A LevelMathsAQAPureExam QuestionsAlgebra & FunctionsQuadratics

Quadratics (AQA A Level Maths: Pure): Exam Questions

Exam code: 7357

3 hours42 questions
Easy
Medium
Hard
Very Hard
12 marks

The equation

x squared plus k x plus 3 equals 0

has no real roots.

Show that

k squared less than 12

How did you do?

23 marks

Find the value of the discriminant of the following expressions:

(i)  x squared minus 3 x plus 4

(ii)  4 x plus 3 minus 2 x squared

(iii)  5 minus 8 x plus 2 x squared

How did you do?

33 marks

Expand and simplify the following expressions:

(i)  left parenthesis x plus 4 right parenthesis left parenthesis 2 x minus 3 right parenthesis

(ii)  left parenthesis 3 x minus 4 right parenthesis left parenthesis 3 x plus 4 right parenthesis

(iii)  left parenthesis 2 x plus 1 right parenthesis squared

How did you do?

43 marks

Factorise the following expressions:

(i)  x squared plus 5 x minus 14

(ii)  25 x squared minus 36

(iii)  2 x squared plus 11 x plus 12

How did you do?

53 marks

Complete the square for the following expressions:

(i)  x squared plus 8 x minus 4

(ii)  2 x squared plus 12 x minus 5

(iii)  5 x squared minus 3 x plus 2

How did you do?

63 marks

Find the solutions of the following equations:

(i)  x squared plus 8 x minus 9 equals 0

(ii)  3 x squared minus 13 x plus 4 equals 0

(iii)  4 x squared minus 6 x minus 5 equals 0

How did you do?

7a3 marks

A curve has the equation

y equals 2 x squared plus 5 x minus 3

(i) Write down the coordinates of the point at which it crosses the y-axis.

(ii) Find the x-intercepts of the curve.

How did you do?

7b3 marks

Sketch the graph of y equals 2 x squared plus 5 x minus 3.

Label clearly any points where the graph meets the coordinate axes.

How did you do?

8a2 marks

Express

x squared plus 10 x plus 24

in the form

left parenthesis x plus a right parenthesis squared plus b

where a and b are integers to be found.

How did you do?

8b1 mark

Hence write down the coordinates of the minimum point on the curve with equation

y equals x squared plus 10 x plus 24

How did you do?

92 marks

A function is defined by

straight f left parenthesis x right parenthesis equals k x squared plus 2 k x minus 3

The equation straight f open parentheses x close parentheses equals 0 has two distinct real roots.

Show that 

4 k left parenthesis k plus 3 right parenthesis greater than 0

How did you do?

103 marks

Sketch the graph of

y equals left parenthesis 2 x minus 5 right parenthesis squared

Label clearly any points where the graph meets the coordinate axes.

How did you do?

11a3 marks

The curve C has equation

y equals x squared minus 3 x plus 2

Find the coordinates of all points where C meets the coordinate axes.

How did you do?

11b2 marks

Sketch the graph of C.

Label clearly all points where the curve meets the coordinate axes.

How did you do?

12a2 marks

Express the equation of the curve

y equals x squared plus 8 x minus 9

 in the form

y equals left parenthesis x plus b right parenthesis squared plus c

where b and c are integers to be found.

How did you do?

12b1 mark

Hence write down the coordinates of the vertex on the curve.

How did you do?

12c4 marks

Sketch the graph of

y equals x squared plus 8 x minus 9

Label clearly the coordinates of

  • any turning points

  • any points where the graph meets the coordinate axes

How did you do?

133 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis.

The coordinates of the turning point and the points where the graph meets the x-axis have been labelled.

2-2-edexcel-alevel-maths-pure-q9medium

Sketch the graph of y equals straight f left parenthesis x plus 2 right parenthesis.

Label clearly the coordinates of

  • any turning points

  • any points where the graph meets the x-axis

How did you do?

1a3 marks

A function is defined by

straight g left parenthesis x right parenthesis equals 12 plus 4 x minus x squared

The curve y equals straight g open parentheses x close parentheses meets the y-axis at the point P.

(i) Find all values of x for which straight g open parentheses x close parentheses equals 0.

(ii) Write down the coordinates of P.

How did you do?

1b3 marks

(i) Express straight g left parenthesis x right parenthesis in the form a minus left parenthesis x minus b right parenthesis squared, where a and b are constants to be found.

(ii) Hence write down the coordinates of the turning point on the graph of space y equals straight g left parenthesis x right parenthesis.

How did you do?

1c2 marks

Sketch the graph of y equals straight g left parenthesis x right parenthesis.

Label clearly the coordinates of

  • any turning points

  • any points where the graph meets the coordinate axes

How did you do?

2a3 marks

A function is given by

straight f left parenthesis x right parenthesis equals x squared plus 4 x plus 5

By expressing the function in the form

straight f open parentheses x close parentheses equals open parentheses x plus a close parentheses squared plus b

where a and b are integers to be found, find

(i) the minimum value of the function

(ii) the value of x for which the function is at its minimum value.

How did you do?

2b1 mark

Hence prove that the equation

straight f open parentheses x close parentheses equals 0

 has no real roots.

How did you do?

33 marks

A function is given by

straight f left parenthesis x right parenthesis equals k x squared plus 2 k x minus 3

The equation straight f open parentheses x close parentheses equals 0 has two distinct real roots.

Find the possible values of k.

How did you do?

43 marks

The equation

2 x squared minus 4 x plus 3 minus 2 k equals 0

has real roots.

Find the possible values of k.

How did you do?

52 marks

The equation

x squared plus p x plus q equals 0

has no real roots.

Show that

space p squared less than 4 q

How did you do?

63 marks

Solve the equation

x to the power of 4 minus 13 x squared plus 36 equals 0

How did you do?

74 marks

Solve the equation

x to the power of 2 over 5 end exponent plus x to the power of 1 fifth end exponent equals 6

How did you do?

81 mark

A function is defined by

straight f left parenthesis x right parenthesis equals left parenthesis a x minus b right parenthesis squared

where a and b are non-zero constants.

A teacher claims that the quadratic expression must have a discriminant of zero.

Without expanding the brackets, explain why this must be true.

How did you do?

9a3 marks

Express

y equals 4 x squared plus 8 x minus 5

 in the form

y equals a left parenthesis x plus b right parenthesis squared plus c

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

How did you do?

9b1 mark

Hence write down the coordinates of the minimum point on the curve.

How did you do?

9c3 marks

Sketch the graph of y equals 4 x squared plus 8 x minus 5.

Label clearly the coordinates of

  • any turning points

  • any points where the graph meets the coordinate axes

How did you do?

10a2 marks

Find the solutions to the equation

2 x squared plus x minus 6 equals 0

How did you do?

10b4 marks

A curve has the equation

y equals 2 x squared plus x minus 6

By expressing the equation of the curve in the form

y equals a open parentheses x plus b close parentheses squared plus c

where a, b and c are constants to be found, find the coordinates of the turning point on the curve.

How did you do?

10c2 marks

Sketch the graph of y equals 2 x squared plus x minus 6

Label clearly the coordinates of

  • any turning points

  • any points where the graph meets the coordinate axes

How did you do?

11
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Sketch the graph of y equals 12 x squared minus 5 x minus 72.

Label clearly the coordinates of any points where the curve meets the coordinate axes. 

How did you do?

123 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis.

The coordinates of the turning point and the points where the graph meets the coordinate axes have been labelled.

2-2-edexcel-alevel-maths-pure-q10hard

Sketch the graph of y equals straight f left parenthesis x plus 3 right parenthesis.

Label clearly the coordinates of

  • any turning points

  • any points where the graph meets the coordinate axes

How did you do?

133 marks

The figure below shows the curve y equals straight f left parenthesis x right parenthesis.

The coordinates of any points where the curve meets the coordinate axes are shown.

The coordinates of the maximum point are also shown.

2-2-edexcel-alevel-maths-pure-q10vhard

The graph of y equals straight f left parenthesis x right parenthesis plus 6 meets the x-axis at two points, one of which has coordinates open parentheses negative 2 comma space 0 close parentheses.

Sketch the graph of y equals straight f left parenthesis x right parenthesis plus 6.

Label clearly the coordinates of

  • the maximum point on the curve

  • any points where the curve meets the coordinate axes

How did you do?

15 marks

The equation 0 equals k x squared plus 2 k x minus 3 has two distinct real roots.

The equation 0 equals k x squared plus 4 k x minus 16 has no real roots.

Find the possible values of k.

How did you do?

27 marks

The curve C has the equation y equals x squared minus 3 x plus 2

The line l has the equation y equals 3 x minus 7

On the same coordinate axes, sketch C and l.

Label clearly the coordinates of

  • any points of intersection between C and l

  • all points where l meets the coordinate axes

  • all points where C meets the coordinate axes

  • any turning points on C

How did you do?

3a2 marks

The curve with equation

y equals 3 x squared plus 2 p x plus 4 q

where space p and q are non-zero constants does not meet the x-axis.

Show that 

space p squared less than k q

where k is a constant to be found.

How did you do?

3b3 marks

Given that the curve

y equals 3 x squared plus 2 p x plus 4 q

passes through the points with coordinates left parenthesis negative 2 comma space 6 right parenthesis space and left parenthesis 2 comma space 6 right parenthesis, find the values of space p and q.

How did you do?

4a3 marks

The equation

2 k minus 3 k x minus x squared equals 0

where k is a negative constant has has two distinct real roots.

Find the possible values of k.

How did you do?

4b3 marks

In the case where k equals negative 1, sketch the graph of y equals 2 k minus 3 k x minus x squared.

Label clearly the coordinates of all points where the graph meets the coordinate axes.

How did you do?

5a3 marks

A function is given by

straight f left parenthesis x right parenthesis equals x squared plus 8 x plus c

where c is a constant.

By expressing the function in the form

straight f open parentheses x close parentheses equals open parentheses x plus a close parentheses squared plus b

find, in terms of c where necessary,

(i) the minimum value of the function

(ii) the value of x for which the function is at its minimum value

How did you do?

5b1 mark

Find the possible values of c for which the equation straight f open parentheses x close parentheses equals 0 has no real roots.

How did you do?

6a
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A model for the arch of a bridge over a river is given by

y equals 4 minus x squared over 8

The minimum water level of the river under the bridge is represented by the x-axis and all measurements are in metres.

The width of the river at its minimum water level is the distance between the two x-intercepts.

2-2-edexcel-alevel-maths-pure-q6vhard

The maximum water level is given by the line y equals 0.5

Determine whether the width of the river under the bridge at its maximum water level exceeds 11 metres.

How did you do?

6b
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A canal boat is modelled as a cuboid above water level with a cross-section measuring 6 m wide and 2.5 m tall.

Determine whether it is possible for the canal boat to fit underneath the bridge when the river is at its minimum water level.

How did you do?

6c2 marks

To support the bridge, the arch will continue 2.5 m vertically below the minimum water level.

Find the exact distance between the ends of the base of the arch.

How did you do?

7a3 marks

Solve the equation

2 to the power of 2 x end exponent plus 64 equals 20 left parenthesis 2 to the power of x right parenthesis

How did you do?

7b3 marks

Solve the equation 

5 square root of x plus 3 equals 2 x

How did you do?

7c3 marks

Solve the equation

x to the power of 2 over 3 end exponent plus 2 x to the power of 1 third end exponent equals 8

How did you do?

8a1 mark

A stone is thrown vertically upwards from the top of a cliff. The stone lands in the sea vertically below. 

The path of the stone is modelled by

straight h left parenthesis t right parenthesis equals 24 plus 2 t minus 0.5 t squared space space space space space space space space space space space space space space t greater or equal than 0

where

  • straight h is the height, in metres, of the stone above sea level

  • t is the time in seconds since the stone was thrown

Write down the vertical height above sea level from which the stone was thrown.

How did you do?

8b3 marks

Find the maximum height the stone reaches above sea level.

How did you do?

8c
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Find how long it takes for the stone to reach the sea.

Give your answer in seconds to 1 decimal place.

How did you do?

9a3 marks

Express 

y equals negative 6 x squared plus 8 x minus 5

in the form 

y equals a minus b left parenthesis x plus c right parenthesis squared

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

How did you do?

9b1 mark

Hence write down the exact coordinates of the maximum point on the curve with equation

space y equals negative 6 x squared plus 8 x minus 5

How did you do?

9c3 marks

Sketch the graph of

y equals negative 6 x squared plus 8 x minus 5

Label clearly the coordinates of

  • the maximum point of the curve

  • any points where the curve meets the coordinate axes

How did you do?

102 marks

The minimum point on the curve with equation 

y equals x squared plus p x plus q

has coordinates open parentheses 3 comma space 1 close parentheses.

Find the values of space p and q.

How did you do?

11a2 marks

The equation

4 k minus 6 k x minus x squared equals 0

where k less than 0, has two distinct real roots, alpha and beta, where 0 less than alpha less than beta.

Sketch the graph of y equals 4 k minus 6 k x minus x squared.

Label clearly the points where the graph meets the coordinate axes.

How did you do?

11b3 marks

Find the possible values of k.

How did you do?

12a3 marks

Solve the equation

8 square root of x equals 48 minus x

How did you do?

12b3 marks

Solve the equation

2 to the power of 4 x end exponent plus 64 equals 20 left parenthesis 2 to the power of 2 x end exponent right parenthesis

How did you do?

133 marks

Given that

x squared plus 6 x y plus 9 y squared equals 0

find and simplify a relationship between x and y.

How did you do?

1a3 marks

The functions straight f open parentheses x close parentheses and straight g left parenthesis x right parenthesis are defined by

table row cell straight f left parenthesis x right parenthesis end cell equals cell left parenthesis k minus 1 right parenthesis x squared minus left parenthesis k minus 2 right parenthesis x minus 2 k space space space space space space space space space space space space space x element of straight real numbers end cell row cell straight g left parenthesis x right parenthesis end cell equals cell left parenthesis k minus 1 right parenthesis x squared minus 3 k x plus k plus 1 space space space space space space space space space space space space space space space space x element of straight real numbers end cell end table

 where k is a non-zero constant and k squared not equal to 1.

When the curve y equals straight f open parentheses x close parentheses and the curve y equals straight g open parentheses x close parentheses are plotted on the same set of coordinate axes, they intersect once only.

Find, in terms of k, the x-coordinate of the point of intersection.

How did you do?

1b2 marks

Hence, in the case when k equals 3, find the exact coordinates of the point of intersection.

How did you do?

26 marks

The equation

k squared x squared minus 4 x plus 5 equals k squared

has two distinct real roots.

Find the possible values of k.

How did you do?

3a3 marks

Show that the equation 

a x squared plus b x plus c equals 0

can be written in the form

a open parentheses x plus fraction numerator b over denominator 2 a end fraction close parentheses squared minus open parentheses fraction numerator b squared minus 4 a c over denominator 4 a end fraction close parentheses equals 0

where a, b and c are constants and a not equal to 0.

How did you do?

3b3 marks

Hence show that

x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

How did you do?