2023 Paper 1

Given $\displaystyle y = 7x \tan 2x$, find $\displaystyle \frac{dy}{dx}.$

Express $\displaystyle \frac{3x^2 - x - 14}{(x + 3)(x - 1)^2}$ in partial fractions.

A system of equations is defined by
$\displaystyle x - 3y + z = -1$
$\displaystyle 3x - 2y + 4z = 11$
$\displaystyle x + 4y + 2z = 15$
Use Gaussian elimination to determine whether the system shows redundancy, inconsistency or has a unique solution.

Use integration by parts to find $\displaystyle \int x^4 \ln x \, dx$, $\displaystyle x > 0.$

Find the particular solution of the differential equation
$\displaystyle \frac{d^2y}{dx^2} - 4\frac{dy}{dx} - 5y = 10x^2 + 11x - 23$
given that $\displaystyle y = 2$, $\displaystyle \frac{dy}{dx} = 14$ when $\displaystyle x = 0.$

(a)Express $\displaystyle z = 1 + \sqrt{3}i$ in polar form.

(b)Hence, or otherwise, show that $\displaystyle z^3$ is real.

(a)Find an expression for $\displaystyle \sum_{r=1}^{n} (r^2 + 3r)$ in terms of $\displaystyle n.$
Express your answer in the form $\displaystyle \frac{1}{3}n(n + a)(n + b).$

(b)Hence, or otherwise, find $\displaystyle \sum_{r=11}^{20} (r^2 + 3r).$

(a)Consider the statement:
For all integers $\displaystyle a$ and $\displaystyle b$, if $\displaystyle a < b$ then $\displaystyle a^2 < b^2.$
Find a counterexample to show that the statement is false.

(b)Let $\displaystyle n$ be an odd integer.
Prove directly that $\displaystyle n^2 - 1$ is divisible by 4.

(a)State the matrix $\displaystyle A$, associated with an anti-clockwise rotation of $\displaystyle \frac{\pi}{2}$ radians about the origin.

The matrix $\displaystyle B$ is given by $\displaystyle B = \begin{pmatrix} -\dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\[7pt] -\dfrac{1}{2} & -\dfrac{\sqrt{3}}{2} \end{pmatrix}.$
The matrix $\displaystyle AB$ is associated with an anti-clockwise rotation of $\displaystyle \alpha$ radians about the origin.

(b)(i) Determine $\displaystyle AB.$

(b)(ii) Find the value of $\displaystyle \alpha.$

(c)Determine the least positive integer value of $\displaystyle n$ such that $\displaystyle (AB)^n = I$, where $\displaystyle I$ is the $\displaystyle 2 \times 2$ identity matrix.