A binary operation ♦ is defined on the set R, of real numbers by \(a ♦ b = \frac{ab}{4}\). Find the value of \(\sqrt{2} ♦ \sqrt{6}\).
Answer
(C)
\(\frac{\sqrt{3}}{2}\)
5
If \((x - 3)\) is a factor of \(2x^{3} + 3x^{2} - 17x - 30\), find the remaining factors.
Answer
(D)
(2x + 5)(x + 2)
6
Two functions f and g are defined by \(f : x \to 3x - 1\) and \(g : x \to 2x^{3}\), evaluate \(fg(-2)\).
Answer
(A)
-49
7
Given that \(\frac{1}{8^{2y - 3y}} = 2^{y + 2}\).
Answer
(C)
1
8
Given that \((\sqrt{3} - 5\sqrt{2})(\sqrt{3} + \sqrt{2}) = p + q\sqrt{6}\), find q.
Answer
(B)
-4
9
If \(f(x) = \frac{1}{2 - x}, x \neq 2\), find \(f^{-1}(-\frac{1}{2})\).
Answer
(A)
4
10
Find the coefficient of \(x^{4}\) in the binomial expansion of \((1 - 2x)^{6}\).
Answer
(B)
240
11
Find the equation of the line passing through (0, -1) and parallel to the y- axis.
Answer
(C)
x = 0
12
The roots of the equation \(2x^{2} + kx + 5 = 0\) are \(\alpha\) and \(\beta\), where k is a constant. If \(\alpha^{2} + \beta^{2} = -1\), find the values of k.
Answer
(C)
\(\pm 4\)
13
Find the sum of the exponential series \(96 + 24 + 6 +...\)
The mean age of n men in a club is 50 years. Two men aged 55 and 63 years left the club, and the mean age reduced by 1 year. Find the value of n.
Answer
(B)
20
17
A committee of 4 is to be selected from a group of 5 men and 3 women. In how many ways can this be done if the chairman of the committee must be a man?
Answer
(D)
175
18
Simplify \(\frac{^{n}P_{4}}{^{n}C_{4}}\)
Answer
(A)
24
19
Which of the following is a singular matrix?
Answer
(A)
\(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
20
Simplify \(8^{n} \times 2^{2n} \div 4^{3n}\)
Answer
(A)
\(2^{-n}\)
21
The area of a sector of a circle is 3\(cm^{2}\). If the sector subtends an angle of 1.5 radians at the centre, calculate the radius of the circle.
Answer
(C)
2 cm
22
A particle of mass 2.5 kg is moving at a speed of 12 m/s. If a force of magnitude 10 N acts against it, find how long it takes to come to rest.
Answer
(B)
3.0 s
23
Age(in years)
1 - 5
6 - 10
11 - 15
Frequency
3
5
2
Calculate the standard deviation of the distribution.
Answer
(C)
3.50
24
In a firing contest, the probabilities that Kojo and Kwame hit the target are \(\frac{2}{5}\) and \(\frac{1}{3}\) respectively. What is the probability that none of them hit the target?
Answer
(B)
\(\frac{2}{5}\)
25
The equation of the line of best fit for variables x and y is \(y = 19.33 + 0.42x\), where x is the independent variable. Estimate the value of y when x = 15.
Answer
(C)
25.63
26
Find the coordinates of the point on the curve \(y = x^{2} + 4x - 2\), where the gradient is zero.
Answer
(D)
(-2, -6)
27
Find the least value of the function \(f(x) = 3x^{2} + 18x + 32\).
Answer
(A)
5
28
A force of 32 N is applied to an object of mass m kg which is at rest on a smooth horizontal surface. If the acceleration produced is 8\(ms^{-2}\), find the value of m.
Answer
(D)
4
29
Given that \(\begin{pmatrix} 1 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} -6 \\ P \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}\), find the value of P.
Answer
(D)
-3
30
Find the coordinates of the centre of the circle \(4x^{2} + 4y^{2} - 5x + 3y - 2 = 0\).
Answer
(C)
\((\frac{5}{8}, -\frac{3}{8})\)
31
A and B are two independent events such that \(P(A) = \frac{2}{5}\) and \(P(A \cap B) = \frac{1}{15}\). Find \(P(B)\).
Answer
(C)
\(\frac{1}{6}\)
32
The parallelogram PQRS has vertices P(-2, 3), Q(1, 4), R(2, 6) and S(-1,5). Find the coordinates of the point of intersection of the diagonals.
Answer
(C)
\((0, 4\frac{1}{2})\)
33
Find, in surd form, the value of \(\cos 165\).
Answer
(D)
\(-\frac{1}{4}(\sqrt{6} + \sqrt{2})\)
34
The mean and median of integers x, y, z and t are 5 and z respectively. If x < y < z < t and y = 4, find (x + t).
Answer
(A)
12
35
If \(a = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and \(b = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\), find a vector c such that \(4a + 3c = b\).
