2022 Paper 2

Express $\displaystyle \frac{3x^{2}-3x+5}{x(x^{2}+5)}$ in partial fractions.

Find the exact value of $\displaystyle \int_{0}^{3}\frac{4}{2x+1}dx.$

Use the Euclidean algorithm to find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 634a+87b=1.$

Use integration by parts to find $\displaystyle \int(x+2)(2x+7)^{\frac{1}{2}}dx.$

Matrix $\displaystyle A$ is given by $\displaystyle A=\begin{pmatrix}1&3&1\\2&k&3\\k&18&-7\end{pmatrix}$, where $\displaystyle k\in\mathbb{R}.$
Find the values of $\displaystyle k$ so that the matrix $\displaystyle A$ is singular.

The first three terms of a sequence are defined algebraically by $\displaystyle x+5$, $\displaystyle 3x+2$, $\displaystyle 5x-1$, where $\displaystyle x\in\mathbb{N}.$

(a)Show that these three terms form the start of an arithmetic sequence.

(b)Find a simplified expression for the $\displaystyle 15^{th}$ term of this sequence.

(c)Given that the sum of the first 20 terms of this sequence is 1130, find the value of $\displaystyle x.$

The complex number $\displaystyle z=3+i$ is a root of $\displaystyle z^{2}-6z+a=0$ where $\displaystyle a$ is a real number.

(a)State the second root of $\displaystyle z^{2}-6z+a=0.$

(b)Hence, or otherwise, find the value of $\displaystyle a.$

The expression $\displaystyle z^{2}-6z+a$ is a factor of $\displaystyle z^{3}-z^{2}-20z+b$ where $\displaystyle b$ is a real number.

(c)Find the value of $\displaystyle b.$

(a)Differentiate $\displaystyle x \ln x - x$ with respect to $\displaystyle x.$

(b)Hence find the general solution of the differential equation $\displaystyle \frac{dy}{dx} + y \ln x = x^{-x}.$

The matrix $\displaystyle A$ is given by $\displaystyle A=\begin{pmatrix}3&-2\\0&1\end{pmatrix}.$
Prove by induction that $\displaystyle A^{n}=\begin{pmatrix}3^{n}&1-3^{n}\\0&1\end{pmatrix}, \forall n\in \mathbb{N}.$

Solve the differential equation $\displaystyle \frac{d^{2}y}{dx^{2}}-4\frac{dy}{dx}+4y=9 \sin x+13 \cos x$ given that $\displaystyle y=5$ and $\displaystyle \frac{dy}{dx}=0$ when $\displaystyle x=0.$

A curve defined parametrically has the following properties:
$\displaystyle x = \tan^{-1} 2t$
$\displaystyle \frac{dy}{dx} = 6t(1+4t^{2})$
$\displaystyle y = 5$ when $\displaystyle t = 1.$
Find $\displaystyle y$ in terms of $\displaystyle t.$

Let $\displaystyle z = \cos \theta + i \sin \theta.$

(a)Use de Moivre's theorem to state an expression for $\displaystyle z^{4}.$

(b)State and simplify the binomial expansion of $\displaystyle (\cos \theta + i \sin \theta)^{4}.$

(c)Hence show that:
(i) $\displaystyle \cos 4\theta = 8\cos^{4}\theta - 8\cos^{2}\theta + 1.$
(ii) $\displaystyle \sin \theta \cot 4\theta$ can be written in terms of $\displaystyle \cos \theta$ only.

A security spotlight is situated 10 metres from a straight fence. The spotlight rotates at a constant speed and makes one full revolution every 12 seconds. $\displaystyle L$ is the spotlight, $\displaystyle G$ is the nearest point on the fence, $\displaystyle P$ is where the light hits the fence, $\displaystyle \theta$ is the angle between $\displaystyle LG$ and $\displaystyle LP$, and $\displaystyle x$ is the distance in metres from $\displaystyle G$ to $\displaystyle P.$

(a)Show that:
(i) $\displaystyle \frac{d\theta}{dt} = \frac{\pi}{6}$ radians per second
(ii) $\displaystyle \frac{dx}{dt} = \frac{5\pi}{3}\sec^{2}\theta$ metres per second.

(b)Prove that $\displaystyle 1+\tan^{2}\theta=\sec^{2}\theta.$

(c)Hence, or otherwise, find the exact value of $\displaystyle \frac{dx}{dt}$ when $\displaystyle P$ is 5 metres from $\displaystyle G.$