A-level · Coordinate geometry

Straight linesExam-style questions

The gradient of a line as a constant rate of change, the three forms of a line's equation — y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0 — and how to move between them.

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10 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A company buys a new machine for its factory.

Two years after it was bought, the machine was valued at £31 200

Six years after it was bought, the machine was valued at £19 600

The value of the machine, £ $V$ thousand, $t$ years after it was bought is modelled by the equation

$V = a + bt$

where $a$ and $b$ are constants.
(a) Find the value of $a$ and the value of $b$ .

(3 marks)

The company plans to sell the machine when its value falls to £7500. According to the model, this happens exactly $T$ years after the machine was bought.

Making your method clear,
(b) find the value of $T$ , giving your answer to one decimal place.

(Solutions relying entirely on calculator technology are not acceptable.)

(2 marks)
Level 2 · 5 marksTry it
2
A property website records the average price of a house in a town at the start of each year.

At the start of 2017 the average price was £182 400

At the start of 2022 the average price was £223 900

The website models the average price, £ $H$ thousand, $t$ years after the start of 2015 by the equation

$H = pt + q$

where $p$ and $q$ are constants.
(a) Find the value of $p$ and the value of $q$ .

(3 marks)

According to the model, the average price reaches £260 000 exactly $T$ years after the start of 2015.
(b) Find the value of $T$ , giving your answer to one decimal place.

(2 marks)
Level 2 · 5 marksTry it
3
A driver can park in either of two car parks near a railway station. Car park P charges £3 on entry and then 60p for each hour of parking. Car park Q has no entry charge and the first 2 hours of parking are free; after that, Q charges £1.20 for each hour.
(a) Find, in terms of $t$ , an expression for the cost, in pounds, of parking for $t$ hours in each car park, where $t > 2$ .

(2 marks)
(b) Hence find the length of stay for which the two car parks charge the same amount.

(2 marks)
Level 2 · 4 marksTry it
4
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The points $P$ and $Q$ have coordinates $(-\sqrt{5},\ 2)$ and $(3\sqrt{5},\ 10)$ respectively.
(a) Show that the gradient of the line segment $PQ$ can be written as $p\sqrt{5}$ , where $p$ is a rational number to be found.

You must show each step of your working.

(2 marks)

The line $l$ is the perpendicular bisector of $PQ$ .
(b) Find an equation for $l$ , giving your answer in the form $y = mx + c$ , where $m$ and $c$ are constants.

(3 marks)
Level 3 · 5 marksTry it
5
Show that the points $A(-1, -4)$ , $B(2, 2)$ and $C(4, 6)$ are collinear (lie on the same straight line).
Level 3 · 3 marksTry it

5 more questions on straight lines

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