GCSEFurther MathsAQATopic Questions2. Algebra & FunctionsAlgebra Toolkit

Algebra Toolkit (AQA GCSE Further Maths)

Topic Questions

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

Give your answer to 3 significant figures.

You must show your working.

[4]

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2
Sme Calculator3 marks

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

[3]

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3
Sme Calculator3 marks

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

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

[3]

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4
Sme Calculator2 marks

5m is decreased by 40%
The answer is (m + 1)
Work out the value of m.

[2]

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5
Sme Calculator3 marks

The rectangle and triangle shown have equal areas.

q3-paper2-nov2021-aqa-gcse-furthermaths

Work out the value of   d over e

Give your answer in its simplest form.

[3]

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6
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 972pi

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

Work out the value of k.

[3]

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7
Sme Calculator3 marks

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

[3]

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8
Sme Calculator6 marks

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

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

a space equals.......... space b space equals.......... space space a space equals.......... space b space equals.......... space open square brackets 6 close square brackets

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9
Sme Calculator4 marks

p left parenthesis x space – space 1 right parenthesis space plus space 2 left parenthesis 3 x space plus space k right parenthesis space identical to space 4 left parenthesis x space plus space 2 right parenthesis space space spacewhere space p spaceand k are integers.

Work out the values of p and k.

[4]

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

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10
Sme Calculator3 marks

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

[3]

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11
Sme Calculator5 marks

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

The curved surface area is 60pi cm2
l space equals space 3 r

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.

[5]

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12
Sme Calculator2 marks

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

[2]

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13
Sme Calculator3 marks

The diagram shows a solid hemisphere.

The diameter is 12a cm

The volume is 486pi cm3

qp12-2019-paper-2-aqa-gcse-further-maths

Work out the value of a.

[3]

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14
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 space plus space a right parenthesis left parenthesis x space plus space b right parenthesis where a and b are positive integers.

[5]

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15
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.

[3]

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16
Sme Calculator4 marks

Pspace equals space 4 x   and   Q equals space 7 x

P increases by 25%
Q decreases by 40%

Now, P is 28 greater than Q.

Work out the value of x.

[4]

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17
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.

[3]

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18
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.

[4]

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19a
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 space equals space – 5.5

[1]

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19b
Sme Calculator3 marks

Work out the values of m when p space equals space 2

[3]

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20
Sme Calculator2 marks

PQRS is a trapezium.

q7-2018-paper-2-aqa-gcse-further-maths

The area of the trapezium is 63 square units.

Work out the value of a.

[2]

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21
Sme Calculator3 marks

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

[3]

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22
Sme Calculator2 marks

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

[2]

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23
Sme Calculator3 marks

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

Give your answer in its simplest form.

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

[3]

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24
Sme Calculator2 marks

Factorise fully   48 space – space 75 x squared

[2]

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25
Sme Calculator4 marks

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

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

[4]

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26
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.

[4]

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

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

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27
Sme Calculator3 marks

Expand and simplify   space open parentheses x minus 5 close parentheses cubed

[3]

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28a
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.

[2]

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

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

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28b
Sme Calculator1 mark

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

[1]

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28c
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

[1]

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29
Sme Calculator2 marks

Factorise fully   x to the power of 4 minus 81

[2]

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30
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

[2]

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31
Sme Calculator3 marks

The x2 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.

[3]

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32
Sme Calculator3 marks

A cylinder has base radius space x spacecm and height y cm

A hemisphere has radius 6y cm

The cylinder and hemisphere have equal volumes.

Work out the value of x over y
You must show your working.

[3]

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