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‘O’ Level Power Revision Series
Additional Mathematics
EVALUATION TEST PAPER
REAL EXAMINATION QUESTIONS
for Secondary 4
Name: ______________________
Time Start: ___________
Date: ______________________
Time End: ____________
Total Marks:
/ 100
16 questions
Total time: 120 min
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.
FOLLOW ALL INSTRUCTIONS CAREFULLY.
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1.
Solve the simultaneous equations
x + 2y = 1
2
3 x + 5 xy − 2 y = 10
2
[4]
2.
(a)
(b)
[2]
Find the range of values of x for which x(6 − x) ≤ 5 .
Find the values of k for which the line 2 x − y = k and the curve
xy + x 2 + 3 = 0 do not intersect.
[3]
3.
(a)
Solve the equation
(b)
Without using a calculator, evaluate (log 3)(log 8).
4.
5.
6.
2 x +1
3
x
+ 18(3 ) − 81 = 0 .
[3]
2
9
Solve the following equations:
3
1
(i) 32 5 x −1 − (64 x ) = 0
4
x
x
(ii) e (2 e − 1) = 10
Without using tables or calculators, find the value of k such that
 5 3 243
10  6
x

= 2+k 3
−
+

 5
5
45
180


[2]
[3]
[3]
[3]
(a) Solve the equation:
log 2 2 − log 2 ( x + 4) = 2 − 2 log 2 x
(b) Evaluate, without the se of calculator, the expression:
log3 81 − log5 125
[3]
7.
Find the range of values of x for which 9(1 − x) p 2 x( x − 6) .
[2]
8.
(a) If the roots of the equation 3 x 2 + (k − x) x + 2 − k 2 = 0 are real, find the
range of values of k. Hence deduce the number of points at which the
[3]
line y = 3 x − 2 intersects the curve y = 2 x 2 − 9 .
[3]
(b) The equation 2 x 2 + 7 x + a = 0 has roots α and α, and 4 x 2 + bx + 16 = 0
has roots α 2 and α 2 . Calculate the possible values of a and b. [4]
9.
It is given that 2 x3 + k x 2 − 18 x + 8 is exactly divisible by 2 x − 1 .
i) Show that k = 3
ii) Factorise 2 x3 + 3 x 2 − 18 x + 8
iii) Hence solve the equation 2 x6 − 3 x4 − 18 x2 − 8 = 0
[2]
[3]
[3]
1
O Level Power Revision Series
Additional Mathematics
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10.
4 − 2x
in partial fractions. Hence, find the partial fractions of
(x − 1)(x2 + 7 )
2 x 4 − 3 x3 + 15 x 2 − 23 x + 11
4 + 2x
(b)
[8]
2
(x + 1)(x2 + 7 )
(x − 1)(x + 7 )
Express
(a)
11.
In the expansion of (1-3x)n, the sum of the coefficient of x and x2 is 75.
(i) Find the value of n, where n is a positive integer.
[4]
2
(ii) Using the value of n found in (i), find the coefficient of x in the
2
n

expansion of  x −  (1 − 3 x ) .
[3]
x

12.
Solutions to this question by accurate drawing will not be accepted.
ABCD is a rhombus where A=(-3,5) and C=(3,1). Given that AB is parallel
to the line 7y=4x, find
(i)
the equation of AB,
[2]
(ii)
the equation of the perpendicular bisector of AC
[2]
(iii)
the coordinates of B and of D
[4]
13.
(a) It is known that variables x and y are related by the equation,
2
x + py + qxy = 0 , where p and q are constants. When the graph of
1
x
against is drawn, the resulting line passes through the point (2,7) and
y
x
5
[3]
has a gradient of . Calculate the value of p and of q .
2
(b)
The table shows experimental values of two variables, x and y .
X
Y
1
2.82
2
3.70
3
4.85
4
5.24
5
8.32
It is believed that one of the experimental values of y is abnormal and also
that x and y are related by an equation of the form
y
3
2x
= a b , where
a and b are constants. It is possible to represent the above equation on a
straight line graph by plotting lg y as the vertical axis and choosing a
suitable variable for the horizontal axis. Explain how this can be done and
hence draw the straight line graph for the given data.
[4]
Use your graph
(i) estimate the value of a and of b ,
[2]
(ii) identify the abnormal reading of y and estimate its correct value. [2]
2
O Level Power Revision Series
Additional Mathematics
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14.
(a) Prove the identity (sin θ + cos θ) (tan θ + cot θ) ≡ sec θ + cosec θ.
[3]
(b) Find all the angles between 0° and 360° inclusive, which satisfy the
equations
(i) 3 tan2 y = 5 sec y – 1
[3]
(ii) 4 sin (2x + 30°) + 3 = 0
[3]
(c) Find all the values of θ between 0 and 6 for which 2 sin
θ
4
cot
θ
4
=3
[2]
15.
Find all the angles between 0° and 360° which satisfy the following
equations.
(a) 4 cos 2x + 2 sin x =3
16.
Given that tan A =
(b) sin x + sin 2x + sin 3x = 0
[6]
3
, where 180° < A < 270°, evaluate, without using
4
tables or a calculator,
(a) sec A
(b) sin 2A
(c) sin 4A
(d) cos (30° – A)
4
Given further that tan (A – B) = , find the exact value of tan B.
5
[6]
3
O Level Power Revision Series
Additional Mathematics
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Answer Key:
1.
(a)
(b)
x ≤ 1 or x ≥ 5
-6 < k < 6
2.
(a)
x=1
(b)
1
3.
(a)
x=1
4.
(i)
1
(ii)
-2
(iii)
−
1
2
(b)
1
1
2
2
or 2
3
5.
k = -8.8
6.
(a)
x=4
(b)
2
1
2
3
or x > 3
2
7.
x< −
8.
(a)
k≤ −
(b)
a = 4, b = -33; or a = -4, b = -65
(i)
k=3
(ii)
(2x-1) (x+4) (x-2)
(iii)
x = 2, -2
9.
4
4
or k ≥ ; 2
3
3
4
O Level Power Revision Series
Additional Mathematics
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10.
x+9
1
−
4( x − 1) 4(x 2 + 7 )
x+9
1
−
4( x − 1) 4(x 2 + 7 )
(a)
2x −1 +
(b)
1
x−9
−
4( x + 1) 4(x 2 + 7 )
11.
(i)
n=5
12.
(i)
7y = 4x + 47
(ii)
13.
(a)
p= −
1
2
q = -2
(b)
plot y vs x, grad = −
(bi)
60°, 360°
(bii)
x = 99.3°, 140.7°, 279.3°, 320.7°
(c)
0
(a)
30°, 150°, 194.5°, 345.5°
(b)
90°, 120°, 180°, 240°, 270°
(a)
−
14.
15.
16.
5
4
(ii)
(b)
24
25
525
y=
3
x+3
2
(iii)
B = (4, 9); D = (-4, -3)
3
1
lg b, lg-intercept = lg a
2
3
(c)
336
625
(d)
−
3+ 4 3
1
;−
10
32
5
O Level Power Revision Series
Additional Mathematics