Past Exam Questions

The function \(f\) is defined, for \(0^\circ\leq x\leq360^\circ\), by \(f(x)=a+b\sin cx\), where \(a\), \(b\) and \(c\) are constants with \(b\gt 0\) and \(c\gt 0\). The graph of \(y=f(x)\) meets the \(y\)-axis at the point \((0,-1)\), has a period of \(120^\circ\) and an amplitude of \(5\).

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

(ii) Write down the value of each of the constants \(a\), \(b\) and \(c\).

(i) Write down the amplitude of \(4\sin 3x-1\).

(ii) Write down the period of \(4\sin 3x-1\).

(iii) Sketch the graph of \(y=4\sin 3x-1\) for \(-90^\circ\leq x\leq90^\circ\).

The function \(f\) is defined, for \(0^\circ\leq x\leq360^\circ\), by \(f(x)=4+3\sin2x\).

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

(ii) State the period of \(f\).

(iii) State the amplitude of \(f\).

(i) Sketch the graph of

\(y=2\cos3x-1\quad\text{for }-90^\circ\leq x\leq90^\circ.\)

(ii) Write down the amplitude of \(2\cos3x-1\).

(iii) Write down the period of \(2\cos3x-1\).

(i) Sketch the graph of

\(y=5\cos4x-3\quad\text{for }-90^\circ\leq x\leq90^\circ.\)

(ii) Write down the amplitude of \(y\).

(iii) Write down the period of \(y\).

The figure shows part of the graph of \(y=p+q\cos rx\). Find the value of each of the integers \(p\), \(q\) and \(r\).

(a) (i) State the amplitude of \(15\sin2x-5\).

(ii) State the period of \(15\sin2x-5\).

(b) The diagram shows the graph of \(y=f(x)\), where \(f(x)\) is a trigonometric function.

(i) Write down two possible expressions for the trigonometric function \(f(x)\).

(ii) State the number of solutions of the equation \(f(x)=1\) for \(-180^\circ\leq x\leq180^\circ\).

(i) The curve

\(y=a+b\sin cx\)

has an amplitude of \(4\) and a period of \(\dfrac{\pi}{3}\). Given that the curve passes through the point \(\left(\dfrac{\pi}{12},2\right)\), find the value of each of the constants \(a\), \(b\) and \(c\).

(ii) Using your values of \(a\), \(b\) and \(c\), sketch the graph of \(y=a+b\sin cx\) for \(0\leq x\leq\pi\) radians.

It is given that \(y=1+\tan3x\).

(i) State the period of \(y\).

(ii) Sketch the graph of \(y=1+\tan3x\) for \(0^\circ\leqslant x\leqslant180^\circ\).

(i) The curve

\(y=a+b\sin cx\)

has an amplitude of \(4\) and a period of \(\dfrac{\pi}{3}\). Given that the curve passes through the point \(\left(\dfrac{\pi}{12},2\right)\), find the value of each of the constants \(a\), \(b\) and \(c\).

(ii) Using your values of \(a\), \(b\) and \(c\), sketch the graph of \(y=a+b\sin cx\) for \(0\leq x\leq\pi\) radians.

(a) Sketch the graph of \(y=3\cos2x-1\), for \(0^\circ\leq x\leq360^\circ\).

(b) Given that \(y=4\sin6x\), write down

(i) the amplitude of \(y\),

(ii) the period of \(y\).

The diagram shows part of the curve \(y=a+4\cos bx\), where \(a\) and \(b\) are positive constants. The curve meets the \(y\)-axis at the point \((0,6)\) and the \(x\)-axis at the point \(\left(\dfrac{\pi}{6},0\right)\). The curve meets the \(x\)-axis again at the point \(P\) and has a minimum at the point \(M\).

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

Using your values of \(a\) and \(b\), find

(ii) the exact coordinates of \(P\),

(iii) the exact coordinates of \(M\).

Given that \(y=3+4\cos 9x\), write down

(i) the amplitude of \(y\),

(ii) the period of \(y\).

(a) Given that \(y=7\cos10x-3\), where the angle \(x\) is measured in degrees, state

(i) the period of \(y\),

(ii) the amplitude of \(y\).

(b) Find the equation of the curve shown, in the form \(y=ag(bx)+c\), where \(g(x)\) is a trigonometric function and \(a\), \(b\) and \(c\) are integers to be found.

The graph of \(y=a\sin(bx)+c\) has an amplitude of \(4\), a period of \(\dfrac{\pi}{3}\), and passes through the point \(\left(\dfrac{\pi}{12},2\right)\). Find the value of each of the constants \(a\), \(b\), and \(c\).

Given that \(y=2\operatorname{sec}^2\theta\) and \(x=\tan\theta-5\), express \(y\) in terms of \(x\).

The graph of \(y=a\cos(bx)+c\) has an amplitude of \(3\), a period of \(\dfrac{\pi}{4}\), and passes through the point \(\left(\dfrac{\pi}{12},\dfrac52\right)\). Find the value of each of the constants \(a\), \(b\), and \(c\).

(i) On the axes below, sketch the graph of

\(y=3\sin x-2\)

for \(0^\circ\le x\le360^\circ\).

(ii) Given that

\(0\le |3\sin x-2|\le k\)

for \(0^\circ\le x\le360^\circ\), write down the value of \(k\).

(a) It is given that

\(2+\cos\theta=x,\qquad 1\lt x\lt 3\)

and

\(2\operatorname{cosec}\theta=y,\qquad y\gt 2.\)

Find \(y\) in terms of \(x\).

(b) Solve the equation

\(3\cos\frac{\phi}{2}=\sqrt3\sin\frac{\phi}{2}\)

for

\(-4\pi\lt \phi\lt 4\pi.\)

Solve the equation

\(3\sin\left(2x+\frac{\pi}{4}\right)=\sqrt3\cos\left(2x+\frac{\pi}{4}\right),\)

for \(0\leq x\leq\pi\).