Advanced level 2026 North west regional mock further mathematics 2

1. A function $y(x)$ satisfies $\frac{d^{2}y}{dx^{2}} + \frac{2}{x}\frac{dy}{dx} + y = 0, x > 0$

   a) Using the substitution $y = \frac{u}{x}$, show that the differential equation reduces to $u” + u = 0$. (4 marks)

   Hence solve the differential equation completely, giving the general solution for $y$ (4 marks)

2. a) Express $\frac{2x^{2} + 5x + 3}{(x – 1)^{2}(x^{2} + 4)}$ into partial fractions.

   Hence, show that $\int_{0}^{2} \frac{2x^{2} + 5x + 3}{(x – 1)^{2}(x^{2} + 4)} dx = -4 – \frac{1}{2}\ln 2 – \frac{\pi}{8}$ (8 marks)

   b) Solve the hyperbolic equation $2 \sinh x + 3 \cosh x = 5$ giving the exact solutions in logarithmic form. (3 marks)

3. a) Prove by mathematical induction that for every positive integer $n$, $5^{2n} – 1$ is divisible by 24. (4 marks)

   b) Let $G = \{1, \omega, \omega^{2}, \omega^{3}, \omega^{4}, \omega^{5}\}$ where $\omega = \cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3})$, and the operation on $G$ is complex multiplication.

   i. Show that $G$ is a group under multiplication. (3 marks)

   ii. Given that $G$ is a cyclic group, state its generator and the order (2 marks)

4. a) The points $A, B$ and $C$ lie in the plane $\Pi$. Relative to a fixed origin $O$, their position vectors are $\vec{a} = 2\mathbf{i} + \mathbf{j} – 3\mathbf{k}, \vec{b} = -\mathbf{i} + 4\mathbf{j} + \mathbf{k}, \vec{c} = 3\mathbf{i} – 2\mathbf{j} + 5\mathbf{k}$

   i. Find the vector product $\vec{AB} \times \vec{AC}$ and hence, write a cartesian equation of the plane $\Pi$. (3 marks)

   The point $D$ has position vector $\vec{d} = 4\mathbf{i} + \mathbf{j} + 2\mathbf{k}$.

   ii. Show that the perpendicular distance from the point $D$ to the plane $\Pi$ is given by $\frac{51}{\sqrt{23}}$ units. (2 marks)

   iii. Calculate the volume of the tetrahedron $ABCD$. (2 marks)

   b) Consider the curve with polar equation $r = \frac{4}{1 + \cos \theta}$.

   i. Show that the Cartesian equation of the curve is $y^{2} = 16 – 8x$. (2 marks)

   ii. Find $\frac{dr}{d\theta}$ and hence or otherwise, determine the equation of the tangent to the curve at $\theta = \frac{\pi}{2}$ in Cartesian form. (3 marks)

5. a) The linear transformation $T: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is represented by the matrix $A = \begin{pmatrix} 1 & -2 \\ 2 & 1 \end{pmatrix}$.

   The line $l$ has vector equation $\vec{r} = \begin{pmatrix} 1 \\ -1 \end{pmatrix} + t\begin{pmatrix} 2 \\ 1 \end{pmatrix}, t \in \mathbb{R}$.

   Find the Cartesian equation of the image line $T(l)$. (4 marks)

   b) A linear transformation $S: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is defined by $S\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2x – y \\ 4x – 2y \end{pmatrix}$.

   i. Find the kernel of $S$ (3 marks)

   ii. Hence, determine the dimension of $\ker(S)$ (1 mark)

6. Let $f(x) = x^{3} – 3x – 1$.

   a) Show that the equation $f(x) = 0$ has at least one real root in the interval $(1, 2)$ (2 marks)

   b) Show that there exists $c \in (0, 2)$ such that $f'(c) = \frac{f(2) – f(0)}{2 – 0}$ (2 marks)

   c) Hence, determine the exact value of $c$ (1 mark)

7. Let $v_{n} = 3^{n}$ for $n \geq 1$. Define a second sequence $(u_{n})$ by $u_{n} = \frac{2v_{n}}{v_{n} + 1}, n \geq 1$.

   a) Show that $(v_{n})$ is a geometric progression. (2 marks)

   b) Prove algebraically that $(u_{n})$ is strictly increasing for all $n \geq 1$. (3 marks)

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