A LevelMathsEdexcelPureExam QuestionsCoordinate GeometryEquation of a Straight Line

Equation of a Straight Line (Edexcel A Level Maths: Pure): Exam Questions

Exam code: 9MA0

3 hours39 questions
Easy
Medium
Hard
1
3 marks

The equation of a straight line is y equals 2 x minus 6.

Write down:

(i) the gradient of the line

(ii) the coordinates of the point where the line crosses the y-axis

(iii) the coordinates of the point where the line crosses the x-axis

How did you do?

2
3 marks

Find the coordinates of the midpoint of the straight line connecting the following points:

(i) (2 , 4) and (6 , 10)

(ii) (-3 , 6) and (5 , 9)

(iii) (0 , -8) and (3 , 2)

How did you do?

3
3 marks

Find the exact length of the straight-line segments connecting the following points:

(i) (2 , 4) and (5 , 8)

(ii) (3 , - 6) and (-2, -14)

(iii) (5, -13) and (2, -7)

How did you do?

4
6 marks

Find the equations of the following straight lines, given their gradient, m, and a point that lies on the line, P left parenthesis x comma space y right parenthesis.

Give your answers in the form y equals m x plus c.

(i) m equals 2 and P left parenthesis 3 space comma space 5 right parenthesis

(ii) m equals negative 2 and P open parentheses negative 1 comma space 3 close parentheses

(iii) m equals 1 half and P open parentheses 5 comma space minus 2 close parentheses

How did you do?

5
3 marks

Given that a straight line passes through the points Pand  Q shown, find the gradient of the following lines:

(i) P open parentheses 2 comma space 6 close parentheses comma space Q open parentheses 4 comma space 12 close parentheses

(ii) P open parentheses negative 3 comma space 4 close parentheses comma space Q left parenthesis negative 8 comma space 24 right parenthesis

(iii) P open parentheses 1 comma space minus 3 close parentheses comma space Q left parenthesis negative 3 comma space 6 right parenthesis

How did you do?

6
3 marks

Rewrite the following equations of straight lines into the form a x plus b y plus c equals 0, where a, b and c are integers.

(i) y equals 3 x minus 5

(ii) y equals 1 half x plus 7

(iii) 1 third y equals 1 over 6 x minus 1 over 9

How did you do?

7
3 marks

A straight line passes through the points (4 , 8) and (-4 , 10).

(i) Find the gradient of the straight line.

(ii) Hence, find the equation of the straight line, giving your answer in the form y equals m x plus c.

How did you do?

8a
1 mark

A gardener is modelling the rate at which a shrub grows using the equation h equals 3 t plus 5, where

  • h is the height of the shrub, in cm

  • t is the number of weeks after the shrub was first planted

Write down the height of the shrub when it was first planted.

How did you do?

8b
1 mark

Find the height of the shrub predicted by the model after six weeks.

How did you do?

8c
2 marks

According to the model, how many weeks should it take for the shrub to reach a height of 29 cm?

How did you do?

9a
1 mark

An electrician charges a fixed fee of £50, plus £20 per hour.

Using h for the number of hours a job takes, and P for the total cost of a job in pounds, write down an equation linking h and P.

How did you do?

9b
2 marks

The electrician quotes a customer a price of £200 to complete a job.

How long is the electrician expecting the job will take?

How did you do?

9c
1 mark

A rival electrician charges a fixed fee of £38, plus £24 per hour.

Write down an equation in P and h for the total cost of a job from the rival electrician.

How did you do?

9d
2 marks

Determine which electrician would be the cheapest for a job taking 4 hours.

How did you do?

1
3 marks

The straight line L is parallel to the straight line with equation 2 x plus y minus 5 equals 0, and passes through the point open parentheses 1 comma space 1 close parentheses.

Find the equation of L.

How did you do?

2
3 marks

The straight line L is perpendicular to the straight line with equation y minus 1 third x plus 2 over 3 equals 0, and passes through the origin.

Find the equation of L.

How did you do?

3
2 marks

Find the equation of the straight line with a gradient of -2 that passes through the point (7, -3), giving your answer in the form y equals m x plus c.

How did you do?

4a
3 marks

The straight line l passes through the points (3, 4) and (9, 2).

Find the equation of l, giving your answer in the form y equals m x plus c.

How did you do?

4b
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1 mark

Write down the gradient of a line perpendicular to l.

How did you do?

5
4 marks

Find the equation of the straight line that passes through the points (-2, 3) and (3, 7).

Give your answer in the form

a x plus b y plus c equals 0

where a, b and c are integers to be found.

How did you do?

6
3 marks

Find the equation of the straight line parallel to y equals 2 x plus 3, that passes through the point (3, 12).

Give your answer in the form y equals m x plus c.

How did you do?

7a
2 marks

The coordinates of the endpoints of a diameter of a circle are (-3, 5) and (3, -3).

Find the length of the diameter.

How did you do?

7b
2 marks

Find the coordinates of the centre of the circle.

How did you do?

8a
2 marks

Three points A, B and C, have coordinates (-5,-11), (1, 1) and (4, 7) respectively.

Find the gradient of the line segment A B.

How did you do?

8b
2 marks

Find the gradient of the line segment B C.

How did you do?

8c
1 mark

Determine whether the line segments A B and B C are parallel or perpendicular.

How did you do?

8d
1 mark

Determine whether the line segments A Band B C lie along the same straight line.

How did you do?

9a
3 marks

A dog breeder is measuring the rate at which a puppy grows by measuring its back length, the distance from the base of the neck to the base of the tail.

  • At 2 weeks old, the puppy's back length measured 5 cm.

  • Six weeks later, the puppy’s back length had increased by 4.2 cm.

Find a linear model that links L, the back length of the puppy in cm, to w, the age of the puppy in weeks.

How did you do?

9b
1 mark

Use the model to determine the back length of the puppy at birth.

How did you do?

9c
2 marks

Use the model to find the age of a puppy that has a back length of 23.9 cm.

How did you do?

9d
2 marks

This particular breed of dog is fully grown after 40 weeks.

Find the back length of the puppy at 40 weeks and comment on the suitability of the linear model beyond 40 weeks.

How did you do?

10a
2 marks

The line l has equationspace 2 x minus y plus 3 equals 0.

l crosses the x-axis at point A and crosses the y-axis at point B.

Find the coordinates of points A and B.

How did you do?

10b
2 marks

Hence, find the area of the triangle O A B, where O is the origin.

How did you do?

11a
2 marks

A straight line passing through the origin O, is perpendicular to the straight line with equation x plus y equals 16.

The two lines meet at the point R.

Find the coordinates of R.

How did you do?

11b
2 marks

P is a point such that O P colon P R space equals space 3 colon 1

Find the coordinates of P.

How did you do?

12
3 marks

Straight lines l subscript 1 and l subscript 2 are both parallel to the straight line with equation 3 x minus y plus 4 equals 0.

  • l subscript 1 passes through the origin

  • l subscript 2 passes through the point (-4, 7). 

Find the equations of l subscript 1and l subscript 2, giving your answers in form y equals m x plus c.

How did you do?

13
4 marks

The line l passes through the points (4, 2) and (8, 5).

Find the equation of the line l, giving your answer in the form a x plus b y plus c equals 0
where a,b and c are integers to be found.

How did you do?

14
4 marks

The line segment A B is the diameter of a circle.

Ahas coordinates (-7,-9) and B has coordinates (9, 3).

Find the coordinates of the centre of the circle and the length of the diameter.

How did you do?

15a
3 marks

Three points A, B and C, have coordinates (-8, 1), (4, 4) and (12, 6) respectively.

Find the gradients of the line segments A B and B C.

How did you do?

15b
1 mark

Determine whether the line segments A B and B C lie along the same straight line.

How did you do?

16
4 marks

Find the equation of the straight line perpendicular to 2 x plus 3 y minus 4 equals 0 spacethat passes through the point (-1, -1).

Give your answer in the form a x plus b y plus c equals 0 comma where a, b and c are integers.

How did you do?

1a
3 marks
Coordinate plane with lines l_1 and l_2 intersecting at point C. Points A (the x-intercept of l_1), B (the x-intercept of l_2), and O (the origin) are marked on axes.
Figure 4

The line l subscript 1 has equation y equals 3 over 5 x plus 6

The line l subscript 2 is perpendicular to l subscript 1 and passes through the point B open parentheses 8 comma space 0 close parentheses, as shown in the sketch in Figure 4.

Show that an equation for line l subscript 2 is

5 x plus 3 y equals 40

How did you do?

1b
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5 marks

Given that

• lines l subscript 1 and l subscript 2 intersect at the point C

• line l subscript 1 crosses the x-axis at the point A

find the exact area of triangle A B C, giving your answer as a fully simplified fraction in the form p over q

How did you do?

2a
4 marks

The distance a particular car can travel in a journey starting with a full tank of fuel was investigated.

  • From a full tank of fuel, 40 litres remained in the car’s fuel tank after the car had travelled 80 km

  • From a full tank of fuel, 25 litres remained in the car’s fuel tank after the car had travelled 200 km

Using a linear model, with V litres being the volume of fuel remaining in the car’s fuel tank and d km being the distance the car had travelled, find an equation linking V with d.

How did you do?

2b
3 marks

Given that, on a particular journey

  • the fuel tank of the car was initially full

  • the car continued until it ran out of fuel

find, according to the model,

(i) the initial volume of fuel that was in the fuel tank of the car,

(ii) the distance that the car travelled on this journey.

How did you do?

2c
1 mark

In fact the car travelled 320 km on this journey.

Evaluate the model in light of this information.

How did you do?

3
4 marks

The line l subscript 1 has equation 3 x minus 2 y plus 4 equals 0 and crosses the x-axis at point A.

The line l subscript 2 has equation y equals 5 minus x and crosses the y-axis at point B.

Find the area of the triangle O A B, where O is the origin.

How did you do?

4
4 marks

Two perpendicular lines l subscript 1 and l subscript 2 intersect at point P open parentheses 2 comma space 5 close parentheses.

l subscript 2 crosses the x-axis at point Q open parentheses negative 3 comma space 0 close parentheses.

Find an equation of l subscript 1, giving your answer in the form y equals a plus b x.

How did you do?

5
4 marks

The points of intersection of the following straight lines form a parallelogram.

Find the coordinates of all four vertices of the parallelogram.

table row cell 2 y end cell equals cell 3 x plus 12 end cell row blank blank blank row cell 2 y end cell equals cell 8 minus x end cell end table      table row cell 3 x minus 2 y minus 12 end cell equals 0 row cell 1 half x plus y plus 2 end cell equals 0 end table

How did you do?

6a
2 marks

The line l subscript 1 has equation 5 x minus 2 y plus 12 equals 0.

The line l subscript 2 has equation 2 x plus 5 y plus 28 equals 0.

Determine if the lines l subscript 1 and l subscript 2 are parallel, perpendicular, or neither.

How did you do?

6b
3 marks

l subscript 1 and l subscript 2 intersect at the point P.

Find the coordinates of P.

How did you do?

6c
3 marks

l subscript 1 meets the y-axis at the point Q.

Point R lies on the line P Q such that P R colon R Q =3 :1.

Find the coordinates of R.

How did you do?

7a
3 marks

The line l passes through the points open parentheses p comma space 2 p close parentheses and open parentheses 3 p comma space 9 p close parentheses.

Find an equation for the line l.

How did you do?

7b
1 mark

The line l intersects the y-axis at the point (0, 3).

Find the value of p.

How did you do?

8
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4 marks

A line segment A B is tangent to a circle at point M.

The endpoints of A B have coordinates (-5, 16) and (5, 14).

M is the midpoint of A B.

The line M N is a diameter of the circle.

N has coordinates (-4, -5).

Find the coordinates of the centre of the circle, C.

Find also the area of the circle, giving your answer to 3 significant figures.

How did you do?

9
4 marks

Three points A, B and C, have coordinates (-4,-16), (2, 5) and (10, 33) respectively.

Show that A, B and C lie on the same straight line.

How did you do?

10
5 marks

The line l subscript 1 has equation 3 x minus 2 y plus 10 equals 0 and crosses the x-axis at the point A.

The line l subscript 2 is perpendicular to l subscript 1 and crosses the x-axis at (9, 0).

l subscript 2 crosses the y-axis at point B.

Find the area of the triangle O A B, where O is the origin.

How did you do?

11a
2 marks

VAT is a tax that is added to goods and services.

A plumber charges a fixed fee of £27.50, plus £21 per hour. The plumber then charges VAT on top of the total cost, at a rate of 20%.

Let P be the amount of money made by the plumber after h hours of work.

Find a linear model that links P and h.

How did you do?

11b
3 marks

A rival plumber charges a fixed fee of £37.80, plus £24 per hour. These prices already include VAT.

Find the number of hours for which both plumbers would charge the same amount.

How did you do?

12
5 marks

A quadrilateral has four vertices with coordinates (-1, 6), (-3, 2), (0, -4) and (2, 0).

Find the equation for each of the four lines that form the quadrilateral.

State the mathematical name of the quadrilateral formed.

How did you do?

13
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3 marks

Two perpendicular lines intersect at (-4,-4). One of the lines also passes through the point (0, 6), the other passes through the point (0,-5.6).

A kite is formed by these two lines and two others. The kite has a line of symmetry along the y-axis. 

Find the area of the kite.

How did you do?

14
6 marks

The tangent to a circle passes through the points A open parentheses negative 8 comma space minus 1 close parentheses and B open parentheses 16 comma space 7 close parentheses.

The tangent touches the circle at the point N, where A N colon N B space equals space 5 colon 3.

Find the equation of the straight line along which the diameter of the circle at point N lies.

Give your answer in the form a x plus b y plus c equals 0, where a, b and care integers to be found.

How did you do?