Advanced level 2026 Eta college mock further mathematics 1

Here is the extracted text from the eighth image, which covers Further Mathematics and Linear Algebra topics:

1. Using the substitution $v = \frac{dy}{dx}$, where $v$ is a function in $x$, show that the differential equation

   $$(x^2 + 1)\frac{d^2y}{dx^2} + 2x\frac{dy}{dx} + \frac{6}{1+x^2} = 0$$

   can be transformed to the differential equation $\frac{dv}{dx} + \frac{2xv}{x^2+1} = -\frac{6}{(x^2+1)^2}$. (3 marks)

   Hence, or otherwise, show that if $y = 2, \frac{dy}{dx} = 0$ when $x = 0$, then $y = 2 – 3(\tan^{-1}x)^2$. (5 marks)

2. Given that $f(x) = \frac{x^4 + 2x^2 + x + 3}{(x+2)(x^2 + 1)^2}$,

   (a) Show that $(x^2+1)^2 + x + 2 \equiv x^4 + 2x^2 + x + 3$ (1 mark)

   Hence, or otherwise,

   (b) express $f(x)$ in partial fractions. (2 marks)

   (c) show that $\int_{0}^{1} f(x) dx = \ln \left| \frac{3}{2} \right| + \frac{1}{8}(2+\pi)$. (5 marks)

3. (i) Prove by mathematical induction, or otherwise, that $7^n(2n+1) + 11$ is divisible by 4, for all $n \in \mathbb{N}, n \ge 1$. (5 marks)

   (ii) The set $G = \{1, 5, 8, 12\}$ forms a group under multiplication modulo 13.

   (a) Draw up the operation table for the group. (2 marks)

   (b) Find $5^4(\text{mod } 13)$, hence state a generator of the group. (3 marks)

4. (i) Using the Intermediate Value Theorem, show that the equation $e^x + 4x – 2 = 0$ has a unique solution in the interval $[0, 1]$. (4 marks)

   (ii) Prove that $\coth^{-1}(2x) = \frac{1}{2} \ln \left( \frac{2x-1}{2x+1} \right)$ (3 marks)

   Hence, investigate the parity of $f(x) = \coth^{-1} \left( \frac{2x}{3} \right)$, and evaluate $\int_{-1}^{1} \coth^{-1} \left( \frac{2x}{3} \right) dx$. (3 marks)

5. (i) A linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^n$ is defined by the matrix $\mathbf{M} = \begin{pmatrix} 3 & 1 & -1 \\ 6 & 2 & -2 \\ 3 & 1 & -1 \end{pmatrix}$

   (a) Show that the transformation maps $\mathbb{R}^3$ onto a line and find the equation of the line. (5 marks)

   (b) Find the kernel of $T$ in the form $\{\lambda\mathbf{a} + \mu\mathbf{b}\}, \mathbf{a}, \mathbf{b} \in \mathbb{R}^3$. (3 marks)

   (ii) Prove that the vectors $\mathbf{a} = -\mathbf{j} – 4\mathbf{k}$ and $\mathbf{b} = 5\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ are linearly independent. (2 marks)

Would you like me to show the proof for the divisibility in question 3(i) or help you construct the group table for question 3(ii)?