If \(\frac{1}{5^{-y}} = 25(5^{4-2y})\), find the value of y.
Answer
(B)
2
2
Simplify: \((1 - \sin \theta)(1 + \sin \theta)\).
Answer
(D)
\(\cos^{2} \theta\)
3
Given that \(3x + 4y + 6 = 0\) and \(4x - by + 3 = 0\) are perpendicular, find the value of b.
Answer
(B)
3
4
Given that \(x * y = \frac{x + y}{2}, x \circ y = \frac{x^{2}}{y}\) and \((3 * b) \circ 48 = \frac{1}{3}\), find b, where b > 0.
Answer
(C)
5
5
If \(f(x) = 3x^{3} + 8x^{2} + 6x + k\) and \(f(2) = 1\), find the value of k.
Answer
(A)
-67
6
If \(8^{x} ÷ (\frac{1}{4})^{y} = 1\) and \(\log_{2}(x - 2y) = 1\), find the value of (x - y).
Answer
(A)
\(\frac{5}{4}\)
7
Simplify \(\frac{1 + \sqrt{8}}{3 - \sqrt{2}}\).
Answer
(D)
\(1 + \sqrt{2}\)
8
Using the binomial expansion \((1+x)^{6} = 1 + 6x + 15x^{2} + 20x^{3} + 15x^{4} + 6x^{5} + x^{6}\), find, correct to 3 dp, the value of \((1.98)^{6}\).
Answer
(C)
60.255
9
If \((x + 2)\) and \((3x - 1)\) are factors of \(6x^{3} + x^{2} - 19x + 6\), find the third factor.
Answer
(A)
\(2x - 3\)
10
If \(2, (k+1), 8,...\) form an exponential sequence (GP), find the values of k.
Answer
(D)
-5 and 3
11
A box contains 5 red and k blue balls. A ball is selected at random from the box. If the probability of selecting a blue ball is \(\frac{2}{3}\), find the value of k.
Answer
(D)
10
12
If \(\frac{x + P}{(x - 1)(x - 3)} = \frac{Q}{x - 1} + \frac{2}{x - 3}\), find the value of (P + Q).
Answer
(C)
0
13
Find the derivative of \(\sqrt[3]{(3x^{3} + 1}\) with respect to x.
Answer
(B)
\(\frac{3x^{2}}{\sqrt[3]{(3x^{3} + 1)^{2}}}\)
14
If \(T = \begin{pmatrix} -2 & -5 \\ 3 & 8 \end{pmatrix}\), find \(T^{-1}\), the inverse of T.
