GCSEFurther MathsAQAExam Questions2. Algebra & FunctionsAlgebra Toolkit

Algebra Toolkit (AQA GCSE Further Maths)

Exam Questions

2 hours32 questions
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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

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1b
Sme Calculator3 marks

Work out the values of m when p equals 2

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2
Sme Calculator2 marks

Factorise fully 48 – 75 x squared

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3
Sme Calculator3 marks

Expand and simplify open parentheses x minus 5 close parentheses cubed

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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

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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.

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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.

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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

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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 =.......................

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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.

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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.

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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

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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 = .........................

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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

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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.

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1
Sme Calculator3 marks

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

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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.

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3
Sme Calculator2 marks

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

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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.

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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

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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.

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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.

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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.

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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

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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

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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 = .....................

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11b
Sme Calculator1 mark

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

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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

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12
Sme Calculator2 marks

Factorise fully x to the power of 4 minus 81

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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.

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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.

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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

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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.

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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.

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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.

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