Revision NotesExam QuestionsPast Papers
GCSEFurther MathsAQAExam Questions2. Algebra & FunctionsAlgebra Toolkit

Algebra Toolkit (AQA GCSE Further Maths)

Exam Questions

2 hours32 questions
Medium
Hard
Very Hard
1a
Sme Calculator1 mark

p equals fraction numerator m plus 2 over denominator m squared plus 1 end fraction

Work out the value of p when m equals – 5.5

How did you do?

1b
Sme Calculator3 marks

Work out the values of m when p equals 2

How did you do?

Did this page help you?

2
Sme Calculator2 marks

Factorise fully 48 – 75 x squared

How did you do?

Did this page help you?

3
Sme Calculator3 marks

Expand and simplify open parentheses x minus 5 close parentheses cubed

How did you do?

Did this page help you?

4
Sme Calculator3 marks

Expand and simplify fully left parenthesis 2 x – 5 right parenthesis left parenthesis 3 x – 4 right parenthesis left parenthesis x plus 2 right parenthesis

How did you do?

Did this page help you?

5
Sme Calculator2 marks

5 m is decreased by 40 percent sign.

The answer is open parentheses m plus 1 close parentheses

Work out the value of m.

How did you do?

Did this page help you?

6
Sme Calculator3 marks

The rectangle and triangle shown have equal areas.

Diagram showing a rectangle and a right triangle. Rectangle sides are 2d and 12de. Height of the triangle is 9d and base of the triangle is 8e², the hypotenuse is unlabelled. Diagram labelled, "Not drawn accurately."

Work out the value of   d over e

Give your answer in its simplest form.

How did you do?

Did this page help you?

7
Sme Calculator3 marks

Expand and simplify fully left parenthesis 5 x plus 3 y squared right parenthesis left parenthesis 4 x – y squared right parenthesis

How did you do?

Did this page help you?

8
Sme Calculator4 marks

p left parenthesis x – 1 right parenthesis plus 2 left parenthesis 3 x plus k right parenthesis identical to 4 left parenthesis x plus 2 right parenthesis where pand k are integers.

Work out the values of p and k.

p =....................... , k =.......................

How did you do?

Did this page help you?

9
Sme Calculator3 marks

In the expansion and simplification of left parenthesis x minus 3 right parenthesis left parenthesis x squared space plus space 5 x space plus space k right parenthesis the coefficient of x squared is equal to the coefficient of x.

k is a constant.

Work out the value of k.

How did you do?

Did this page help you?

10
Sme Calculator2 marks

P Q R S is a trapezium.

A set of axes with four points marked on it, P(2, 0), Q(10, 0), R(9, a) and S(5, a). Points P, Q, R and S are connected and form a trapezium.

The area of the trapezium is 63 square units.

Work out the value of a.

How did you do?

Did this page help you?

11
Sme Calculator3 marks

Expand and simplify fully left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis left parenthesis x plus 4 right parenthesis

How did you do?

Did this page help you?

12
Sme Calculator4 marks

m x plus 4 minus 2 open parentheses x plus p close parentheses identical to 6 open parentheses x plus 1 close parentheses where m and p are integers.

Work out the values of m and space p.

m = .........................

p = .........................

How did you do?

Did this page help you?

13
Sme Calculator2 marks

Factorise fully w to the power of 5 x cubed y squared space plus space w squared x to the power of 6 y cubed

How did you do?

Did this page help you?

14
Sme Calculator3 marks

Expand and simplify 5 left parenthesis 2 x – 1 right parenthesis plus 4 left parenthesis 11 – x right parenthesis

Give your answer in the form space a left parenthesis b x plus c right parenthesis where a comma space b and c are integers greater than 1.

How did you do?

Did this page help you?

1
Sme Calculator3 marks

Factorise fully 6 x squared plus 26 x y – 20 y squared

How did you do?

Did this page help you?

2
Sme Calculator4 marks

Write 2 x squared minus 16 x plus 13 in the form a left parenthesis x space plus space b right parenthesis squared plus c where a comma space b and c are integers.

How did you do?

Did this page help you?

3
Sme Calculator2 marks

Factorise fully  12 p q cubed r – 18 p q squared r squared plus 24 p q squared r

How did you do?

Did this page help you?

4
Sme Calculator3 marks

The radius of a sphere, in cm, is fraction numerator 3 k over denominator 2 end fraction

The volume of the sphere, in cm3, is 972 pi

Volume of a sphere = 4 over 3 pi r cubed where r is the radius

Work out the value of k.

How did you do?

Did this page help you?

5
Sme Calculator2 marks

Factorise fully open parentheses p plus 6 close parentheses to the power of 11 minus open parentheses p plus 6 close parentheses to the power of 10

How did you do?

Did this page help you?

6
Sme Calculator3 marks

The diagram shows a solid hemisphere.

The diameter is 12 a cm

The volume is 486 pi cm3

Diagram of a hemisphere with a diameter of 12a cm.

Volume of a sphere = 4 over 3 pi r cubed where r is the radius

Work out the value of a.

How did you do?

Did this page help you?

7
Sme Calculator5 marks

Show that open parentheses x plus 1 close parentheses open parentheses x plus 3 close parentheses open parentheses x plus 4 close parentheses minus x open parentheses x squared plus 7 x plus 11 close parentheses

can be written in the form left parenthesis x plus a right parenthesis left parenthesis x plus b right parenthesis where a and b are positive integers.

How did you do?

Did this page help you?

8
Sme Calculator4 marks

P equals 4 x and Q equals 7 x

P increases by 25%
Q decreases by 40%

Now, P is 28 greater than Q.

Work out the value of x.

How did you do?

Did this page help you?

9
Sme Calculator3 marks

Factorise fully  6 left parenthesis y plus 3 right parenthesis to the power of 5 plus 4 left parenthesis y plus 3 right parenthesis to the power of 4

Give your answer in its simplest form.

Do not attempt to expand left parenthesis y plus 3 right parenthesis to the power of 5 space or space left parenthesis y plus 3 right parenthesis to the power of 4

How did you do?

Did this page help you?

10
Sme Calculator4 marks

A equals 2 minus 5 x space space space space space space space space space space space B equals 3 x minus 1 space space space space space space space space space space space space C equals x squared

Show that left parenthesis 2 A plus 3 B right parenthesis squared identical to A plus B plus C

How did you do?

Did this page help you?

11a
Sme Calculator2 marks

You are given that x squared plus 6 x plus 2 identical to open parentheses x plus h close parentheses squared plus k

Work out the values of h and k.

h = .....................

k = .....................

How did you do?

11b
Sme Calculator1 mark

Write down the coordinates of the minimum point on the curve y equals x squared plus 6 x plus 2

How did you do?

11c
Sme Calculator1 mark

Solve the equation x squared plus 6 x plus 2 equals 0

Give your answers in the form a plus-or-minus square root of b

How did you do?

Did this page help you?

12
Sme Calculator2 marks

Factorise fully x to the power of 4 minus 81

How did you do?

Did this page help you?

13
Sme Calculator3 marks

The x squared term in the expansion of open parentheses 3 x plus 4 close parentheses open parentheses x squared plus p x plus 5 close parentheses is negative 23 x squared

Work out the value of p.

How did you do?

Did this page help you?

14
Sme Calculator3 marks

A cylinder has base radiusspace xcm and height y cm

A hemisphere has radius 6 y cm

The cylinder and hemisphere have equal volumes.

Volume of a sphere = 4 over 3 pi r cubed where r is the radius

Work out the value of x over y

You must show your working.

How did you do?

Did this page help you?

1
Sme Calculator6 marks

3 x squared plus 2 b x plus 8 a can be written in the form 3 left parenthesis x plus a right parenthesis squared plus b plus 2

Work out the two possible pairs of values of a and b.

a equals.......... space b equals.......... space space a equals.......... space b equals.......... space

How did you do?

Did this page help you?

2
Sme Calculator4 marks

By multiplying both sides of the equation by x to the power of 1 half end exponent

Solve   2 x to the power of 3 over 2 end exponent minus 3 x to the power of 1 half end exponent equals 7 x to the power of negative 1 half end exponent    for  x greater than 0

Give your answer to 3 significant figures.

You must show your working.

How did you do?

Did this page help you?

3
Sme Calculator3 marks

Solve 4 open parentheses x minus 5 close parentheses squared equals k squared where k is a constant.

Give your answers in their simplest form in terms of k.

How did you do?

Did this page help you?

4
Sme Calculator5 marks

A cone has base radius r cm, perpendicular height h cm and slant height l cm

The curved surface area is 60 pi cm2

l equals 3 r

Curved surface area of a cone = pi r l

where r is the radius and l is the slant height

Work out the value of h.

Give your answer in the form  a square root of 10 where a is an integer greater than 1 You must show your working.

How did you do?

Did this page help you?