2024 Paper 1

Differentiate the following with respect to $\displaystyle x$:

(a)$\displaystyle y = \cot 3x$

(b)$\displaystyle f(x) = 5x(4x - 7)^{\frac{1}{2}}$

A complex number is defined by $\displaystyle z = 1 + i.$

(a)Express $\displaystyle z$ in polar form.

(b)Use de Moivre's theorem to evaluate $\displaystyle z^8.$

A geometric sequence of positive terms has third term 36 and fifth term 16.

(a)Calculate the value of the common ratio.

(b)Calculate the value of the first term.

(c)State why the associated geometric series has a sum to infinity.

(d)Find the value of this sum to infinity.

Matrix $\displaystyle A$ is defined by $\displaystyle A = \begin{pmatrix} 6 & 1 \\ 11 & 3 \end{pmatrix}.$

(a)Find $\displaystyle A^{-1}$, the inverse of matrix $\displaystyle A.$

Matrix $\displaystyle B$ is defined by $\displaystyle B = \begin{pmatrix} -4 & 3 \\ -5 & 2 \end{pmatrix}.$

(b)Find the matrix $\displaystyle M$ such that $\displaystyle AM = B.$

The function $\displaystyle f(x)$ is defined by $\displaystyle f(x) = x^3 - x$, $\displaystyle x \in \mathbb{R}.$

(a)Determine whether $\displaystyle f(x)$ is even, odd or neither.

(b)Show that the graph of $\displaystyle y = f(x)$ has a point of inflection.

(a)Find the $\displaystyle 2 \times 2$ matrix, $\displaystyle A$, associated with a reflection in the $\displaystyle x$-axis.

(b)Describe the transformation associated with the matrix $\displaystyle B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$

(c)Find the $\displaystyle 2 \times 2$ matrix, $\displaystyle C$, associated with a reflection in the $\displaystyle x$-axis followed by the transformation associated with $\displaystyle \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$

A curve is defined by the equation $\displaystyle x^2y + 4xy^2 = -32$, $\displaystyle y > 0.$

(a)Use implicit differentiation to find an expression for $\displaystyle \frac{dy}{dx}.$

The curve has only one stationary point.

(b)Find the coordinates of the stationary point.

Use the substitution $\displaystyle u = \tan 2x$ to evaluate

$\displaystyle \int_{0}^{\frac{\pi}{8}} \frac{\sqrt{\tan 2x}}{\cos^2 2x} \,dx.$