If \(P = {x : -2 < x < 5}\) and \(Q = {x : -5 < x < 2}\) are subsets of \(\mu = {x : -5 \leq x \leq 5}\), where x is a real number, find \((P \cup Q)\).
Svar
(A)
\({x : -5 < x < 5}\)
2
Express \(\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}}\) in the form \(p\sqrt{3} + q\sqrt{2}\).
Svar
(B)
\(7\sqrt{2} - \frac{17\sqrt{3}}{3}\)
3
An operation * is defined on the set, R, of real numbers by \(p * q = p + q + 2pq\). If the identity element is 0, find the value of p for which the operation has no inverse.
Svar
(A)
\(\frac{-1}{2}\)
4
Consider the statements:
p : Musa is short
q : Musa is brilliant
Which of the following represents the statement "Musa is short but not brilliant"?
Svar
(C)
\(p \wedge \sim q\)
5
If \(f(x) = \frac{4}{x} - 1, x \neq 0\), find \(f^{-1}(7)\).
Svar
(C)
\(\frac{1}{2}\)
6
If \(y = 4x - 1\), list the range of the domain \({-2 \leq x \leq 2}\), where x is an integer.
If the solution set of \(x^{2} + kx - 5 = 0\) is (-1, 5), find the value of k.
Svar
(B)
-4
9
The remainder when \(x^{3} - 2x + m\) is divided by \(x - 1\) is equal to the remainder when \(2x^{3} + x - m\) is divided by \(2x + 1\). Find the value of m.
Svar
(C)
\(\frac{1}{8}\)
10
If (2t - 3s)(t - s) = 0, find \(\frac{t}{s}\).
Svar
(A)
\(\frac{3}{2}\) or \(1\)
11
Solve for x in the equation \(5^{x} \times 5^{x + 1} = 25\).
Svar
(C)
\(\frac{1}{2}\)
12
If \(\log_{10}y + 3\log_{10}x \geq \log_{10}x\), express y in terms of x.
Svar
(D)
\(y \geq \frac{1}{x^{2}}\)
13
Simplify \(\frac{^{n}P_{5}}{^{n}C_{5}}\).
Svar
(D)
120
14
Given n = 3, evaluate \(\frac{1}{(n-1)!} - \frac{1}{(n+1)!}\)
Svar
(D)
\(\frac{11}{24}\)
15
Find the coefficient of \(x^{3}\) in the expansion of \([\frac{1}{3}(2 + x)]^{6}\).
Svar
(D)
\(\frac{160}{729}\)
16
Find the fourth term in the expansion of \((3x - y)^{6}\).
Svar
(A)
\(-540x^{3}y^{3}\)
17
The 3rd and 6th terms of a geometric progression (G.P.) are \(\frac{8}{3}\) and \(\frac{64}{81}\) respectively, find the common ratio.
Svar
(B)
\(\frac{2}{3}\)
18
Given that \(-6, -2\frac{1}{2}, ..., 71\) is a linear sequence , calculate the number of terms in the sequence.
Svar
(D)
23
19
If \(\begin{vmatrix} m-2 & m+1 \\ m+4 & m-2 \end{vmatrix} = -27\), find the value of m.
Find the gradient to the normal of the curve \(y = x^{3} - x^{2}\) at the point where x = 2.
Svar
(A)
\(\frac{-1}{8}\)
28
Find the minimum value of \(y = 3x^{2} - x - 6\).
Svar
(B)
\(-6\frac{1}{12}\)
29
The radius of a circle increases at a rate of 0.5\(cms^{-1}\). Find the rate of change in the area of the circle with radius 7cm. \([\pi = \frac{22}{7}]\)
Svar
(B)
22\(cm^{2}s^{-1}\)
30
Find an expression for y given that \(\frac{\mathrm d y}{\mathrm d x} = x^{2}\sqrt{x}\)
Svar
(C)
\(\frac{2x^{\frac{7}{2}}}{7} + c\)
31
Given that \(n = 10\) and \(\sum d^{2} = 20\), calculate the Spearman's rank correlation coefficient.
Svar
(C)
0.879
32
Find the variance of 11, 12, 13, 14 and 15.
Svar
(A)
2
33
A fair coin is tossed 3 times. Find the probability of obtaining exactly 2 heads.
Svar
(B)
\(\frac{3}{8}\)
34
A box contains 14 white balls and 6 black balls. Find the probability of first drawing a black ball and then a white ball without replacement.
Svar
(B)
0.22
35
Given that \(r = 3i + 4j\) and \(t = -5i + 12j\), find the acute angle between them.
