Pure Mathematics 2 — Variant 2

The polynomial $p(x)$ is defined by $p(x) = ax^{3} + bx - 10,$ where $a$ and $b$ are constants. It is given that $(x + 2)$ is a factor of $p(x)$ and that the remainder is $-55$ when $p(x)$ is divided by $(x + 3)$.

Find the values of $a$ and $b$.

Hence factorise $p(x)$ completely.

Sketch, on the same diagram, the graphs of $y = x + 3$ and $y = |2x - 1|$.

Solve the equation $x + 3 = |2x - 1|$.

Find the value of $y$ such that $5^{\frac{1}{2}y} + 3 = |2 \times 5^{\frac{1}{2}y} - 1|$. Give your answer correct to 3 significant figures.

The curve with equation $y = 5x - 2\tan 2x$ has exactly one stationary point in the interval $0 \leqslant x < \frac{1}{4}\pi$. Find the coordinates of this stationary point, giving each coordinate correct to 3 significant figures.

Given that $\int_{a}^{a+14} \frac{1}{3x} \, \mathrm{d}x = \ln 2$, find the value of the positive constant $a$.

A curve has equation $x^{2} + 4x\cos 3y = 6$. Find the exact value of the gradient of the normal to the curve at the point $(\sqrt{2}, \frac{1}{12}\pi)$.

By sketching a suitable pair of graphs on the same diagram, show that the equation $\ln x = 2\mathrm{e}^{-x}$ has exactly one root.

Verify by calculation that the root lies between $1.5$ and $1.6$.

Show that if a sequence of values given by the iterative formula $x_{n+1} = \mathrm{e}^{2\mathrm{e}^{-x_{n}}}$ converges, then it converges to the root of the equation in part (a).

Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

Prove that $4\sin x \sin(x + \frac{1}{6}\pi) \equiv \sqrt{3} - \sqrt{3}\cos 2x + \sin 2x$.

Find the exact value of $\int_{0}^{\frac{5}{6}\pi} 4\sin x \sin(x + \frac{1}{6}\pi) \, \mathrm{d}x$.

Find the smallest positive value of $y$ satisfying the equation $4\sin(2y)\sin(2y + \frac{1}{6}\pi) = \sqrt{3}$. Give your answer in an exact form.