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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 + 2 y = 1
2
3x 2 + 5xy
− 2y
= 10
[4]
2. (a) Find the range of values of x for which x ( 6 − x)
≤ 5 . [2]
(b) Find the values of k for which the line 2 x − y = k and the curve
2
xy + x + 3 = 0 do not intersect. [3]
3. (a) Solve the equation 3 2 x+ 1 x
+ 18(
3 ) − 81 = 0 . [3]
(b) Without using a calculator, evaluate 3)( 8).
[2]
( log log
2
9
4. Solve the following equations:
3
x −1
1
x
(i) 325
− (64 ) = 0
4
[3]
x x
(ii) e ( 2e
−1)
= 10
[3]
5. Without using tables or calculators, find the value of k such that
⎛ 5 3 243 10 ⎞ 6
⎜
⎟
− +
x = 2 + k
⎝ 5 45 180 ⎠ 5
3
[3]
6. (a) Solve the equation:
log2 2 − log2
( x + 4) = 2 − 2 log2
x
[3]
(b) Evaluate, without the se of calculator, the expression:
log381− log5
125
[3]
7. Find the range of values of x for which 9(1
− x)
p 2x(
x − 6)
. [2]
8.
2 2
(a) If the roots of the equation 3 x + ( k − x)
x + 2 − k = 0 are real, find the
range of values of k. Hence deduce the number of points at which the
line y = 3x
− 2 intersects the curve y = 2 x
2 − 9 . [3]
(b) The equation 2 x 2 + 7x
+ 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.
3 2
It is given that 2 x + k x −18x
+ 8 is exactly divisible by 2x −1.
i) Show that k = 3
[2]
3 2
ii) Factorise 2 x + 3 x −18x
+ 8
[3]
4 2
iii) Hence solve the equation 2 x 6 − 3 x −18
x − 8 0
[3]
O Level Power Revision Series
Additional Mathematics
1

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10. Express ( x − )( 2
+ 7)
(a)
2 x
4
4 − 2x
in partial fractions. Hence, find the partial fractions of
1 x
3 2
− 3 x + 15 x − 23x
+ 11
4 + 2x
(b)
2
( x −1)( x + 7)
( 1)( 2
[8]
x + x + 7)
11. In the expansion of (1-3x) n , the sum of the coefficient of x and x 2 is 75.
(i) Find the value of n, where n is a positive integer. [4]
(ii) Using the value of n found in (i), find the coefficient of x 2 in the
⎛ 2 ⎞
expansion of ⎜ x − ⎟( 1−
3x)
n . [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
x 1 against is drawn, the resulting line passes through the point (2,7) and
y x
5
has a gradient of . Calculate the value of p and of q . [3]
2
(b) The table shows experimental values of two variables, x and y .
X 1 2 3 4 5
Y 2.82 3.70 4.85 5.24 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
b
3
= a 2x
, where
a andb 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 ofb , [2]
(ii) identify the abnormal reading of y and estimate its correct value. [2]
O Level Power Revision Series
Additional Mathematics
2

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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 tan 2 y = 5 sec y – 1 [3]
(ii) 4 sin (2x + 30°) + 3 = 0 [3]
θ θ
(c) Find all the values of θ between 0 and 6 for which 2 sin cot = 3
4 4
[2]
15. Find all the angles between 0° and 360° which satisfy the following
equations.
(a) 4 cos 2x + 2 sin x =3 (b) sin x + sin 2x + sin 3x = 0 [6]
16. Given that tan A = 4
3 , where 180° < A < 270°, evaluate, without using
tables or a calculator,
(a) sec A (b) sin 2A (c) sin 4A (d) cos (30° – A)
Given further that tan (A – B) = 5
4 , find the exact value of tan B. [6]
O Level Power Revision Series
Additional Mathematics
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Answer Key:
1. (a) x ≤ 1or x ≥ 5
(b) -6 < k < 6
2. (a) x = 1
(b)
1
1 2
3. (a) x = 1 (b)
1
1
2
4. (i) 1
(ii) -2
(iii)
2
− or 2
3
5. k = -8.8
6. (a) x = 4
(b) 2 2
1
7. x <
3
− or x > 3
2
8. (a) k ≤
4 4
− or k ≥ ; 2
3 3
(b) a = 4, b = -33; or a = -4, b = -65
9. (i) k = 3
(ii) (2x-1) (x+4) (x-2)
(iii) x = 2, -2
O Level Power Revision Series
Additional Mathematics
4

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10.
1 x + 9
−
4( x −1)
4 x
(a)
(b)
( 2
+ 7)
1 x + 9
2x
−1+
−
4
2
x +
4
( x −1) 4( 7)
1 x − 9
−
+ x
( x 1) 4( 2
+ 7)
11. (i) n = 5 (ii) 525
3
12. (i) 7y = 4x + 47 (ii) y = x + 3 (iii) B = (4, 9); D = (-4, -3)
2
13. (a) p =
1
− q = -2
2
(b) plot y vs x, grad =
3
1
− lg b, lg-intercept = lg a
2
3
14. (bi) 60°, 360°
(bii) x = 99.3°, 140.7°, 279.3°, 320.7°
(c) 0
15. (a) 30°, 150°, 194.5°, 345.5°
(b) 90°, 120°, 180°, 240°, 270°
16. (a)
5
− (b)
4
24
25
(c)
336
625
(d)
3+
4
−
10
3
; −
1
32
O Level Power Revision Series
Additional Mathematics
5