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1
Answer all questions
1. Find the set of values of x such that 2x
3x [4]
2. Using the laws of the algebra of sets, show that ')'(')( QQPQP [4]
3. Show that )3( x is a factor of .63234 x x x x State another linear factor
Hence,write down one quadratic factor of .63234 x x x x , and find a second
quadratic factor of this polynomial. Find the set of values of x so that 0)( x f . [10]
4. a) The real roots of the quadratic equation 02 cbxax are and .
Prove thata
b and
a
c . [5]
b) If and are roots of the equation 0322 x x , find a quadratic equation with
the roots 12 and 12 , expressing your answer in the form 0
2 cbyay
where a, b and c are integers. [6]
5. The functions f and g are defined by
0,,:
0,,ln:
x x x xh
x x x x f
i) Show that the composite function fh( x) exists and find the function. State the range
of fh( x). [5]
ii) Determine whether the composite function hf ( x) exists. [3]
iii) Sketch the graph of fh( x) and hence determine whether fh( x) is one-to-one. [4]
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2
6.. . If ,
211
121
112
A find 2 A [2]
find the values of m and n such that 02 nI mA A , where I is 3 x 3 identity
matrix and O is 3 x 3 null matrix. [3]
Hence find 3 A and [4]
7. a) Find the complex numbers z in the form bia if i z 24102 . [6]
b) Hence, find the arguments of . z [3]
8., Express4
12 x
in the form of partial fractions. Hence, show that
.2
3ln
4
1
4
16
4
2
dx x
[7]
9. The region R, in the first quadrant, is bounded by the line x y 3 and the curve
x y
1
3 between their points of intersection. Find the coordinates of these points of
intersection, and draw a sketch showing the region R. [4]
Find the area of R. [4]
Find the volume of the solid formed when R is rotated through one revolution about
the x-axis . [4]
10. The function f is defined by f : xk e x x , and k is a positive constant.
a) State the range of f
b) Find f ( ln k ) and simplify your answer
c) Find the inverse function of f and state its domain
d) On the same axes, sketch the curves of f and 1 f . [7]
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3
11. Express
2
1
2
)x21(
)x31(
in ascending powers of x until and including the term in .2 x
State the range of values of x such that the expansion is valid. [4]
12. i) Express)1r 2)(1r 2(
2
in partial fractions [3]
ii) Show that 1n2
n2
)1r 2)(1r 2(
2n
1r
[3]
iii) Find the sum of the series3129
1
75
1
53
1
31
1
and find
1r )1r 2)(1r 2(
1 [5]
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4
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1
1 Express 4 sin x + cos x in the form r sin (x + α), where r > 0 and 0o < α < 90o.
Hence solve the equation 4 sin x + cos x = 2 for 0 o < x < 360o. (6 marks)
2 Given that y = , prove that y= - tan2 . Find the exact value of
tan2 15o in the form p+q , where p, q and r are integers. (6 marks)
3 The forces F1= (5i + 3j)N, F2= (4i -6j)N, F3= (-2i + 7j)N act at a point.
(a)
Calculate the magnitude of the resultant force.
(b)
Using the scalar product, calculate the angle between the resultant force
and the force F4= (5i + 3j)N
(2 marks)
(4 marks)
4 A certain substance evaporates at a rate which is proportional to the amount of
substance left. Given that the initial amount of the substance is A and the amount
which has evaporated at time t is x,
(a)
write a differential equation to show the rate of evaporation.
(b) Solve the differential equation and sketch the graph of x against t.
(c) Given that it took ln seconds for half the amount to be evaporated, find
how long it takes for of the initial amount to be evaporated.
(1 marks)
(6 marks)
(4 marks)
5 In triangle ABC, the point X divides BC internally in the ratio m:n, where m+n=1.
Express AX2 in terms of AB, BC, CA, m and n. (5 marks)
6 In the triangle ABC, the point P lies on the side AC such that BPC = ABC.Show that the triangles BPC and ABC are similar.
If AB=4 cm, AC=8 cm and BP=3 cm, find the area of the triangle BPC.
(3 marks)
(4 marks)
7 The probability that it rains in a certain area is 0.2. The probability that an
accident occurs at a particular corner of a road is 0.05 if it rains and 0.02 if it does
not rain. Find the probability that it rains if an accident occurs at the corner. (5 marks)
8 (a) X is a random variable such that X~B(n,p). If E(X) = 2 and Var (X) = ,
find the values of n and p, and P(X=3).
(b)
On average, the number of books read by an adult is 2 books per year.Using the Poisson distribution, find, correct to 4 decimal places, the
probability that
(i)
an adult reads exactly 3 books per year.
(ii)
an adult reads more than 5 books in 2 years.
(5 marks)
(5 marks)
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2
9 The following data presented as a stem plot are the weekly expenditure of a group
of college students.
2
3
4
5
6
7
8
2 5
0 3 4 6 8
1 3 3 7 9
1
2
2
5
0 4
3
5
Key 3| 3 means RM33
(a) Find the percentage of students who spend less than RM50.
(b) Find the mean and standard deviation of the students’ expenditure.
(c) Find the median and semi-interquartile range of the students’ expenditure.
(d) Construct a boxplot for the above data and state the shape of the data.
(2 marks)
(4 marks)
(5 marks)
(4 marks)
10 Table below shows a probability distribution of a discrete random variable, X.X 0 3 6 9 12
P(X=x) 0.2 p q q 0.1
a) Given that E(X) = 6, find the value of p and q.
b) Calculate the variance of X.
(4 marks)
(3 marks)
11 X is the random variable of a normal distribution where X~N(µ, ). If P(X>120)
= 0.0415 and P(X<90) = 0.2114, find the values of µ and
(8 marks)
12 The cumulative distribution function of a continuous random variable is given by
a) Determine the value of k.
b) Determine the lower quartile of X
c) Find the probability density function of X and sketch the graph of y = f(x).
d) Find Var (2X – 1)
(2 marks)
(2 marks)
(4 marks)
(7 marks)
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1
Answer all questions
1. 2x
3x
02x
3x M1
0322
x
x x
0)1)(3(
x
x x A1
Using a table for the sign of x
x x )1)(3(
The set values of x is ),3()0,1( x A1 [4]
2. Using the laws of the algebra of sets, show that ')'(')( QQPQP [4]
)'(')( QPQP =
')
'(
')( QPQP B1
= )'()''( QPQP M1
= )'('PPQ M1
= 'Q
=
'Q
A1
[4]
‐
‐
+
‐
+
‐
+
+
(x + 1)
(x – 3)
x
)3x)(1x(
x
30-1
M1
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2
3. 3 x , 6)3(3)3()3()3()3( 234 f M1
= 6333339
= 0
A1
Another linear factor is 3 x B1
Quadratic factor is )3)(3( x x = 32 x B1
)2)(3()( 22 x x x x f M1
The second quadratic factor is 22 x x A1
0)( x f
0)2)(3( 22 x x x
24
1)
2
1()2( 22 x x x M1
=
4
7)
2
1( 2 x > 0 , For all values of . x A1
0)3( 2 x M1
0)3)(3( x x
.,33, x A1 [10]
4. Equation with the roots and can be written as
0))(( x x M1
0)(2 x x … (1) A1
Quadratic equation is 02 cbxax
02 a
c x
a
b x (2) B1
Comparing the equation (1) & (2),
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3
a
b )( M1 ( comparing the coefficient x and constant )
a
c A1 (for each correct answers) [5]
OR
Roots are
a
acbb x
2
42
Let a
acbb
2
42 and
a
acbb
2
42 B1
a
b
a
acbb
a
acbb
)
2
4()
2
4(
22
M1A1
a
c
a
acbb
a
acbb
)2
4)(2
4(
22
M1A1
[5]
b) For 0322 x x ; 2 , 3 B1
11 22 = 222
= 22)(2
M1
= 4 A1
)1)(1(22 = 1
2222
= 12)()(22 M1
= 12 A1
A quadratic equation with the roots 12 and 12 is
01242 y y A1 [6]
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4
5. i) Range of h is ,0 B1
Domain of f is ,0 B1
Range of h = domain of f )( x fh exists. B1 (with reason)
0,,ln)( x x x x f x fh B1
Range of fh is , B1
ii) Range of f is , B1
Domain of
h is
,00,
B1
Range of f domain of g hf does not exists. B1 (with reason)
iii)
The function is not one‐to‐one B1
because any line that is drawn parallel to the x‐axis intersects the graph at 2
points
( or there are 2 values of x for the same value of y ) B1 reason) [12]
x
y
11
G1 (correct
shape)
G1 ( all correct )
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5
6.. . 2 A
211
121
112
211
121
112
=
655
565
556
M1 A1
02
nI mA A
000
000
000
00
00
00
2
2
2
655
565
556
n
n
n
mmm
mmm
mmm
M1
05 m
5m
B1
026 nm
4n A1
A A A
I A A
I A A
45
45
045
23
2
2
=
211
121
112
4
655
565
556
5 M1
=
222121
212221
212122
A1
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6
1
43
41
41
4
1
4
3
4
14
1
4
1
4
3
211
121
112
500
050
005
4
1
1)5(41
)5(4
1
)5(4
1
45
045
1
2
2
2
A
M A I A
I A I A
I A A
I A A
I A A
[9]
7. a i z 24102
ibia 2410)(2 M1
iabiba 2410222
)2........(..........12
)1......(..........1022
ab
ba M1
From (2) a
b12
, Subt. in (1)
102
2122 a
a M1
01442104 aa
0)82)(182( aa A1
182 a , .82 a
a is real , 182 a
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7
23a A1(either one)
23a , 2223
12
b
2223 z i or 2223 i A1 [6]
b) 2223 z i
Arg. 23
22tan 1 z M1
= 0.588
rad.
A1
2223 z i
Arg.
23
22tan
1 z
= ‐2.55 rad. A1
8 Let 224
12
x
B
x
A
x
:. 1 = A(x+2) + B(x‐2) B1
x = 2, 1 = 4A ; A =
4
1
X = ‐2 1 = ‐4B ; B =
4
1
)2(4
1
)2(4
1
4
12
x x x
A1
M1
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8
6
4
6
4
2)2(4
1
)2(4
1
4
1 dx x xdx
x B1
=
6
4
)2ln(4
1)2ln(
4
1
x x M1
=
6
42
2ln
4
1
x
x
Shown
2
3ln
4
1
4
6ln
4
1
2
6
8
4ln
4
1
6
2ln
4
1
8
4ln
4
1
9. x x
31
3 M1
02 x x
A1
x = 0 , 2 A1
x y 3
3
(2,1) 3
y
x
G1 curve correct
M1
A1 [7]
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9
Area of R = dx x
x
2
01
33 M1
= 2
0
2
1ln32
3
x
x x A1
= 3ln326 M1
= 3ln34 A1
Volume = dx x
dx x
222
01
33
M1
=
2
0
12
0
32
1
19
339
x x
x x A1
= 3
8 M1 (subst.) A1 [12]
10. a) ,k B1
b) k k ek f k 2ln ln B1
c) yk e x M1
k y x ln
k x x f
ln:1
A1
domain is ,k B1
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10
d)
11. 2
1
2 )x21()x31(
= ...)x2(
21
)2
3)(
2
1(
)x2)(
2
1(1[)x31( 22
] M1
= ...])x4(8
3x1)[x9x61( 22
= ...]2
31)[961( 22 x x x x M1
= ...x9x6x6x2
3x1 222
= ...2
3371 2 x x A1
)2
1,
2
1( x B1 [4]
k
k
k+1
k+1
f
1 f
D1 correct graph for f
D1 correct
graph
for
f 1
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11
12.i))1r 2)(1r 2(
2
=
1r 2
B
1r 2
A
=)1r 2)(1r 2(
B)1r 2( A)1r 2(
B1
When2
1r ; 2 A2 1 A
When2
1r ; 2B2 1B
Therefore1r 2
1
1r 2
1
)1r 2)(1r 2(
2
A1
ii)
n
1r )1r 2)(1r 2(2 =
n
1r 1r 21
1r 21 M1
=
n
1r 1r 2
1
1r 2
1
= )0(f )n(f M1
=1n2
11
=1n2
n2
A1
iii)3129
1
75
1
53
1
31
1
=
15
1r )1r 2)(1r 2(
1 M1
=
15
1r )1r 2)(1r 2(2
21
=
1n2
n2
2
1 M1
=31
15 A1
M1
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12
1r )1r 2)(1r 2(
1 =
1r
n )1r 2)(1r 2(
1lim M1
=
1n2
nlimn
=
n
12
1limn
=2
1 A1
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13
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1
1.
=
r2 = 42 + 12
4 sin
x + cos
x =
=2
X+14.04O = 29.02O, 150.98O
X = 15.0O, 1370O
B1
M1
M1
A1
M1
A1
2.
=
=
= ‐tan2 ( )
Let α= 30o
, ‐ tan2
( )=
=
tan 15o= 7‐4
M1
M1
A1
M1
M1
A1
3.a) FR= (5i+3j) + (4i‐6j) + (‐2i +7j) = 7i + 4j
|FR| = = 8.062 N
B1
B1
3.b) cos θ=
=
=
M1
M1
M1
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3
In ΔABC, by using the cosine rule
In ΔABX, by using the cosine rule
=
=
=
=
=
=
M1
M1
M1
A1
6.
In ΔBPC and ΔABC
given
common angle
remaining angle
are similar AAA. Shown.
B1
B1
B1
M1
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4
By using Heron’s formula,
Area of ΔBPC = = 6.54cm2
M1
M1
A1
7 Let R: Event that it rains
Let A: Event that an accident occurs
P(R/A) = = M1 M1
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5
=
= = 0.3846
M1 M1
A1
8.a) X
8C3
0.2076
B1(both)
M1
A1(both)
M1
A1
8. b)i Let X= number of books read per year
X
= 0.1804
M1
A1
8.b)ii Let Y = number of books read in two years
Y
M1
A1
9.a) % students who spent less than RM50 =
= 60%
M1
A1
9.b) Mean =
Var = = 237.1
Std Dev = 15.4
M1A1
M1
A1
9.c) Median=
Semi interquartile range = = 9.25
B1
M1A1
9.d)
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6
box , median, Q 1, Q3 correct
whiskers seen and drawn on graph paper all correct
skewed to the left (negative)
D1
D1 D1
B1
10.a) E(X)=6 = 3p +15q + 1.2
1.6 = p+5q ......(1)
p+2q=0.7 ......(2)
p=0.1
q=0.3
M1
M1
A1
A1
10.b) Var (X) = E(X2)‐E2(X)
= 50.4‐36 = 14.4
B1M1 A1
11. P(X>120) = 0.0415
From table, P(Z>1.733) = 0.0415
From table, P(Z>0.802) = 0.2114
(1)/(2)
= ‐194.49 + 2.161
3.161
From (1)....
=11.83
B1
M1
B1
M1
M1
M1
A1
A1
12.a) k(6‐3) =1
k=
M1
A1
12.b) Q 1= 3+( =3.75 M1A1
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7/18/2019 [Edu.joshuatly.com] Gerak Gempur Perak STPM 2012 Maths T [5A930C99]
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7
12.c)
f(x) =
On the graph,
values
horizontal line
M1A1
D1
D1
12.d) E(X2) =
=
=
= 21
E(X) =
=
=
= 4.5
Var(X) = 21‐
=0.75
Var(2X‐1) = 4 Var(X) = 4x0.75
M1
M1
A1
M1 A1
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