Past Exam Questions

Given that \(x = \sin^{-1}\left(\frac{2}{5}\right)\), find the exact value of

(i) \(\cos^2 x\),

(ii) \(\tan^2 x\).

The function \(f : x \mapsto 5 \sin^2 x + 3 \cos^2 x\) is defined for the domain \(0 \leq x \leq \pi\).

1. Express \(f(x)\) in the form \(a + b \sin^2 x\), stating the values of \(a\) and \(b\).
2. Hence find the values of \(x\) for which \(f(x) = 7 \sin x\).
3. State the range of \(f\).

(a) Solve the equation \(3 \tan^2 x - 5 \tan x - 2 = 0\) for \(0^\circ \leq x \leq 180^\circ\).

(b) Find the set of values of \(k\) for which the equation \(3 \tan^2 x - 5 \tan x + k = 0\) has no solutions.

(c) For the equation \(3 \tan^2 x - 5 \tan x + k = 0\), state the value of \(k\) for which there are three solutions in the interval \(0^\circ \leq x \leq 180^\circ\), and find these solutions.

The function \(f : x \mapsto 3 \cos^2 x - 2 \sin^2 x\) is defined for \(0 \leq x \leq \pi\).

(i) Express \(f(x)\) in the form \(a \cos^2 x + b\), where \(a\) and \(b\) are constants.

(ii) Find the range of \(f\).

The function \(f : x \mapsto p \sin^2 2x + q\) is defined for \(0 \leq x \leq \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = f(x)\).

(i) In terms of \(p\) and \(q\), state the range of \(f\).

(ii) State the number of solutions of the following equations.

(a) \(f(x) = p + q\)

(b) \(f(x) = q\)

(c) \(f(x) = \frac{1}{2}p + q\)

(iii) For the case where \(p = 3\) and \(q = 2\), solve the equation \(f(x) = 4\), showing all necessary working.

The equation of a curve is \(y = 3 \cos 2x\) and the equation of a line is \(2y + \frac{3x}{\pi} = 5\).

(i) State the smallest and largest values of \(y\) for both the curve and the line for \(0 \leq x \leq 2\pi\).

(ii) Sketch, on the same diagram, the graphs of \(y = 3 \cos 2x\) and \(2y + \frac{3x}{\pi} = 5\) for \(0 \leq x \leq 2\pi\).

(iii) State the number of solutions of the equation \(6 \cos 2x = 5 - \frac{3x}{\pi}\) for \(0 \leq x \leq 2\pi\).

Angle x is such that \(\sin x = a + b\) and \(\cos x = a - b\), where a and b are constants.

(i) Show that \(a^2 + b^2\) has a constant value for all values of x.

(ii) In the case where \(\tan x = 2\), express a in terms of b.

The diagram shows part of the graph of \(y = a + \tan bx\), where \(x\) is measured in radians and \(a\) and \(b\) are constants. The curve intersects the \(x\)-axis at \(\left(-\frac{1}{6}\pi, 0\right)\) and the \(y\)-axis at \((0, \sqrt{3})\). Find the values of \(a\) and \(b\).

Find the exact value of \(\int_{0}^{6} \frac{x(x+1)}{x^2+4} \, dx\).

(i) Using the expansions of \(\cos(3x + x)\) and \(\cos(3x - x)\), show that \(\frac{1}{2}(\cos 4x + \cos 2x) = \cos 3x \cos x\).

(ii) Hence show that \(\int_{-\frac{1}{6}\pi}^{\frac{1}{6}\pi} \cos 3x \cos x \, dx = \frac{3}{8}\sqrt{3}\).

It is given that \(x = \ln(1-y) - \ln y\), where \(0 < y < 1\).

(i) Show that \(y = \frac{e^{-x}}{1 + e^{-x}}\).

(ii) Hence show that \(\int_0^1 y \, dx = \ln \left( \frac{2e}{e+1} \right)\).

(i) Prove the identity \(\tan 2\theta - \tan \theta \equiv \tan \theta \sec 2\theta\).

(ii) Hence show that \(\int_{0}^{\frac{1}{6}\pi} \tan \theta \sec 2\theta \, d\theta = \frac{1}{2} \ln \frac{3}{2}\).

(a) Find \(\int (4 + \tan^2 2x) \, dx\).

(b) Find the exact value of \(\int_{\frac{1}{4}\pi}^{\frac{1}{2}\pi} \frac{\sin(x + \frac{1}{6}\pi)}{\sin x} \, dx\).

(i) Prove that \(\cot \theta + \tan \theta \equiv 2 \csc 2\theta\).

(ii) Hence show that \(\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} \csc 2\theta \, d\theta = \frac{1}{2} \ln 3\).

(i) Prove the identity \(\cos 4\theta + 4 \cos 2\theta \equiv 8 \cos^4 \theta - 3\).

(ii) Hence

(a) solve the equation \(\cos 4\theta + 4 \cos 2\theta = 1\) for \(-\frac{1}{2}\pi \leq \theta \leq \frac{1}{2}\pi\),

(b) find the exact value of \(\int_0^{\frac{1}{4}\pi} \cos^4 \theta \, d\theta\).

It is given that \(f(x) = 4 \cos^2 3x\).

(i) Find the exact value of \(f'(\frac{1}{9}\pi)\).

(ii) Find \(\int f(x) \, dx\).

(i) Prove the identity \(\cos 3\theta \equiv 4 \cos^3 \theta - 3 \cos \theta\).

(ii) Using this result, find the exact value of \(\int_{\frac{1}{3}\pi}^{\frac{1}{2}\pi} \cos^3 \theta \, d\theta\).

(i) Using the expansions of \(\cos(3x-x)\) and \(\cos(3x+x)\), prove that \(\frac{1}{2}(\cos 2x - \cos 4x) \equiv \sin 3x \sin x\).

(ii) Hence show that \(\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} \sin 3x \sin x \, dx = \frac{1}{8}\sqrt{3}\).

(i) Prove the identity \(\cos 4\theta - 4 \cos 2\theta + 3 \equiv 8 \sin^4 \theta\).

(ii) Using this result find, in simplified form, the exact value of \(\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} \sin^4 \theta \, d\theta\).

(a) Using the expansions of \(\sin(3x + 2x)\) and \(\sin(3x - 2x)\), show that \(\frac{1}{2}(\sin 5x + \sin x) \equiv \sin 3x \cos 2x\).

(b) Hence show that \(\int_0^{\frac{1}{4}\pi} \sin 3x \cos 2x \, dx = \frac{1}{5}(3 - \sqrt{2})\).