Advanced level 2026 Littoral regional mock further mathematics 3

Here is the extracted data from the second image, organized by question number.

Question 1

A system of three forces $\mathbf{F_1}$, $\mathbf{F_2}$, and $\mathbf{F_3}$ acts at points with position vectors $\mathbf{r_1}$, $\mathbf{r_2}$, and $\mathbf{r_3}$ respectively.

$\mathbf{F_1} = (\mathbf{i} + \mathbf{j} + \mathbf{k})N, \quad \mathbf{r_1} = (\mathbf{j} + \mathbf{k})m$

$\mathbf{F_2} = (-\mathbf{i} + \mathbf{j} + 2\mathbf{k})N, \quad \mathbf{r_2} = (-\mathbf{i} – \mathbf{j} + 2\mathbf{k})m$

$\mathbf{F_3} = (2\mathbf{i} – \mathbf{j} + \mathbf{k})N, \quad \mathbf{r_3} = (\mathbf{i} – \mathbf{j} + \mathbf{k})m$

• (i) Show that the system of forces does not reduce to a single force. (7 marks)

• A fourth force $\mathbf{F_4}$ is added to the system such that the system of four forces reduces to a couple $\mathbf{G} = (\mathbf{i} + 6\mathbf{j} + \mathbf{k})Nm$.

• (ii) Find $\mathbf{F_4}$. (2 marks)

• (iii) Show that the line of action of $\mathbf{F_4}$ has cartesian equation given by

  $$l_4 : \frac{x-1}{-2} = \frac{y}{-1} = \frac{z+1}{-4}$$

  (4 marks)

Question 2

A particle of mass $m$ is attached to a spring with spring constant $k$ and the system is subjected to a resistive force of magnitude $|2\lambda v|N$ throughout its motion, where $v$ is the velocity of the particle and $x(t)$ is its displacement from a fixed point, $O$ at any time $t$.

• (i) Show that the differential equation governing the motion of the particle is

  $$m\frac{d^2x}{dt^2} + 2\lambda\frac{dx}{dt} + kx = 0$$

  (3 marks)

• (ii) Show further that the displacement, $x(t)$ of the particle at any time, $t$ for which the motion of the particle is damped harmonic is given by

  $$x(t) = Ae^{-\mu t} \cos(\omega t + \phi)$$

  Where $\mu$ and $\omega$ are to be determined. (7 marks)

• (iii) Hence, State the damping factor and find the period of the motion. (3 marks)

Question 3

A uniform rod $AB$ of mass $3m$ and length $2a$ has its center at $O$.

• (i) Show by integration that $I_A = 4ma^2$, where $I_A$ is the moment of inertia of the rod about an axis perpendicular to the plane containing the rod and passing through $A$. (4 marks)

• The rod is hinged at $A$ and held horizontal initially at rest. It is then allowed to perform oscillations such that at any time, $t$ it makes an angle $\theta$ with the horizontal.

• (ii) Show also that $2a\left(\frac{d\theta}{dt}\right)^2 = 3g \sin \theta$. (4 marks)

Question 4

(a) A discrete random variable $Y$ has cumulative probability distribution function, $F$ given by:

• (i) Determine the value of the constant $a$. (2 marks)

• (ii) Determine the mode, median and mean of $Y$. (6 marks)

(b) A continuous function $y = f(x)$ satisfies the differential equation

• (i) Prove that $y(x) = 2 + 2x + \frac{3}{4}x^2 + \dots$ is the Maclaurin series expansion of $y = f(x)$ in ascending powers of $x$ up to and including the term in $x^2$. (3 marks)

• (ii) Evaluate $y(-1)$. (1 mark)

• Hence, use the approximation $y_{n+1} \approx y_{n-1} + 2h\left(\frac{dy}{dx}\right)_n$ with a step length of $h = 1$.

• (iii) Find the value of $y$ when $x = 1$. (2 marks)

• (iv) Hence compute $\int_{-1}^{1} f(x) dx$, using Simpsons rule. (3 marks)