Pure Mathematics 3 — Variant 3

Solve the equation $\ln(2x - 1) = 2\ln(x + 1) - \ln x$. Give your answer correct to 3 decimal places.

Expand $\sqrt{\frac{1 + 2x}{1 - 2x}}$ in ascending powers of $x$, up to and including the term in $x^{2}$, simplifying the coefficients.

Find the exact value of $\int_{0}^{\frac{1}{4}\pi} x\sec^{2}x \, \mathrm{d}x$.

The parametric equations of a curve are $x = 2t - \tan t, \quad y = \ln(\sin 2t),$ for $0 < t < \frac{1}{2}\pi$. Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \cot t$.

On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $|z + 2| \leqslant 2$ and $\operatorname{Im} z \geqslant 1$.

Find the greatest value of $\arg z$ for points in the shaded region.

Solve the quadratic equation $(1 - 3i)z^{2} - (2 + i)z + i = 0$, giving your answers in the form $x + iy$, where $x$ and $y$ are real.

Show that the equation $\sqrt{5}\sec x + \tan x = 4$ can be expressed as $R\cos(x + \alpha) = \sqrt{5}$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Give the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.

Hence solve the equation $\sqrt{5}\sec 2x + \tan 2x = 4$, for $0^\circ < x < 180^\circ$.

The curve with equation $y = \frac{x^{3}}{e^{x} - 1}$ has a stationary point at $x = p$, where $p > 0$.

Show that $p = 3(1 - e^{-p})$.

Verify by calculation that $p$ lies between $2.5$ and $3$.

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

With respect to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix} 4 \\ -3 \\ -2 \end{pmatrix}$. The midpoint of $AC$ is $M$ and the point $N$ lies on $BC$, between $B$ and $C$, and is such that $BN = 2NC$.

Find the position vectors of $M$ and $N$.

Find a vector equation for the line through $M$ and $N$.

Find the position vector of the point $Q$ where the line through $M$ and $N$ intersects the line through $A$ and $B$.

A gardener is filling an ornamental pool with water, using a hose that delivers 30 litres of water per minute. Initially the pool is empty. At time $t$ minutes after filling begins the volume of water in the pool is $V$ litres. The pool has a small leak and loses water at a rate of $0.01V$ litres per minute. The differential equation satisfied by $V$ and $t$ is of the form $\frac{\mathrm{d}V}{\mathrm{d}t} = a - bV$.

Write down the values of the constants $a$ and $b$.

Solve the differential equation and find the value of $t$ when $V = 1000$.

Obtain an expression for $V$ in terms of $t$ and hence state what happens to $V$ as $t$ becomes large.

Let $\mathrm{f}(x) = \frac{5 - x + 6x^{2}}{(3 - x)(1 + 3x^{2})}$.

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

Find the exact value of $\int_{0}^{1} \mathrm{f}(x) \, \mathrm{d}x$, simplifying your answer.