2025 Paper 2

Triangle ABC has vertices $\displaystyle A(-9,-14),$ $\displaystyle B(9,20)$ and $\displaystyle C(21,-24).$

(a) Find the equation of the altitude through B.
(b) Find the equation of the median through A.
(c) Determine the point of intersection of the altitude through B and the median through A.

The diagram shows the graph of $\displaystyle y=x^{2}-2x+3.$

Calculate the shaded area.

A function, g, is defined by $\displaystyle g(x)=(x-4)^{3},$ where $\displaystyle x\in\mathbb{R}.$
Find the inverse function, $\displaystyle g^{-1}(x).$

(a) Show that the points $\displaystyle A(-3,2,-1),$ $\displaystyle B(6,-1,5)$ and $\displaystyle C(12,-3,9)$ are collinear.
(b) State the ratio in which B divides AC.

(a) Express $\displaystyle 5~\cos~x-9~\sin~x$ in the form $\displaystyle k~\cos(x+a)$ where $\displaystyle k \gt 0$ and $\displaystyle 0 \lt a \lt 2\pi.$
(b) Hence solve
$\displaystyle 5~\cos~x-9~\sin~x=7$
for $\displaystyle 0\le x \lt 2\pi.$

E,ABCD is a rectangular-based pyramid as shown.
$\displaystyle \vec{AD}=6i+4j+2k$
$\displaystyle \vec{DC}=2i-4j+2k$
$\displaystyle \vec{DE}=-4i-3j+4k$

Express $\displaystyle \vec{BE}$ in terms of i, j and k.

A sequence satisfies the recurrence relation $\displaystyle u_{n+1}=mu_{n}+4$, where m is a constant.
(a) The sequence approaches a limit of 10 as $\displaystyle n\rightarrow\infty.$ Determine the value of m.
(b) Given that $\displaystyle u_{1}=19$, calculate the value of $\displaystyle u_{0}.$

A hotel owner is designing signs showing the room numbers. Each sign is a rectangle with a right-angled triangle above it. The length and breadth of the rectangle are 5x centimetres and y centimetres respectively. The shorter sides of the triangle are 3x centimetres and 4x centimetres.

The area of the sign is 150 square centimetres.
(a) Show that the perimeter, P cm, of the sign is given by $\displaystyle P=9.6x+\frac{60}{x}.$
Each sign will be lit using a lighting strip placed around its perimeter.
The hotel owner requires the perimeter, P, of the sign to be as small as possible.
(b) Find the minimum value of P.

Functions f and g are defined on the set of real numbers by:
$\displaystyle f(x)=x^{5}+3$
$\displaystyle g(x)=1-x^{3}.$
(a) Find an expression for $\displaystyle h(x)$, where $\displaystyle h(x)=f(g(x)).$
(b) Find $\displaystyle h^{\prime}(x)$

A radioactive substance, which has been collected, decays over time.
The mass of the radioactive substance remaining is modelled by
$\displaystyle M=150e^{-0.0054t}$
where M is the mass, in micrograms, t years after the radioactive substance was collected.
(a) Determine the initial mass of the radioactive substance.
(b) Calculate the time taken for the mass of the radioactive substance to decay to 120 micrograms.

Circle $\displaystyle C_{1}$ has equation $\displaystyle (x+5)^{2}+(y-6)^{2}=9.$
(a) State the centre and radius of $\displaystyle C_{1}.$
Circle $\displaystyle C_{2}$ has equation $\displaystyle x^{2}+y^{2}-14x+6y+54=0.$
(b) State the centre and radius of $\displaystyle C_{2}.$
Circles $\displaystyle C_{1}$ $\displaystyle C_{2}$ and $\displaystyle C_{3}$ are touching as shown in the diagram. The centre of circle $\displaystyle C_{3}$ lies on the line joining the centres of $\displaystyle C_{1}$ and $\displaystyle C_{2}.$

(c) Determine the equation of $\displaystyle C_{3}.$