Pure Mathematics 2 — Variant 2

Solve the equation $|5x - 2| = |4x + 9|$.

A curve has equation $y = 7 + 4\ln(2x + 5)$. Find the equation of the tangent to the curve at the point $(-2, 7)$, giving your answer in the form $y = mx + c$.

The variables $x$ and $y$ satisfy the equation $y = 3^{2a}a^{x}$, where $a$ is a constant. The graph of $\ln y$ against $x$ is a straight line with gradient $0.239$.

Find the value of $a$ correct to 3 significant figures.

Hence find the value of $x$ when $y = 36$. Give your answer correct to 3 significant figures.

Show that $\sin 2\theta \cot \theta - \cos 2\theta \equiv 1$.

Hence find the exact value of $\sin \frac{1}{6}\pi \cot \frac{1}{12}\pi$.

Find the smallest positive value of $\theta$ (in radians) satisfying the equation $\sin 2\theta \cot \theta - 3\cos 2\theta = 1$.

Given that $y = \tan^{2}x$, show that $\frac{\mathrm{d}y}{\mathrm{d}x} = 2\tan x + 2\tan^{3}x$.

Find the exact value of $\int_{\frac{1}{4}\pi}^{\frac{1}{3}\pi} (\tan x + \tan^{2}x + \tan^{3}x) \, \mathrm{d}x$.

The polynomial $p(x)$ is defined by $p(x) = 4x^{3} + 16x^{2} + 9x - 15$.

Find the quotient when $p(x)$ is divided by $(2x + 3)$, and show that the remainder is $-6$.

Find $\int \frac{p(x)}{2x + 3} \, \mathrm{d}x$.

Factorise $p(x) + 6$ completely and hence solve the equation $p(\operatorname{cosec} 2\theta) + 6 = 0$ for $0^\circ < \theta < 135^\circ$.

A curve has equation $\mathrm{e}^{2x}y - \mathrm{e}^{y} = 100$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{2\mathrm{e}^{2x}y}{\mathrm{e}^{y} - \mathrm{e}^{2x}}$.

Show that the curve has no stationary points. It is required to find the $x$-coordinate of $P$, the point on the curve at which the tangent is parallel to the $y$-axis.

Show that the $x$-coordinate of $P$ satisfies the equation $x = \ln 10 - \frac{1}{2}\ln(2x - 1)$.

Use an iterative formula, based on the equation in part (c), to find the $x$-coordinate of $P$ correct to 3 significant figures. Use an initial value of $2$ and give the result of each iteration to 5 significant figures.