2024 Paper 1

A line passes through the point (0,4) and makes an angle of $\displaystyle 30^{\circ}$ with the positive direction of the x-axis as shown in the diagram.

Determine the equation of the line.

A sequence is defined by the recurrence relation
$\displaystyle u_{n+1}=\frac{1}{5}u_{n}+12$
with $\displaystyle u_{1}=20$
(a)  Calculate the value of $\displaystyle u_{2}.$
(b)  (i) Explain why this sequence approaches a limit as $\displaystyle n\rightarrow\infty$.
      (ii) Calculate this limit.

Given that $\displaystyle y=(5x^{2}+3)^{7}$
find $\displaystyle \frac{dy}{dx}.$

P and Q have coordinates (-6, 1, 2) and (-1, 11, -8) respectively.
Find the coordinates of the point R which divides PQ in the ratio 2:3.

A function, h, is defined by
$\displaystyle h(x)=2x^{3}-7$
where $\displaystyle x\in\mathbb{R}.$
Find the inverse function, $\displaystyle h^{-1}(x).$

The right-angled triangle in the diagram is such that $\displaystyle \sin p=\frac{1}{\sqrt{5}}$ and $\displaystyle 0 \lt p \lt \frac{\pi}{4}.$

(a)  Determine the value of:
      (i) $\displaystyle \sin 2p$
      (ii) $\displaystyle \cos 2p$
(b)  Hence determine the value of $\displaystyle \sin 4p$.

The line $\displaystyle y=2x$ is a tangent to the circle with equation
$\displaystyle x^{2}+y^{2}-14x-8y+45=0$
Determine the coordinates of the point of contact.

The equation $\displaystyle x^{2}+(m-4)x+(2m-3)=0$ has no real roots.
Determine the range of values for m.
Justify your answer.

Express
$\displaystyle \log_{a}5+\log_{a}80-2\log_{a}10$
in the form $\displaystyle \log_{a}k$ where k is a positive integer.

(a)  Show that $\displaystyle (x-1)$ is a factor of
$\displaystyle 2x^{4}+3x^{3}-4x^{2}-3x+2$
(b)  Hence, or otherwise, factorise $\displaystyle 2x^{4}+3x^{3}-4x^{2}-3x+2$ fully.

(a)  Express $\displaystyle \cos x^{\circ}+\sqrt{3}\sin x^{\circ}$ in the form $\displaystyle k\cos(x-a)^{\circ}$, where $\displaystyle k \gt 0$ and $\displaystyle 0 \lt a \lt 360.$
(b)  Hence, or otherwise, sketch the graph with equation
$\displaystyle y=\cos x^{\circ}+\sqrt{3}\sin x^{\circ}$
for $\displaystyle 0\le x\le360$.

The function f is given by $\displaystyle f(x)=12\sqrt[3]{x},$ $\displaystyle x \gt 0.$
When $\displaystyle x=a$ the rate of change of f with respect to x is 1.
Determine the value of a.

P and Q are the points (4, 10) and (6, 2) respectively.
(a)  Find the equation of the perpendicular bisector of PQ.
The point R has coordinates (12, 2).
A circle passes through the points P, Q and R. The chord QR is horizontal.

(b)  Find the equation of the circle.