2025 Paper 1

Use the binomial theorem to expand $\displaystyle \left(\frac{1}{x} - 3x\right)^4.$
Simplify your answer.

Given $\displaystyle f(x) = \frac{2x^3 + x}{3 + 2x}$, find $\displaystyle f'(x).$ Simplify your answer.

Two complex numbers are defined as $\displaystyle z = 11 + 10i$ and $\displaystyle w = 3 - 2i.$
Find $\displaystyle \frac{z}{w}$ in the form $\displaystyle a + bi$, where $\displaystyle a, b \in \mathbb{R}.$

Matrices $\displaystyle A$ and $\displaystyle B$ are defined by $\displaystyle A = \begin{pmatrix} -3 & 2 \\\ 0 & 1 \end{pmatrix}$ and $\displaystyle B = \begin{pmatrix} 2 & 2 \\\ 5 & \lambda \end{pmatrix}$ where $\displaystyle \lambda \in \mathbb{R}.$

(a)Find $\displaystyle 3A + 2B.$

(b)(i) Find $\displaystyle A'B$, where $\displaystyle A'$ is the transpose of $\displaystyle A.$

(b)(ii) Find an expression for the determinant of $\displaystyle A'B.$

(b)(iii) Determine the value of $\displaystyle \lambda$ such that $\displaystyle A'B$ is singular.

Find $\displaystyle \int \frac{1}{1 + 4x^2} dx.$

On a suitable domain a curve is given by the equation $\displaystyle y = f(x)$, where
$\displaystyle f(x) = \frac{x^2 + x + 5}{x - 2}.$

(a)Express $\displaystyle f(x)$ in the form $\displaystyle Ax + B + \frac{C}{x - 2}$, where $\displaystyle A, B$ and $\displaystyle C$ are constants.

(b)State the equations of the vertical and non-vertical asymptotes of the curve.

Solve the differential equation

$\displaystyle \frac{dy}{dx} = \frac{y}{2x - 1}$, $\displaystyle x, y > 1$,

given that $\displaystyle y = 12$ when $\displaystyle x = 5.$ Express $\displaystyle y$ in terms of $\displaystyle x.$

Three planes are defined by

$\displaystyle \pi_1:\quad x - y + 4z = -6$
$\displaystyle \pi_2:\quad 2x + 3y + z = 15$
$\displaystyle \pi_3:\quad 3x + 2y - 2z = 16$

(a)Use Gaussian elimination to find T, the point of intersection of the three planes.

The line $\displaystyle L_1$ is defined by $\displaystyle \frac{x + 4}{3} = \frac{y - 7}{2} = \frac{z - 4}{1}.$

(b)Find P, the point of intersection of the line $\displaystyle L_1$ and the plane $\displaystyle \pi_3.$

The line $\displaystyle L_2$ passes through points T (the point of intersection of the three planes) and P (the point of intersection of the line $\displaystyle L_1$ and the plane $\displaystyle \pi_3$).

(c)Find, in parametric form, the equations of the line $\displaystyle L_2.$