2018 Paper 1

Expand and simplify $\displaystyle (3x+1)(x-1)+2(x^2-5)$.

Solve, algebraically, the system of equations

$$\begin{aligned} 4x + 5y &= -3 \\ 6x - 2y &= 5 \end{aligned}$$

Two vectors are given by $\displaystyle \mathbf{u}=\begin{pmatrix}1\\ 5\\ 1\end{pmatrix}$ and $\displaystyle \mathbf{u}+\mathbf{v}=\begin{pmatrix}6\\ -4\\ 3\end{pmatrix}$.
Find vector $\displaystyle \mathbf{v}$.
Express your answer in component form.

Solve $\displaystyle x^{2}-11x+24=0.$

Part of the graph of $\displaystyle y=a\cos bx^{\circ}$ is shown in the diagram.

State the values of $\displaystyle a$ and $\displaystyle b.$

The cost of a journey with Tom's Taxis depends on the distance travelled. The graph below shows the cost, $\displaystyle P$ pounds, of a journey with Tom's Taxis against the distance travelled, $\displaystyle d$ miles.

Point A represents a journey of 8 miles which costs £14.
Point B represents a journey of 12 miles which costs £20.

(a)  Find the equation of the line in terms of $\displaystyle P$ and $\displaystyle d$. Give the equation in its simplest form.
(b)  Calculate the cost of a journey of 5 miles.

Determine the nature of the roots of the function $\displaystyle f(x)=2x^{2}+4x+5$.

In the diagram shown below, ABCDEFGHJK is a regular decagon.
•  Angle KLJ is 17°.
•  AKL is a straight line.

Calculate the size of shaded angle KJL.

In triangle XYZ:
$\displaystyle XZ=10$ centimetres
$\displaystyle YZ=8$ centimetres
$\displaystyle \cos Z = \frac{1}{8}$

Calculate the length of XY.

Express $\displaystyle \frac{9}{\sqrt{6}}$ with a rational denominator.
Give your answer in its simplest form.

Given that $\displaystyle \cos 60^{\circ}=0.5$, state the value of $\displaystyle \cos 240^{\circ}.$

The diagram shows a triangular prism, ABCDEF, relative to the coordinate axes.

$\displaystyle A(4,0,5)$, $\displaystyle E(2,0,0)$.
$\displaystyle AD=AE$, $\displaystyle DC=8$ units.
Edges EF, DC and AB are parallel to the y-axis.
Write down the coordinates of B and C.

Change the subject of the formula $\displaystyle y=g\sqrt{x}+h$ to $\displaystyle x$.

Remove the brackets and simplify $\displaystyle (\frac{2}{3}p^{4})^{2}$.

Sketch the graph of $\displaystyle y=(x-6)(x+4)$.
On your sketch, show clearly the points of intersection with the x-axis and the y-axis, and the coordinates of the turning point.

A square based pyramid is shown in the diagram below.

The square base has length 6 centimetres.
The volume is 138 cubic centimetres.
Calculate the height of the pyramid.

Express $\displaystyle \sin x^{\circ}\cos x^{\circ}\tan x^{\circ}$ in its simplest form.
Show your working.

(a)  (i) Express $\displaystyle x^{2}-6x-81$ in the form $\displaystyle (x-p)^{2}+q$.
      (ii) Hence state the equation of the axis of symmetry of the graph of $\displaystyle y=x^{2}-6x-81$.
(b)  The roots of the equation $\displaystyle x^{2}-6x-81=0$ can be expressed in the form $\displaystyle x=d \pm d\sqrt{e}$. Find, algebraically, the values of $\displaystyle d$ and $\displaystyle e$.