Advanced level 2026 Centre mock further mathematics 2

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Question 1

1. Consider the differential equation

   $$(E) : \frac{dy}{dx} – 2y = 2(e^{2x} – 1)$$

a. Show that the function $h$ defined on $\mathbb{R}$ by $h(x) = 2xe^{2x} + 1$ is a solution to $(E)$. (2 marks)

Consider a well defined and derivable functions $f$, in $\mathbb{R}$ such that $y = f + h$

b. Show that $y$ is a solution to the differential equation $(E)$ if and only if $f$ is a solution to $(E’) : f’ – 2f = 0$. (3 marks)

c. Find the solutions to $(E’)$ and $(E)$. (3 marks)

Question 2

2. A binary operation $\star$ is defined on a set $G$ by $x \star y = x + y + xy$ for all $x, y \in G$

a. Show that $\star$ is associative on $G$. (2 marks)

b. Prove by induction or otherwise that $(1 \star 2) \star 3 \star 4 \star \dots \star n = (n + 1)! – 1$. (4 marks)

c. Hence evaluate $1 \star (2 \star (3 \star 4 \star \dots \star (99 \star 100) \dots ))$. (2 marks)

Question 3

3. Consider the polar curve $r_1 = 3 + 2\sin\theta$ and $r_2 = a\sin 3\theta$ for $-\pi < \theta \leq \pi$

a. Determine the equation of the tangents at the pole to $r_2$ and state why $r_1$ has no tangents at the pole. (3 marks)

b. Sketch the polar curve $r_2$ and find the area enclosed by one of its loops. (5 marks)

Question 4

4. Consider the integral $I_n$ where $n$ is a positive integer defined by

   $$I_n = \int_{0}^{\infty} \frac{dx}{(1 + x^2)^n}$$

a. Find $I_1$ and show that for all $n \in \mathbb{Z}^+$

(4 marks)

b. Prove by mathematical induction or otherwise that

(3 marks)

c. Hence, deduce that

(3 marks)

Hint: You may deduce a telescopic product in (a) above.