Past Exam Questions

In the triangle ABC, AB = 12 cm, angle BAC = 60° and angle ACB = 45°. Find the exact length of BC.

In the diagram, \(\triangle ABC\) is a triangle in which \(AB = 4 \text{ cm}\), \(BC = 6 \text{ cm}\) and angle \(\angle ABC = 150^\circ\). The line \(CX\) is perpendicular to the line \(ABX\).

(i) Find the exact length of \(BX\) and show that angle \(CAB = \tan^{-1} \left( \frac{3}{4 + 3\sqrt{3}} \right)\).

(ii) Show that the exact length of \(AC\) is \(\sqrt{52 + 24\sqrt{3}} \text{ cm}\).

In the diagram, ABED is a trapezium with right angles at E and D, and CED is a straight line. The lengths of AB and BC are \(2d\) and \(2\sqrt{3}\,d\) respectively, and angles BAD and CBE are \(30^\circ\) and \(60^\circ\) respectively.

1. Find the length of CD in terms of d.
2. Show that angle CAD = \(\tan^{-1}\!\left(\frac{2}{\sqrt{3}}\right)\).

In the diagram, triangle ABC is right-angled and D is the mid-point of BC. Angle DAC is \(30^\circ\) and angle BAD is \(x^\circ\). Denoting the length of AD by l,

1. express each of AC and BC exactly in terms of l, and show that \(AB = \tfrac{1}{2}l\sqrt{7}\),
2. show that \(x = \tan^{-1}\!\left(\frac{2}{\sqrt{3}}\right) - 30^\circ\).

The diagram shows part of the graph of \(y = \sin(a(x + b))\), where \(a\) and \(b\) are positive constants. The graph is plotted with the x-axis ranging from \(-\frac{2}{3}\pi\) to \(2\pi\) and the y-axis ranging from -1 to 1. State the value of \(a\) and one possible value of \(b\).

The diagram shows part of the curve with equation \(y = p \sin(q\theta) + r\), where \(p, q\) and \(r\) are constants.

(a) State the value of \(p\).

(b) State the value of \(q\).

(c) State the value of \(r\).

The diagram shows part of the graph of \(y = a \cos(bx) + c\).

(a) Find the values of the positive integers \(a\), \(b\) and \(c\).

(b) For these values of \(a\), \(b\) and \(c\), use the given diagram to determine the number of solutions in the interval \(0 \leq x \leq 2\pi\) for each of the following equations.

(i) \(a \cos(bx) + c = \frac{6}{\pi} x\)

(ii) \(a \cos(bx) + c = 6 - \frac{6}{\pi} x\)

The diagram shows part of the graph of \(y = a \tan(x - b) + c\).

Given that \(0 < b < \pi\), state the values of the constants \(a\), \(b\), and \(c\).

The diagram shows part of the graph of \(y = a + b \sin x\). Find the values of the constants \(a\) and \(b\).

The diagram shows part of the graph of \(y = a + b \sin x\). State the values of the constants \(a\) and \(b\).

The diagram shows the graph of \(y = a \sin(bx) + c\) for \(0 \leq x \leq 2\pi\).

(i) Find the values of \(a, b\) and \(c\).

(ii) Find the smallest value of \(x\) in the interval \(0 \leq x \leq 2\pi\) for which \(y = 0\).

A curve has equation \(y = 3 \cos 2x + 2\) for \(0 \leq x \leq \pi\).

(a) State the greatest and least values of \(y\).

(b) Sketch the graph of \(y = 3 \cos 2x + 2\) for \(0 \leq x \leq \pi\).

(c) By considering the straight line \(y = kx\), where \(k\) is a constant, state the number of solutions of the equation \(3 \cos 2x + 2 = kx\) for \(0 \leq x \leq \pi\) in each of the following cases.

(i) \(k = -3\)

(ii) \(k = 1\)

(iii) \(k = 3\)

The function f is defined by f(x) = a + b cos 2x, for 0 ≤ x ≤ π. It is given that f(0) = -1 and f(\(\frac{1}{2}\pi\)) = 7.

1. Find the values of a and b.
2. Find the x-coordinates of the points where the curve y = f(x) intersects the x-axis.
3. Sketch the graph of y = f(x).

(i) Sketch and label, on the same diagram, the graphs of \(y = 2 \sin x\) and \(y = \cos 2x\), for the interval \(0 \leq x \leq \pi\).

(ii) Hence state the number of solutions of the equation \(2 \sin x = \cos 2x\) in the interval \(0 \leq x \leq \pi\).

(i) Sketch the graph of the curve \(y = 3 \sin x\), for \(-\pi \leq x \leq \pi\).

The straight line \(y = kx\), where \(k\) is a constant, passes through the maximum point of this curve for \(-\pi \leq x \leq \pi\).

(ii) Find the value of \(k\) in terms of \(\pi\).

(iii) State the coordinates of the other point, apart from the origin, where the line and the curve intersect.

The function \(f\), where \(f(x) = a \sin x + b\), is defined for the domain \(0 \leq x \leq 2\pi\). Given that \(f\left(\frac{1}{2}\pi\right) = 2\) and that \(f\left(\frac{3}{2}\pi\right) = -8\),

(i) find the values of \(a\) and \(b\),

(ii) find the values of \(x\) for which \(f(x) = 0\), giving your answers in radians correct to 2 decimal places,

(iii) sketch the graph of \(y = f(x)\).

Functions \(f\) and \(g\) are defined by

\(f : x \mapsto 2 - 3 \cos x\) for \(0 \leq x \leq 2\pi\),

\(g : x \mapsto \frac{1}{2} x\) for \(0 \leq x \leq 2\pi\).

(i) Solve the equation \(fg(x) = 1\).

(ii) Sketch the graph of \(y = f(x)\).

(i) Solve the equation \(2 \cos x + 3 \sin x = 0\), for \(0^\circ \leq x \leq 360^\circ\).

(ii) Sketch, on the same diagram, the graphs of \(y = 2 \cos x\) and \(y = -3 \sin x\) for \(0^\circ \leq x \leq 360^\circ\).

(iii) Use your answers to parts (i) and (ii) to find the set of values of \(x\) for \(0^\circ \leq x \leq 360^\circ\) for which \(2 \cos x + 3 \sin x > 0\).

(i) Sketch, on the same diagram, the curves \(y = \sin 2x\) and \(y = \cos x - 1\) for \(0 \leq x \leq 2\pi\).

(ii) Hence state the number of solutions, in the interval \(0 \leq x \leq 2\pi\), of the equations

(a) \(2 \sin 2x + 1 = 0\),

(b) \(\sin 2x - \cos x + 1 = 0\).

(i) Sketch, on the same diagram, the graphs of \(y = \sin x\) and \(y = \cos 2x\) for \(0^\circ \leq x \leq 180^\circ\).

(ii) Verify that \(x = 30^\circ\) is a root of the equation \(\sin x = \cos 2x\), and state the other root of this equation for which \(0^\circ \leq x \leq 180^\circ\).

(iii) Hence state the set of values of \(x\), for \(0^\circ \leq x \leq 180^\circ\), for which \(\sin x < \cos 2x\).