GCE 2026 South west regional mock mathematics
GCE 2026 South west regional mock mathematics
Question 2
a) Find as a series in ascending powers of $x$ up to and including the term in $x^3$, the solution of the differential equation:
Given that $y = 1$ and $\frac{dy}{dx} = 2$ when $x = 0$. Use the solution to find an approximate value of $y$ when $x = \frac{1}{2}$.
(6 marks)
b) Use the approximation $y_{n+1} \approx y_{n-1} + 2h\left(\frac{dy}{dx}\right)_n$ with $h = 0.1$ to find the approximation to the value of $y$ when $x = 0.4$ given that $\frac{dy}{dx} = x^2 + y^2$ and $y = 1$ when $x = 0$.
Use Simpson’s rule to approximate $\int_0^{0.4} y \, dx$.
(6 marks)
Question 3
a) A particle moves along a straight-line $ox$ such that at time $t$ seconds its displacement is $x \text{ m}$ from a fixed point $D$ of the line is given by $\frac{d^2x}{dt^2} + 9x = 0$. Show that the motion is oscillatory and state the period.
(5 marks)
b) Two identical springs AB and BC each of natural length $2a$ are joined together at B and carry a particle of mass $m$ at B. The ends A and C are fixed to two points in a vertical line with A at a distance $8a$ above C. In the position of equilibrium $AB = 5a$. The particle is given a small downward displacement from its position of equilibrium and is then released from rest. Find:
i) the modulus of elasticity of the spring
ii) the period of the oscillations.
(10 marks)
Question 4
a) A car of mass $m$ moves along a straight level road with its engine switched off. The resistance to its motion at speed $v$ is proportional to $v^2 + U^2$, where $U$ is a constant. It comes to rest from speed $U$ in time $T$. Show that the resistance at zero speed is $\frac{m\pi U}{4T}$.
(7 marks)
b) The car now starts from rest and moves, under the same law of resistance, with its engine exerting a constant force of magnitude $\frac{m\pi U}{T}$. Show that it reaches speed $U$ after travelling a distance $\frac{2UT}{\pi} \ln\left(\frac{3}{2}\right)$. State the limiting speed under this force.
(6 marks)
