2016 Paper 2

PQR is a triangle with vertices $\displaystyle P(0,-4)$, $\displaystyle Q(-6,2)$ and $\displaystyle R(10,6).$
(a)  (i) State the coordinates of M, the midpoint of QR.
      (ii) Hence find the equation of PM, the median through P.
(b)  Find the equation of the line, L, passing through M and perpendicular to PR.
(c)  Show that line L passes through the midpoint of PR.

(a)  (i) Show that $\displaystyle (x+1)$ is a factor of
$\displaystyle 2x^{3}-9x^{2}+3x+14$
      (ii) Hence solve the equation $\displaystyle 2x^{3}-9x^{2}+3x+14=0.$
(b)  The diagram below shows the graph with equation
$\displaystyle y=2x^{3}-9x^{2}+3x+14$
The curve cuts the x-axis at A, B and C.

(i) Write down the coordinates of the points A and B.
(ii) Hence calculate the shaded area in the diagram.

Circles $\displaystyle C_{1}$ and $\displaystyle C_{2}$ have equations
$\displaystyle (x+5)^{2}+(y-6)^{2}=9$
and
$\displaystyle x^{2}+y^{2}-6x-16=0$
respectively.

(a)  Write down the centres and radii of $\displaystyle C_{1}$ and $\displaystyle C_{2}.$
(b)  Show that $\displaystyle C_{1}$ and $\displaystyle C_{2}$ do not intersect.

The picture shows a model of a water molecule.

Relative to suitable coordinate axes, the oxygen atom is positioned at point $\displaystyle A(-2,2,5).$ The two hydrogen atoms are positioned at points $\displaystyle B(-10,18,7)$ and $\displaystyle C(-4,-6,21)$ as shown in the diagram.

(a)  Express $\displaystyle \vec{AB}$ and $\displaystyle \vec{AC}$ in component form.
(b)  Hence, or otherwise, find the size of angle BAC.

Scientists are studying the growth of a strain of bacteria. The number of bacteria present is given by the formula
$\displaystyle B(t) = 200e^{0.107t}$
where $\displaystyle t$ represents the number of hours since the study began.

(a)  State the number of bacteria present at the start of the study.
(b)  Calculate the time taken for the number of bacteria to double.

A council is setting aside an area of land to create six fenced plots where local residents can grow their own food. Each plot will be a rectangle measuring $\displaystyle x$ metres by $\displaystyle y$ metres as shown in the diagram.

(a)  The area of land being set aside is $\displaystyle 108\text{ m}^{2}.$ Show that the total length of fencing, $\displaystyle L$ metres, is given by
$\displaystyle L(x)=9x+\frac{144}{x}$
(b)  Find the value of $\displaystyle x$ that minimises the length of fencing required.

(a)  Express $\displaystyle 5\cos x-2\sin x$ in the form $\displaystyle k\cos(x+a)$, where $\displaystyle k \gt 0$ and $\displaystyle 0 \lt a \lt 2\pi.$
(b)  The diagram shows a sketch of part of the graph of $\displaystyle y=10+5\cos x-2\sin x$ and the line with equation $\displaystyle y=12.$ The line cuts the curve at the points P and Q.

Find the x-coordinates of P and Q.

For a function $\displaystyle f$, defined on a suitable domain, it is known that:
$\displaystyle f'(x)=\frac{2x+1}{\sqrt{x}}$ and $\displaystyle f(9)=40$
Express $\displaystyle f(x)$ in terms of $\displaystyle x.$

(a)  Given that $\displaystyle y=(x^{2}+7)^{\frac{1}{2}}$, find $\displaystyle \frac{dy}{dx}.$
(b)  Hence find
$\displaystyle \int\frac{4x}{\sqrt{x^{2}+7}}dx.$

(a)  Show that
$\displaystyle \sin 2x \tan x = 1 - \cos 2x$
where $\displaystyle \frac{\pi}{2} \lt x \lt \frac{3\pi}{2}.$
(b)  Given that $\displaystyle f(x)=\sin 2x \tan x$, find $\displaystyle f'(x).$