Advanced level 2026 south west regional mock further mathematics 2
Advanced level 2026 south west regional mock further mathematics 2
Further Mathematics Mock Exam – Paper 2 (Section Extract)
1. (i) Using the substitution $u = x – y$, find the general solution of the differential equation
(6 marks)
(ii) Given the differential equation $\frac{d^2y}{dx^2} + 4y = 6x$, find
(a) The complementary function. (2 marks)
(b) The particular integral of the differential equation. (2 marks)
2. Given two vectors $\vec{p} = 3\vec{i} + a\vec{j} – \vec{k}$, $\vec{q} = -\vec{i} + \vec{j} + 2b\vec{k}$, $a, b \in \mathbb{Z}$, and that $\vec{p} \times \vec{q} = 5\vec{i} – 11\vec{j} + 4\vec{k}$,
(a) Find the values of the real constants $a$ and $b$. (3 marks)
(b) Are the vectors $\vec{p}$ and $\vec{q}$ linearly independent? (2 marks)
(c) Find the Cartesian equation of the plane that contains the vectors $\vec{p}$ and $\vec{q}$ and passes through the point with position vector $2\vec{i} + \vec{j} – \vec{k}$. (2 marks)
Given also the vector $\vec{r} = 3\vec{i} + \vec{j} – \vec{k}$,
(d) compute $(\vec{p} \times \vec{q}) \cdot \vec{r}$ and interpret your result geometrically. (2 marks)
3. (i) Let $G$ be the set of matrices of the form $\begin{pmatrix} a & 0 \\ c & \frac{1}{a} \end{pmatrix}$ where $a \in \mathbb{R} – \{0\}$, $c \in \mathbb{R}$. Determine whether or not the set $G$ forms a group under matrix multiplication (Assume associativity).
(5 marks)
(ii) Find the general solution to the linear congruence $3x + 7 \equiv 2 \pmod{13}$.
(Give your answer in the form $x = a + bt$, where $a, b \in \mathbb{N}$, $t \in \mathbb{Z}$.)
(5 marks)
4. Express $f(x) = 13\cosh x + 12\sinh x$ in the form $r\cosh(x + y)$, where $r > 0$ and $y > 0$.
(Leave your answer in terms of natural logarithms.)
(6 marks)
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© TRU/RPI-Sc/SWAMT/PMS/0770/P2/MOCK 2026
