2024 Paper 1

Given that $\displaystyle f(x)=(x+3)^2$, evaluate $\displaystyle f(7).$

Expand and simplify $\displaystyle (x+1)(x^2-4x+5).$

Given $\displaystyle \mathbf{a}=\begin{pmatrix}3\\ 4\\ -1\end{pmatrix}$ and $\displaystyle \mathbf{b}=\begin{pmatrix}5\\ 3\\ 2\end{pmatrix}$, find the resultant vector $\displaystyle 3\mathbf{a}+\mathbf{b}.$
Express your answer in component form.

The prices, in pounds (£), of the cameras on display in a shop are listed below.
155  160  190  210  230  240

(a)  Calculate the median and the interquartile range of these prices.

On a website, a sample of camera prices have a median of £195 and an interquartile range of £73.
(b)  Make two valid comments comparing the prices of the cameras in the shop and on the website.

Solve, algebraically, the system of equations

$$\begin{aligned} 2p - 7r &= 11 \\ 3p + 2r &= 4 \end{aligned}$$

The graph of $\displaystyle y=a\cos bx^{\circ}$, $\displaystyle 0\le x\le360$, is shown.

(a)  State the value of $\displaystyle a.$
(b)  State the value of $\displaystyle b.$

In a car rally, competitors start at different times.
The scattergraph shows the relationship between the length of time they have been driving, $\displaystyle T$ minutes, and the distance to the finishing line, $\displaystyle D$ kilometres.

A line of best fit has been drawn.
Point A represents a competitor who has been driving for 3 minutes and is 26 kilometres from the finishing line.
Point B represents a competitor who has been driving for 10 minutes and is 12 kilometres from the finishing line.

(a)  Find the equation of the line of best fit in terms of $\displaystyle D$ and $\displaystyle T.$ Give the equation in its simplest form.
(b)  Use your equation from part (a) to estimate the distance the competitor is from the finishing line after 7 minutes.

The diagram below shows a circle centre O.

• AC is a tangent to the circle at the point B.
• CE is a tangent to the circle at the point D.
• DG and BF are diameters of the circle.
• Angle DFE is 125°.
Calculate the size of shaded angle BCD.

A straight line has equation $\displaystyle x+4y-24=0.$
Find the gradient of this line.

(a)  Express $\displaystyle x^2-6x+8$ in the form $\displaystyle (x-a)^2+b.$
(b)  Hence, or otherwise, state the coordinates of the turning point of the graph of $\displaystyle y=x^2-6x+8.$

The diagram shows the graph of $\displaystyle y=x^2-6x+8.$

A line PQ has been drawn parallel to the x-axis, where:
• P lies on the y-axis
• P and Q lie on the graph of $\displaystyle y=x^2-6x+8.$
(c)  Find the coordinates of Q.

In the diagram, triangles ABC and ADE are mathematically similar.

• BC = 3 centimetres
• DE = 7 centimetres
• AD = 10.5 centimetres
Calculate the length of BD.