Past Exam Questions

(a) Solve \(6\sin^2x-13\cos x=1\) for \(0^\circ\leq x\leq360^\circ\).

(b) (i) Show that, for \(-\dfrac{\pi}{2}\lt y\lt \dfrac{\pi}{2}\), \(\dfrac{4\tan y}{\sqrt{1+\tan^2y}}\) can be written in the form \(a\sin y\), where \(a\) is an integer.

(ii) Hence solve \(\dfrac{4\tan y}{\sqrt{1+\tan^2y}}+3=0\) for \(-\dfrac{\pi}{2}\lt y\lt \dfrac{\pi}{2}\) radians.

(a)(i) Use the factor theorem to show that \(2x-1\) is a factor of \(p(x)\), where \(p(x)=4x^3+9x-5\).

(ii) Write \(p(x)\) as a product of linear and quadratic factors.

(b)(i) Show that

\(13\tan x\operatorname{sec}x-4\sin x-5\operatorname{sec}^2x=0\)

can be written as

\(4\sin^3x+9\sin x-5=0.\)

(ii) Using your answers to part (a)(ii) and part (b)(i), solve

\(13\tan x\operatorname{sec}x-4\sin x-5\operatorname{sec}^2x=0\)

for \(0\lt x\lt 2\pi\) radians.

(i) Show that

\(\frac{\tan x}{1+\operatorname{sec}x}+\frac{1+\operatorname{sec}x}{\tan x}\equiv\frac{2}{\sin x}.\)

(ii) Hence solve

\(\frac{\tan x}{1+\operatorname{sec}x}+\frac{1+\operatorname{sec}x}{\tan x}=1+3\sin x\)

for \(0^\circ\lt x\lt180^\circ\).

(i) Show that

\(\frac{\operatorname{cosec}x-\operatorname{cot} x}{1-\cos x}\equiv\operatorname{cosec}x.\)

(ii) Hence solve

\(\frac{\operatorname{cosec}x-\operatorname{cot} x}{1-\cos x}=2\)

for \(0^\circ\lt x\lt180^\circ\).

(a) Solve

\(3\operatorname{cot}^2\left(y-\frac{\pi}{4}\right)=1\)

for \(0\lt y\lt\pi\) radians.

(b) Solve

\(7\operatorname{cot} z+\tan z=7\operatorname{cosec}z\)

for \(0^\circ\leq z\leq360^\circ\).

(a) (i) Show that \(\dfrac{(1-\sin A)(1+\sin A)}{\sin A\cos A}=\operatorname{cot}A\).

(ii) Hence solve \(\dfrac{(1-\sin3x)(1+\sin3x)}{\sin3x\cos3x}=\dfrac12\) for \(0^\circ\leq x\leq180^\circ\).

(b) Solve \(10\tan^2y-\operatorname{sec}y-1=0\) for \(0\leq y\leq2\pi\) radians.

(a) Solve \(3\cos^2\theta+4\sin\theta=4\) for \(0^\circ\leqslant\theta\leqslant180^\circ\).

(b) Solve \(\sin2\phi=\sqrt3\cos2\phi\) for \(-\dfrac{\pi}{2}\leqslant\phi\leqslant\dfrac{\pi}{2}\) radians.

(a) Solve

\(10\cos^2x+3\sin x=9\)

for \(0^\circ\lt x\lt360^\circ\).

(b) Solve

\(3\tan2y=4\sin2y\)

for \(0\lt y\lt\pi\) radians.

(a) Solve

\(10\cos^2x+3\sin x=9\)

for \(0^\circ\lt x\lt360^\circ\).

(b) Solve

\(3\tan2y=4\sin2y\)

for \(0\lt y\lt\pi\) radians.

Solve

\(\operatorname{sec} x=\operatorname{cot} x-5\tan x\)

for \(0^\circ\lt x\lt 360^\circ\).

(a) Solve

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

for \(0\lt x\lt \pi\) radians.

(b) Solve

\(3\operatorname{sec} y=4\operatorname{cosec}y\)

for \(0^\circ\lt y\lt 360^\circ\).

(c) Solve

\(7\operatorname{cot} z-\tan z=2\operatorname{cosec}z\)

for \(0^\circ\lt z\lt 360^\circ\).

Solve the equation

(i) \(4\sin\left(3x-\dfrac{\pi}{4}\right)=3\) for \(0\leqslant x\leqslant \dfrac{\pi}{2}\) radians,

(ii) \(2\tan^2 y+\operatorname{sec}^2 y=14\operatorname{sec} y+3\) for \(0^\circ\leqslant y\leqslant 360^\circ\).

Solve the equation

(a) \(2\lvert \sin x\rvert=1\) for \(-\pi\le x\le \pi\) radians,

(b) \(3\tan(2y+15^\circ)=1\) for \(0^\circ\le y\le 180^\circ\),

(c) \(3\operatorname{cot}^2 z=\operatorname{cosec}^2 z-7\operatorname{cosec} z+1\) for \(0^\circ\le z\le 360^\circ\).

(a) Solve \(3\operatorname{cosec}2x-4\sin2x=0\) for \(0^\circ\leq x\leq180^\circ\).

(b) Solve \(3\tan\left(y-\dfrac{\pi}{4}\right)=\sqrt3\) for \(0\leq y\leq2\pi\) radians, giving your answers in terms of \(\pi\).

(a) Solve \(2\operatorname{cot}(z+35^\circ)=5\) for \(0^\circ\leq z\leq360^\circ\).

(b) (i) Show that \(\dfrac{\operatorname{sec}\theta}{\operatorname{cot}\theta+\tan\theta}=\sin\theta\).

(ii) Hence solve \(\dfrac{\operatorname{sec}3\theta}{\operatorname{cot}3\theta+\tan3\theta}=-\dfrac{\sqrt3}{2}\) for \(-\dfrac{\pi}{2}\leq\theta\leq\dfrac{\pi}{2}\), giving your answers in terms of \(\pi\).

(a) Show that

\(\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=2\operatorname{cosec}x.\)

(b) Solve the following equations.

(i) \(\operatorname{cot}^2y+\operatorname{cosec}y-5=0\), for \(0^\circ\le y\le360^\circ\).

(ii) \(\cos\left(2z+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt3}{2}\), for \(0\le z\le\pi\) radians.

(a) The diagram shows an equilateral triangle \(ABC\) with side \(a\). \(M\) is the midpoint of \(AC\) and angle \(AMB=90^\circ\).

Use the diagram to find \(\sec30^\circ\).

(b) Show that \(\dfrac{1}{\sec x-1}+\dfrac{1}{\sec x+1}\) can be written as \(2\operatorname{cosec}x\cot x\).

For \(-1\leqslant x\leqslant1\), it is given that \(x=2-3\sin\theta\) and \(y=3\cot^2\theta\).

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

(a) Given that \(0\leqslant\theta\lt \frac{\pi}{2}\), show that \(\frac{\sin\theta}{\sqrt{\operatorname{cosec}^2\theta-1}}+\frac{1}{\sqrt{1+\tan^2\theta}}\) can be written as \(\sec\theta\).

(b) Given that \(\sec x=\alpha\), where \(\frac{3\pi}{2}\lt x\leqslant2\pi\), find \(\sin x\) in terms of \(\alpha\).

(a) Show that \((2 \tan \theta+\sec \theta)(2 \tan \theta-\sec \theta)=3 \tan ^{2} \theta-1\).

(b) Hence solve the equation \((2 \tan \theta+\sec \theta)(2 \tan \theta-\sec \theta)=1\) for \(0^{\circ} \leqslant \theta \leqslant 180^{\circ}\).