Past Exam Questions

(a) The function f, defined by \(f : x \mapsto a + b \sin x\) for \(x \in \mathbb{R}\), is such that \(f\left(\frac{1}{6}\pi\right) = 4\) and \(f\left(\frac{1}{2}\pi\right) = 3\).

1. Find the values of the constants \(a\) and \(b\).
2. Evaluate \(ff(0)\).

(b) The function g is defined by \(g : x \mapsto c + d \sin x\) for \(x \in \mathbb{R}\). The range of g is given by \(-4 \leq g(x) \leq 10\). Find the values of the constants \(c\) and \(d\).

The equation of a curve is \(y = 2 \cos x\).

(i) Sketch the graph of \(y = 2 \cos x\) for \(-\pi \leq x \leq \pi\), stating the coordinates of the point of intersection with the \(y\)-axis.

Points \(P\) and \(Q\) lie on the curve and have \(x\)-coordinates of \(\frac{\pi}{3}\) and \(\pi\) respectively.

(ii) Find the length of \(PQ\) correct to 1 decimal place.

The line through \(P\) and \(Q\) meets the \(x\)-axis at \(H (h, 0)\) and the \(y\)-axis at \(K (0, k)\).

(iii) Show that \(h = \frac{5}{9} \pi\) and find the value of \(k\).

The diagram shows the graphs of \(y = \tan x\) and \(y = \cos x\) for \(0 \leq x \leq \pi\). The graphs intersect at points \(A\) and \(B\).

(i) Find by calculation the \(x\)-coordinate of \(A\).

(ii) Find by calculation the coordinates of \(B\).

(a) Solve the equation \(\sin^{-1}(3x) = -\frac{1}{3}\pi\), giving the solution in an exact form.

(b) Solve, by factorising, the equation \(2 \cos \theta \sin \theta - 2 \cos \theta - \sin \theta + 1 = 0\) for \(0 \leq \theta \leq \pi\).

The diagram shows part of the graph of \(y = a \, \cos x - b\), where \(a\) and \(b\) are constants. The graph crosses the \(x\)-axis at the point \(C(\cos^{-1} c, 0)\) and the \(y\)-axis at the point \(D(0, d)\). Find \(c\) and \(d\) in terms of \(a\) and \(b\).

Solve the equation \(\sin^{-1}(4x^4 + x^2) = \frac{1}{6}\pi\).

It is given that \(\alpha = \cos^{-1}\left(\frac{8}{17}\right)\).

Find, without using the trigonometric functions on your calculator, the exact value of \(\frac{1}{\sin \alpha} + \frac{1}{\tan \alpha}\).

A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, \(h\), in meters, of a passenger above the ground is given by the formula \(h = 60(1 - \cos kt)\). In this formula, \(k\) is a constant, \(t\) is the time in minutes that has elapsed since the passenger started the ride at ground level and \(kt\) is measured in radians.

(i) Find the greatest height of the passenger above the ground.

One complete revolution of the wheel takes 30 minutes.

(ii) Show that \(k = \frac{1}{15}\pi\).

(iii) Find the time for which the passenger is above a height of 90 m.

Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for

1. \(\cos \theta\),
2. \(\tan \theta\),
3. \(\sin(\theta + \pi)\).

Find the value of x satisfying the equation \(\sin^{-1}(x - 1) = \arctan(3)\).

The reflex angle \(\theta\) is such that \(\cos \theta = k\), where \(0 < k < 1\).

(i) Find an expression, in terms of \(k\), for

(a) \(\sin \theta\),

(b) \(\tan \theta\).

(ii) Explain why \(\sin 2\theta\) is negative for \(0 < k < 1\).

(a) Find the possible values of x for which \(\sin^{-1}(x^2 - 1) = \frac{1}{3}\pi\), giving your answers correct to 3 decimal places.

(b) Solve the equation \(\sin(2\theta + \frac{1}{3}\pi) = \frac{1}{2}\) for \(0 \leq \theta \leq \pi\), giving \(\theta\) in terms of \(\pi\) in your answers.

Given that \(\cos x = p\), where \(x\) is an acute angle in degrees, find, in terms of \(p\),

1. \(\sin x\),
2. \(\tan x\),
3. \(\tan(90^\circ - x)\).

It is given that \(a = \\sin \theta - 3 \\cos \theta\) and \(b = 3 \\sin \theta + \\cos \theta\), where \(0^\circ \leq \theta \leq 360^\circ\).

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

(ii) Find the values of \(\theta\) for which \(2a = b\).

The functions f and g are defined for \(-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi\) by

\(f(x) = \frac{1}{2}x + \frac{1}{6}\pi\),

\(g(x) = \cos x\).

Solve the following equations for \(-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi\).

1. \(gf(x) = 1\), giving your answer in terms of \(\pi\).
2. \(fg(x) = 1\), giving your answers correct to 2 decimal places.

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

(ii) Use this result to explain why \(\tan \theta > \sin \theta\) for \(0^\circ < \theta < 90^\circ\).

The function \(f : x \mapsto a + b \cos x\) is defined for \(0 \leq x \leq 2\pi\). Given that \(f(0) = 10\) and that \(f\left( \frac{2}{3}\pi \right) = 1\), find

1. the values of \(a\) and \(b\),
2. the range of \(f\),
3. the exact value of \(f\left( \frac{5}{6}\pi \right)\).

(a) Express the equation \(3 \cos \theta = 8 \tan \theta\) as a quadratic equation in \(\sin \theta\).

(b) Hence find the acute angle, in degrees, for which \(3 \cos \theta = 8 \tan \theta\).

The function \(f\) is such that \(f(x) = 2 \sin^2 x - 3 \cos^2 x\) for \(0 \leq x \leq \pi\).

(i) Express \(f(x)\) in the form \(a + b \cos^2 x\), stating the values of \(a\) and \(b\).

(ii) State the greatest and least values of \(f(x)\).

(iii) Solve the equation \(f(x) + 1 = 0\).

The acute angle x radians is such that \(\tan x = k\), where \(k\) is a positive constant. Express, in terms of \(k\),

1. \(\tan(\pi - x)\),
2. \(\tan\left(\frac{1}{2}\pi - x\right)\),
3. \(\sin x\).