Pure Mathematics 3 — Variant 3

Find, in terms of $a$, the set of values of $x$ satisfying the inequality $2|3x + a| < |2x + 3a|$, where $a$ is a positive constant.

Solve the equation $\cos(\theta - 60^\circ) = 3\sin\theta$, for $0^\circ \leqslant \theta \leqslant 360^\circ$.

Show that the equation $\log_{3}(2x + 1) = 1 + 2\log_{3}(x - 1)$ can be written as a quadratic equation in $x$.

Hence solve the equation $\log_{3}(4y + 1) = 1 + 2\log_{3}(2y - 1)$, giving your answer correct to 2 decimal places.

The curve $y = \mathrm{e}^{-4x}\tan x$ has two stationary points in the interval $0 \leqslant x < \frac{1}{2}\pi$.

Obtain an expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ and show it can be written in the form $\sec^{2}x(a + b\sin 2x)\mathrm{e}^{-4x}$, where $a$ and $b$ are constants.

Hence find the exact $x$-coordinates of the two stationary points.

The complex number $3 - \mathrm{i}$ is denoted by $u$.

Show, on an Argand diagram with origin $O$, the points $A$, $B$ and $C$ representing the complex numbers $u$, $u^{*}$ and $u^{*} - u$ respectively. State the type of quadrilateral formed by the points $O$, $A$, $B$ and $C$.

Express $\frac{u^{*}}{u}$ in the form $x + \mathrm{i}y$, where $x$ and $y$ are real.

By considering the argument of $\frac{u^{*}}{u}$, or otherwise, prove that $\tan^{-1}\left(\frac{3}{4}\right) = 2\tan^{-1}\left(\frac{1}{3}\right)$.

The parametric equations of a curve are $x = \frac{1}{\cos t}$, $y = \ln\tan t$, where $0 < t < \frac{1}{2}\pi$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\cos t}{\sin^{2}t}$.

Find the equation of the tangent to the curve at the point where $y = 0$.

Let $\mathrm{f}(x) = \frac{5x^{2} + 8x - 3}{(x - 2)(2x^{2} + 3)}$.

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

Hence obtain the expansion of $\mathrm{f}(x)$ in ascending powers of $x$, up to and including the term in $x^{2}$.

At time $t$ days after the start of observations, the number of insects in a population is $N$. The variation in the number of insects is modelled by a differential equation of the form $\frac{\mathrm{d}N}{\mathrm{d}t} = kN^{\frac{3}{2}}\cos 0.02t$, where $k$ is a constant and $N$ is a continuous variable. It is given that when $t = 0$, $N = 100$.

Solve the differential equation, obtaining a relation between $N$, $k$ and $t$.

Given also that $N = 625$ when $t = 50$, find the value of $k$.

Obtain an expression for $N$ in terms of $t$, and find the greatest value of $N$ predicted by this model.

With respect to the origin $O$, the point $A$ has position vector given by $\overrightarrow{OA} = \mathbf{i} + 5\mathbf{j} + 6\mathbf{k}$. The line $l$ has vector equation $\mathbf{r} = 4\mathbf{i} + \mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$.

Find in degrees the acute angle between the directions of $OA$ and $l$.

Find the position vector of the foot of the perpendicular from $A$ to $l$.

Hence find the position vector of the reflection of $A$ in $l$.

The constant $a$ is such that $\int_{1}^{a} x^{2}\ln x \, \mathrm{d}x = 4$.

Show that $a = \left(\frac{35}{3\ln a - 1}\right)^{\frac{1}{3}}$.

Verify by calculation that $a$ lies between 2.4 and 2.8.

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