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. \(4(6 + 5\pi)\)
- B. \(4(6 + 2\pi)\)
- C. \(4(3 + 3\pi)\)
- D. \(4(3 + 5\pi)\)
Find the coordinates of the point which divides the line joining P(-2, 3) and Q(4, 9) internally in the ratio 2 : 3.
- A. \((5\frac{2}{3}, \frac{2}{5})\)
- B. \((\frac{2}{5}, 5\frac{2}{5})\)
- C. \((\frac{2}{5}, 2\frac{2}{5})\)
- D. \((\frac{-2}{5}, 5\frac{2}{5})\)
Evaluate \(\int_{0}^{2} (8x – 4x^{2}) \mathrm {d} x\).
- A. \(-16\)
- B. \(\frac{-16}{3}\)
- C. \(\frac{16}{3}\)
- D. \(16\)
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. \([g = 10ms^{-2}]\).
- A. 0 and 1
- B. 0 and 4
- C. 0 and 5
- D. 1 and 4
Given that \(P = \begin{pmatrix} y – 2 & y – 1 \\ y – 4 & y + 2 \end{pmatrix}\) and |P| = -23, find the value of y.
- A. -4
- B. -3
- C. -1
- D. 2
Given that \(\frac{\mathrm d y}{\mathrm d x} = \sqrt{x}\), find y.
- A. \(2x^{\frac{3}{2}} + c\)
- B. \(\frac{2}{3}x^{\frac{3}{2}} + c\)
- C. \(\frac{3}{2}x^{\frac{3}{2}} + c\)
- D. \(\frac{2}{3}x^{2} + c\)
Find the range of values of x for which \(x^{2} + 4x + 5\) is less than \(3x^{2} – x + 2\)
- A. \(x > \frac{-1}{2}, x > 3\)
- B. \(x < \frac{-1}{2}, x > 3\)
- C. \(\frac{-1}{2} \leq x \leq 3\)
- D. \(\frac{-1}{2} < x < 3\)
The fourth term of an exponential sequence is 192 and its ninth term is 6. Find the common ratio of the sequence.
- A. \(\frac{1}{3}\)
- B. \(\frac{1}{2}\)
- C. \(2\)
- D. \(3\)
Differentiate \(x^{2} + xy – 5 = 0\).
- A. \(\frac{-(2x + y)}{x}\)
- B. \(\frac{(2x - y)}{x}\)
- C. \(\frac{-x}{2x + y}\)
- D. \(\frac{(2x + y)}{x}\)
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).
- A. y + 5x + 3 = 0
- B. 2y - 5x - 9 = 0
- C. 5y + 2x - 8 = 0
- D. 5y - 2x - 12 = 0
Given that \(f(x) = 2x^{3} – 3x^{2} – 11x + 6\) and \(f(3) = 0\), factorize f(x).
- A. (x - 3)(x - 2)(2x + 2)
- B. (x + 3)(x - 2)(x - 1)
- C. (x - 3)(x + 2)(2x -1)
- D. (x + 3)(x - 2)(2x - 1)
If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} – 6x + 5 = 0\), evaluate \(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\).
- A. \(\frac{24}{5}\)
- B. \(\frac{8}{5}\)
- C. \(\frac{5}{8}\)
- D. \(\frac{5}{24}\)
If \(\sqrt{x} + \sqrt{x + 1} = \sqrt{2x + 1}\), find the possible values of x.
- A. 1 and -1
- B. -1 and 2
- C. 1 and 2
- D. 0 and -1
Find the third term in the expansion of \((a – b)^{6}\) in ascending powers of b.
- A. \(-15a^{4}b^{2}\)
- B. \(15a^{4}b^{2}\)
- C. \(-15a^{3}b^{3}\)
- D. \(15a^{3}b^{3}\)
If \(f(x) = x^{2}\) and \(g(x) = \sin x\), find g o f.
- A. \(\sin^{2} x\)
- B. \(\sin x^{2}\)
- C. \((\sin x)x^{2}\)
- D. \(x \sin x\)
Express \(\log \frac{1}{8} + \log \frac{1}{2}\) in terms of \(\log 2\).
- A. 3 log 2
- B. 4 log 2
- C. -3 log 2
- D. -4 log 2
Given that \(a^{\frac{5}{6}} \times a^{\frac{-1}{n}} = 1\), solve for n.
- A. -6.00
- B. -1.20
- C. 0.83
- D. 1.20
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 \(x \in R\).
- A. \(\frac{-x}{1 - x}, x \neq 1\)
- B. \(\frac{1}{1 - x}, x \neq 1\)
- C. \(\frac{-1}{1 - x}, x \neq 1\)
- D. \(\frac{x}{1 - x}, x \neq 1\)
Express (14N, 240°) as a column vector.
- A. \(\begin{pmatrix} -7 \\ -7\sqrt{3} \end{pmatrix}\)
- B. \(\begin{pmatrix} 7\sqrt{3} \\ 7\sqrt{3} \end{pmatrix}\)
- C. \(\begin{pmatrix} -7\sqrt{3} \\ -7 \end{pmatrix}\)
- D. \(\begin{pmatrix} 7 \\ -7\sqrt{3} \end{pmatrix}\)
Evaluate \(\frac{\tan 120° + \tan 30°}{\tan 120° – \tan 60°}\)
- A. \(\sqrt{3} + \sqrt{2}\)
- B. \(\frac{2}{3}\)
- C. \(\frac{1}{3}\)
- D. \(-2\sqrt{3}\)