Past Exam Questions

(i) Sketch, on a single diagram, the graphs of \(y = \cos 2\theta\) and \(y = \frac{1}{2}\) for \(0 \leq \theta \leq 2\pi\).

(ii) Write down the number of roots of the equation \(2\cos 2\theta - 1 = 0\) in the interval \(0 \leq \theta \leq 2\pi\).

(iii) Deduce the number of roots of the equation \(2\cos 2\theta - 1 = 0\) in the interval \(10\pi \leq \theta \leq 20\pi\).

(i) Sketch the curve \(y = 2 \sin x\) for \(0 \leq x \leq 2\pi\).

(ii) By adding a suitable straight line to your sketch, determine the number of real roots of the equation \(2\pi \sin x = \pi - x\). State the equation of the straight line.

The equation of a curve is \(y = 3 \cos 2x\). The equation of a line is \(x + 2y = \pi\). On the same diagram, sketch the curve and the line for \(0 \leq x \leq \pi\).

The function \(f\) is such that \(f(x) = a - b \cos x\) for \(0^\circ \leq x \leq 360^\circ\), where \(a\) and \(b\) are positive constants. The maximum value of \(f(x)\) is 10 and the minimum value is \(-2\).

1. Find the values of \(a\) and \(b\).
2. Solve the equation \(f(x) = 0\).
3. Sketch the graph of \(y = f(x)\).

Another curve, with equation \(y = f(x)\), has a single stationary point at the point \((p, q)\), where \(p\) and \(q\) are constants. This curve is transformed to a curve with equation

\(y = -3f\left(\frac{1}{4}(x + 8)\right).\)

For the transformed curve, find the coordinates of the stationary point, giving your answer in terms of \(p\) and \(q\).

Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leq x \leq \frac{1}{2}\pi\) to the graph of \(y = f(x)\), where \(f(x) = 3 + 2 \sin \frac{1}{4}x\), making clear the order in which the transformations are applied.

The curve \(y = \sin x\) is transformed to the curve \(y = 4 \sin\left(\frac{1}{2}x - 30^\circ\right)\).

Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations are applied.

The curve \(y = \\sin 2x - 5x\) is reflected in the \(y\)-axis and then stretched by scale factor \(\frac{1}{3}\) in the \(x\)-direction.

Write down the equation of the transformed curve.

Functions f, g and h are defined for \(x \in \mathbb{R}\) by

\(f(x) = 3 \cos 2x + 2\),

\(g(x) = f(2x) + 4\),

\(h(x) = 2f\left(x + \frac{1}{2}\pi\right)\).

(d) Describe fully a sequence of transformations that maps the graph of \(y = f(x)\) on to \(y = g(x)\). [2]

(e) Describe fully a sequence of transformations that maps the graph of \(y = f(x)\) on to \(y = h(x)\). [2]

In the diagram, the lower curve has equation \(y = \cos \theta\). The upper curve shows the result of applying a combination of transformations to \(y = \cos \theta\).

Find, in terms of a cosine function, the equation of the upper curve.

Functions f and g are such that

\(f(x) = 2 - 3 \sin 2x \quad \text{for} \; 0 \leq x \leq \pi,\)

\(g(x) = -2f(x) \quad \text{for} \; 0 \leq x \leq \pi.\)

(a) State the ranges of f and g.

The diagram below shows the graph of \(y = f(x)\).

(b) Sketch, on this diagram, the graph of \(y = g(x)\).

The function h is such that

\(h(x) = g(x + \pi) \quad \text{for} \; -\pi \leq x \leq 0.\)

(c) Describe fully a sequence of transformations that maps the curve \(y = f(x)\) on to \(y = h(x)\).

The diagram shows the graph of \(y = f(x)\), where \(f(x) = \frac{3}{2} \cos 2x + \frac{1}{2}\) for \(0 \leq x \leq \pi\).

(a) State the range of \(f\).

A function \(g\) is such that \(g(x) = f(x) + k\), where \(k\) is a positive constant. The x-axis is a tangent to the curve \(y = g(x)\).

(b) State the value of \(k\) and hence describe fully the transformation that maps the curve \(y = f(x)\) on to \(y = g(x)\).

(c) State the equation of the curve which is the reflection of \(y = f(x)\) in the x-axis. Give your answer in the form \(y = a \cos 2x + b\), where \(a\) and \(b\) are constants.

The diagram shows the graph of \(y = f(x)\) where the function \(f\) is defined by

\(f(x) = 3 + 2 \sin \frac{1}{4}x\) for \(0 \leq x \leq 2\pi\).

(a) On the diagram above, sketch the graph of \(y = f^{-1}(x)\). [2]

(b) Find an expression for \(f^{-1}(x)\). [2]

(c) The diagram above shows part of the graph of the function \(g(x) = 3 + 2 \sin \frac{1}{4}x\) for \(-2\pi \leq x \leq 2\pi\).

Complete the sketch of the graph of \(g(x)\) on the diagram above and hence explain whether the function \(g\) has an inverse. [2]

The function \(f\) is such that \(f(x) = 3 - 4 \cos^k x\), for \(0 \leq x \leq \pi\), where \(k\) is a constant.

(i) In the case where \(k = 2\),

(a) find the range of \(f\), [2]

(b) find the exact solutions of the equation \(f(x) = 1\). [3]

(ii) In the case where \(k = 1\),

(a) sketch the graph of \(y = f(x)\), [2]

(b) state, with a reason, whether \(f\) has an inverse. [1]

A function f is defined by \(f : x \mapsto 3 - 2 \tan\left(\frac{1}{2}x\right)\) for \(0 \leq x < \pi\).

1. State the range of \(f\). [1]
2. State the exact value of \(f\left(\frac{2}{3}\pi\right)\). [1]
3. Sketch the graph of \(y = f(x)\). [2]
4. Obtain an expression, in terms of \(x\), for \(f^{-1}(x)\). [3]

The function \(f : x \mapsto 4 - 3 \sin x\) is defined for the domain \(0 \leq x \leq 2\pi\).

(i) Solve the equation \(f(x) = 2\). [3]

(ii) Sketch the graph of \(y = f(x)\). [2]

(iii) Find the set of values of \(k\) for which the equation \(f(x) = k\) has no solution. [2]

The function \(g : x \mapsto 4 - 3 \sin x\) is defined for the domain \(\frac{1}{2}\pi \leq x \leq A\).

(iv) State the largest value of \(A\) for which \(g\) has an inverse. [1]

(v) For this value of \(A\), find the value of \(g^{-1}(3)\). [2]

The function f is defined by \(f : x \mapsto 5 - 3 \sin 2x\) for \(0 \leq x \leq \pi\).

(i) Find the range of \(f\). [2]

(ii) Sketch the graph of \(y = f(x)\). [3]

(iii) State, with a reason, whether \(f\) has an inverse. [1]

A function f is defined by \(f : x \mapsto 3 - 2 \sin x\), for \(0^\circ \leq x \leq 360^\circ\).

(i) Find the range of \(f\). [2]

(ii) Sketch the graph of \(y = f(x)\). [2]

A function \(g\) is defined by \(g : x \mapsto 3 - 2 \sin x\), for \(0^\circ \leq x \leq A^\circ\), where \(A\) is a constant.

(iii) State the largest value of \(A\) for which \(g\) has an inverse. [1]

(iv) When \(A\) has this value, obtain an expression, in terms of \(x\), for \(g^{-1}(x)\). [2]

A curve has equation \(y = 2 + 3 \, \sin \frac{1}{2}x\) for \(0 \leq x \leq 4\pi\).

(a) State greatest and least values of \(y\). [2]

(b) Sketch the curve. [2]

(c) State the number of solutions of the equation \(2 + 3 \, \sin \frac{1}{2}x = 5 - 2x\) for \(0 \leq x \leq 4\pi\). [1]

The function f is defined by \(f(x) = 2 - 3 \cos x\) for \(0 \leq x \leq 2\pi\).

1. State the range of \(f\). [2]
2. Sketch the graph of \(y = f(x)\). [2]

The function \(g\) is defined by \(g(x) = 2 - 3 \cos x\) for \(0 \leq x \leq p\), where \(p\) is a constant.

1. State the largest value of \(p\) for which \(g\) has an inverse. [1]
2. For this value of \(p\), find an expression for \(g^{-1}(x)\). [2]