Pure Mathematics 1 — Variant 2

A curve with equation $y = f(x)$ is such that $f'(x) = 2x^{-\frac{1}{3}} - x^{\frac{1}{3}}$. It is given that $f(8) = 5$. Find $f(x)$.

A curve has equation $y = x^{2} + 2cx + 4$ and a straight line has equation $y = 4x + c$, where $c$ is a constant. Find the set of values of $c$ for which the curve and line intersect at two distinct points.

The first term of a geometric progression and the first term of an arithmetic progression are both equal to $a$. The third term of the geometric progression is equal to the second term of the arithmetic progression. The fifth term of the geometric progression is equal to the sixth term of the arithmetic progression. Given that the terms are all positive and not all equal, find the sum of the first twenty terms of the arithmetic progression in terms of $a$.

Express $2x^{2} - 8x + 14$ in the form $2[(x - a)^{2} + b]$. The functions f and g are defined by $f(x) = x^{2}$ for $x \in \mathbb{R}$, $g(x) = 2x^{2} - 8x + 14$ for $x \in \mathbb{R}$.

Describe fully a sequence of transformations that maps the graph of $y = f(x)$ onto the graph of $y = g(x)$, making clear the order in which the transformations are applied.

Show that $\frac{\sin\theta + 2\cos\theta}{\cos\theta - 2\sin\theta} - \frac{\sin\theta - 2\cos\theta}{\cos\theta + 2\sin\theta} \equiv \frac{4}{5\cos^{2}\theta - 4}$.

Hence solve the equation $\frac{\sin\theta + 2\cos\theta}{\cos\theta - 2\sin\theta} - \frac{\sin\theta - 2\cos\theta}{\cos\theta + 2\sin\theta} = 5$ for $0^\circ < \theta < 180^\circ$.

Functions f, g and h are defined as follows: $f : x \mapsto x - 4x^{\frac{1}{2}} + 1$ for $x \geqslant 0$, $g : x \mapsto mx^{2} + n$ for $x \geqslant -2$, where $m$ and $n$ are constants, $h : x \mapsto x^{\frac{1}{2}} - 2$ for $x \geqslant 0$.

Solve the equation $f(x) = 0$, giving your solutions in the form $x = a + b\sqrt{c}$, where $a$, $b$ and $c$ are integers.

Given that $f(x) \equiv gh(x)$, find the values of $m$ and $n$.

It is given that a curve has equation $y = k(3x - k)^{-1} + 3x$, where $k$ is a constant.

Find, in terms of $k$, the values of $x$ at which there is a stationary point. The function f has a stationary value at $x = a$ and is defined by $f(x) = 4(3x - 4)^{-1} + 3x$ for $x \geqslant \frac{3}{2}$.

Find the value of $a$ and determine the nature of the stationary value.

The function g is defined by $g(x) = -(3x + 1)^{-1} + 3x$ for $x \geqslant 0$. Determine, making your reasoning clear, whether g is an increasing function, a decreasing function or neither.