CIE A Level Maths: Pure 1

Topic Questions

2.1 Equation of a Straight Line

1
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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 intercepts the y-axis,

(ii)
the coordinates of the point where the line intercepts the x-axis.

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2
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5 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).

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3
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5 marks

Find the 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).

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4
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6 marks

Find the equations of the following straight lines, given the gradient, straight m, and a point space P left parenthesis x space comma space y right parenthesis spacethat each line passes through.

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

(i)
straight m = 2, P left parenthesis 3 space comma space 5 right parenthesis,
(ii)
straight m equals negative 2 comma space space P open parentheses negative 1 space comma space 3 close parentheses comma
(iii)
straight m equals 1 half comma space space P open parentheses 5 space comma space minus 2 close parentheses.

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5
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6 marks

Given that a straight line passes through the points space P open parentheses x subscript 1 space comma space y subscript 1 space close parenthesesand  Q left parenthesis x subscript 2 space comma space y subscript 2 right parenthesis, work out the gradient of the following lines:

(i)
P open parentheses 2 space comma space 6 close parentheses comma space space Q open parentheses 4 space comma space 12 close parentheses comma
(ii)
P open parentheses negative 3 space comma space 4 close parentheses comma space space Q left parenthesis negative 8 space comma space 24 right parenthesis comma
(iii)

P open parentheses 1 space comma space minus 3 close parentheses comma space space Q left parenthesis negative 3 space comma space 6 right parenthesis.

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6
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5 marks

Write the equations of the straight lines below in 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 comma
(ii)
y equals 1 half x plus 7 comma
(iii)

1 third y equals 1 over 6 x minus 1 over 9.

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7
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3 marks
(i)
Write down an equation of a straight line that is parallel to y equals 4 x plus 3.
(ii)
Write down an equation of a straight line that is perpendicular to y equals 8 x minus 5.

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8
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3 marks

The line L is parallel to the line with equation 2 x plus y minus 3 equals 0, and passes through the point (1 , 1).

Find the equation of the line L.

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9
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3 marks

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

Find the equation of the line L.

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10
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4 marks

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

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

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11a
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1 mark

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

h is the height of the shrub in centimetres t weeks after planting.

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

11b
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2 marks

Work out the height of the shrub after six weeks.

11c
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2 marks

How long should it take the shrub to reach a height of 29 cm?

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1
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2 marks

Lines l subscript 1 and l subscript 2 are both parallel to the line with equation 3 x minus y plus 4 equals 0. l subscript 1 passes through the origin and 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.

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2
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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.

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3
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3 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.

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4a
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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.

4b
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1 mark

Explain why your answer to part (a) shows that A B and B C are collinear.

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5a
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4 marks

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

At 1.5 weeks old the puppy measured 2.3 cm and at 6.5 weeks old it measured 6.3 cm.

Using a linear model, find an equation linking L, the length of the puppy in centimetres, to w, the age of the puppy in weeks.

5b
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1 mark

Use the model to find the length of the puppy at age 20 weeks. 

5c
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2 marks

Use the model to find the age of a puppy that has a back-length of 11.9 cm.

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6
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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.

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7a
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4 marks

Two perpendicular lines l subscript 1 and l subscript 2 intersect at point P(2, 5).

l subscript 2 crosses the x-axis at point Q(-3, 0).

Find an equation for the line l subscript 1.

7b
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4 marks

l subscript 1 crosses the x-axis at the point R.
Find the area of the triangle P Q R.

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8
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4 marks

The intersections 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

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9a
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3 marks

An electrician charges a fixed fee of £45 plus £22 per hour.

Defining suitable variables, write down an equation to represent the charges made by the electrician.

9b
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4 marks

A rival electrician charges a fixed fee of £36 and £25 per hour. Determine which electrician is cheapest for a job taking 4 hours?

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10a
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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.

10b
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4 marks

l subscript 1 and l subscript 2 intersect at the point P.
Find the coordinates of P.

10c
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4 marks

l subscript 1 meets the y-axis at point Q.
Point R lies on the line P Q such that P R colon R Q =3 :1.
Find the coordinates of R.

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1
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2 marks

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

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2
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4 marks

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

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3
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2 marks

Find the equation of a line parallel to y equals 2 x plus 3 and passing through the point (3, 12).

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4a
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2 marks

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

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

4b
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1 mark

Write down the gradient of a line perpendicular to l.

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5a
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2 marks

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

Find the length of the diameter.

5b
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2 marks

Find the centre of the circle.

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6a
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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.

6b
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1 mark

Find the gradient of the line segment B C.

6c
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1 mark

What do you notice about the gradients of the lines A B and B C?

6d
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1 mark

Explain why your answers to (a) and (b) show that the lines A Band B C are collinear?

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7a
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4 marks

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

At 3 weeks old the puppy measured 6 cm and at 6 weeks old it measured 8 cm

Find an equation in the form L equals m w plus c, linking L, the length of the puppy in centimetres to w, the age of the puppy in weeks. m and c are constants to be found.

7b
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1 mark

The equation found in part (a) is to be used as a model for the puppy’s growth.

How many centimetres per week does the model suggest the puppy grows?

7c
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1 mark

Use the model to find the length of the puppy after 15 weeks.

7d
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2 marks

Use the model to find the age of a puppy that has a back-length of 17 cm.

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8a
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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.

8b
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2 marks

Find the area of the triangle OAB, where O is the origin.

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9a
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2 marks

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 connecting h and P.

9b
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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?

9c
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2 marks

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

Write down an equation for the total cost of a job from the rival electrician.

9d
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3 marks

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

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10a
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3 marks

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

The two lines meet at point R. P is a point such that O P space colon P R space equals space 3 space colon 1.

Find the equation of the perpendicular line and hence, the co-ordinates of point R.

10b
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2 marks

Find the coordinates of P.

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1
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3 marks

Find the equation of a line perpendicular to space 2 x plus 3 y minus 4 equals 0 spacethat passes through the point (-1, -1), giving your answer in the form a x plus b y plus c equals 0 comma where ab and c are integers.

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2a
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3 marks

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

Find an equation for the line l

2b
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2 marks

l intercepts the y-axis at (0, 3).

Find the value of p.

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3
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4 marks

A line segment AB is the tangent to a circle at point M. The end points of AB have coordinates (-5, 16) and (5, 14) respectively and M is the midpoint of AB.

The line MN is the diameter of the circle and N has coordinates (-4,-5).

Find the coordinates of the centre of the circle C and the area of the circle to three significant figures.

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4
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4 marks

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

Show that the lines A B and B C are collinear.

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5a
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3 marks

A dog breeder is measuring the rate at which a puppy grows by measuring its “back-length”— that is the distance from the base of the neck to the base of the tail. At 2 weeks old the puppy measured 5 cm. Six weeks later the puppy’s back-length had increased by 4.2 cm.

Using a linear model, find an equation linking L, the length of the puppy in centimetres, to w, the age of the puppy in weeks.

5b
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1 mark

What was the back-length of the puppy at birth?

5c
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1 mark

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

5d
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2 marks

This particular breed of dog is fully grown after 40 weeks.
Find the dog’s length at 40 weeks and comment on the suitability of the model.

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6
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5 marks

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

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7
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5 marks

The point P open parentheses negative 5 comma negative 2 close parentheses lies on the line l subscript 1, l subscript 1 crosses the x-axis at the point R.

Another line, l subscript 2, is perpendicular to l subscript 1 at the point P and crosses the x-axis at the point Q(-1, 0).

Find the area of the triangle P Q R.

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8a
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2 marks

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%.

Defining suitable variables write down an equation to represent the charges, including VAT, made by the plumber.

8b
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4 marks

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

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

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9
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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 and state its mathematical name.

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10
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4 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.

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11
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6 marks

The tangent to a circle passes through the points A(-8, -1) and B(16, 7).

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

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

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