2023 Paper 2

Triangle PQR has vertices $\displaystyle P(5,-1)$, $\displaystyle Q(-2,8)$ and R(13, 3).

(a) Find the equation of the altitude from P.
(b) Calculate the angle that the side PR makes with the positive direction of the x-axis.

The diagram shows the cubic graph of $\displaystyle y=f(x)$ with stationary points at (2, 0) and (0,-2).

On the diagram in your answer booklet, sketch the graph of $\displaystyle y=2f(-x)$.

A function, f, is defined by $\displaystyle f(x)=(3-2x)^{4}$ where $\displaystyle x\in\mathbb{R}.$
Calculate the rate of change of f when $\displaystyle x=4.$

A function $\displaystyle f(x)$ is defined by
$\displaystyle f(x)=\frac{2}{x}+3$ , $\displaystyle x \gt 0$
Find the inverse function, $\displaystyle f^{-1}(x)$

The diagram shows part of the curve with equation $\displaystyle y=x^{3}-2x^{2}-4x+1$ and the line with equation $\displaystyle y=x-5$.
The curve and the line intersect at the points where $\displaystyle x=-2$ and $\displaystyle x=1.$

Calculate the shaded area.

(a) Express $\displaystyle 7\cos x^{\circ}-3\sin x^{\circ}$ in the form $\displaystyle k\sin(x+a)^{\circ}$ where k > 0, $\displaystyle 0 \lt a \lt 360$.
(b) Hence, or otherwise, find:
(i) the maximum value of $\displaystyle 14\cos x^{\circ}-6\sin x^{\circ}$
(ii) the value of x for which it occurs where $\displaystyle 0\le x \lt 360$

Determine the range of values of x for which the function
$\displaystyle f(x)=2x^{3}+9x^{2}-24x+6$
is strictly decreasing.

Circle $\displaystyle C_{1}$ has equation $\displaystyle (x-4)^{2}+(y+2)^{2}=37.$
Circle $\displaystyle C_{2}$ has equation $\displaystyle x^{2}+y^{2}+2x-6y-7=0$.
(a) Calculate the distance between the centres of $\displaystyle C_{1}$ and $\displaystyle C_{2}$.
(b) Hence, show that $\displaystyle C_{1}$ and $\displaystyle C_{2}$ intersect at two distinct points.

A curve, for which
$\displaystyle \frac{dy}{dx}=8x^{3}+3$
passes through the point (-1, 3).
Express y in terms of x.

A patient is given a dose of medicine.
The concentration of the medicine in the patient's blood is modelled by
$\displaystyle C_{t}=11e^{-0.0053~t}$
where:
•  t is the time, in minutes, since the dose of medicine was given
•  $\displaystyle C_{t}$ is the concentration of the medicine, in $\displaystyle mg/l,$ at time t.
(a) Calculate the concentration of the medicine 30 minutes after the dose was given.
The dose of medicine becomes ineffective when its concentration falls to $\displaystyle 0.66~mg/l.$
(b) Calculate the time taken for this dose of the medicine to become ineffective.

A net of an open box is shown.
The box is a cuboid with height h centimetres.
The base is a rectangle measuring 3x centimetres by 2x centimetres.

(a) (i) Express the area of the net, $\displaystyle A~cm^{2}$ in terms of h and x.
(ii) Given that $\displaystyle A=7200~cm^{2}$ show that the volume of the box, $\displaystyle V~cm^{3},$ is given by
$\displaystyle V=4320x-\frac{18}{5}x^{3}.$
(b) Determine the value of x that maximises the volume of the box.

The line $\displaystyle x+3y=17$ is a tangent to a circle at the point (2, 5).

The centre of the circle lies on the y-axis.
Find the coordinates of the centre of the circle.