A binary operation * is defined on the set of real numbers, R, by \(x * y = x + y - xy\). If the identity element under the operation * is 0, find the inverse of \(x \in R\).
Answer
(A)
\(\frac{-x}{1 - x}, x \neq 1\)
2
Solve: \(\sin \theta = \tan \theta\)
Answer
(D)
0°
3
Given that \(a^{\frac{5}{6}} \times a^{\frac{-1}{n}} = 1\), solve for n.
Answer
(D)
1.20
4
Express \(\log \frac{1}{8} + \log \frac{1}{2}\) in terms of \(\log 2\).
Answer
(D)
-4 log 2
5
If \(f(x) = x^{2}\) and \(g(x) = \sin x\), find g o f.
Answer
(B)
\(\sin x^{2}\)
6
Find the third term in the expansion of \((a - b)^{6}\) in ascending powers of b.
Answer
(B)
\(15a^{4}b^{2}\)
7
If \(\sqrt{x} + \sqrt{x + 1} = \sqrt{2x + 1}\), find the possible values of x.
Answer
(D)
0 and -1
8
If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} - 6x + 5 = 0\), evaluate \(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\).
Answer
(B)
\(\frac{8}{5}\)
9
Given that \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\) and \(f(3) = 0\), factorize f(x).
Answer
(C)
(x - 3)(x + 2)(2x -1)
10
Find the equation of the line that is perpendicular to \(2y + 5x - 6 = 0\) and bisects the line joining the points P(4, 3) and Q(-6, 1).
Answer
(D)
5y - 2x - 12 = 0
11
Differentiate \(x^{2} + xy - 5 = 0\).
Answer
(A)
\(\frac{-(2x + y)}{x}\)
12
The fourth term of an exponential sequence is 192 and its ninth term is 6. Find the common ratio of the sequence.
Answer
(B)
\(\frac{1}{2}\)
13
Find the range of values of x for which \(x^{2} + 4x + 5\) is less than \(3x^{2} - x + 2\)
Answer
(B)
\(x < \frac{-1}{2}, x > 3\)
14
Given that \(\frac{\mathrm d y}{\mathrm d x} = \sqrt{x}\), find y.
Answer
(B)
\(\frac{2}{3}x^{\frac{3}{2}} + c\)
15
Given that \(P = \begin{pmatrix} y - 2 & y - 1 \\ y - 4 & y + 2 \end{pmatrix}\) and |P| = -23, find the value of y.
Answer
(B)
-3
16
An object is thrown vertically upwards from the top of a cliff with a velocity of \(25ms^{-1}\). Find the time, in seconds, when it is 20 metres above the cliff. \([g = 10ms^{-2}]\).
Find the coordinates of the point which divides the line joining P(-2, 3) and Q(4, 9) internally in the ratio 2 : 3.
Answer
(B)
\((\frac{2}{5}, 5\frac{2}{5})\)
19
The angle subtended by an arc of a circle at the centre is \(\frac{\pi}{3} radians\). If the radius of the circle is 12cm, calculate the perimeter of the major arc.
A committee consists of 5 boys namely: Kofi, John, Ojo, Ozo and James and 3 girls namely: Rose, Ugo and Ama. In how many ways can a sub-committee consisting of 3 boys and 2 girls be chosen, if Ozo must be on the sub-committee?
Answer
(C)
18
22
Forces 50N and 80N act on a body as shown in the diagram. Find, correct to the nearest whole number, the horizontal component of the resultant force.
Answer
(A)
13N
23
The sales of five salesgirls on a certain day are as follows; GH¢ 26.00, GH¢ 39.00, GH¢ 33.00, GH¢ 25.00 and GH¢ 37.00. Calculate the standard deviation if the mean sale is GH¢ 32.00.
Answer
(B)
GH¢ 5.66
24
A circular ink blot on a piece of paper increases its area at the rate \(4mm^{2}/s\). Find the rate of the radius of the blot when the radius is 8mm. \([\pi = \frac{22}{7}]\).
Answer
(B)
0.08 mm/s
25
Express \(\frac{x^{2} + x + 4}{(1 - x)(x^{2} + 1)}\) in partial fractions.
Answer
(B)
\(\frac{3}{1 - x} + \frac{2x + 1}{x^{2} + 1}\)
26
Two bodies of masses 3kg and 5kg moving with velocities 2 m/s and V m/s respectively in opposite directions collide. If they move together after collision with velocity 3.5 m/s in the direction of the 5kg mass, find the value of V.
Answer
(B)
6.8 m/s
27
The equation of a circle is \(x^{2} + y^{2} - 8x + 9y + 15 = 0\). Find its radius.
Answer
(C)
\(\frac{1}{2}\sqrt{85}\)
28
A particle is acted upon by two forces 6N and 3N inclined at an angle of 120° to each other. Find the magnitude of the resultant force.
Answer
(D)
\(3\sqrt{3}\) N
29
If \(s = 3i - j\) and \(t = 2i + 3j\), find \((t - 3s).(t + 3s)\).
Find the upper quartile of the following scores: 41, 29, 17, 2, 12, 33, 45, 18, 43 and 5.
Answer
(B)
41
32
Given that \(P = \begin{pmatrix} 3 & 4 \\ 2 & x \end{pmatrix}; Q = \begin{pmatrix} 1 & 3 \\ -2 & 4 \end{pmatrix}; R = \begin{pmatrix} -5 & 25 \\ -8 & 26 \end{pmatrix}\) and PQ = R, find the value of x.
Answer
(D)
5
33
Two out of ten tickets on sale for a raffle draw are winning tickets. If a guest bought two tickets, what is the probability that both tickets are winning tickets?
Answer
(B)
\(\frac{1}{45}\)
34
P and Q are the points (3, 1) and (7, 4) respectively. Find the unit vector along PQ.
Answer
(C)
\(\begin{pmatrix} 0.8 \\ 0.6 \end{pmatrix}\)
35
If \(g(x) = \frac{x + 1}{x - 2}, x \neq -2\), find \(g^{-1}(2)\).
