Past Exam Questions

Variables \(x\) and \(y\) are related by the equation

\(y=2+\tan(1-x),\)

where \(0\leq x\leq\frac{\pi}{2}\). Given that \(x\) is increasing at a constant rate of \(0.04\) radians per second, find the corresponding rate of change of \(y\) when \(y=3\).

A curve has equation

\(y=\cos\frac{x}{4},\)

where \(x\) is in radians. The normal to the curve at the point where \(x=\frac{4\pi}{3}\) cuts the \(x\)-axis at the point \(P\). Find the exact coordinates of \(P\).

The function \(f\) is defined by \(f(x)=3\sin^2x-2\cos x\) for \(2\le x\le4\), where \(x\) is in radians.

(a) Find the \(x\)-coordinate of the stationary point on the curve \(y=f(x)\).

(b) Solve the equation \(f(x)=1-3\cos x\).

You are given that

\(y=\frac1{\cos2x}.\)

(a) Show that

\(\frac{dy}{dx}=\frac{k\sin2x}{\cos^22x},\)

where \(k\) is a constant to be found.

(b) Find the values of \(x\) such that

\(\frac{dy}{dx}=\frac5{\sin2x}\)

for \(0\lt x\lt \frac{\pi}{2}\).

The curve \(y=3\tan^2 x\) is defined for \(0^\circ\lt x\lt 360^\circ\).

(a) Show that

\(\frac{dy}{dx}=m\tan x\operatorname{sec}^2x,\)

where \(m\) is a constant to be found.

(b) Find all the values of \(x\) for which

\(\frac{dy}{dx}=3\operatorname{sec}x\operatorname{cosec}x.\)

It is given that \(y=\dfrac{\tan 3x}{\sin x}\).

(a) Find the exact value of \(\dfrac{dy}{dx}\) when \(x=\dfrac{\pi}{3}\).

(b) Hence find the approximate change in \(y\) as \(x\) increases from \(\dfrac{\pi}{3}\) to \(\dfrac{\pi}{3}+h\), where \(h\) is small.

(c) Given that \(x\) is increasing at the rate of \(3\) units per second, find the corresponding rate of change in \(y\) when \(x=\dfrac{\pi}{3}\), giving your answer in its simplest surd form.

Differentiate \(\tan 3x\cos\dfrac{x}{2}\) with respect to \(x\).

Given that \(y=2\sin 3x+\cos 3x\), show that

\(\displaystyle \frac{d^2y}{dx^2}+\frac{dy}{dx}+3y=k\sin 3x,\)

where \(k\) is a constant to be determined.

The curve \(y=4+5\sin 3x\) passes through the point \(P\), where \(x=\dfrac13\pi\).

(i) Find \(\dfrac{dy}{dx}\).

(ii) Find the equation of the tangent to the curve at \(P\), giving your answer in the form \(y=mx+c\).

(i) Differentiate \(1+\tan\left(\dfrac{x}{3}\right)\) with respect to \(x\).

(ii) Hence find \(\displaystyle\int \operatorname{sec}^2\left(\dfrac{x}{3}\right)\,dx\).

In this question, the units of \(x\) are radians and the units of \(y\) are centimetres.

It is given that \(y=(1+\cos3x)^{10}\).

(i) Find the value of \(\dfrac{dy}{dx}\) when \(x=\dfrac{\pi}{2}\).

Given also that \(y\) is increasing at a rate of \(6\text{ cm s}^{-1}\) when \(x=\dfrac{\pi}{2}\),

(ii) find the corresponding rate of change of \(x\).

A curve has equation

\(y=\frac{x^3}{\sin 2x}.\)

Find

(i) \(\dfrac{dy}{dx}\),

(ii) the equation of the tangent to the curve at the point where \(x=\dfrac{\pi}{4}\).

(i) Differentiate \((\cos x)^{-1}\) with respect to \(x\).

(ii) Hence find \(\dfrac{dy}{dx}\), given that \(y=\tan x+4(\cos x)^{-1}\).

(iii) Using your answer to part (ii), find the values of \(x\) in the range \(0\leq x\leq2\pi\) such that \(\dfrac{dy}{dx}=4\).

(i) Without using a calculator, solve the equation \(6c^3-7c^2+1=0\).

It is given that \(y=\tan x+6\sin x\).

(ii) Find \(\dfrac{dy}{dx}\).

(iii) If \(\dfrac{dy}{dx}=7\), show that \(6\cos^3x-7\cos^2x+1=0\).

(iv) Hence solve the equation \(\dfrac{dy}{dx}=7\), for \(0\leq x\leq\pi\) radians.

A curve has equation \(y=\dfrac{\mathrm{e}^{x^2}}{x-2}\) for \(x\lt 2\).

(a) Find the value of \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) when \(x=0\).

When \(x=0\), \(y\) is increasing at the rate of \(0.5\) units per second.

(b) Find the corresponding rate of change of \(x\).

It is given that \(y=\frac{\ln(2x^2+1)}{x+2}\).

(a) Find \(\frac{\mathrm dy}{\mathrm dx}\).

(b) Given that \(x\) increases from \(2\) to \(2+h\), where \(h\) is small, find the approximate change in \(y\).

(c) Given that \(y\) is decreasing by \(0.4\) units per second, find the corresponding rate of change in \(x\) when \(x=2\).

The equation of a curve is \(y=\frac{\mathrm{e}^{-3 x+2}}{x+1}\) where \(x\lt -1\). (a) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{e}^{-3 x+2}}{(x+1)^{2}}(A x+B) \quad\) where \(A\) and \(B\) are integers to be found. (b) Hence show that there is only one stationary point on the curve and find its exact coordinates.

It is given that \(y=\frac{\ln \left(3 x^{2}-1\right)}{x+2}\), for \(x\gt \frac{1}{\sqrt{3}}\). When \(x=1, y\) is increasing at the rate of \(h\) units per second. Find, in terms of \(h\), the corresponding rate of change in \(x\), giving your answer in exact form.

The diagram shows part of the curve \(y=x-\frac{x^{2}}{4}\) and the line \(y=-4\). The curve and the line intersect at the point \(A\). (a) The maximum point on the curve is at a perpendicular distance \(h\) from the line \(y=-4\). Find the value of \(h\).

(b) Find the exact \(x\)-coordinate of \(A\).

(c) Find the acute angle between the tangent to the curve at \(A\) and the line \(y=-4\).

A curve has equation \(y=\frac{\left(3 x^{2}-5\right)^{\frac{1}{3}}}{x+4}\). (a) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) can be written in the form \(\frac{A x^{2}+B x+C}{\left(3 x^{2}-5\right)^{\frac{2}{3}}(x+4)^{2}}\), where \(A, B\) and \(C\) are integers. (b) Hence find the \(x\)-coordinates of the stationary points on the curve. Give your answers in their simplest exact form.