Pure Mathematics 3 — Variant 1

Solve the equation $3\mathrm{e}^{2x} - 4\mathrm{e}^{-2x} = 5$. Give the answer correct to 3 decimal places.

Sketch the graph of $y = |2x + 3|$.

Solve the inequality $3x + 8 > |2x + 3|$.

Find the coefficient of $x^{3}$ in the binomial expansion of $(3 + x)\sqrt{1 + 4x}$.

Show that the equation $\sin 2\theta + \cos 2\theta = 2\sin^{2}\theta$ can be expressed in the form $\cos^{2}\theta + 2\sin\theta\cos\theta - 3\sin^{2}\theta = 0$.

Hence solve the equation $\sin 2\theta + \cos 2\theta = 2\sin^{2}\theta$ for $0^\circ < \theta < 180^\circ$.

The equation of a curve is $x^{2}y - ay^{2} = 4a^{3}$, where $a$ is a non-zero constant.

Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2xy}{2ay - x^{2}}$.

Hence find the coordinates of the points where the tangent to the curve is parallel to the $y$-axis.

Relative to the origin $O$, the points $A$, $B$ and $C$ have position vectors given by $\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} 4 \\ 3 \\ 2 \end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix} 3 \\ -2 \\ -4 \end{pmatrix}$. The quadrilateral $ABCD$ is a parallelogram.

Find the position vector of $D$.

The angle between $BA$ and $BC$ is $\theta$. Find the exact value of $\cos\theta$.

Hence find the area of $ABCD$, giving your answer in the form $p\sqrt{q}$, where $p$ and $q$ are integers.

The variables $x$ and $y$ satisfy the differential equation $\cos 2x \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{4\tan 2x}{\sin^{2}3y}$, where $0 \leqslant x < \frac{1}{4}\pi$. It is given that $y = 0$ when $x = \frac{1}{6}\pi$. Solve the differential equation to obtain the value of $x$ when $y = \frac{1}{6}\pi$. Give your answer correct to 3 decimal places.

Let $\mathrm{f}(x) = \dfrac{3 - 3x^{2}}{(2x + 1)(x + 2)^{2}}$.

Express $\mathrm{f}(x)$ in partial fractions.

Hence find the exact value of $\int_{0}^{4} \mathrm{f}(x) \, \mathrm{d}x$, giving your answer in the form $a + b\ln c$, where $a$, $b$ and $c$ are integers.

The constant $a$ is such that $\int_{0}^{a} x\mathrm{e}^{-2x} \, \mathrm{d}x = \frac{1}{8}$.

Show that $a = \frac{1}{2}\ln(4a + 2)$.

Verify by calculation that $a$ lies between 0.5 and 1.

Use an iterative formula based on the equation in (a) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

The polynomial $x^{3} + 5x^{2} + 31x + 75$ is denoted by $\mathrm{p}(x)$.

Show that $(x + 3)$ is a factor of $\mathrm{p}(x)$.

Show that $z = -1 + 2\sqrt{6}\mathrm{i}$ is a root of $\mathrm{p}(z) = 0$.

Hence find the complex numbers $z$ which are roots of $\mathrm{p}(z^{2}) = 0$.