Fl - gT; � z; = .,fi.g © G

•

I

f

.... .. -� -; j

.t(

-7{

iv) tZ>:iC'-' Ti DZJ VD Q lfo�f> S5�€ c) oto€zf 8t8DZ'l GJO=die, wDw�mt8 q�6i e-oe<tee! ots�ae,r:.:, :Gd

=(u_�T2)Ti 0 \T2;(J'i�}) __ �;_g(�_-1)G)

(b) D6:ic., P • .tntD S, e-@:fo8D B

9@f:>(B, S) = 9<:Jt:i(B, P) + geD(P, S) 0

I+

u2 ( ,.--=---_ .Ji;-l '2g ·. . .

' p

"'- ·

ts' (i) B1 ?8: 90Dwc.,"2n1e,mmi6e6'2n c�oB IS cnm:f �c.,zm e,o�8. \..:./

(ii) B2 tB g(gDmc.,, c�ec:fan �tew:>znt'BoD 15° e21S>::f-61<..:lt.:sf e,:i<i8. G)

SI

S c> e.:,:1e�21ii1!!1D B'lf, @c,?S>

1�.(al�-- 'fI6>: . .:��, ABC:���� ��')ff{ A-. ·&,��-��"1��-��ed

p:· . '.

so--� ��let�: ·

. .. ·

�:��:e;r����- ···. 2.:M+.Jm,--lm�a ·· ·· · ·· · · · ·

(b>ABCiJ:�aitO-··-�'��,d� �:··�:_�m. ��:AB_�­�- elt:· 11co �- �dc.!Lt{�f -'�t.tdi',ti• -�. �-� -�i�{iilisl m$ -� ·BD

��=t�-:r· :·$D·�··�··�·-

I>

,·�:�:e,� � �,,tc;��UA.��:��:�-�:�'.:���.

(a)

�!)c)� e�e:Jii3 i1"'®"' e��e@�:

A

p

mg

Iii• F

P cf0GeeJD, ED �@Gf: +-

T = m(J; - F) . . . . . . . . . .. . . . ... (1)@

Q et0aeDu, DB�ecrJ: 'jYmgsina-T =m(Ji -Fcosa) ................ (2)

S cf0GJD.O, AC �@m: � m m . @ -gcosa=-(f

2 +Fsma) .............. {3) 10

2 2 · --

�1Q��0, BC�e�: -----. 0=MF+m(F-f

1)+m(F-J

1 cosa)+; (F+ J

2 sina) ............... : (4) @

(I)+ (2)m gsina=2f

1 -F(l+cosa)

=> fi = gsina+�(l+cosa) G

(3) 251,

/2 =gcosa-Fsina

(4) => 0 = F{M + S�n }-mfi(l+cosa)+; /2sina

0= � (2M+5m)-;(1+cosa){gsina+F(l+cosa)}+; sina(gcosa ....:. Fsina)@ mgsina = F�M +5m-m(1+cosa}2-msin2 a}

=F{2M +3m_-2mcosa} -0F= mgsma

2M + 3m - 2mcoosa(b) _________ p __ ..... T.-- .. ii.... R

'\.-:- . • ... -ol ..

mg

e)3: (S:L + e;, ©. =2S1c;.:)l5)�, ©al�©@:zn:

l·.··2-3m• - �2m(Cl f?)_ +

2(aB)2 -3mgaQ-cosB) = O "-.:)

10 � \!V . 2 ... • 3g f) .. = ---(1-cosB)2a

0 c

·d(J 3 ··

G 20- = g sin& s

dB 2a.· · .

•• 3g => aB=-sinb'4

P es):, Q qaae e,�oo::i Z1D825"! e-'(Dtz3 i3-�®"' @(.O�G'@zf:

PB �@ro P qo<3eDc> "'

T=ma8G =>T=m 3: sine: (v

= 2il25ll5J@D q:i��"'

. QO �em Q qaaef)c> 7 3mgcos8-

_R = 3ma BG R = 3mgcos8-3ma �! (1-cosB) G

. • · 9mg· 9riig = 3mgcos0--- + --· cosu 2 2

= 3mg (5cosB-3)f":\

2 . 0

- -- ---- - ---

tl. � .�<P a� So>� �c,� ·2mitd �!ri{d� 9$�� �)Cl' c)� e�.Jd q!le A�6 ·'°'la �tA? ·.,n� A·@.$-� -�· �m e� .il·� e®D �ebd c!ltJt ilmf� � q�d. �� 4'•� a� G'!��o m-f� :P4� � � qi:m. Ali US a l:)21, qz:,x!.

BA oB qtsJ ·aoc, ci@!P .. ��-1') ---,� ; Oh!)._·q�·l l'�,�\li�d�� �� tl1aibeoc! D OJ$jf/:i) u ::=,-lf ���¢ �� f� ��- �� t62:t. �D �l.':S;��iO o� .x �le-��:tl;�·','�,�����:.r.·��o.�,>�·a«i-� mtml $V 0��2n; �63 .X�x-·; �o ,i,2 :::¥et).�:� r.382:lio� ���:1�. {" ;caf(A2��x2)��� &�d o:1�=� �,,:;�. �e q.m�)ts�l �etl �:,,,1Q .A:� &D C!ic:nrl'fh. �

· ¢ri» '9ffl ewtrl ® 863a@ Jf E (."MlS-�. tHS:lCZmtfl..

e:ioe ��sf.>� Cq�r.,·ee �4ti�.� ao ·��al. oo tite�.

q�Oie·�· ea� t:>1���:qtif=�Zi'f:� .. P��-�o·fJe� � f'Ctia �'6

�t®a .� �-Jt{1-cosd(i}} � �•isho.m�.tsr •. P,� �·� &S&,�_§H�:Pi � ¢$),�,�,g,• S=.1:1>�.e,,j'-��·-,,n_e��·¢�'.�-��UJ...

. .

P qoCleDD F � ma. <5>�:i��

J mg-T=m; G) T = 2mg ( x) • · .. ·. 92:i'.l:l:>:>C)Z)J WoGJ�WC., = 2mg

aG A T "l;D'IJ! zlBBe®zrl' t0:> m D82rl' ©@�@@?5'.i

•• 2g g=x+-x a0

X+oiX=O,

·· 2g(· a) . x+-;;-

x-2

=0

0

T mg

���������--���----------

- ao,Oc, <!'lJDl.5a!o, E>�""'ol(SHM) C <>..ri>fs<>: X = 0 =>& = BC

SHM •,®2:))6-$ue, -f ei r-'82Sl 8e,s®, X 2 = a/ (A2 -X2),

@®w A co� e) zn<:tc;o � m11Qco �D-

5a-2 . . a7 2 '·9a2

--+-=A =--6 4- 1_6 -

.;a

G) A wm Z1�:> , A = - , 5 ' 4 . '

_Q . -\:V

X -

a 3a. m251c-,=> X ==0 ==>-. =

A,�® ..x-::-:-- �------- --- =>- - - - - ·. -- '• 2 ·4· .Sa

x=-.-

· 4

a\_---

-

21�

c�-- -

{

A\

Of!J?Sf� 8�o@c) �«>:i �®c) qo�c) CDZJZ):> 2Sl:iCc., t1

,

mt1 =fl-a :::::::> t

1 =-.!..(,r-a)@<63� e-�� ci�.

0 {J) 0 ��e t, ��{n-cas-'(})} 0

-/ .-. a 5a e,c,8® �2S>zSlc.:i =-+A= - .

2 4

. '5't)2:S) 25.1 2g)®"' 2ld

...___. ·------�·- -.-, ---···-----·

B

T

T

mg

X = X-; = a CQS{J)f + /3 sin mt .. (i) G'®tfl m2 = 2: , �2:s>�dGd t)e.,�®zsf C,OWGl!!l2S:lc.:1 2ID6�.

q>DWCZS)GC.,251 X =X = -amsinaz+ ftcocosaz, ......................... (ii)

�o®w@c&� ( t = 0 D.2:S'l !l8 ), x = 0 :::> .:i: = u = � Sga e-0.

0 8 8

�2:S)® P = : G a => -2@ ID:> U=/Jm;

.�e,b@: x = �(1- cosmt) + !:!_ sinmt, �. 2 . {J) . . .

am . . f7'\ .

::::;, x =. i sm@t+u cosmt. 0

I t

r,.i :::).�.JiJ�,('·�� � q;(�.� zR;J.;_;-�1t:.t;�· � ·�±:;� c_:!"c ·'-"°Y·-�es;s-=.:.')� iS1¢:, ':'1�;J-B�·} �":t-�:�. t>� c..:-!1-:;5;:.:,:;;:� :::,::;�.:::<;i �:�·(��Z"!) f58-� �- �tt�.

�(!;J e:;d��=G c;-€<ii'Dct<-� I files-d Ox, Oy �·:::)�;_} -��=�-------�c . . . �{2,�� ·_ - ·--+ ��L�\2, __ �i -----· -·-------,,�

I c H,�) _____ L_ ______ , __ F, = (). n _J .L {O, 1)

: -.- - -- - -� }.� = (6.- 12)·----- I

·""' {o. 6) -----------· f ___ · ·---- � F:� = (9, 3) _ ___ lr------

L 1. i} F

I i:P� Fi �c e-�$a§J o �ec.:,; ::v.; A cl, I) (':.=�ir,;;�� 0C:-) �ow ::em.0 �iz; 5Jtr {iA��'J.. �.

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60 N m !!;;�1e:dv�1� 8� e·tzi @'D e-ezt�:r3.:::.,.

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(iii) &c:: e.°iw® (0, �) C.�:i:16..::;'63 �dQ mdm ::;)£ :--)cs;:;:;�d m:-. ��� G: -� gd�ii\tst 9Bd0:>bzt:lc

;:s,d'i[) ete�. (';1 � c;,;,Ja, ®c6J-:.:n. �£! Ei!ccs-.:.:1 @o:.:io e>!!J&v 0(¥,-!)�:6.:.'.! rud.?� ..... � .6):;.:;;-c::;;:clS�-

/\U//UC tD:>PD J.AC e.:i632:.i1DABCD@88c.'.}©2sl.

8(2, 3)

()

ill;= �ec = �( 21- 1·)(!)

DA=f+j

DB= 2!: + 3j

C(4,2) oc = 4{ + 2j

BC=BO+OC G = -(2f + 3j) + 4{ + 2j

D(}.,µ) - 2£-L'.G)

_,, ___ , ---'-' il.-1 -1 !ID //BC===> - =!:!_�A - 1 = 2(1 - µ)

2 -1

,l + 21-1 = 3 ............. (l)

pi)' .l AC ==:>PD.AC= 0.

C0 AC= 3£ + j

(!) -

� {(;-'-il}i+(!'""':µ t}>(3f+J :,:� o

9X+ 3µ = 3 ............... (2,

( 1) z! � 9.A + 18µ � i1

15µ = -5 � µ --2 ?;\3.�

.,. n....., _,OD 11 . i .

t:.J25'.!W:OJ = - l - -J. 3 - 3-

3

(0,6) 9

y R

N(O,b)--+-

12

L(O, 1) 6

A

0

R

-R

.4

8 · 8 - 3.A + - --"- µ. = 0

3

� \..J'

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· y 83 BOCtsl:>D,

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