skema add maths trial spm 2013 penang k12 w
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MAJLIS PENGETUA SEKOLAH MALAYSIACAWANGAN PI]LAU PTNANG
MODUL LATIHAN BERFOKUS SPM 2013
ADD ITIONAL I\{ATHEIVIATIC S
KERTAS 1
1VIARK SCTTN'ME
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)a
ADDITIONAL MATHEMATICS PAPER I 2OI3
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Question Solution and Mark Scheme SubMarlur
TotalMark
I (a)
(b)
64
f:*-)r' OR f(x)=tr'
I
I 2
2 (a)
(b)
7
1
81: l-4n--3
I
2 3
3 (a)
(b)
x+3
-30
B2:
BI
2-L=85
t(S1 :3
I
3 4
42.907 and - A.5734
- (-7) *B2:.
BI: 3x2
2(3)
-'lx-5 = 0
- 4(3)(-5)
3 3
s (a)
o)
P=-1, Q=L6
r=-1
)
I 3
w\M/v. rnysch oolch i I d ren . co m
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6I
x < --3
or ,> I2
B2
BI (3x+r)(2x-3)>0
3 3
7-:
3
: 3n+ 5(n +2) -2n
: 33' or 3s(n+2)
B2
BI or 32"
3 3
I+ or t,-'{*) or i-lrl2, (tog, p-logrt)"H
Bl : logo p -logo5 or #
3 3
e (a)
o)
36
54
B1: +1- -
3
I
2 3
10 (a)
o)
1456
972
B2:
B1 :
l4s6 -
2b'-r
hb' - r)] or re =+(gf-'
) or a=4andr=3
I
3 4
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11 (a)
(b)
P:7
217
Br , +[zt:l
+ (r r - r)4]- f,frrr,
+ (4- r)41
OR
i?re)+(7-l)41
19+23+27+31 +35+39+43
I
2 3
12 (a)
o)
!' :-5x + lox
p = 2, q: 5 (both)
BZ i p:2 or q:5
Bl , ?-o --s Il-p {=-5(l)+10 0=-5p+10
t
3 4
13 (a)
o)
3
i6y+8x+7 =0
v-l= -*A+z)B1 . Z J
I
2 3
t4 48.19", 180", 31 1.81" / 48ol 1"180p, 31 l"4g'
B3 : 48.19o, 180" (both)
7BZ : cos 0 = a or cosd = -t or 48.19" 148" 1l' or 180"
3
Bl : cosa = 3(t-cos' e)-t
4 4
ls (a)
(b)
1.A47 rad
45.50
B2' ltd (z.ors)+loxuysin6oo
Br , *Af(z.ols) or ]trxulsin6oo
I
3 4
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347Zlt
16 (a)
(b)
(i)
3i_+4j
5
BI
I
2 3
t7 (a)
(b)
sI -81
t0 40+-y7- 7--
BI 8I * lr.tl-81)
I
2 3
18 (1, 6)
B2 : 2x+3=5 or x=1 or f=6
Bl , i dr=2x+3\, &:
3 3
19' 72n
B2
B1
3On x 2
hf
3 3
20 4
3
BZ: -l+ *b4=-3m
B1mx'
2
44
[,* &-2[ s@) ell
3 3
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2t
n=-2
Br'ffi.i")
n---2
FJZ; m-1 or5
3m---)
3 3
22 (a)
o)
-7
367 ( qt\'6 -[eJ
m+92
BI:
3.804
B1
2
2 4
23 (a)
(b)
"crx"cr+"co
27405
44s
B1
I
23
(a)
o)
24
t56t-I ::96
80 70x-360 360
100 I st-360 I 18
7
162
BI
I
2 3
(a)
(b)
25
11z+m+m+ - =l
1
8
3
8
B1
I
2 3
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341U1ADDITIONAL MATHEMATIC S
Kertas 2September
^l/. - ram2J
MAJLIS PENGETUA SEKOLAH MALAYSIACAWANGAN PULAU PINANG
MODUL LATIHAN BERFOKUS SPM 2OI3
MARI( SCHE,ME
ADDITIONAL MATHEMATIC S
Paper 2
Two hours and thirty minutes
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Mark Scheme
2
SE,CTION A ( 40 MARKS )
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v:or
v
5-xor.x:5-y
Eliminate y or x
x*(5-x)+x*7=0
or
*(5- y)y+*(5-y)+ 7:0or
*(-r-7\x+ I l=5
\x)
x--1r7or
Y:6,-2OR
!:6r-2or
x: - | ,7
Solve the Quadratic Equation
Us-ins factorisation
(x+ lXx-7):0OR
(y+2)(y-6):0
Using formula
20)
OR
- (-4) r2(r)
Comnleting the square
(r- 3)2 -32 -7 -oOR
b,-2)' -22 -12 = o
- (-6) t Jc-otz - qe)(t)
- 4(1X-12)
OW-l if steps to solve quadratic equation are not shown.
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Mark Scheme
8, 20, 32
d: 12
*8 + (n- 1)*12:176
Note : Accept complete listing of the terrns
Substitute x:1 into px+ q*and equate to 2
p(1) + q(l)2 :2 Substitnte x:3 into Px + q*and equate to 0
p(3)+q(3)2:g
p_3or
q- -lq: .-,1 .
orp_3
Integrate *3x + 'r1- 1)x2 wrt x--.-.-
42 35XX1'-- --+c'23
Substitute (3, 4) into *y
to find c
3x
3y= 3x2
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Mark Scheme
L : 20.5
* 20.5 +
s.5(6) + ls.s(13) +zs.s * (s) + 3s.5 * (6) + 4s.5(10)
Modal class ll - 20
Use rqid-points
2(-4) + 3(6) 2(0) + 3(s) )' 3+2 )
D(2,3)
Usezlxtnz--1y _ *3 : *-2(i -_*2)
or*3:*(_2)(*2)+c
=!
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Mark Scheme
6 (a) Shape of cosine
2cyclesfor0 <x<2n
Amplitude: 3
Modulus
trtrtrtr
Accept
sine
graph
v-L+l7t
Sketch the straight line
..Number of solutions : 8
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strcTroNB(40MALKS)
Mark Scheme
x-l 2 3 4 5 6 7
logroy 0.90 0.83 0.78 0.72 4.67 0.60
Plot logrqy against
6 *points plotted correctly
Line of best fit
logroy: loglsh - (x- l) logls& trUse 't = log16ft
Use*z:-log1s&
k- 1.15 10.47
;
Note : SS-l if part of the scale is not uniform or not usingthe scales given or not using the graph paper
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loB,o
1.
jlit,li;jii
0t234567 :.
,,.
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0.
0
v
0
8
.7
.4
.3
.t
0.
1
i1
t
{
t:tI
IIllt.ltj
{Ititt
!
Ii;1
,
:Jrll:it:Ilt.
.taI
!t
i?
.i I'iill!it
,lli" i ,it'.;l,II,ii ..i
Nit^'i-.*_-"-!..",-.*1.",*_.*.-,..*..".-l -t -*..
..
iiIIt;II
,I
t{.;"
{
ii
i!I
::c
!
ItI
iI
....t"
't' ; Iiii.i!l,...,.r -. [ ,.1.,-.':'l, .l: IIiil,!r,r. t Iit\
T.,iI1.,,t
i\i:l il
"1'I,l
\ *.i:-'i -"
t,.l'iiili,
i:-.-,.1"- -.-L.:;-j.- -- l.I,l:..,,',.',...'i,l
I
, ,. ' ,l
i,.1 \{ :i:\,li it:
l: I
:" i"-*l-.r -. , ,?, :- -:-,--l
;,1.'.-.-r->
i--.---...-t.-:T---1- I t. --.--J*-.:"'
., .,|,
-'-!: -l
:x-*--L
.lIxi--t -:-?t-i
:IJLI :-- -fj --*j
- r- -i:::li,l
I
i-1---1
,I
,l:lr.-i.,t
,i--:_--li
i: 't,."-.iI ;,t
i:lIIiltr l:ii
::l.l
-i-t-t-I
*1----i-
II:l ai-*f, -, -,
"',- -*-:-'..',*...":.
:!!r
'-:::ir----.ia-
i i -ii i{.': ) i!t-i....l-.:
- r'--.-"i- -:--
Ii i: '-++-;
I'I-l-_ -- r- 1
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341 212
Mark Scheme
(i) & (ii) RS or QR
R3: -y 3*-y
4
A-
PT=PQ+=QR5'
or equivalent
t4-x * - v
tJ) )-
(i) PU n(l t* ! v)) )- @(ii) P(J=PR+fRS
+ ( 3 \ptr=Z*k[-2.;r)
Equate the coefficient of xand y and solve simultaneouslinear equations
,1ls-- or4
. 15h- - ot16
.15h: -16
k=!4
Use Triangle Law for
Ful IMarI
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I
I
Ii
I
I
I
I
i
i
Ques Mark SchemeSub
Ma rksTotalMark
9 (a
(b
(c)
Find lL!:"
dY -zx-6
dx
-4
(1,-5) trr)
Y--44x-
_$_o-lyg/_: -61-:_0_x:0r 6OR
Solve simultaneous equations
v -= -6!-:t-qll{ y-=:.x =.0, 'l
-EEerl"- .A-:-6:llF-lt -x3 612Ar - 32
Use limi, tI[ . *( f or2)rn [;- r)
OR
I it{ -t["- se_q -olt1riete lq
Az: *"-rrrr*(7)
Ai
U rntlttzt6
l7
2
3
5 l0
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t0 341212
Ques Mark SchemeSub
MarksFull
Mark
r0 (a)
(b)
(c)
(d)
trUse s : r0 to fihd arc
lglell,-qr- 4P-Arc AB - (*6 + l2)(1.05)
18.9
Use cos I "05' : OD o, sin I .g5' - I__t_?_ __ --r__ :?
OD : 5.971 ; CD: 10.409
49 -: -g P- : !-P -! -v-"- 4-n-*6 + *10.409 + *12.029 + *18.9
!?-:(l-611?) --:?.2BD: 12.029
4
Usel =%lO to
find area of sector
AOB
Ar :l t-u + t2)2(l .os)'2\ ',
*Ar_ l%*(oD)*(cD)l
139.02 10
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ll 3 47 212
Ques vrww. myschoolchi ld ren.com Mark SchemeSub
MarksFull
I
Markl
1l (a) (i) 8c, 10.+s)s (o.s s)3
0. 1719
(ii) n (0.45)(0.55) : 940.5
(b) (i) 255 - 240
m
10
Y -240(ii)User:fff:r-{-:-2-3-9-9-?19-
-l<z<l
Use 1 -2P(Z> *1)
l - 2(o.l s87)
0,6826 ll 0,6827
. 1t I
(700 x *0.6826) 10
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t2
SECTION C (20 MARKS)
3 412t2
Ques Mark SchcmeSub
MarksFull
Mark
l2 (a) 5 -2t <0
5t>2
(b)Irt:g-vr i:-?t-ylv- st -f + c Substitute t:0and
v: -4 in*y to
l,:r4 -g-c=-4Fl_t_u-"_ _r_ _:_q -&l-r_
(,-l)0-4):0
4
(c) fl'-'g::r ! ! :?! -' - -1- Y1-'
st2 f.r - ---4t23
Find total distance travelled
Substitutet: *1 or t=*4 into *s
OR
Use t' o,r fl in's
l[iudtlor ti"at,
re t oL tt 6.3333310
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Mark Scheme
ur, &xlooQo
x : 1.50
r20(l2s) + I 37.{45) + * l2s(170) + 135(80)
125+45+170+80
126.76il126.8
ur" ?)? x loo - *126 .76s00 @
J(RM) 633.8 // 634.00
llg. B7 .s100
( RM ) 3.20 z:125 tr
l s9.s
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l4
' "r
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Ques Mark Scheme,' SubMarks
FullMark
l4 (a)
(b)
(c)
(d)
ZPSR : 60o
Y-': :"_':l:'j'l:
PRz -82 +g2
i:_tffl:lf'ri lI- 2(8)(9) cos 60"
PR: 8 544 t/ 8.54
Use sine rule in MQS to find PQ
Pg=gsin 120' sin 48"
PQ : 9.323 ll9.32
Ar : % *(9.323X8) sin l2'
OR
Use sine rule in LPQSto find p.S
OR
Use sine rule or cosine ruleto find Z SPR
Az: Y, (8)(9) sin 60"
OR
*(9.323X9 + *Qs)sin 48"
aR
* 19 .323)*(8. 544)sin (12+* zP)
38.933
sin 60" - 4oR8
%x9 x h: * Area of APSft
l0
38.90
6.928/t 6.93
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l5 341212
Ques Mark Scheme SubMarks
Fu llMark
15 (a)
(b)
(c)
3y
20
80 or equivalent
equivalent
equivalentor
trtr
Draw correctly at least one straight linefrom the *inequalities which involvesxand y
Draw correctly all the three*straight linesNote : Accept dashed lines
The correct region shaded
(i) zo < x < 4o @(ii) Maximum point: (20, 60) tr
Use 30 000x + 50 000yfor point in the *region R
l0
Note:ss-r ifin (a), the symbol ":" is not used at all ormore than three inequalities are given
in (b), does not use the scale given or '
does not use graph paper orinterchange between x-axis and y-axis
.
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