2017 Paper 2

Triangle ABC is shown in the diagram below. The coordinates of B are $\displaystyle (3,0)$ and the coordinates of C are $\displaystyle (9,-2)$. The broken line is the perpendicular bisector of BC.

(a)  Find the equation of the perpendicular bisector of BC.
(b)  The line AB makes an angle of $\displaystyle 45^{\circ}$ with the positive direction of the x-axis. Find the equation of AB.
(c)  Find the coordinates of the point of intersection of AB and the perpendicular bisector of BC.

(a)  Show that $\displaystyle (x-1)$ is a factor of
$\displaystyle f(x)=2x^{3}-5x^{2}+x+2$
(b)  Hence, or otherwise, solve $\displaystyle f(x)=0$.

The line $\displaystyle y=3x$ intersects the circle with equation
$\displaystyle (x-2)^{2}+(y-1)^{2}=25$

Find the coordinates of the points of intersection.

(a)  Express $\displaystyle 3x^{2}+24x+50$ in the form $\displaystyle a(x+b)^{2}+c$.
(b)  Given that
$\displaystyle f(x)=x^{3}+12x^{2}+50x-11$
find $\displaystyle f'(x)$.
(c)  Hence, or otherwise, explain why the curve with equation $\displaystyle y=f(x)$ is strictly increasing for all values of $\displaystyle x$.

In the diagram, $\displaystyle \vec{PR}=9\mathbf{i}+5\mathbf{j}+2\mathbf{k}$ and $\displaystyle \vec{RQ}=-12\mathbf{i}-9\mathbf{j}+3\mathbf{k}$.

(a)  Express $\displaystyle \vec{PQ}$ in terms of $\displaystyle \mathbf{i}, \mathbf{j}$ and $\displaystyle \mathbf{k}$.
The point S divides QR in the ratio 1:2.
(b)  Show that $\displaystyle \vec{PS}=\mathbf{i}-\mathbf{j}+4\mathbf{k}$.
(c)  Hence, find the size of angle QPS.

(a)  Find the x-coordinate of the stationary point on the curve with equation
$\displaystyle y=6x-2\sqrt{x^{3}}$
(b)  Hence, determine the greatest and least values of $\displaystyle y$ in the interval $\displaystyle 1\le x\le9$.

Sequences may be generated by recurrence relations of the form
$\displaystyle u_{n+1}=ku_{n}-20$, $\displaystyle u_{0}=5$
where $\displaystyle k\in\mathbb{R}$.
(a)  Show that $\displaystyle u_{2}=5k^{2}-20k-20.$
(b)  Determine the range of values of $\displaystyle k$ for which $\displaystyle u_{2} \lt u_{0}.$

Two variables, $\displaystyle x$ and $\displaystyle y$, are connected by the equation $\displaystyle y=kx^{n}$. The graph of $\displaystyle \log_{2}y$ against $\displaystyle \log_{2}x$ is a straight line as shown.

Find the values of $\displaystyle k$ and $\displaystyle n$.

(a)  Show that the points $\displaystyle A(-7,-2)$, $\displaystyle B(2,1)$ and $\displaystyle C(17,6)$ are collinear.
Three circles with centres A, B and C are drawn inside a circle with centre D as shown.

The circles with centres A, B and C have radii $\displaystyle r_{A}, r_{B}$ and $\displaystyle r_{C}$ respectively.
$\displaystyle r_{A}=\sqrt{10}$, $\displaystyle r_{B}=2r_{A}$, $\displaystyle r_{C}=r_{A}+r_{B}$.
(b)  Determine the equation of the circle with centre D.

(a)  Show that
$\displaystyle \frac{\sin 2x}{2\cos x}-\sin x \cos^{2}x=\sin^{3}x$
where $\displaystyle 0 \lt x \lt \frac{\pi}{2}.$
(b)  Hence, differentiate
$\displaystyle \frac{\sin 2x}{2\cos x}-\sin x \cos^{2}x$
where $\displaystyle 0 \lt x \lt \frac{\pi}{2}.$