Past Exam Questions

The curve with equation \(y=a\sin bx+c\), where \(a\), \(b\) and \(c\) are constants, passes through the points \((4\pi,11)\) and \(\left(-\frac{4\pi}{3},5\right)\). It is given that \(a\sin bx+c\) has period \(16\pi\).

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

(b) Using your answer to part (a), find the coordinates of the minimum point on the curve for \(0\le x\le16\pi\).

Sketch the graph of

\(y=\left|4\cos 2x\right|\)

for \(0\leq x\leq \pi\), giving the coordinates of the points where the graph meets the axes.

The diagram shows the graph of \(y=a\sin bx+c\), where \(a\), \(b\) and \(c\) are integers, for \(-180^{\circ}\le x\le 180^{\circ}\). Find the values of \(a\), \(b\) and \(c\).

The graph of \(y=a\tan bx+c\) has asymptotes at \(x=-4\pi\) and \(x=4\pi\), and passes through the points \(P(0,3)\) and \(Q(2\pi,7)\).

(a) Find the period of the graph.

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

(a) On the axes, sketch the graph of

\(y=5\sin\frac{x}{2}+1\)

for \(-2\pi\leq x\leq 2\pi\).

(b) Write down the amplitude of \(5\sin\frac{x}{2}+1\).

(c) Write down the period of \(5\sin\frac{x}{2}+1\).

The diagram shows the graph of \(y=a\sin bx+c\), where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).

On the axes, sketch the graph of

\(y=4\sin3x-2\)

for \(-\frac{\pi}{3}\leq x\leq\frac{\pi}{3}\).

The diagram shows the graph of \(y=a\sin bx+c\), where \(x\) is in radians and \(-2\pi\leqslant x\leqslant2\pi\), and where \(a\), \(b\) and \(c\) are positive constants.

Find the value of each of \(a\), \(b\) and \(c\).

The graph shows the curve

\(y=a\cos bx+c,\)

for \(0\leq x\leq2.8\), where \(a\), \(b\) and \(c\) are constants and \(x\) is in radians. The curve meets the \(y\)-axis at \((0,3)\) and the \(x\)-axis at the point \(P\) and point \(R\left(\frac{5\pi}{6},0\right)\).

The curve has a minimum at point \(Q\). The period of \(a\cos bx+c\) is \(\pi\) radians.

(a) Find the value of each of \(a\), \(b\) and \(c\).

(b) Find the coordinates of \(P\).

(c) Find the coordinates of \(Q\).

The graph of

\(y=a+2\tan bx,\)

where \(a\) and \(b\) are constants, passes through the point \((0,-4)\) and has period \(480^\circ\).

(a) Find the value of \(a\) and of \(b\).

(b) On the axes, sketch the graph of \(y\) for values of \(x\) between \(0^\circ\) and \(480^\circ\).

The function \(\mathrm f\) is defined, for \(0^\circ\leq x\leq810^\circ\), by

\(\mathrm f(x)=-2+\cos\frac{2x}{3}.\)

(a) Write down the amplitude of \(\mathrm f\).

(b) Find the period of \(\mathrm f\).

(c) On the axes, sketch the graph of \(y=\mathrm f(x)\).

The diagram shows the graph of \(y=a\sin\frac{x}{b}+c\) for \(-360^\circ\leq x\leq 360^\circ\), where \(a\), \(b\) and \(c\) are integers.

(a) Write down the period of \(a\sin\frac{x}{b}+c\).

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

Find the possible values of the constant \(c\) for which the line \(y=c\) is a tangent to the curve

\(y=5\sin\frac{x}{3}+4.\)

(a) The curve has equation

\(y=a\cos bx+c,\)

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

(b) Another curve has equation

\(y=2\sin3x+4.\)

Write down:

(i) the amplitude,

(ii) the period in radians.

The diagram shows the graph of \(y=f(x)\), where

\(f(x)=a\cos bx+c\)

for \(0\leq x\leq \frac{8\pi}{3}\).

Explain why \(f\) is a function, state the range of \(f\), and find the values of \(a\), \(b\) and \(c\).

(a) Write down the period of

\(2\cos\frac{x}{3}-1.\)

(b) Sketch the graph of

\(y=2\cos\frac{x}{3}-1\)

for \(-360^\circ\leq x\leq360^\circ\).

(a) Solve

\(\tan3x=-1\)

for \(-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}\) radians, giving your answers in terms of \(\pi\).

(b) Use your answers to part (a) to sketch the graph of

\(y=4\tan3x+4\)

for \(-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}\) radians. Show the coordinates of the points where the curve meets the axes.

(a) The curve \(y=a\sin bx+c\) has a period of \(180^\circ\), an amplitude of \(20\) and passes through the point \((90^\circ,-3)\). Find the value of each of the constants \(a\), \(b\) and \(c\).

(b) The function \(g\) is defined, for \(-135^\circ\leq x\leq135^\circ\), by

\(g(x)=3\tan\frac{x}{2}-4.\)

Sketch the graph of \(y=g(x)\), stating the coordinates of the point where the graph crosses the y-axis.

(a) Write down the amplitude of \(1+4\cos\frac{x}{3}\).

(b) Write down the period of \(1+4\cos\frac{x}{3}\).

(c) Sketch the graph of \(y=1+4\cos\frac{x}{3}\) for \(-180^\circ\leq x\leq180^\circ\).

The function \(\mathrm{f}\) is given by

\(\mathrm{f}(x)=2\cos\frac{x}{3}-1.\)

(a) Write down the amplitude of \(\mathrm{f}\).

(b) Write down the period of \(\mathrm{f}\).

(c) Sketch the graph of \(y=\mathrm{f}(x)\) for \(-\pi\leq x\leq3\pi\).