2019 Paper 1

Given that $\displaystyle f(x)=5x^{3}$, evaluate $\displaystyle f(-2).$

Evaluate $\displaystyle \frac{3}{8} \times 1\frac{5}{7}.$
Give your answer in its simplest form.

Expand and simplify $\displaystyle (x+5)(2x^{2}-7x-3).$

The diagram below shows a sector of a circle, centre C.

The radius of the circle is 30 centimetres.
Calculate the length of the major arc AB.
Take $\displaystyle \pi=3\cdot14.$

The midday temperatures in Grantford were recorded over a nine day period.
The temperatures, in °C, were
4  7  4  3  6  10  9  5  3

(a)  Calculate the median and semi-interquartile range for these temperatures.

Over the same nine day period the midday temperatures in Endoch were also recorded.
The median temperature was 8 °C, and the semi-interquartile range was 1.5 °C.

(b)  Make two valid comments comparing the midday temperatures of Grantford and Endoch during this period.

The fuel consumption of a group of cars is recorded.
The scattergraph shows the relationship between the fuel consumption, $\displaystyle F$ kilometres per litre, and the engine size, $\displaystyle E$ litres, of the cars.

A line of best fit has been drawn.

(a)  Find the equation of the line of best fit in terms of $\displaystyle F$ and $\displaystyle E.$ Give the equation in its simplest form.

Amaar’s car has an engine size of 1.1 litres.
(b)  Use your equation from part (a) to estimate how many kilometres per litre he should expect to get.

The area of a trapezium is given by the formula
$\displaystyle A=\frac{1}{2}h(x+y).$
Make $\displaystyle x$ the subject of the formula.

John bought 7 bags of cement and 3 bags of gravel.
The total weight of these bags was 215 kilograms.
(a)  Write down an equation to illustrate this information.

Shona bought 5 bags of cement and 4 bags of gravel.
The total weight of her bags was 200 kilograms.
(b)  Write down an equation to illustrate this information.

(c)  Calculate the weight of one bag of cement and the weight of one bag of gravel.

The graph shows a parabola.

The maximum turning point has coordinates (4, 20) as shown in the diagram.

(a)  Write down the equation of the axis of symmetry of the graph.

The equation of the parabola is of the form $\displaystyle y=b-(x+a)^{2}.$
(b)  State the values of (i) $\displaystyle a$ and (ii) $\displaystyle b.$

In triangle PQR, $\displaystyle \overrightarrow{PR}=\begin{pmatrix}6\\ -4\end{pmatrix}$ and $\displaystyle \overrightarrow{RQ}=\begin{pmatrix}-1\\ 8\end{pmatrix}.$

(a)  Express $\displaystyle \overrightarrow{PQ}$ in component form.

M is the midpoint of PR.
(b)  Express $\displaystyle \overrightarrow{MQ}$ in component form.

Pam is designing a company logo.
She starts by drawing a regular pentagon ABCDE.
The vertices of the pentagon lie on the circumference of a circle with centre O.

She then adds to the design as shown in the diagram below.

AF is a diameter of the circle.
Calculate the size of angle OFB.

Express $\displaystyle \frac{\sqrt{2}}{\sqrt{40}}$ as a fraction with a rational denominator.
Give your answer in its simplest form.

Part of the graph of $\displaystyle y=3\cos(x+45)^{\circ}$ is shown in the diagram.

The graph has a minimum turning point at A.
State the coordinates of A.

Solve the equation $\displaystyle \frac{x}{2}-1=\frac{3-x}{5}.$

A ball is kicked from a clifftop.

The height, $\displaystyle h$ metres, of the ball relative to the clifftop after $\displaystyle t$ seconds is given by $\displaystyle h=12t-5t^{2}.$
(a)  Calculate the height of the ball above the clifftop after 2 seconds.

The graph below represents the height, $\displaystyle h$ metres, of the ball relative to the clifftop after $\displaystyle t$ seconds.

The sea is 17 metres below the clifftop.
(b)  After how many seconds will the ball hit the sea?