Advanced level 2026 North West mock further mathematics 3

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Question 1

Three forces F₁, F₂ and F₃ act on a rigid body at points with position vectors r₁, r₂ and r₃ respectively, relative to the origin O, given by:

• F₁ = i + j + 3k, r₁ = i – j + 2k

• F₂ = i – j + 2k, r₂ = 2i – k

• F₃ = 2i + j – k, r₃ = 3i + j

i) Show that the system of forces is not in equilibrium.

A fourth force F₄ is added to the system at the point r₄ with position vector r₄ = i – j + k to form a couple. Find:

ii) The force F₄

iii) The vector moment of the couple

iv) Equation of line of action of the force F₄ (11 marks)

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)

Question 5

Two smooth uniform spheres, A and B, have equal radii. The mass of A is $3m$ and the mass of B is $4m$. The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, A is moving with speed $3u$ at $30^\circ$ to the line of centers of the spheres and B is moving with speed $2u$ at $30^\circ$ to the line of centers of the spheres. The direction of motion of B is turned through an angle of $90^\circ$ by the collision.

(No marks listed on visible snippet for Q5)