a em,x)m 10 x 25 250 eq)x)m 750...author: subject: combined maths marking schemes keywords
TRANSCRIPT
�· m� P �· «l3 •� AJlll'1 Q� 8� itim v �. d� �aeo. eSc)�ea,Md$ai�!OD®��itilif•�a0.��·c,s,P�v �·�Q�,.��(Q���.t:lSl!>�PimQ(JJrogm:im,8��
o{Oi§l"'c> l = �(M!:) --+ �,�
O=(Amu-mv)-(mu-lmv)
⇒ 0=(..t-l)u+(A.-l)v
⇒ 0=(,'l-IXu+v)
⇒1=1.
(u+v)=e(u+v)
:. e=J.
0
0
2. .,.a:,·�� d4)C il(g� � t=O� � em �uf ��� r::i1t'i �f:IFxk£ttd!A� Sc1d & ... £ICm(o � ·..SJ<6stJ, � t= T@ � �� Sdz41€nd .,ed·& �.� ecm el). ,�()'8') �.iS co6e em =cb �ae, .,_ � �G4cd.� � �a,e �·� � �. T. f e0:1 g�� $B>Gil.oe6$e.o�..
V
A
T fT
e,06® cm = OAB �'"1lif..,� �"e"' = !(t + f T)xfT.. . 2 . g
. =:\f+g)
3
3. ot� PA-BCD�� 3{)"i!M� �·� �.hl ••�e·•�mfd �.en��-����.Ata���sg&)e.��·t;·��·mfiO� �id·tc c:iB. Jj� qfJo � M,�� �� &) qu,d AB� CD &w. � �e .cnSf> ��� ae �. �aa. � � � toE>e ��� e,W����.�����·�eo�
*F· 2rng
x+ 2y= zfkom�25'! ⇒i+2y=0 ⇒2y=-x. 0
E. = mg_ / for P: T-mgsin3o· = mf
F : mg .J. for 2mg: 2mg- 2'f = 2m F
0
0
4. � Mkg� � � -� 111.lgj � a,e tic1J c:,adm_ � q;i�� =D oh!ffl��eoa��-��-����@d�el)ll����:�1.Jl.-�.tm�---..i(>-0)�.�� �f):,Jdood���-eitm�P-�W·� qadfJl5J� s,:, �
�G.' V ms;-t C!tl>. � � � �qm&, � tff Sf) �
nn +- [:11---••-=r-........,,r!i?J-+ F
. IOOOP r;-"\ 9111�•�@eco F=-v-N ......... ,. •.. (1) • "-V .
0
E.=mg_ ➔ forM: F-J.M-T�Mf ·.u ......... (2) · 0\E_=mg_ ➔ form: T-hn=mf '."·--------� (3)
(D j
. · IOOOP · · ·. ·�t� (l), (2) 10:> (3)⇒.-. -. -AM -T =¥I
. . V
.
. 1000.P
. . · M > ,;_ . ⇒-·-.-)41-T=-(T-lm):;
.v: ··· m, ',;,
5
5. ·� (fo� • .,.i,..2j�?ai+l:Q =ll �· O,,t)e ��A 8:13 Bez:i!e:i,���ore�; c,111$ «{>O).� �· � �. AOB =� ae�zdis,. . .
. . ,4
C�.·o,tcB�-�t�c.,,.13�.0C�y���esc>Sd.(l s· . e£lQ ���-
--
a,isJ �&ec.,: OA ·OB= (-[+2l)·(2at+af)
;:,;-2a+2a =0
0 :.AOB=;-©
OC=OA+AC
=OA+OB
== (-J +2!!H +(2+a)J:. CD
CD
0
6. 0Aa»OB�r&:l���.WOqE:le�"dem�. �m,2al:Q:lacSW�AB�������' G� '48$��e-e, AOB=:f � oJB·=a@e ,-cnm.
· .AOG = aQlcl �. � � � $&&".u,tt,,· ·
0
B CD
A· w
• :r . . .
AOB =2
, @iEJ2ff A, 0 cos, Bw6� (,o3) �-®ex.,AB � e:lasm� �ttdmt� G �-
:.AG= OG.
�AOG=OAG=a. CD
0 Ti =Wcosa. 0
T2
=Wsina. 0
---·-----
w
B
7
7. A�Bo�O&:c��·��Q!iSG:i���;P(AUB)=¾•
P(A'UB')=¾ u» PCB I A)=¼ as�•�t11. P�)� P(B)�� ..
A'vB'=(Ar.B)', �� P((AnBY)=I-P(AriB) 0E).
5 1 �5 .:.P(AriB)=1-6
=6. �
P(BIA) P(Br.A) ⇒ P(A)= 6 =�. �
0(A) :¼ 3 \._::_J
s. �. � is,e� i;e�. �- �� 1 ·� �«riQ tl1J<$ qt8. •� � �2��� §� �*1:11· �•·� �-,&t@iif·.� ®e�� �
�� eLGie.
(i) ·�mc>.mm�·������:,,__(ii) � cna! ◄ � � �� .� C!J� !lee,
��
5 I 5 (i)8��6=-X-=-.
0 9 2 18 s 0
( .. ) Si;i 4 3 2 I n e:-,!:1J6=-x-x-=-.
9 8 7 21 0
9, �- a>azs, �c.,:i,J a,.Q._b, b� !Sl!I Y�t �-o.;I;,_Jflipo i�• 00> $c � � .a<b � <N)® � � � � lr'
<il®e��a'.Xl�-�Xt9111�,�-qv:2!.�lli��1'� !a!) �.a�b�
0
:.3a+3b+ab=21 --------·······- (l) 0 -�
l&, Zt,Z:2,; .. ,Xfo� � � � � �-� toa 9�. Xjg �El�
� � .. �- � �� .� Wal> �qp.,llJM' � t!I� -!W�
�qsmm.
0 .-. I: x,2 = 990.
1 .. , 0
0
11.(a) ce, a" S6c:t �ui ���em j. � Z.aem
��c���·�•�ai:i� -4" �,����u����ciel:lf��fl
ii� er@$, <� o>�) � �� � � ·� �+i
----'!ooi·�----+ CSoRr,6.-&l.C��ilMd�•,r---rfR-Ja =- 0 z.a .� ®&lm· R � et,u, ss ��
u. a� g��·��. ,; > t:�a&,«>e·�� •�se � ll!ldzmn.(Ji) S �tc)t.Sf c,c,�· � a 1ml h·
1 ·� � 1'1{� � � z:s,d$.
B ·ea-.iof!Vll!.l Sc>�· �.-0.� ltnr,p JO»I� � � � �t'>; ��c) *� <l� � � v.km.11-l� �=de� eomo G'.l� �; � usin8 < I'< ii 41!). �Zib�·� i9& m:,.ec,m � � G'.l•m 0tS c�� �. �.:� � • ioia eom � �m!f!® �tt)O � �· � �.� '9 � �-
�oo �s � ��� � � �� �· ,z._2a cile �2'>; @®@ a=sin--t {ii5?'9) �-
a
C R
tr Ut· 0 R;:::ucos-·i=-----( 5 . . 4 .Ji
-a= usin�t-_!_gt2 --------(2(2!)4 2
1 2R2
(1) aJa (2) ⇒ -a=R--g-2 u 2
0
0 ⇒ gR2 -u1R-u2a=0
--------- ·---
B
.'. R = u2 ±./u4 +4u2ag.
2g
⇒ R>2a.
:. G:>C �o:>��0 C!>2S'lat.J-r�.
0
------------------------------------
. (b) f_(S.E) =➔ u 0[(B,E)=v 0 [(B,S)=: J?.0
f_(B, S) = ru�. E) + K.(E,:R©
= f_(E, S) + f_(B, E)- -
=PQ+·QR
=PR.
. .·
tfDG)� e?b)l�'-'= R1QR2 0
usin0<v<u
=,r-2a; �53, QR2R1 =a.(�
u
• _ • • -1 (usinB) .. a-sm -- .
----------------------------
/ 2ucos0:. ti + t 2 =--=--�---,,---,--
u2 cos2 0-(v2 -u2 sin2 0)
2/ucosO 2 • � =2 2 (·: cos 0 + sm" 0 = 1)
u -v .
CD
----·------------------------
U.(�)d��I\BCD�.�2mQ�� g �·�=lo�-��-6:td-�AD!lliO 1· :.:. C
(a)
BC���-�-AB���flll� p,
I c� � � e&. � � AB;::2ti�_ BAI}= a � ate,c,$i}0<a<f �.oosa=¾�M:4(ld��IM� ..4_ D �Ill! ec qio}D � �- eaa � 1 (> 211) �- ��
. tf&)mr- t1>�2:I Bf3 800. q� s®O � � c= � � � <t� cd=z,JQQ m 9 P q,,qfld i. � � -� tti� fl �- � {l � � � �i». <1{� ��tn�P�AB��� .. a�-BCb._i:n&,��4i&>� � &l ii?- � �
-� � �� � ffi �-� �-� p� ��-
_1'.lEJ '< P.� A1.1>03 �-� � �11: ��.(b) �t.t ci:m::s, � m Jl A.UlO B��� l(>�) �dg:
�:a�� �:�:�-211!�.Cq�� ,®l):z, � (Jt� CtaG. �;� � qd'G,· a� ��®i:> � ¢�� e-lll!dl. C-�-!fc:,$e <; _. ID'J B q,.q O� 9 &3.:tcd me=�������-��r::, � l:llClb lft2». Clld'e �-� A� Cllll$D � eib� C. "<><eOO -� ®l:!I � -Sdtd mo-d � � �� �ei�. c;."" � � � �- ·fr'=..l(i-oostn &e00� .. ! e®!i! 8� OC lllt� � � �- a
..L
T T RI c ·
R· I
mg
F
R2 _....
A D
2mg
e A
���
®
g:{P,Block)=f / g (Q,Block) = f ..-
lSleJ� [(Block, E) = F �
. '
tr�
10 � � ®� u <©�118 �> 1 e,[email protected]. <e-cteG> � - 2011·1 � �� � � � �- 13
0=2mF +m(F- f)+m(F- f cos a)
3⇒0=4F-f-fx5
5F � :. f =:=2 -----------(1) �
P qoc.QDcJ ,I mg sin a -T = m(f -Fcosa) ------ ------ (2)
T =m(f-F) -------�-----(3)
0 &- ·,��
_(2) + (3) ⇒ mgx1=m(f-F)+m(f-Fx¾)
⇒ 4g=5J-5F+5f-3F
⇒ 4g=l0f-8F 0�3'f (l) ⇒ 4g= 25F-8F
4 ⇒ F=Ug. 0(1)⇒/=]0g
17
l s=ut+-at2
:
0
0
@ri--f-o
® c-1--o
�
------·-----·---------------
(b)
k
A
:o,
P.E.=0 :--{ -------------- ------�-2mg
mg
l . · 2 1 · 2 -x2mx(a0) +2x-xmx(a0) +2mgacos0-mg(k-a0)-mg(k+a0) =-2mgk+2mga2 2 �
::;. 2a02 =-2gcos0_+2g ®
02 g . 0 :. =--(I-cos ). a
_F=mg:
C etJc;roo/'; R-2mgcosB = -2m • a82
⇒ R = 2mgcos0- 2mg(I-cos0) .
�1PEJO KE 10 Equation 5
0
13. �!::.\) �m a a,:, sc��m:i � mg� «l'{q� � -�zxt ¢mf 0�d a®":ilod � 3i:z co:i:sr � � o� � � �$ -� tf<'• <Kil�•� m·� (1� 4't'I? .<fti:D. q� o ��-../ii�� c> � �c, z:w� er:ee.c,i;rl� �m xomlS.l, a-s:x< 3¢ � i+.!(x-2a'.)=0� �m � ae� <II®®.
. a .
c,de q�iS � � � em�· c®m �- lll� •W tf�� � �o �trxi mllll'@ e:,oe!!'a$. t}� �m
. a S X < 3a �.00. f = ! ( 4ax - ,I) cile ecl�!b. . ..
X=x-2a�@ «>� qoi=n�:-a�X<a�G» "x2 ·=!(it -x2 )�u1 9�=<Sm$; � A �- ��- -� � � 19l>. -�·�.������:��-fi'"e �-�a®.�� -� * �- -oS! � 09 � �<:! � ezm:4� c§ � ��-!l:SX�a� ;f =!(r�.:x-2:)� � <'ttz:Di � B ��me)
l:DQ� q� � � i:iie.� � -��� ll� tsi6� e� ¢l!j qaf �- �� ¢- Qfe· it�-� "e> � fl6· t1rJe0.Sa � .. .,,s.� 6'fg � �o.x-,;o,m.
a�x<3a: �!.03
T=mg(x-a) 0 a
{j
f._ =mg_: '1te-'�0@zi' 0 m o,� ,l,;mg-T:::mx ®
mg(
· ) ...⇒ mg-- x-a = i"l'JX
⇒ x+ g (x-2a)=0. fs\a .
\...J
mg. 3a
PE 10
·--------------------
16
·---�-X:::x-2a⇒X=x © me>� aSx<3a � -a�X <a.
X2 "." ! {4o(X + 2a)-(X +2a)2 } 0
=!{402 -X2 } for-a�X<a 0 :.A=2a.(D--------
---------
--------·------�--,.---------
---------·-------·-------.----.----------------·------------
z,!)c)ir,' <$rs! sm�"'� lfl"'®�Z'l' eoiQ!:>c) c,9 s�m� i= � (·.-e = l) .. _kl =K(iJ2 -X2 )
X=a0�6� X=� eO.
ga=K(B2 -a2 ) �a \...V
⇒B=✓2a .(D
a
I, -= -, z3CSJ t, = - - @O.
ft tel _
� 2 - 2 g
---------------------------------- ---------------- . �
-------- ------�
14.(a) A� B m_@)m m,eCmG, AOE>a Q �fl<, §le<m' �� A� B e63:ms, e=e:c �=>Z:1:> 853� @(W�lsi� 8� a� J, �- O�� C �l¼M:SI � ��- c = (l- J.) a+ lb c.:tcf3 GiB@� d&l O<l < L et&.
A(.;-too CB �� a,b1D1>•,t �� .� � 6 � C�:i,c:, A1J� � �- aa;ei,, aed .AC� CB=.). :_(1-A) a6m.t cte�mZI).
��, oc� AIJB� ��m QcS Sm§. lbl(•��) � laf(b·c)@i) c,e� daiW,,l.�
t: E
(b) Ol� A:IJCD '°'13 AS= l m too ,ac = .... 1m � � C);,> qti>c:5 /\ cmicfl ���-�-&:>em Sw 2/§,3, 4/i,.P. io:> .!J �
ae ag� P.A. DA,.DC. (JJ;t •. CEl!O-J .lJEt.r � � ��€>t0 D .· . C ���c) �;�-&e-� �-� <!il), P== 4� Q-=8� �. �a�� �-�z!. BAroo DA �ear�-��-�-�,�t��e:i,-� .J5m
01ea. t1.lf> �QBd � �$ z.fii �� � ��� � •. ¢'1�-� ���- A'----'B
� � $C� �-� �-ms BA�� A-Ba tn.6l r,,<S Im
i�aro��.
i.e. AC:CB=A:(1-A) CD
19
a·c b·c 0⇒ =-===-=- s
If! I !QI
A
10 - � GJ� IT (�f@:11!�· o0oil'.)a)) I q.00),CI; (C,.�) � • 2017 I·�� � � � ,crz;tS!. Z\) .1
(b) E
3
A
1fll
er!{E::Jilc.o �a:f®c.o2Sl8 C'!ltnts">c.o D2n �o»,
➔ 3-5+Qcos60° -Pcos60° =0
⇒ P-Q=-4, CD
C
B
CD
t -2✓3 -4✓3 +Qsin60°
+Psin60° =0 CD
. ⇒ P+Q=12 0 :. P = 4 and Q = 8� � . � \...::,_I
·--------------------
10 ".�®�II (�d �) I ;.!'O).m. (C;@os) � - 2017 .1 �· C)o@(9�a1 �d !.!>S � qi;t». · 21 . . . • 1 •
H a
➔ X=5+3+8cos60° -4cos60�·=10 0t Y =2✓3 -4✓
3+8sin60° +4�in60° =4✓3
R=✓100+48 =2-fri 0
E
s
1-m
0
H �m �z:sf zs,e BA, e,l!/992:shs>� �a1a e�cibD ;,o�e)2S'> emf1t:i,c., c.,1.8 a,z3§.
ii) . .-6./3-+2,/3(i+a)+✓3(3+4-2)=0 @� -6a+2 + 2a+S =0
a=im, 0 ---------------·---------------
C
15.(a) Q/6 Wa:>:1 c�ZS> � 2a El� c1t>i� c:1ml!Mt.'i'l06 �z;rl o� Socl @�®d ®m mQl� lfil:ll. i:ilO 2W to> �CD 2a §e"f�3� AB i;""'1e.)2Sl A <S'l:S)eru 6xSd � �-�9®0 �Cl qe:iE> l:S)0 �B qtl>68 (;ti@� �Z'I'� 9®c>Soc:! @�ZS>ll ()@<5� �� �!:Sf�e-d m@:i �m. 't.fJiffl cr),-:'.f@cl c.:)25) Bde:f z:oea, �fud � Bottt �-eoma e®a>
B
2a
Q
elm qm6 �O.mo 0®�� oc>�. (<f'?S So.d _'---___ _.._ ______,_ <'Oddz:ile) cori;w� Otoe., @e�-) iz�z:ood �& e!lO � A tnue:f @<D@® qzro e:1tf-s �= µ�. µ � Jj oo@otrlD::rl�.
(b) 0-��Doeiam �eot:d Cl� � e'( .AB,Br, AD, BD WJ CD r:iim-rd� .. � ow$2dr:i®m�m 6�� 01.iilc!e� <Sto@d ���-AB = ©ad a �� BC= @)c>c:S .'ia � � . p Q
(a)
r
BAD= BDA =BCD= 30°eE>.�c;;�Ba'!� 150N1mDio�-300Ntnl<1��m.l!>-o AB a>:> BC li36d � c,� �§m A !00c � � �� ei; Poo:i Q &1Jsie �
. qJWJ6 �2R C1ol Sod me� mi� (flm.P = 250 N @el @otrltmfts). @&f qo?S>� '°��m 'gt»:z,:iac �0i:ozm::if � c5oCW. � e .. �e go>l)JQ/e � c,-c},J qJmzil.. @zrog® � �2!'$21) �� z:romm.
A
t0S B -4---'-
0 2 I 12W • R
:z
X F
2a
a
s
�_...,
aRI
w
z:\lDor., CO�!:OJ t R, +R
2 =W
0 2,:j1D8r., Cl�?:D:l _.
0.F=S
(b)
µ> IFI-(R, +Rj
µ;?;✓3W w
µ�✓3
®
0
D
l50x2a+300(2a-f)-P.3a=0 (D
⇒P=250N 0
300N
-----------------------········-···-----------------··---------------
0 6011
1
0
0
0
r
0
200✓3
100✓3
0rf-----,,,C.....--40-0 __ ___:___�
·250✓350 60
°
0
l�f;) a:JtllzBCA e-tllo=® AB 2S0../3N
0 _;.
BC 200../3N ®�--
CD 400N
® DA SOON
® DB 100../3 N
®
25
16. 0zs!msa1 c 10, qoc:i a� �o c>amu,:iz@.?O �-w,f>0com gm � 6� �QZS) mclO @zs!�o Ic @c) 2a �6Bzf �i5l @El eomE>�.
:,r;
p
c.:1:>@1, c:5ioea!&> PQ, PR .oo ST a,m, �52:l> �a,2:1> ezitrlcjc., P � � �·w®8}o1Sm �c,:, wm ede � i::5>1:a>8 tUm- PQ ?SYJ PR 13Jl.f$� �P �� � c:ieiltflS>�c> �-! � c:,s Q ?Oil Rem� , sr a
-a
s,......::;..._ _ _.__�,...,rOJ�� qi1Sl. PQ = PR:. a, ST,= 2a � PO = 2 @E) � ,ws,; 61®� 0 0� QR a:>J ST c:itn �@� ® ®{.).D ezzf'5l!Q @El. me> Ii SUT om e$tm:.)�<D1.5l �mwc., kp §1 Zl?� d� �w � com @tlf� 0 eooqom asJ � c)� �; �- k(>O)� @El. sur�
c:,:l@� c5toc;,d � (fl.S � L � �. � �=6 �oc:) r4®c> Sod Q}� � Bdel:i tl:I®- Di��� t5ld® � B6cf oo,&,®l!i) e:k,b°� @t)®z5, � �- @� cSQ)c) e®r:t}� qt.tD. LtiJm p1llX5m Q)e � JS)d k > ¼ cW �-
�m k= I cr8 w�. Pez:sfe� � � m0m � Lc>� t:.11$ 09 � 'i'Oi5> 8153� ® � ��� <:IS. m < 3pa Q)f) �mvffl2i).
0
--�=---L.><.-.....-----lc----"""\,_--
c --G X
A
u
T
!:. . 0
J apacos0d0
c8c> /lm:a(A0)p �a, x= -2-�---
1,ap
·0
=: Jcos0d0
2
a=-•sm tr
2a =-;;·0
0
---------------------------------·----------
PR
PQ
ST
sur··
ti,oamfm. Do,'��- .·. _ _ \_·.
2ap
Sdfd ,6, P 9c) drm:sfw ! -m !!:c.e0
a
0 a
0-·
a
0 a 2a
0-+-
tr
xi
-- a trak (Ds. ⇒(4+1rk)x1 =-+a+--+2ak 2 2
� (7lk+4k+3)a ⇒xi=-----.
n-k+4 2
. (.iik+4k+3JE._ !!_ 0I.e. > .n-k+4 2 2
0
s
0
0
p
¢:> 1lk+4k +3 > trk+ 4. r-------------'��--1-----� R
i
,,.,•'
25
------------·-------'------------------..!=::==
R,
s
p
mg t-----��=--1-- - ________ ....,_ R,
�50 [(4ap+1l'a p)+m]x2 =(4ap+,ra p)x1• (D <::> [(4ap+1ra p)+m]x2 =(4ap+1ra p)(1f+7_):!..
,, ,'. 1r+4 ,2
<::} [(4ap+1ra p)+m]x2 =ap(1r+7)� 2
- ap(1r+7) a (D ¢:;> X2 =r L(4ap+1ra p)+m] 2
. ap(n+7) a a i.e. [(4ap +;a p)+ m] 2 >2
<=> ap(Tt+7)>4ap+1ra p+m
<=>m<3ap. 0 ---·----------·--•-. ·------
0
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e®®8�8Q:z,�p·�mMm�4 ��-
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P(X) = P(X I A) P(A) + P(X I B) P(B) +�(XI C) P(C). •• --- --- • •• • - ··(I)
(I) ⇒ 2._ = �x.!.+�x.!.+ (m+I) x.!_'tc)'i
18 15 3 15 3 2(2m + I) 3 "--V
5 8 (m+ 1) ⇒---=--'----'--6 15 2(2m + I)
⇒ 3(2m +I)= 5(m + l)⇒ m=2
_0
0
----------------------------------------------
3 0·m=2⇒P(XIC)=-. 5 10
P(CIX)= P(XIC)P(C) � P(X) \._y
· 3 I -x� �=J.Q__]_ \...::__,J
5 18
9
. 25 0 --------------·------
(hJ
c�� o6:i�C.5 f ®:IQ:x) c=f•»
x2
X
0-2 IS
2-4 25 3 9
4-6 40 5 25
· 6-8 15 7 49
8-10 5 9 81
Ii =100
= I.rx = 440 =4.4 (D · µ Lf 100
--------· ...... ,._ ... ..,,. _______________________ _
------------·-----·----------------
fx fx2
15 15
75 225
200 1000
105 735
45 405
r1x =44o .f x2 = 2380
?� �3� . 7j
= 2380 -(44)2
fs' 100 JO \....::_)
=✓23.8-19.36 0
=✓4.44
�2.11. 0 ----.... ------------·---·---------------------·------ ------------- ·--- -------·--·--------------------
. 10 0M=4+-x2 S 40
==4.5. 0-·
K= 3(4.4-4.5) 02.11
0.3 ==--
2.11�- 0.14. 0
·········---------------···--···-··-····---·---·-·····-------·---------·-·-·-··