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CONFIDENTIAL*
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAANPERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAANPERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAANPERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNPPEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNPPEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PEPERIKSAAN PERCUBAAN
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
NEGERI PAHANG DARUL MAKMUR 2010
Instructions to candidates:
Answer allquestions.
Answers may be written in either English or Bahasa Malaysia.
All necessary working should be shown clearly.
Non-exact numerical answers may be given correct to three significant figures,
or one decimal place in the case of angles in degrees, unless a different level of accuracy
is specified in the question.
Mathematical tables, a list of mathematical formulae and graph paper are provided.
This question paper consists of 5 printed pages.
950/1, 954/1 STPM 2010
Three hours
MATHEMATICS S
PAPER 1
MATHEMATICS T
PAPER 1
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CONFIDENTIAL* 2
Mathematical Formulae for Paper 1 Mathematics T / Mathematics S :
Logarithms :
a
xx
b
ba
log
loglog
Series :
)1(2
1
1
nnrn
r
)12)(1(6
1
1
2
nnnrn
r
22
1
3 )1(4
1
nnrn
r
Integration :
dxdxdu
vuvdxdx
dvu
cxfdxxf
xf )(ln)(
)('
ca
x
adx
xa
1
22 tan
11
ca
xdx
xa
1
22sin
1
Series:
Nnwhere
,
21)( 221 nrrnnnnn bba
r
nba
nba
naba
1,!
)1()1(
!2
)1(1)1( 2
xx
r
rnnnx
nnnxx rn where
Coordinate Geometry :
The coordinates of the point which divides the line joining (x1,y1) and (x2,y2) in the
ratio m: nis
nm
myny
nm
mxnx 2121 ,
The distance from ),( 11 yx to 0 cbyax is
22
11
ba
cbyax
Numerical Methods :
Newton-Raphson iteration for 0)( xf :
)('
)(1
n
nnn
xf
xfxx
Trapezium rule :
b
a nn yyyyyhdxxf ])(2[
2
1)( 1210
n
abhrhafyr
andwhere )(
Trigonometry :
BAAABA sincoscossin)sin(
BABABA sinsincoscos)cos(
BA
BABA
tantan1
tantan)tan(
AAAAA 2222 sin211cos2sincos2cos AAA 3sin4sin33sin AAA cos3cos43cos 3
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CONFIDENTIAL* 3
1. Given thati
aiz
34
5
where 1, 2 iRa ,
(a) simplify z into the form yix , wherexandyare real numbers, [2 marks]
(b) find the value of a if z is a real number. [2 marks]
2. The real function f is defined by
1,45
1,
1,
)( 2
22
xxk
xk
xxk
xf .
Find the values of k if f is continuous everywhere. [4 marks]
3.(a) Evaluate 2253
5
2
55 loglogloglog yyyy ify= 5. [2 marks]
(b) Find the value of y in surd form if 3)(log)(loglog 352
55 yyy . [3 marks]
4.(a) Solve the equation 14 xx . [2 marks]
(b) On the same axes, sketch the graphs of xy 4 and 1 xy . [2 marks]
(c) Use the information in (a) and (b), find the solution set of the inequality 14
1
x
x.
[3 marks]
5. It is given that matrix
311
21
302
kA , where3
7, kRk .
(a) Show thatAis non-singular. [2 marks]
(b) Find the inverse of A if k=1 . [4 marks]
(c) Use the result in (a), solve the system of linear equations
73 zyx
xzy 2
123 xz . [3 marks]
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CONFIDENTIAL* 4
6. The real functions f and g are defined as follows :
0,:
0,ln2
1:
xxxg
xxxf
(a)Use the sketch graph of function g, explain why the inverse function 1g exists.[3 marks]
(b) Find 1g and state its domain. [2 marks]
(c) Show that f is an increasing function. [2 marks]
(d) State with reason, whether the composite function fg
is defined. [2 marks]
7.(a) Given )34ln(32 2 xey x , find
dx
dy. [2 marks]
(b) The equation of a curve is given by xyxy 4ln2 .
(i) Show that the first derivative of y with respect to x is
)2(2
14
xyx
xy
. [3 marks]
(ii) Find the equation of the normal to the curve at the point (1 , 4). [4 marks]
8. (a) Show that, for all values of p, the point )4,2( 2 ppP lies on the parabola xy 82 .
[2 marks]
(b) Find the equation of the tangent to the parabola xy 82 at the point )4,2( 2 ppP [3 marks]
(c) The tangent to the parabola xy 82 at the point )4,2( 2 ppP meets they-axis at Q.The quadrilateral PQRS is a parallelogram. Given R is the fixed point (-2 , 0),
(i) find the coordinates of S in terms of p, [2 marks](ii) show that as pvaries, the locus of the point S is the parabola )2(22 xy .
[3 marks]
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CONFIDENTIAL* 5
9. (a) By sketching the graph of xy sin3 for 44 x and another suitable graph
on the same axes, show that the equation xecx
cos3 , x in radian , has only one
positive root. [4 marks]
(b) Verify that the positive root of xecx
cos3 lies between 2.2 and 2.3. [2 marks]
(c) Use Newton-Raphson method, find the positive root of xecx
cos3 correct to
three significant figures. [4 marks]
10.(a) Show that
321
16
1
8
1
4
1
2
1
)2( xxxx . [2 marks]
(b) Use the result in (a), expressxx 31)2(
4
in ascending power of x up to and
including the terms in 2x [3 marks]
(c) State the range of values of x for which the expansion in (b) is valid. [2 marks]
(d) By substituting27
1x , find the approximate value of 2 correct to three
decimal places. [3 marks]
11.(a) Determine dxxx
21
2 )2(2 . [2 marks]
Hence, use integration by parts, find dxx
x
22
3
. [3 marks]
(b) Diagram 1 shows the region R, bounded by the curve
xy sec , the x-axis, the y-axis and the line3
x .
(i) Use the trapezium rule with three ordinates to estimatethe area of region R correct to three decimal places.
[3 marks]
(ii) Find the exact volume of the solid generated when theregion R is revolves about the x-axis. [3 marks]
y
xR
3
O
y= sec x
Diagram 1
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CONFIDENTIAL* 6
12. The polynomial nxmxxxf 8)( 23 , where mand nare constants, has a
remainder 2 when divided by x + 2.
Given )(' xf is the first derivative of )(xf with respect to x and 2 is the zero of
)(' xf ,
(a) show that m= 5 and find the value of n, [4 marks]
(b) determine whetherx+ 3 is a factor of )(xf , [2 marks]
(c) show that 0)( xf has only one real root, [3 marks]
(d) express)('
1
xfas a sum of partial fractions. [3 marks]
END OF QUESTION PAPER
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CONFIDENTIAL*
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN
PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN
JPNP PEPERIKSAAN PERCUBAAN JPNP PEPERIKSAAN PERCUBAAN JPNP
PEPERIKSAAN PERCUBAAN
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
NEGERI PAHANG DARUL MAKMUR 2010
Instructions to candidates:
Answer allquestions. Answers may be written in either English or Malay.
All necessary working should be shown clearly.
Non-exact numerical answers may be given correct to three significant figures, or one
decimal place in the case of angles in degrees, unless a different level of accuracy is
specified in the question.
Mathematical tables, a list of mathematical formulae and graph paper are provided.
This question paper consists of printed pages.
954/2 STPM 2010
Three hours
MATHEMATICS T
PAPER 2
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CONFIDENTIAL* 3
Mathematical Formulae for Paper 2 Mathematics T :
Numerical Methods :
Newton-Raphson iteration for 0)( xf :
)('
)(1
n
nnn
xf
xfxx
Trapezium rule :
b
a nn yyyyyhdxxf ])(2[
2
1)( 1210
n
abhrhafyr
andwhere )(
Correlation and regression :
Pearson correlation coefficient:
22
yyxx
yyxxr
ii
ii
Regression line of yon x:
y= a + bx
where
2i
ii
xx
yyxxb
xbya
Trigonometry
BAAABA sincoscossin)sin( BABABA sinsincoscos)cos(
BA
BABA
tantan1
tantan)tan(
AAAAA 2222 sin211cos2sincos2cos AAA 3sin4sin33sin AAA cos3cos43cos 3
2
BAcos
2
BAsin2Bsi nAsin
2
BAsin
2
BAcos2Bsi nAsin
2
BAcos
2
BAcos2BcosAcos
2
BAsin
2
BAsin2BcosAcos
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CONFIDENTIAL* 4
1. (a) Solve the equation
for 0 180 [3]
(b) Express in the form ,where a, band care constants. [4]
2. (a) Express in the form , where R is a
positive constant and an acute angle. [2]
(b) Hence, find the maximum and minimum values for
and their corresponding angles.
Sketch of the graph in the interval 0 360 . [4]
Hence, find the set of values of , with 0 360, which0 . [4]
3. The diagram given shows two intersecting circles ABC and CDEF, XY is a tangent to
the circle ABC at C, while BC is a tangent to CDEF at C.
Prove that , ABC is similar to DCF. [6]
4. The position vectors of three points A , B and C relative to the origin O
are , and respectively.
(a) Find the scalar product of . [2]
(b) Hence , find the ABC and the area of triangle ABC. [4]
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CONFIDENTIAL* 5
5. When a valve is released, water flowed into a large tank that is initially empty.
The volume, v litres , in the tank increase at the rate
where t is measured in hours from the time the valve is released.
(a) What is the initial rate of the water entering the tank ? [1]
(b) Find an expression for v in terms of t. [4]
(c) Determine the value of v when t is 40 minutes. [2]
(d) Find the time taken, to the nearest minute, for the volume of water in the
tank to be 4 litres. [4]
6. Two cyclist, A and B are initially 25 km apart with B on a bearing of N 67E from A.A is moving at 18 km/h in a direction S 20E and B is moving at 12 km/h due south.The velocities of A and B remain unchanged.
(a) Find the velocity of A relative to B. [5]
(b) Find the nearest distance between the two cyclists. [3]
(c) Find the time when the distance between the two cyclists is the nearest. [2]
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1
Marking Scheme
PEPERIKSAAN PERCUBAAN STPM NEGERI PAHANG 2010
Mathematics T Paper 1 (954/1)/ Mathematics S Paper 1 (950/1)
NO. SCHEME MARKS
1(a)
(b)
415
025
415
25
415
25
320
916
341520
)34)(34(
)34)(5(
34
5
a
a
iaa
aaii
ii
iai
i
ai
z
M1use conjugate to
make denominator
real
A1
M1
[A1 [4]
2
14
0)1)(4(
045
)1(45)1(
45limlim
)(lim)(lim
2
22
1
22
1
11
kork
kk
kk
kk
xkxk
xfxf
xx
xx
M1 A1
M1
A1 [4]
3(a)
253
)221)(22(2
1
22...321
5,log...logloglog 2253
5
2
55
yyyyy
M1
A1
(b)
4
4
3
5
5
55
5
5
3
5
2
55
125
5
4
3log
3log4
log33log
3
log1
log
3...)(log)(loglog
y
y
y
y
yyy
y
yyy
M1
A1
A1 [5]
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2
NO. SCHEME MARKS
4(a)
3
1
5
1
0)13)(15(
01215
1216
14
2
22
orx
xx
xx
xxx
xx
M1
A1
(b)
B1
B1
(c)
xx
x
x
41
14
1
From the graph, xx 41 when5
1
3
1 xorx
Solution set: }5
1
3
1:{ xorxx
M1 A1
A1 [7]
5(a)
0..
73
73
37
73
)1(30)5(2
11
13
31
20
31
212
Aei
A
kwhen
k
k
kkA
A is non-singular
M1
A1
0
1
1
x
y xy 4
1 xy
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4
NO. SCHEME MARKS
6(a)
The line y = c, c constant, cuts the graph of xxg )( at only one point
g is one-to-one and onto in 0x 1g exists.
B1
M1
A1
(b)
0,)(
0,
)(
)(
21
2
1
xxxg
xxy
yx
ygx
yxg
B1 B1
(c)
0)('
021
01
0
2
1)('
ln2
1)(
xf
x
xx
xxf
xxf
f is an increasing function.
M1
A1
(d)
gf
gf
DR
DR
),0[),(
),0[),(
function g o f is not defined.
M1
A1 [9]
7(a)
xxe
dx
dy x
34
34 32
2
M1 A1
(b)(i)
)2(2
14
1442
441
2
xyx
xy
dx
dy
x
xy
dx
dyx
dx
dyy
ydx
dyx
xdx
dyy
M1 A1
A1
y
x
xxg )(
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5
NO. SCHEME MARKS
(ii)
17
4
4
17)24)(1(2
1)4)(1(4),4,1(
)2(2
14
normal
T
m
m
dx
dyat
xyx
xy
dx
dy
The equation of normal at (1,4)
072174
446817
)1(17
44
yx
xy
xy
M1
A1
M1
A1 [9]
8(a)
22
222
22
16)2(88
16)4(
)4,2(,8
ppx
ppy
ppPatxy
xy 82 satisfied by P(2p2, 4p)
P lies on the parabola xy 82
M1
A1
(b)
pm
pdx
dym
ppPat
ydx
dy
dx
dyy
xy
T
T
1
4
4
),4,2(
4
82
8
2
2
Equation of tangent
2
22
2
2
24
)2(1
4
pxpy
pxppy
pxp
py
M1
M1
A1
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6
NO. SCHEME MARKS
(c) (i) At Q, x = 0
)2,22(
222
2
4
2
2
2
22
2
2
04,
2
22
2
2'
2
0
)0,2(),2,0(),4,2(),,(
)2,0(
2
2
2
2
2
2
2
2
ppS
pyandpx
ppyand
px
pppyx
MM
RpQppPyxS
pQ
py
ppy
PRQS
M1
A1
(c)(ii) As p varies, locus of P:
)2(2
22
22
2
2222
2
2
2
2
xy
yx
yx
py
pyandpx
B1
M1
A1 [10]
9(a)
xx
xx
ecxx
sin3
sin
13
cos3
y = 3 sin x and y = x
Only one intersection point for 0x
ecxx
cos3 has only one positive root.
B1Correct curve
M1Getting equation of
straight line
A1Correct straight line
seen
A1
-3
3
0
y
4 2 3 4 3 2 x
y = x
y = 3sinx
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7
NO. SCHEME MARKS
(b)
00629.03.2)3.2sin(3)3.2(
02255.02.2)2.2sin(3)2.2(
sin3)(0sin3
sin3
cos3
f
f
xxxfxx
xx
ecxx
Since f(2.2) > 0 and f(2.3) < 0
the positive root lies between 2.2 and 2.3
M1
A1
(c)
28.2
2792.2
8845.2
0842.025.2
25.2('
)25.2(25.2
25.22
3.22.2
1cos3)('
sin3)(
2
1
f
fx
x
xxf
xxxf
28.2
2789.2
9519.2
0010.02792.2
)2792.2('
)2792.2(2792.23
f
fx
The positive root = 2.28 (to 3 s. f.)
M1
A1
A1
A1 [10]
10(a)
...16
1
8
1
4
1
2
1
...8
1
4
1
2
11
2
1
...2!3
)3)(2)(1(
2!2
)2)(1(
21
2
1
212)2(
32
3
32
1
11
xxx
xxx
xxx
xx
M1
A1
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8
NO. SCHEME MARKS
(b)
...4
2322
...2
1
2
3
4
2732
...8
27
2
31...
2
12
...)3(!2
2
3
2
1
)3(2
1
1...16
1
8
1
4
1
2
1
4
)31()2(431)2(
4
2
222
22
232
2
1
1
xx
xxxxx
xxxx
xxxxx
xxxx
M1Expansion of
21
31
x
A1
Expand 21
31 x
correctly
A1
(c) The expansion is valid when
3
1
3
1:
3
1
3
12
1312
xx
x
xandx
xandx
M1
A1
(d)
27
1
x
.).3(414.1
4137.1
81
55
2916
60712
2916
6071
55
2812916
6071
2255
3274
2916
6071
9
8
27
55
4
27
1
4
23
27
122
27
131
27
12
4 2
pd
M1
A1
A1 [10]
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9
NO. SCHEME MARKS
11(a)
cx
cx
dxxx
2
1
2
2
1
2
2
1
2
)2(2
2
1
)2()2(2
cx
xx
dxxxxx
dxxxxx
dxxxx
dxxxdxx
x
2
3
)2(2
)2(22
)()2(2)2(22
1
)2)(2(2
1
)2(2
2
3
222
2
1
222
2
1
22
1
22
2
1
22
2
1
23
2
3
= cxxx 23
222 )2(3
22
M1
A1
M1
A1
A1
(b)(i)
62
03
h
x y = sec x01 x 11y
62
x
2y =1.1547
33
x 3y = 2
Area of R
2
321
3
0
390.1
2)1547.1(2162
1
2
2
1
sec
unit
yyyh
dxx
B1
M1
A1
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10
NO. SCHEME MARKS
(b)(ii) Volume
3
30
3
0
2
2
3
0tan3
tan
tan
sec
unit
x
dxx
dxy
M1
A1
A1 [11]
12(a)
5
08412
08)2(2)2(3)2('
823)('
2
2
m
m
mf
mxxxf
6
216208
2)2(8)2(5)2()2( 23
n
n
nf
M1
A1
M1
A1
(b)
0
6)3(8)3(5)3()3( 23
f
x + 3 is a factor of f(x).
M1
A1
(c) 0)22)(3( 2 xxx
3x or
4
)2)(1(42
4022
2
2
2
acbxx
0222 xx has no real roots
f(x) has only one real root.
M1
M1
A1
(d)
)2(2
1
)43(2
3
)(
1
2
3421,0
2
1)46(1,2
)43()2(1
2431
)2)(43(
1
8103
12
xxxf
ABAx
BBx
xBxA
xB
xA
xxxx
M1
M1
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11
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1
Marking Scheme
PEPERIKSAAN PERCUBAAN STPM NEGERI PAHANG 2010
Mathematics T Paper 2 ( 954 / 2 )
Question Scheme Marks
1(a)
***
,
= 90, 180 , =90= 180
= 90 , 180 , 450 , 540=2543 , 5126 , 12834 , 15417
=2543 , 5126 , 90, 12834 , 15417 , 180.
M1
Factor
Formula
M1
A1
1 (b) M1
M1
M1
A1
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3
2
The solution set is
D1 [ draw
line y=1
and y=0 ]
A1
Question Scheme Marks
3.
Given XY is tangent to the ABC at C.
Given BC is a tangent to CDEF at C.
To prove ABC is similar to CDF.
Construction :
Join points F and D.
3ABC = ACX = [ Angles in the alternate segment. ]ACX = DCF = [ Vertically opposite angles ]ABC = DCF =
ACB = OCD = [ Vertically opposite angles ]OCD = CFD = [ Angles in the alternate segment ]ACB = CFD =
CAB = CDF = 180 ( + )[The sum of angles in a triangle is 180. ]Hence , , ABC is similar to DCF. ( AAA )
M1
A1
M1
A1
M1
A1
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4
Question Scheme Marks
4 (a)
==
=
=
= +
= 36
M1
A1
4.(b)
or
M1
A1
Area of ABC = AB CB sin ABC
=
= or 13.499 or 13.5
M1
A1
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5
Question Scheme Marks
5(a)
Initial time , t = 0 ,
= 5Initial rate is 5 liters per hour.
B1
5 (b)
M1 (separate variables)
M1( correct integration)
M1 ( use limits / find C )
A15 (c) t = 40 minutes , t = , t = hour
v = 3.8164 litres
M1
A1
5 (d) When v= 4 ,
***
,
t = ln 2 hours
t = 42 minutes
M1 ( Quadratic )M1 ( Factorise )
M1 ( value for )
A1
Question Scheme Marks
6(a)
D1 [ arrows ]
D1 [ shape ]
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6
= 31 24The velocity of A and B is 7.88 km/h in the direction of
S 5124 E .
M1
[cosine rule]
M1
[sine rule]
A1
6(b)
Shortest distance = 25 sin 6136= 21.9912 km or 21.9 km
D1
M1
A1
6(c)Time =
Time = 1.5095 hour
M1
A1
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