· z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo...

22
_ .... ----·· . -·---------- ------·-· · ---·-·-· 10 X 25 = 250 OS x 150 - 750 100/ 10 _ __ ===============a===========

Transcript of  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo...

Page 1:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

_ .... -----·-··· .. --··----·-·---·-·----- -------· ·-··· ... ---·-·-·

10 X 25 = 250

OS x 150 - 750

101)0/ 10

_____ ........, __ ===============a.==============

Page 2:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

-------..--,.---·-

---

1. c>eo· �z:l)ZSJ ro:r;i,mwc., m � A, 8 ro:i C q� z:tit:i)r:1f i:,® 8g®:�gzrl, 9®0 £3ced' (l,'@e-lC;J� @l:S) t56e @l::5@::i0::siz:,,Q°l::, tfti:5), A qo�OO u g@�d @1c� eci>z.rl@cl 6c .B qo�l> e:i®oo,�OC@eo CD'{@D� o� �. A qo�0o®m <i'.Jt.8m 09. B qo(%E> Dcroo � C tra�D �®en �oe @et:i mre<'.l. A 10:i B q�o 90):1:l:xoiB C:)o��l!)�e $-0. �@§ m1.fftl•@m O�:i) B 63 9@6m� �')Q2n2:5).

B ri;i:i C (r,.mo grn.15:irs,63 e.,oaa��"' � e �f). B e5@m rotfj<ll'®m oav C Ei3 e�Dmc., So:i .... ,�f,&.!:::n.

_hl.�0-�0"'--

0 = mv + mw.- mu 0 =:, V + \V = U --- (i)

v - w = eu --,--- (ii) 0 .'. (i) + (ii) =*" V

(1 + e) ::;: ..--2- u 0

A B C

A B

I'

-�-----B C

I J 0 = --(1 + e)· u 5

------·---------------------------

2. :63e;d en:i 8dd t:5cde)� eg�0g,:::>J � C0:l .J6ga o63o90�m�&.! £-de!. �'<DB®z:si' 6)rn fd O c��]:)c.,l".S) ·&:i �a(l?,Dd gdG!'irb.:>1:(}0� CtG,& 61�0d t;tclc?-tJm ee�; b8�ml:1'lD aBc5d fidinzrl 8&3BC� a ro:> b Q 8dd �,de @<t�r.s>D OJzrlm@m �rogzrl c;o�£l 08.c� a � ln:ide6 er.3 �d� EiD qo�@0 gc.ft'.lro@O@ Bod c,,df)!:;)c,2.[;; ,;i)c) @omi)zrl�.

b. Sa 6Jf) �nD1,:oO::rl @omi0.mm.2 �·

-------- ------------·-·

©

b

--f'----''-,---�-----0 .Jg;

-a-•

Page 3:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

0 ca-.. A ,_ l 2 2 2 wv t';l'SltiJ, 1' v = u + as : ,

0 v·· = 6ga - 2g . a = 4ga ©:. v s.: 2& ©

o = &;.r. ©=-, I -,.J

r--

. 7'- _I g7.,2 © UN J - a = ,_.v ga

2'

:. b = a + 2a - !:!..._2

62,,®, h = Sa 2 ©

0

3. c5io@c:.,S3 A, B too Co� cli::nzrlw 8�.@cigm m, m !:O:i M � qo� �D. Al!):i B qo� -W'l?!lH�� q5:mw�� mznttjt:)tw-5 �®aww zs,d qlt:5), 9@0 &5cl©<:?lt.>c.:ii:m ®mg Cqo�, �"<!t·ro<!l'd (il6i:Jc) cl3!:.'0 q-i.& 9®a �� z:i:>d8�e.aw �� ::;;:,Dcl e'.ll<'.PlC� q5m?V:C .o:>zrl�D!Sm B O qi�:i cnm, qo� W:>

m.:rlqJ Sode@ o�® So-ct O'.>Cc.:,l:11 8&x>li m�l:lJ etrx@OtdO (rt63Dt:i1D53CJ 253�ucox,tJecl &:> @1ro ro5'� Cl0>ill. A :roo B �:i ZSl62SJ ml!5'5-t:iJ@Oi:pmz:3c.:, &3b-6o izlB®e> g@:H,.60,d e,�·ZS16� t§c:,:i (il'.55Dmm.

R

M

A c,'<;wJ t mg - T = mf © Mg

T + ms - 71

= n,j;©

T1 = Mf

©

V

B�-,--

r;; b

__ L

0

...

m B

m .)A

lT, * f

fB m.r,

TA

!1 EJ

Page 4:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

.. ·-··· �-,--,:.:... ...,.;.,·· ' ·-·--···-----·�---------·-··- ..4. dzsi25.iwa:i l\4 k.g ,m P kW i:1c:irn eEloiilzri 8� tsi:,_60:::n £.Se..:,c; a ©-l:Sld�c.'.l:iSlm tp�l:l> Cll$ ®ob©orl �@©'

eiti.,� E.lt:::6'Jc:J i;;,O. 6&3 ve�oo R(> Mg·sin a) N �c::r.n !;:rn&d:iw'-!>d <:ft;::.). e,z:;fmo.:, a€1J@�xio.'itSl � zsi:>dedde)O-eli",c.'.l a ms-2 @C. �® ©B·:i�'l!O:>@m � tll:id�f get;irow �:::.dm. - .ifi!>Jt5C.')<;!!- �l?d O:!.">�D ,�:,(jc.:,c) vcmo � �itS ��w-eDmo . :IOO,O�

)

ms-l 00 � 2!1'10m25>.

g:o::;)t;12� &e<:::; F = l 000 P I-' 0

/ F + Afg sin a - R = Ma. @

JOOOP �.¢- --- + Mg sin a - R_ ·- Ma

I' lOOOP

R .. ·A1g sin a+ Ma CD

·· R-A1gsma , :,;.,

: s

F

1��)Ck.3 Bc.:cri <2t•·U(00·-�::,;')· £.,e:.t�c.:::; l:JbJ VD a = 0 f;·.;!") e::ne!: 9c.:?.�J �-o�.,�·c.1 q.©�

" ::: 1000 P

R - Mg si,1 a 0

R

5. �)Z) 6Zl'J1.S) e1ZD25.iwo m 8 A� B q.o1� ��l:ili:rl'. ttf>e 9©c; zsidSq:,rd ®iS:tm G:)25) t::;\l'°lC� . q.5rn:i5':i:ts tmrl.ey)Z5l. @1te!S'l�0oo cr1�:i Q®i:tl8rn:;'000ed l!>(:!@etl. A c> 8d.d0 a �jdtsi w�gd � c��:'!!lc:,Bzrl 253dtien:i:,E>®cl Bc:i §,;, r;:,,5� e{ dt!ldQc.:i m ® $1 c �D:> e;@oDi:rl' �d1d'DaJ o6@d i3�a'>@d ue:mo i!'} A o®m m,:., i:o, et:i. (6ioo @em<5).) A 10/J C <ra,d ®lB® Bh 0� �:>�f!l:>@cl � mm�@!l tpei-!l�� -� �!!Oz:5l m'@�)m @.®:i$�1o�r.:n£) oe B coo ©t11Z)� �$-0me!l � �6t.'.l £�€'Q g©J�Dtrl e:>�r:.,o-6 8<,.');) t:;m0:':i")w.

v2 = u2 + 2as J @w�c;,'®l:5:i,

(I rod' 8,oe®� C (3Qb (5)1:5)1:i)J g08(l)c..:J II = & ·0

H e<?,a?J j J = mv. 0 ,-\ ;;JJ C t.'.l'<,<>)J t -J = (m + m) v - mu. @ •

,---�-�-,� ........ .i;:1, - I :.: ·) nn> - rn ,! 2!.1a .

J Vt

J

l2sJ

m ·

a

m m

Page 5:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

6. 88·.:51& qo2:l>mecm, 0 qc:ic; §ec.:i�o q2a@<weow Am:> B G�lif:1) e't:;:z:ill.':i) 883® ecr;,��i:n [email protected] + j ro:i 3i-j �12 m:83®. A 6c = A OD = I c!'.):'.) oc = OD = ½ AB Eim 06� 8 C (!):'.) D g63m:::n e��:t)Cil'\��� 863Q® (00'·(.;(Ql':cl is'C'.l302:>)z:;),

CUi:S)ZJ) :

OA = 2i + j-

OB = 3i -.i- -

:. AB = J\O + OB

= - (2i + j ) + (3 i - j )

= i - 2j 0:. AB = � = .fs_..

OC = xi + yi cl8 en£@.- -

OA J_ OC £3C'.l:>, (2j + j ) . (xi + yj) = 0

C A

D

:. y =-2x 0 0.

' I

�-· a _, - Dr., -, .. A,-., ' a, e. G'(!;® r5�2::'lu� c.i OJ�>e.;J,:;2:Sl e.:i�;wJ c; v(';c(;"). 0u.. .

6€ £:it:)J, X = ±

� = l

· XX -

l©·, 3 2

y =

=

3

3 ) 0

-2 j :m-..!_i+ 2 j 0u.

3 3 3

B

Page 6:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

-·-··········· -------·---------·-·---- _ .. ___________________ ·-----------------------------------

7. Bot::l tj@(D S,?C?1.::.igd' a 1:0:i .'.!. �z:.:d-0n e:1,·25:i AP ro:i BP3 ( t.'l"teo1c1� q/5��:i:; md;::u e��d ®8255 8J6e! 8Be,®Bm t•('!c;i tt,1.B:r @6 w 8 p qo�&d, 01bGd i;.1�@-f)m C:J� e'..\®�Ge)t):JO@d oua. AP z:,,zrl.::)J�\� F.pn1£c.,, W �:) (I

t?IE:·115·dd° @-t�:iczrl:251.

e;:,@ g@·®t:.'.l�c.'.lz:>i,

T

. JT. s111....:..

T, =

w

. err. J!) sin --a.+ -2 6

w

2 �in (;+a)

(1

@

0

8, �co 2a �J @d W fl �z:;);>Z$l:i6 AB '1'..eGDd bro A G-��00 dt:J 61dd e<l)@®:d @rn <; B @:!Sl�.vo 9€-0 8dd i:bimc.'.lz;:i:>0 6"';j€o0 <; t:.16:1:, qz:::)). wm£\CJ0 c�@ 86d oec.5o .-;-EfiiD e:li!l��m:.:."l:>E.)&d ::si@J (J"(d@:ti-:! A 1,H::.)f�D0� � aJJ::dBC.'.J e<;tilD @<..'.)� o(Q:iedOc., P ui:n 536d aec.::,d ®<£il5'5. ut�@d F ti:t:i R ®.Si:.� 6geE)gm A 63 � ec'.:-e5.::t-tGi c,)ec.:i c'):J qBc�·cl gB@��D i:;d8J mm. B &'i � w::.rlB.iw

,s,.J� q-1rn �o� 9trigc.5::it), S,e:;®'d <!:'OzsiD::i en£ oBt; W c;_

2

Jr

3

i Jr 16

i Ti

\iV

p

w

0

F A p

<;6.D ro:, @-:£ai@J q:::nd :;e!'5a!! -0n eiai:£.4!):';�2Slc.:i l. ·<; Z5)/£1, W s P s 3 W @0 eei��m.4 4 4

BG·t>:fr;::;-i@C.::25) + R - w = 0. 0 .: --- P+F -·

F = W2

IPI s ,uR

0

0 w - = 0.

PG)

©

1w

=

=

···----

- ,PI s l w4

1-

w

4

\V I w s - p s w 2 4

:s; Ps3w

0 4

·- -- - -- �- -------- -·---

Page 7:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

9. A roJ B c.::m fJ 25301.� qDZllJ::Qc:lts) 8<{w G>�tl)� 018 a:,£§. 8Ho\� qo�zs)@c.:Jm, P(A) = ¾, P(A n B) = �­�J P(A' n B) = /

o �D � en�. P (8). 'SJJ P(A' n B') ee:i�c:iis:l'�; @@63 A' 100 B' Dt)zrl 6@..&vg:d A en:,

B 8 et22?f!d� B<w <;i.��1ll.

P(B) = p ((An B) U(A1 n B)) = P(A n B) + P(A1 n B)

= 2 +

To

. ·. P(B) = I

02

P(A1 n B1) =

=

=

=

P(A1 n B1 ) =

P ((AU 8)1 )

- P(A U B)

- [· P(A) + P(B)- P(A n B) j

I

3 10

- [.l_ + __ l_. 5 2

-

0

- ·--15

0

:.,

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

10. ('51::1) ci�cl 5 CJ l.fS Cm mole Orot:;)0 ®:JZ!>C.':l2l1 Gx_;l:S)i:,i (n:ffi q_.!'.JO (r):.'51 6:!l)ti 3 @'el, t:51::)3<,,·d ®W:tl<7Jl:ir.:i l)):J

@0:.tJdc5c..i c.'.lm 0"<;� 3 Cl e:>®J� ©�. 0B·@ mi:Dc-; oco �·CO::>c.'.lcr.lID.

a,a,3,J,4 0 • b, 3,3,4,4 0

650 2u + JO = 15 ; a = ; , # 0

ed' b + 14=15; b 1. (�)

-----··---------,.·--·---·-·--··w-••w_....--.---·-·-----·----.,_ �.....,. ... ,,.,,... . ._,.., . .,�-..., . ..,, .. �. -.... , .-�··•�-_,, .._,.,,. ,r,.,.....,

Page 8:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

·------·------- ·---------"-·---·-· .. --·-

ll. ((J) I' �:i Q e&:lo!'.S· oo @<;O)z:n e::a9 o:x.5d �01:!J· �oc-i z:n't:)66 tt:,63tl'.l0 o::;:i® �(l;J:>0t:l10 vcme!l csO.?DaCc.:J t = 0 @ �· P w gG-b�K..:> u m s-l <.; Q � g@0mc; {u + 9) m s-1 <t 0� .. P ·5' z53c.:itl'.I cl"06�t�

(a)

'\[) -

f m s·-2 �· Q 63 �w;:ri clC-l'��c.=. (1 + _!_·}, m s-2 "i �'�.

10 (i) l 2 0 P"tc!l:J P Cl):J Q §3 VCr.l)�(::U, 6:�)® rJc.Oc.:Ji::il El):J

(ii) r '2 0 t:if(;tf>:i PD e.,o�d��8 Q B e8r..,-:ic:i(), e>tJ:::.:i@ dibc.:i?.S).

g@t·a:·�ui:1:>('; DgDe �f; roDrozri <t�dm.

l:.):JC;C': I == 0 63 � P ©·@JDb' ddc.:! Q 0@),Scib' 6oc:io QD:i i!>Od 2()0 d {g\�6\!\'Clm 138 "1.)t:; .rnt;f:6D.m � q1,�,. P o�l".l'lo c..,,·§D Q @&� i£�9 eGI� .�:,ec: ®tt-:iom�.

:;j,5:i q,{),Jz @b:i § C;dlti.l'.l c.,@it).�ds;::::,� 0t� 0D. dec.:.iO e.-;�10eld�D fr.ice, v(> u) 11:,t::i.::..?\:¾·r..:i2:.i ve�..,� [;;!,') B, �::i B� G·ai:l'C_q @·t.;�� e:i'5l® @•!'.heeo'.l.t,)� . - -· ---:i,. A Ela ·tjt):i@d ©®:d tr:1,J'•!!ib") �d8. B

l �f.idO�)D t�E;.�;0 .AC �(:i{f: C �\815)

---,

@·::n:id '/l!�1b8 CD 'f.11.!."):i:.00 (D,:tr, �12:'«) (j;>l.l)�U D (i�,o� ,CJ,3. B� �.@jc)s::;t:; ��-��El -·-� .. --• " ·- '-·

D C

�-4-,---- ,4---.... '' ,

' ,,,

' ' AB ��:.Do (;I):(� �c� t::i�(;.c} B @81:, <:>i.n::id �:do�l B D �ili..:�:! 1) @'t.W'.) �fa __ ,._,.,t---·-····------+-·-. t"):::i')•'.0 cir-:::',(.;'.1tS), n I 8 A BD C <y_dD:i <; B

2 &1 R @,) D <;bi0:i (i Deb) �·�I'D:)

80'e.>m f?:_;;,;;s);\�;:,:Je (;E, e:if:)::02,j qz;�:f.:r,. A B

\!

. j

i;.; /.//

,,.

9 vzJ. �,6�

0

'J·--------�----... -------�--------¥-------------------------------------------------------

l'.(Q, P) (u + 9) - u = 9.

=(}" 1) 1·-1 a ( Q' P) + 10 - . - l O .

= 200. 0 ===;;,:--�-·

12

Page 9:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

(b)

=

_l T (9 + 9 + _l T) = 2 . 10 0

T 2

+ 180 T - 4000 =

(T - 20) (T + 200) =

T > 0 Q)le3.::cl', T = 20. 0 D C

I I

·/

---·-·�---»----"----

�o�� �

V(B;, £) = /("'-�, �)

V(IV,E) = .-u, 0 V{B;, JV) = 1� for i = 1, 2.

200

0

0

V(B,, £) = V(B;, }V) + V(W, £) @

= VU,V, E) + V(B;, TY)

- -

= PQ + QR; i = 1, 2

= PR. i== 1,2

©

0

\

0

---

R,

\ \ \

@+@ r;;-·J --- - - ------ - ---- ----- ------r---------- - --- ----- - ---------- - - ------ - - - - - -

® ! 1 -

Page 10:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

PQR2

§@:5)1.@f.',ed,

PR2

= JvlR2

- MP = //- (1[ f -/2

= ; ( � 2v 2 � u 2

- u)

--}lo- -

A 80 C �dfo AC �ed ec_s-;�':>o e3&0 ,m aGIU:lD C 86 D c;d'D:i CD �®-d tle;Z5:iw e5@D B,

�.h T = !.:!...::L::.. I PR +

l v-u ·0

--- � A 80 B �do:i AB �Qi'\;) Dezr1c: B®D e0:i u12;(c.0D B 8D D �dD:i BD �0,cs:J' vc".,a)� t'.,,vu B

2

T -·

2

a \I+ u

+ a-/2 PR2

0

·· a/2 (-l_. - _l_ ·). - a(_J_ __ - -

1-) 0\PR_, PR:. V - u ,V + u

. r;;- (PR, - PR2) 2au uy,:., ---- -

J)J) /JR " \, .. " v -11

a.Jf.ffu

l [ , , ' 2· (2v·-- u-)- 1.t"

22au2 0 V - U

2au 2.au 1 2 -

ll V -u

= 0. 0

I 25 I ------------------------------------------------------------------------

®

�-=============---========

Page 11:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

----------------------------------------12.(a) 'oiei@c.:,63 ABCw:1 LMN&'!S136, ACB.=LNM� n: roo ABC=l.°-ON=� fl BCro':1 MN1�€'J ... m

. � .. . . . 3 . . 2 ai, 't" -�

�J�Jelam e®D 5}d,d ©<3)'9@.� ®m . .t))Ei)Zl) C'c 8�®e)@m X �:, Y c;ib0o® 9®0 d?:S:>:l2::t\l6 �i;:;;t�� G·�l!i')::::l)

roo1clD edzrl� rQ@,2:55 fJ Socl eovcll:SlD �. dz:s,ti'.!'Oc:i 3m � X ���c.:, �)ai® @m 0ezsx:i D@D :5:x;imd f;m q�d y ��r.a;c-,: qc>ee) .tl)o)J q1,1:n. AC IDJ LN' A L.®·�ob tr<?@ ���df5e ceiB@ $tfi® e{5o.b ®!). A ro:i L 0 c:,5tDo cn,53 9®e:. �a:i z:s:,d6 �l:i);';'.,1 @)B.i:oi c.nn C>tro1dr:; <tu::.i')m.X> mmqi.01:S> �cuz:s,�t)o d?S>mw 8�@0gzr$ m i?O:i 2m g Po):> Q (fo� ���c) (fl�J cnto, <;z�e;;! q6� (pd(!)wi:n X

80"®® �. mztl�D ¢'m�d1dEl �J j!P = AL = LQ = a em 1 !f.-

3m

e-ee.-:i P ro':I Q �o� 8g00[;;zrl AC-·� LN ®rn �dD.> mao -c...;;'=3=· ==-· =-=-1=8'--. ---...c::;:;:

M ,cru:n. �{w:8ic:J mcl0cm:ic)�� �'x:i c:p6� e1G-@. Y @Om . c:x,®ei X m� GQ)� t'.l):icc:i, a�,:, g cne�omB°ti�"' BB®o g®o6Dd .e:,@�66 c&:i ro�:,).

(b) 6to�d @om0:i {J'V$ e,� 9®0 B8zj ABCDE &O�cl :Sddtl)CG.'l?::il ('JjLJt:i)Ci cno,� �©.2'./5.a -� :A.B f.!J:�PO �a· �Zi), .qmd 0(.'.j B @ � qo� 2a Q .BCD£ t)a:m.o-x,t.,):,d @z:iJoe,a doba;ii::D @D. A i.'!)J £ <ydm O <,lmcl!;c::.D. S6e:ID w�l?,d 86?08. e&zsiclwc.." m ffd P qoCiJ{;.)::s,i A @ � @c)c.:i G:lG ti)� 6,cdDc;.ti'XJO@d Be) 861,@t)m

iJ<r'.l roei� c,e-�. 0A �(S) 0 (; < () < 2:t}�l:S'l16wd ill' ce:>�::n.

88 P q<>�-'P't.1 @0«>,�, v c.,�zs:>., v2= 48a(2-cos0)&)&�d -©-��

(",Q)rn @[) G,-.e,.��f.h, t')@ ©®:i&CO:>C,!HjJ � p qc,\}0 @m @6G'Clro· 1.r1£i1 l::l)\JZ,') gB.ifili::x,t) �,'1('t,JG)2fi'.:h�1.

Am

(a)

Ace of (X, E)

Ace of (Q,E)

T T

= �

th , .

.F

( .. f qe)c:, £00)

Accof(P,.J.') = .f-F/ f-F

�F+·�:. Ace of(P,E) =

P qoC.?,€\0 X /s3 0{3::>S)(:J Cj�cJ)J ; •;:, ·@

' /'-- F . 1�

-+ T = 3 m F + rn (F + ·-2- ) · . :,

X + y + Z £3c.!?.5'l�5.

= f-F

Page 12:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

(b)

Q $ ve.mc:; O�ef.>J

T- mg {I "" m (f- F +;)2 ,.

� 2mg{!-T- 2mfl @

@

( s = ut + l ar - for .A')' 2

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

P.E. = 0

l.,n/ + m� (2a cos0) = 0 + mo. 4a tis\ 2

c. C, �

v2 = 4oa (2 - cos B), � < 0 < 2n fs'0

. 3 �

mg cos0+ R = ;::2

= 2mg(2 - c.osB'J @+ 0

=., R = mg(4 - 3cos0) >·O - (i) 0

:. G@@ g&�..'.i�D O <:n:rl';:-;i'�(..'.) G'Eimc) G·0. � - ---- - - - -- - - ---- - -----·--- -- - -. - <J1:f - -- --------- -- - - - -- - - .. - - - - ---- � - --. :;. ----

Page 13:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

� 6 mg

S-mr;cos�=" 3

S=

m (0)

� 2 0

0

r-1I 20

J ------------------ - - --- ------ --- --- --·----- -- --- -- - --- --------- --- - - - -- - -

l3. ti3<:5eD : �rsd�c.:i£,.z:r! q:;2:)')� �®E> .qDe .o:>e.ot:ri

coB® o>tfl® e:bohD.cl ®m OA = a 1:0:; A.8 = 2a

0� 0� 0 oa:is® e��bO @('::� q1&fl 0, A roo B C�lllll'! �® 8gui'0�25i 8&lD, 4�t5.I. �Oo��e� �� a eDJ !;.o:):l3-1e50t:D) ®�:Jo?:5).:.:l 11/g fuf t'Y('flldf;; 9�1iJ/i0 mm'.�D� C.i;<$ @.z::_)��dd O Cd.4t:ooo ql')J ':rt 63 qmd q@':ncl @ZS)�t)c do<5:iwo m g P q<>(%c)ZDO 1.n�:1 tflto. P l'f,(%0 B ecl(IJ!;iso ZID6::i smo 025) e1mi::rl mzrlqiD OAB (!}bobEi �� q�Y.) ei@Q). ®�er: P

B

qo�D �1££><3t:lX>D0c.d' 80 @<t:-, IDB� Cr@S: B B::) A i:;dDo P ta vern .G'.l·$i:DO-emo, 0 s x � 2a !.'b�el:l:>,

x+f (x.�£)·=0 ®&l::s:l e.:;� c@m @D �ens5D:d:s·;·<:;-0)6°1AP=xe0. (1 2.

11 - x+!!. c.:l18 e-ro� (jlroc:i D.�tlJ. �®oo<eric.:i •' 2 1-:-

- I.; .

@®f.o (V = �! (:l'!J.

�e;:l.t:i) e:icc l'f�Db'B Bez:ueii'c:f @cl:d�.x. C:t:5:)�J ·i ;,: o.>2 ( c2 - y2 ) �go t5'.i:).5��(.'J�), c 5r::kmdc.:i a')J

A �Om smo e)m �O P © gwOmQ @!.:l:>c,z-.im. . 1-�-

o @e:>tn (;im� �� .53 P 0 ge-Dax.e ...;7so @O (!'\'t:i�;om�.

B So o i;23i'DJ vc1')c> ti®o P o6�zj CD<?) cf.;;;::") ZD3eQ ff { cos-1 (½) T' 2k}

f:i·€•6 k = .fl - ../6 e-D.

@owi::N:.::. OC>:> p (f:;l,lO 9�;:!;��i8 �.,��;,.?.S)(;.':, e G-0. q�3ir_s� IOO>;)� @B i$·ed0d.�.

®

Page 14:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

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

0

d T+ mgL = m (-.t - (i)2

@

T mg(�)- (ii) 0 (i) 1))) (ii) 25) => x+� (x+1)=0, Osxs2a.

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

y = X + ; 8e5@®2rl y = X Gi@al. 0

·· a Sa ©y + aty = 0, 2 s y s 2 .

e@6 at = L cttu.

• 2 C e3r:!r.mc5c::,, ec,2,3 e;&25) £'.10c.:?, y = at (c2 - y2)

e®B ((/ = 1L ©-eJ.a

B &3�. Y= �1 Dm BD J; = 0. 0 :. 0 = oJ(c2 _ (5{1 )

2

) � C= s; _:©

( , ·,) 0 0 . a 2 g Sa - _ a - -A$� y = - , u = - (- ) (-) . 5 + .:, . .. 2 a 2 2 => il=V6gcl.0 -------------------------------------------- -- -- ---- ------------------

Page 15:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

v'6ga/

? 0 /' v- = w + 2 s G'c.:l�@®zrl

. (

/ v2 = 6ga +2(�). a 0 :. v2 = 7ga

:.v = V7ga 0 ·-----------·- ---·-· ..... - .. - ·----··-···· --····· -··- - .. ·-···-··· . . -·

\I

/ n mg 6

e:ioc qiavo� v8b,C5 C5cl®m B Bo A c:;zs!v3 P w��3 11

;:;;)JGC-',

a w t, = a. f's\ 'tlzrl cos a=

2 = l . 0\::Y 5a

v = u + at @c:,�e@zrl 0

/ V7gcl = V6ga + � 12

t, - 2,/f (../7 -V6 ) 0. = 2k [.f;. , ��w k = v-r - -v?,- . ,g

t + t = -,� (cos-- 1 ( j__) + 2k. ), 0@;6 k = ·./7 ·- ·./6- ·I 2 g 5 .

v 2 -- u2 + 2as @c.'.l�@®Z>i, 0 J.t 'I'. O = ( ev )2 - 2 (1 )z @)

Page 16:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

O< e < l 0- fi.. 5

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

[�] -···-·-·····--·--·-- - - :: - - '":' - - ::- .-:_-:_=--::-=-:-�--- � - .-. - - - - - - -.- .- - ----- ·--- -----...... -......... --""'- -'!'!--.-- ,� .. -- �-� �-----.. :-------�--- '!'!'!------------�---�--------··-

- 14. (a) OACB c.:l22)· �:>:m1:i)c:h9c::.� c:.i.18 -'t D c� AG ®?:il AD : DC = 2 : 1�m b�, � e.t:n'�:oo c:Jt8 "t©£3@. o q�@<w<?.�w A !!);J B e��;i,De 8�® ��$� 8g@e§?rl la �o. b ®I!>; @®@ l>O 0t'.l.

I

(a)

OC co, BD ®&<;@.t:S), a, b to� A q19<''6� g�il"1 r.:noIDm.

lie�. OC oirl7':l BD o .c@@ �e �18 m�§. 3 jaf ;.2 + 2(a · l�)i-jbf = 0 @e <!}O�eY.i ·

lttl= lhJro, AOB = ; �. A a, (fCOQ �:1aimm. '" .. . .. ,,,_

(b) @dirltc.., 0 et):, �'{�tn� �(O 2a rJ ABCDEF e::,Bw 111iD9(:jts, J!:DCi:>c:>63 §;{ @e :q}m£3w o<D£k�d o@�rl) I;�ll. §Cc:.3 0 ti, "t'j ·Ox-q�-li>G:l OB �@..:!I <; Oy-qdf!lc:J oJi �@<D t'j_ qv530 _@e_ �"):J db:i(i'd @o:> e�e:e, Isr-:101.� qal!S)25:l@ozrl, orol:i)_ 0��� <;d&1 ql.i»; @®€3 H �i;J CD@ ·®D� 6��:i,:;:; �. 1

1_(P 63DO� De,zr.i t'j a ®c.>d Dezrl <f ® . .55� C-c:�.) _ . _., ... _ _ _ .-------,--------.;.:., - , .. -,�� ... _.. . I

0

� :---

�J5a( I 1&�r1 1 ., -- :- �:a:

3.i

__ / ;:�;,tj

JI

C5�0ffio 9m1�WZS'.IU q)(S"l;l ___ [l_.t)_fJ_D_�-ei-���:-8�-®-��d -�6�6 �-'.6-"'�� I t;td, FE �0cl @c:.lY �om &:icdtJ6 6PN� q.£®/'ji;) @ec68!_e®1) �<D£:i<.'.i0 q.·1��d 1!110� C-1®�. I '"" <>(W& C"''"'''-" ,:;,,, m"l o>c,,.d 5�,:,o:,o!"ilc>, �i,,1) .,j ,@"" ..&.l,v ""'""''""'· ..

_j l3

0

),a

=> ()·.a + b) . (Aa - _!_ b) = 0 3

..

uc =

;,,;,;.. oc =

-7 BD --

=

� BD =

6n +!3C

All +b

-� � BC + CD

+ .!_�

,.a CA 3

t,a +1 - 3 b

)..2 \nl2+(1-l)(a.b)11.-l\bl2 =0 0 c·:b.a=a.b) · 3 3

= 3,.2 \a\2 + 2 (a. b)A -lb\2 = 0 0 I\

\a I== lb\ �J AOB - rt lj?i 3 �

0

0

Page 17:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

-

-'-

�_

-----

--- ---

-----

-----·

--

--

----

-

--

-

-

= a . b = I a I I b I cos 2!. 3 0

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

(b)

. ·i��ZS) e:i@�o�®c..58 [email protected]'$-c:25),

.\ . .

3laj2

).2

+ 2 . ..!...ial2

A-ja( :!:: 0 2 ' ;

. : �

3A2 +). -1 =O 0,'; - 1 ± v I+ 12

'A=-----2

'fu A > 0 ahe'.l25i A =

13 - 1' .

2

2V'3P .../Jp 2a

F A 3P

/iri'\ � X = 3P - 3P = 0 V

0

0

X

@c,J ez:::fe;,,isDc Stag@ 00.:;�2::.)

l02:>'1©2SJ,

-

OA =

-,;,­oc =

"""?""

OE=

ai - ·IT aj

ai + VJ oj

') .-,:.(7)

} ,11,f"' o es �® c.::-c.wms .;;;d(;'c.�-;:;:,:) me,:ti 0-/J.

4 Y = '\/JP + v'3P - 2-v3P = 0 Q � . .

� o_) 2 x 3P. a V3P + 2a V3P + (2a). 2v'3P = M = 12a -V3Pj @

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

0 K� ·_ · r;;-3 ·

� - 6P x (2a + z) � +_ ! 2a v3P = 0 ®

�z = 2a �v

0 �

- - - - - - - - - - - - - - - - - - - - - - - - - 7�. - - - - - --·- - - - .. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . - - - ..I ' ,.

Page 18:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

-----·----·----·---- -----

15 (a) i::o:?::.l t�� �c.') 2a Q AB '<S)'.) BC G:c5.l::i�Jd .:;@ ®<'i�� Bro � s@c:> 0'Ceti e.'.l�W �.:5 (r£e), AB <'i�cs-D @6 W � BC �-EhieC ah::5;' 2W. c; �- A e��Do lfDC ccl(l)b�D 9®0 ece qt:.>� 'Z:l>d �7.0). AB a'.>:J BC <;<floB l'tzi B6t:i e::.@(l) Bg(i,-ug� et �:i fJ e2::-:d-E:lii �:i<;@zrl 13'®® e:;{wtib Bod mcc:,:r:;:, e;;@-Q8.mm:iD0d !j')a):J rnm@O), C tS3 � cfioed eo�t):i

q1.& BC Ci c@@ ��:>U ®d&cl ec.::.� -� @cc.:!� ®@�. /3 = i @D

(�H:::25)'i;:i, B l!.i�®d e' AB c . .a[j ®mm BC r,@$£) ®i:si @o:icro' ' 9t '

9Bi§<.:i:i0D8 mdd ro:i Bod �c6D;:i ee:ic:.,�t\').

tan a = .fj @E:iz:rl ©ei;,iO�:?SJ.9.

U.l) 01o@wro @oznf;:i tn,z51 o:i� e:,7£iec dD:i@c.:! -�t5.'l�Odue � 9®0 e·ee:i e:;�w zs:,,b AB, BC, BD, DC:11:i AC t'6troze� t;<g ooo:B� e:;®�5.tl) @D.

A

(a)

0®� AB= CB = a <'i CD= 2a <; BAC = � <; @D � �ni:». cb� �1.5de A @ � qve; c!S:l�J:lc.:il!:S)o 5®D 3'(51:.':> q�I:) 1::\"'ld cn.t,'), J) (-:,znwed � w �'.)QC.:,� t,ec::i, AC Bode � CD BddD � <rztnc; Bod �G'.lr.) 0'.l� oz.tilde c:@gf::,mD m6J'.l cnclG-d C e:;mw@d � AB ,;�·:00 c::i@:i:::rlrnoo diced 0a�1t)J crz_e, �ffi::il0D @.:.:i�

p Q)cccl ®i'.Dta. e@;l qcl:S)25)� tn:>B�ec.:,m D, B W:J C c:::>z:i'523 e:>(;ro:> 9�:i::,:>&e �f.)rom� q��tti. 6�.

(i) Cp:5:lttl � ©tx59@ <; c.:lZS.hS) go·XQ �d@m <;ig o@ro@go,m�c, ID.)

(ii) PB qci;,cee:i:i,.:n�im.

1\

\ I{' . ' �

B j �,. (2a) == 2W. asin /I QV: >·

a

1W-C

w 2

. . . l sin /3 =

2 .

BC�\m

_.._ X == W . cos /J = If W. © e 2 4, -

t ·sc "�"" : Y � 2W - !i sin /J © = � w. ©

w

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

@

Page 19:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

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

(b)

AJ X. 2a cos a - Y 2a sin a - W a sin a = 0 @ => {3 cos a = 9 sin a. ®

p

0

tan a= {3 ·9

(j) B

®

@

®

. D

C

0� 2a

® w

l/

<;-$':jW

AB

I .Be'

�---� . .

<p::;)£ki

4W r·-

,1a1W:Jf·v ..

_ Ar�L_�\/ 8/) I 2W

<:.·::,')6c:;@ '-'

·-

-i - J -

I

I I t.:___.:..;_ ::..:...' ------!---·--·-------� ', 1··· I '- ,.. I

. . . C} _; -y j VI'

I1. ( .

P=up= 4! ··@ ,v3

@

Page 20:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

(?.otll, 5cnm e�ml1tsis,l/l,rl�[) �--------------------------------------------·-

· 0d2$5�w O roJ eek� 2 a 8 �� o::si:ii::n:id q�w @{noe:>t:.):>O l!il®@:ic;zmadib@d <_1d00m o� �CD 2.77:a 8 AB �8 ez::n:iOe:;£� � BD Bi�eo"' ABo e@:·@ um 06�, q.6c..:, a B . .BCD r.;-bw Dae-5tll:i�:io Gl:5):>oCj�zi"! <'t 0€•2i·mti-:ici'o:1�:.id �®�c:.,£:d e,:tl�� GI� ABCD �B @0cl 'tac @'Ce! oB B5�.mi:r>1�1<;d 0:)�J cn;:D. A ezs!.l!_l.llO �bo @ro:5e�c:' (l)(c)c) �O) q.z63 qtn6 OA 0�25)

AB 0 c®a Ci OD �mm AB 6 �®:J�md li et. oD � BCD ozrltn. OABD ro

tD('!@d 8BD:J q-it.>1. qbw @cri3eed ��!:Sl Bbi'.!>O(:;c.:!l:Sl d!!DroCr.5 a <; ;Ee,()$d�� �(f)i:S) d��Oc, aa � ,M!). �1:de< e5?!:l:d°W (:N:im�r.5, OA 80 Oo'.lf;,c)-1 c?- { 8;r. - 2:r2 - I) a £!dl5lm' � 0 �?o D aioe,xi (.:l� @b@:,� B0 2. a $_;66:d'

;.,Jr:\ I 19 <; 89ch:n 6)0 @eimOmtti.de 8c5d @®t:ic.'.l� ®::n, qbD em:lc;J:::nJd daol.:.')� de dt:lb� .:::ndi!hrl, �im�z::i&:i r.z-c.m. �bw @3:iiC:•lSl:.id �a&!�� �,, �®l'.5ai �06 ��*-6i eo�·ESi.?!'lc:i � B .. t\O �r:i.'l:1DC> A 8 � eo·.it;v,:1 e@�) 6\od @cod f!1G3� OD 8,:5�D qiB.S e!ll25:i�c.:11':iiq1f!1.:."ltY1:'J<)Gk.:� t:)t(.j}(.:i /!j,'l'ii5 fiill) s�i,,f;25.fr,).

B

A

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

(' \ \1/ .. .,

)

OG .., x b13 .:vc9§. cuo = .,.J apacosBdB

X

] ap d() -y

asinej�.!!. ----

© =

- �l

© e' ::n::3155. e:!';::;,:biw 0z:s:/:d'�;:j O Bo 2a �oB25i 86308.rt

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

- 25

Page 21:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

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

(ii)

x = ] 2;r.(asinB) a oa cqsB dB

=

=

-7 2.1r(asin0) aa dBI) .

asintl/2

2 "

l

,T

- COS 0 2

©

. .

. ..

· - · -

Dd�E> dz;;)zj'C)� ... ,:.,. OD(-) Bo �o

AB e:la9 G·Z»ooe:i Jf,(12 0 0 2a

BCD q.bw na2o

© a

Daclmotsxio G'tDooc.; --

q.bw G'(i):fc;o.r.»oc::5 ZDG'i:ih13

8na20 0 0

er.,1.::5)'� '19na"a 0 --2- 5 X

y

· - . ------

OA 0) &lo flo

na

2na+En

-a

)'

l9na2 o y2

na2 a. na + na2·cr: ·( · · 7 ) �] 0 2 . 2rta +' ;t + 8na2 o (-a) v'

19n .Y

2

·y

l9na2 o x2

-8rca +2na + a

- 2 (8n - 2n:2 - l)a

19JT,

rr.a2 er 2

a + 8na2 a . 0 @)

= 2a + !!.._2 ---· Sa

:. X

@

Sa

19

--··-

CD

©

CD C

0

Page 22:  · z:,,Q°l::, tfti:5), A qo OO u g@ d @1c eci>z.rl@cl 6c .B qo l> e:i®oo, OC@eo CD'{@D o . A qo 0 o®m

�)625) e:>·,fje (618LScJ' (,oJ©e:3 68,8::ii) J

5 - 15

15 - 25

25 - 35

35 - 45

45 - 55

9

ll

14

JO

6

50

143 so

s

�O:is

e::si<1$l5(.'.) Y

"'

X

JO

20

30

40

so

(7 =

2,fl

It

.J 4001 .Y

50

I

2

3

4

5

I Ta

x

0

yZ

I

4

9

. i6

25

489 50

s

fy

9

22

42

40

30

1.Jx = 143

0

.. . . ·-··--·--·---

µ = 10,t = IO (143

) = 28.6 61B�e c;ro@d 6DJ f's\• ry SO . �

(= 0-[. 28600)

(= 61, 12650)

5

,,,,

.f.l

9

44

126

160

150

1Jx2 = 489

0

EJ ------------------------------------------------------------------------

29172 = (1 +L) 28600 JOO

29172

=*' 286 = 100 + /J

J-t-_ (T ( 2 ) lOO •

51

50

(I+ P)µ 100 X

0 :. p = 2 0

-· -·0 ·------

x 12.65 \:.J

0