2018 Paper 1

PQR is a triangle with vertices $\displaystyle P(-2,4)$, $\displaystyle Q(4,0)$ and $\displaystyle R(3,6)$.

Find the equation of the median through R.

A function $\displaystyle g(x)$ is defined on $\displaystyle \mathbb{R}$, the set of real numbers, by
$\displaystyle g(x)=\frac{1}{5}x-4$
Find the inverse function, $\displaystyle g^{-1}(x)$.

Given $\displaystyle h(x)=3\cos 2x$ find the value of $\displaystyle h^{\prime}\left(\frac{\pi}{6}\right)$.

The point $\displaystyle K(8,-5)$ lies on the circle with equation
$\displaystyle x^{2}+y^{2}-12x-6y-23=0$

Find the equation of the tangent to the circle at K.

$\displaystyle A(-3,4,-7)$, $\displaystyle B(5,t,5)$ and $\displaystyle C(7,9,8)$ are collinear.
(a)  State the ratio in which B divides AC.
(b)  State the value of $\displaystyle t$.

Find the value of
$\displaystyle \log_{5}250-\frac{1}{3}\log_{5}8$

The curve with equation
$\displaystyle y=x^{3}-3x^{2}+2x+5$
is shown on the diagram.

(a)  Write down the coordinates of P, the point where the curve crosses the y-axis.
(b)  Determine the equation of the tangent to the curve at P.
(c)  Find the coordinates of Q, the point where this tangent meets the curve again.

A line has equation
$\displaystyle y-\sqrt{3}x+5=0$
Determine the angle this line makes with the positive direction of the x-axis.

The diagram shows a triangular prism ABC, DEF.
$\displaystyle \vec{AB}=\mathbf{t}$, $\displaystyle \vec{AC}=\mathbf{u}$ and $\displaystyle \vec{AD}=\mathbf{v}$

(a)  Express $\displaystyle \vec{BC}$ in terms of $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{t}$.
M is the midpoint of BC.
(b)  Express $\displaystyle \vec{MD}$ in terms of $\displaystyle \mathbf{t}$, $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$.

Given that
$\displaystyle \frac{dy}{dx}=6x^{2}-3x+4$
and $\displaystyle y=14$ when $\displaystyle x=2$, express $\displaystyle y$ in terms of $\displaystyle x$.

The diagram shows the curve with equation $\displaystyle y=\log_{3}x$.

(a)  On the diagram in your answer booklet, sketch the curve with equation $\displaystyle y=1-\log_{3}x$.
(b)  Determine the exact value of the x-coordinate of the point of intersection of the two curves.

Vectors $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$ are such that
$\displaystyle \mathbf{a}=4\mathbf{i}-2\mathbf{j}+2\mathbf{k}$ and $\displaystyle \mathbf{b}=-2\mathbf{i}+\mathbf{j}+p\mathbf{k}$
(a)  Express $\displaystyle 2\mathbf{a}+\mathbf{b}$ in component form.
(b)  Hence find the values of $\displaystyle p$ for which $\displaystyle |2\mathbf{a}+\mathbf{b}|=7$.

The right-angled triangle in the diagram is such that $\displaystyle \sin x=\frac{2}{\sqrt{11}}$ and $\displaystyle 0 \lt x \lt \frac{\pi}{4}.$

(a)  Find the exact value of:
      (i) $\displaystyle \sin 2x$
      (ii) $\displaystyle \cos 2x$
(b)  By expressing $\displaystyle \sin 3x$ as $\displaystyle \sin(2x+x)$, find the exact value of $\displaystyle \sin 3x$.

Evaluate
$\displaystyle \int_{-4}^{9}\frac{1}{\sqrt[3]{(2x+9)^{2}}}dx$

A cubic function, $\displaystyle f$, is defined on the set of real numbers.
•  $\displaystyle (x+4)$ is a factor of $\displaystyle f(x)$
•  $\displaystyle x=2$ is a repeated root of $\displaystyle f(x)$
•  $\displaystyle f^{\prime}(-2)=0$
•  $\displaystyle f^{\prime}(x)>0$ where the graph with equation $\displaystyle y=f(x)$ crosses the y-axis

Sketch a possible graph of $\displaystyle y=f(x)$ on the diagram in your answer booklet.