Past Exam Questions

Differentiate \(x^2\mathrm{e}^{3x}\) with respect to \(x\).

The tangent to the curve \(y=\mathrm{e}^{x}(2 x+5)^{\frac{1}{2}}\) at the point where \(x=2\) meets the \(x\)-axis at the point \(X\) and the \(y\)-axis at the point \(Y\). Find the coordinates of the mid-point of \(X Y\), giving your answer in exact form.

A curve has equation \(y=5 \mathrm{e}^{2 x-1}+\mathrm{e}\). The tangent to the curve at the point where \(x=1\) cuts the \(x\)-axis at the point \(P\).

Find the equation of the tangent in the form \(y=m x+c\), where \(m\) and \(c\) are exact values, and hence find the \(x\)-coordinate of \(P\).

A curve has equation

\(y=x\mathrm e^{2x}\).

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

(b) Find the equation of the normal to the curve at the point where \(x=1\).

(c) Use your answer to part (a) to find the exact value of

\(\displaystyle \int_0^2 2x\mathrm e^{2x}\,\mathrm dx.\)

The curve

\(y=5e^{2x}-3\)

meets the \(y\)-axis at the point \(A\). The tangent to the curve at \(A\) meets the \(x\)-axis at the point \(B\). Find the length \(AB\).

(a) Given that

\(y=\frac{e^{2x-3}}{x^2+1},\)

find \(\dfrac{dy}{dx}\).

(b) Hence, given that \(y\) is increasing at the rate of \(2\) units per second, find the exact rate of change of \(x\) when \(x=2\).

Show that the curve \(y=x-\ln(x^2+2x)\) has exactly one stationary point.

Find the \(x\)-coordinate of this point.

(a) Differentiate

\(\ln(x^3+3x^2)\)

with respect to \(x\), simplifying your answer.

(b) Hence find

\(\int \frac{x+2}{x(x+3)}\,dx.\)

It is given that

\(y=\frac{\ln(3x^2-5)}{2x+1},\qquad 3x^2\gt 5.\)

(a) Find the equation of the normal to the curve at the point where \(x=\sqrt2\).

(b) Hence find the approximate change in \(y\) as \(x\) increases from \(\sqrt2\) to \(\sqrt2+h\), where \(h\) is small.

Given that

\(y=\ln(\sin x+3\cos x),\qquad 0\lt x\lt \frac{\pi}{2},\)

(a) find \(\dfrac{dy}{dx}\),

(b) solve the equation \(\dfrac{dy}{dx}=-\frac12\).

Given that

\(y=\ln(1+\sin x),\qquad 0\lt x\lt\pi,\)

(a) find \(\dfrac{dy}{dx}\),

(b) find the exact value of \(\dfrac{dy}{dx}\) when \(x=\dfrac{\pi}{6}\), giving your answer in the form \(\dfrac1{\sqrt a}\), where \(a\) is an integer,

(c) solve the equation \(\dfrac{dy}{dx}=\tan x\).

(i) Differentiate

\((x^2+3)\ln(x^2+3)\)

with respect to \(x\).

(ii) Hence find

\(\int x\ln(x^2+3)\,dx.\)

The function \(y\) is defined by \(y=\dfrac{\ln(4x^2-1)}{x+2}\).

(i) State the values of \(x\) for which \(y\) is not defined.

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

(iii) Hence find the approximate increase in \(y\) as \(x\) increases from \(2\) to \(2+h\), where \(h\) is small.

(i) Write \(\ln\left(\dfrac{2x+1}{2x-1}\right)\) as the difference of two logarithms.

A curve has equation \(y=\ln\left(\dfrac{2x+1}{2x-1}\right)+4x\), for \(x\gt \dfrac12\).

(ii) Using your answer to part (i), show that \(\dfrac{dy}{dx}=\dfrac{ax^2+b}{4x^2-1}\), where \(a\) and \(b\) are integers.

(iii) Hence find the \(x\)-coordinate of the stationary point on the curve.

(iv) Determine the nature of this stationary point.

Given that \(y=\dfrac{\ln(3x^2+2)}{x^2+1}\), find the value of \(\dfrac{dy}{dx}\) when \(x=2\), giving your answer as \(a+b\ln14\), where \(a\) and \(b\) are fractions in their simplest form.

(a) Given that \(y=4 \sin 2 x \cos 2 x\), find the value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when \(x=\frac{\pi}{6}\).

(b) A curve has equation \(y=4 \sin 2 x \cos 2 x\).

The normal to the curve at the point where \(x=\frac{\pi}{6}\) meets the \(x\)-axis at the point \(P\). Find the exact coordinates of \(P\).

(a) (i) Given that \(y=3 \sin ^{2} x+\cos x\), show that \(y+\cot x \frac{\mathrm{~d} y}{\mathrm{~d} x}=k\left(1+\cos ^{2} x\right), \quad\) where \(k\) is an integer.

(ii) Using your value of \(k\), solve the equation \(k\left(1+\cos ^{2} x\right)=4\) for \(-\pi \leqslant x \leqslant \pi\). (b) (i) Differentiate \(y=\tan (x-\sqrt{x})\) with respect to \(x\). (ii) Hence find \(\int \frac{2 \sqrt{x}-1}{\sqrt{x} \cos ^{2}(x-\sqrt{x})} \mathrm{d} x\).

Given that \(y=\tan \frac{x}{2}\), find the exact value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when \(x=\frac{\pi}{3}\).

A curve has equation \(y=x\sin2x\).

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

(b) Find the equation of the tangent to the curve at \(x=\frac\pi4\).

(c) Use your answer to part (a) to find the exact value of

\(\displaystyle \int_0^{\pi/6}2x\cos2x\,\mathrm dx.\)

(a) The function f is defined by \(\mathrm{f}(x)=\frac{1+2 \sin ^{2} x}{\cos ^{2} x}\) for \(-\frac{\pi}{2}\lt x\lt \frac{\pi}{2}\). (i) Show that \(\mathrm{f}(x)\) can be written as \(a \tan ^{2} x+b\), where \(a\) and \(b\) are integers. (ii) Hence solve the equation \(\mathrm{f}(x)=4\).

(iii) Hence also find the gradient of the curve \(y=\mathrm{f}(x)\) at each of the points where \(y=4\). (b) Solve the equation \(50 \cos ^{2} \theta=5 \sin \theta+47\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).