Determine the coefficient of \(x^{2}\) in the expansion of \((a + 3x)^{6}\).
- A. \(18a^{2}\)
- B. \(45a^{4}\)
- C. \(135a^{4}\)
- D. \(1215a^{2}\)
Find the equation of a circle with centre (-3, -8) and radius \(4\sqrt{6}\).
- A. \(x^{2} - y^{2} - 6x + 16y + 23 = 0\)
- B. \(x^{2} + y^{2} + 6x + 16y - 23 = 0\)
- C. \(x^{2} + y^{2} + 6x - 16y + 23 = 0\)
- D. \(x^{2} + y^{2} - 6x + 16y + 23 = 0\)
Evaluate \(\frac{1}{1 – \sin 60°}\), leaving your answer in surd form.
- A. \(1 - \sqrt{3}\)
- B. \(2 - \sqrt{3}\)
- C. \(4 - 2\sqrt{3}\)
- D. \(4 + 2\sqrt{3}\)
What percentage increase in the radius of a sphere will cause its volume to increase by 45%?
- A. 13%
- B. 15%
- C. 23%
- D. 25%
The fourth term of a geometric sequence is 2 and the sixth term is 8. Find the common ratio.
- A. \(\pm 1\)
- B. \(\pm 2\)
- C. \(\pm 3\)
- D. \(\pm 4\)
If \(\begin{pmatrix} 3 & 2 \\ 7 & x \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix} \), find x.
- A. 5
- B. 6
- C. 7
- D. 8
The inverse of a function is given by \(f^{-1} : x \to \frac{x + 1}{4}\).
- A. \(f : x \to 4x - 1\)
- B. \(f : x \to 4x + 1\)
- C. \(f : x \to \frac{4x - 1}{4}\)
- D. \(f : x \to \frac{x - 1}{2}\)
Solve \(9^{2x + 1} = 81^{3x + 2}\)
- A. \(\frac{-3}{4}\)
- B. \(\frac{-2}{3}\)
- C. \(\frac{4}{5}\)
- D. \(\frac{3}{2}\)
A line is perpendicular to \(3x – y + 11 = 0\) and passes through the point (1, -5). Find its equation.
- A. 3y - x -14 = 0
- B. 3x + y + 1 = 0
- C. 3y + x + 1 = 0
- D. 3y + x + 14 = 0
If \(y^{2} + xy – x = 0\), find \(\frac{\mathrm d y}{\mathrm d x}\).
- A. \(\frac{1 - y}{2y}\)
- B. \(\frac{1 - 2y}{x}\)
- C. \(\frac{1 - y}{x + 2y}\)
- D. \(\frac{1}{x + 2y}\)
Find \(\lim \limits_{x \to 3} \frac{x + 3}{x^{2} – x – 12}\)
- A. -1
- B. \(\frac{-1}{7}\)
- C. \(\frac{1}{7}\)
- D. 1
\(f(x) = (x^{2} + 3)^{2}\) is defines on the set of real numbers, R. Find the gradient of f(x) at x = \(\frac{1}{2}\).
- A. 4.0
- B. 6.5
- C. 5.0
- D. 10.6
The sum and product of the roots of a quadratic equation are \(\frac{4}{7}\) and \(\frac{5}{7}\) respectively. Find its equation.
- A. \(7x^{2} - 4x - 5 = 0\)
- B. \(7x^{2} - 4x + 5 = 0\)
- C. \(7x^{2} + 4x - 5 = 0\)
- D. \(7x^{2} + 4x + 5 = 0\)
\(f(x) = p + qx\), where p and q are constants. If f(1) = 7 and f(5) = 19, find f(3).
- A. 13
- B. 15
- C. 17
- D. 26
The equation of a circle is \(3x^{2} + 3y^{2} + 6x – 12y + 6 = 0\). Find its radius
- A. 1
- B. \(\sqrt{3}\)
- C. \(\sqrt{11}\)
- D. \(\sqrt{6}\)
Solve \(3x^{2} + 4x + 1 > 0\)
- A. \(x < -1, x < -\frac{1}{3}\)
- B. \(x > -1, x > -\frac{1}{3}\)
- C. \(x > \frac{1}{3}, x < -1\)
- D. \(x < \frac{1}{3}, x > -1\)
Simplify \(\sqrt[3]{\frac{8}{27}} – (\frac{4}{9})^{-\frac{1}{2}}\)
- A. \(\frac{-5}{6}\)
- B. \(-\frac{4}{27}\)
- C. \(0\)
- D. \(\frac{2}{9}\)
A function is defined by \(f(x) = \frac{3x + 1}{x^{2} – 1}, x \neq \pm 1\). Find f(-3).
- A. \(-1\frac{1}{4}\)
- B. \(-1\)
- C. \(\frac{4}{5}\)
- D. \(1\)
Find the remainder when \(5x^{3} + 2x^{2} – 7x – 5\) is divided by (x – 2).
- A. -51
- B. -23
- C. 29
- D. 49
Evaluate \(\cos (\frac{\pi}{2} + \frac{\pi}{3})\)
- A. \(\frac{-2}{\sqrt{3}}\)
- B. \(\frac{-\sqrt{3}}{2}\)
- C. \(\frac{\sqrt{3}}{4}\)
- D. \(\frac{4}{\sqrt{3}}\)
If \(\log_{9} 3 + 2x = 1\), find x.
- A. \(\frac{-1}{2}\)
- B. \(\frac{-1}{4}\)
- C. \(\frac{1}{4}\)
- D. \(\frac{1}{2}\)