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GCSE / End of Key Stage 4 Maths Assessment

In this assessment, students will review their comprehension of various topics across key stage 4. May take approximately 45 minutes to complete.

'GCSE / End of Key Stage 4 Maths Assessment' worksheet

Key stage:  KS 4

Curriculum topic:  End of Key Stage Assessments

Curriculum subtopic:  Starter Assessments

Difficulty level:  

down

Worksheet Overview

QUESTION 1 of 10

In this assessment, you'll review your comprehension of various topics across key stage 4. May take approximately 45 minutes to complete but take all the time you need.

Work out:

 

3 - 0.32

Work out, without a calculator:

 

91/3 × 31/3

Write the following recurring decimal as a fraction in its lowest terms.

 

 

(Write your answer as a reduced fraction in the form a/b.)

Work out:

 

7  -  5 + 7
15 6 10

 

(Give your answer as a fraction, reduced to its lowest terms, e.g. 7/10.)

Place these numbers in ascending order of size, by converting them to decimals first:

 

8.9%     1/11     9 × 10-2     0.01½     0.312

Column A

Column B

1 (smallest)
8.9%
2
0.01½
3
9 × 10-2
4
0.312
5 (largest)
1/11

Work out, without a calculator:

 

122/3 ×   182/3

Add together:

 

4 × 106  +   6 × 104

1 × 1024

4.06 × 106

4.6 × 106

The number n has been rounded to the nearest thousand, to get 73 000.

A and B are the upper and lower bounds for n, such that  A ≤ n < B.

 

Select the correct values for A and B, the lower and upper bounds.

(Write the number without spaces between the numbers.)

 value
A
B

Simplify by rationalising the denominator of this fraction, giving your answer as a surd in the form a√b, where a and b are integers and b is as small as possible, or if the answer is a whole number, just state the whole number.

 

(√20)/4

½√5

(√5)/4

Rearrange the following equation so that it is in the form y = ...

 

2y - 4x = 8

y = 4 + 2x

y = 4 - 2x

y = 2x - 4

Factorise the following quadratic expression.

 

4x2 + 39x - 10

(4x - 5) (x + 2)

(4x + 1) (x - 10)

(x - 2) (4x + 5)

(4x - 1) (x + 10)

Use the quadratic formula to solve for x, giving your answers to 3 sig. figs.

 

17x + 21 = 3x2

x = 6.70

x = -4.04

x = 6.71

x = -1.04

Is this an equation with one solution, an equation with no solutions or an identity?

 

3x + 12 = 3(x + 4)

equation with one solution

equation with no solutions

identity

Select all the possible equations for the curve labelled B below.

y = x² - 5x - 6

y = (x + 2) (x + 3)

y = (x - 2) (x - 3)

y = (x + 3) (x + 2)

y = x² + 5x + 6

Find all the x-intercepts of the graph with the following equation:

(i.e. where the line crosses the x-axis)

 

y = 8x2 - 10x - 3

(1½, 0)

(-1½, 0)

(-4, 0)

(-¼, 0)

Select all the possible equations represented by the red line shown below.

 

y = 4x - 5

x = 4y - 5

y + 5 = 4x

y = ¼x - 5

y = 5x - 5

Find the gradient of a line perpendicular to the line with equation:

 

5x - 10y = 2

Divide the following expression by 4x3.

 

40x7 + 36x6 - 16x4 - 20x3

10x4 + 9x3 - 4x - 5

36x5 + 9x2 - 4x - 5x

10x4 + 9x2 -12x - 5

Find the smallest integer that satisfies the inequality:

 

15 - 5x ≤ 2(1 - x)

Use the quadratic formula to solve for x, giving your answers to 3 sig. figs.

 

3x2 - 19x + 2 = 0

x = -3.23

x = 6.23

x = -4.107

x = 0.107

Solve the following simultaneous equations:

 

18 - y2 = x2

y = 6 - x

 solutions
x
y

You may use a calculator for this question.

 

William invests £50 000 at 3% compound interest p.a. 

He makes no further withdrawals or investments.

 

How much is his investment worth in pounds after five years?

(Just write the number with 2 decimal places.)

You may use a calculator for this question.

 

Given that y is directly proportional to x and that x = 19 when y = 123.5,

find the value of x when y = 29.25.

You may use a calculator for this question.

 

The population of a certain species is expected to fall by x% each year.

After 10 years the population is halved.

 

Calculate x to 1 decimal place.

You may use a calculator for this question.

 

 

Calculate the volume (in cm3) of a cone with radius R equal to 2 cm and height H equal to 6 cm.

(Write your answer to 1 dp and just write the number.)

You may use a calculator for this question.

 

Find the length of the cuboid's diagonal AG of this 4 cm × 5 cm × 10 cm cuboid in cm to 3 sig. figs.

(Just write the number.)

 

You may use a calculator for this question.

 

In triangle PQR, which is not drawn to scale,

PQ = 12 cm, PR = 22 cm, and ∠QPR = 35º.

 

Calculate the length QR in cm to 3 sig. figs.

(Just write the number.)

 

You may use a calculator for this question.

 

Using trigonometry, calculate the angle θº to 3 sig. figs.

(Just write the number.)

 

 

Without using a calculator, state the value of:

 

cos 30°

2/√3

√3/2

√2/3

3/2

3/√2

The tangent SU touches the circle, with centre C, at T.

Calculate the angle TUP in degrees.

(Just write the number.)

 

The diagram shows a circle, centre C and three points A, B and P on the circumference.

Calculate the angle ∠ABC.

(Just write the number.)

 

You may use a calculator for this question.

 

Fred's eye level is 1.6 metres above ground.

He is standing on a cliff which is h metres above sea level and measures the angle of depression to a boat 400 m away as 26°.

 

Calculate the height of the cliff, h, in metres.

(Give your answer as a number to 1 decimal place.)

 

In triangle ABC above,

a = 6.2 cm

b = 7 cm

∠BCA = 125º

 

Calculate the triangle's area in cm2 to 3 sig. figs.

(Just write the number.)

The probability that it will rain at the weekend is 6/24.

What is the probability that it will not rain at the weekend?

(Write your answer as a fraction in the form a/b in its lowest terms.)

A money bag contains 20 × £1 coins, 25 × 50p coins, n × 20p coins, 10 × 10p coins and 5 × 2p coins.

The probability that a 10p coin is selected at random is 2/19.

 

Calculate n, the number of 20p coins in the bag.

There are 16 red balls and 4 white balls in a bag.

Fatima randomly chooses 2 balls, removing each in turn from the bag.

 

What is the probability of choosing one of each colour?

(Give your answer as a reduced fraction in the form a/b.)

Look at these scatter graphs.

Match each situation to the most likely scatter graph.

 

A  

    

B

   

 

C 

Column A

Column B

Outside Temperature and No. of Ice Creams Sold
A
Shoe Size and Maths Test Marks
B
Distance Travelled and Petrol Left in the Tank
C

A survey was carried out to record the marks some students scored in a test.

The marks were as follows:

 

25, 51, 20, 25, 18, 40, 30, 30, 40, 30, 19

 

Which box plot shows this information?

 

A

B

C

none of them

You may use a calculator for this question.

 

Over the course of five days, there were the following number of empty seats at a theatre.

 

12 seats, 14 seats, 10 seats, 20 seats, 24 seats

 

Calculate the standard deviation in the number of empty seats to 3 sig. figs.

  • Question 1

Work out:

 

3 - 0.32

CORRECT ANSWER
2.91
EDDIE SAYS
0.3 × 0.3 = 0.09
3 - 0.09 = 2.91
  • Question 2

Work out, without a calculator:

 

91/3 × 31/3

CORRECT ANSWER
3
EDDIE SAYS
= (9 × 3)1/3
= 271/3
= 3√27
= 3
  • Question 3

Write the following recurring decimal as a fraction in its lowest terms.

 

 

(Write your answer as a reduced fraction in the form a/b.)

CORRECT ANSWER
15/37
EDDIE SAYS
Three digits recur, so 405 over 999, which reduces.
Divide top and bottom by 27 or by 9 and 3 in stages.
  • Question 4

Work out:

 

7  -  5 + 7
15 6 10

 

(Give your answer as a fraction, reduced to its lowest terms, e.g. 7/10.)

CORRECT ANSWER
1/3
EDDIE SAYS
LCM of 15, 6 and 10 is 30.
14/30 - 25/30 + 21/30 = 10/30 = 1/3
  • Question 5

Place these numbers in ascending order of size, by converting them to decimals first:

 

8.9%     1/11     9 × 10-2     0.01½     0.312

CORRECT ANSWER

Column A

Column B

1 (smallest)
8.9%
2
9 × 10-2
3
1/11
4
0.312
5 (largest)
0.01½
EDDIE SAYS
8.9% = 0.089
9 × 10-2 = 0.09
1/11 = 0.0909...
0.312 = 0.0961
0.01½ = √0.01 = 0.1
  • Question 6

Work out, without a calculator:

 

122/3 ×   182/3

CORRECT ANSWER
36
EDDIE SAYS
(12 × 18)1/3
= 2161/3
= 3√216
= 6
6² = 36
  • Question 7

Add together:

 

4 × 106  +   6 × 104

CORRECT ANSWER
4.06 × 106
EDDIE SAYS
4 × 106 = 4 000 000
6 × 104 = 60 000
4 000 000 + 60 000 = 4 060 000 = 4.06 × 106
  • Question 8

The number n has been rounded to the nearest thousand, to get 73 000.

A and B are the upper and lower bounds for n, such that  A ≤ n < B.

 

Select the correct values for A and B, the lower and upper bounds.

(Write the number without spaces between the numbers.)

CORRECT ANSWER
 value
A
B
EDDIE SAYS
The number n lies between 72 500 and 73 500, in order for it to be rounded to 73 000 to the nearest thousand.
  • Question 9

Simplify by rationalising the denominator of this fraction, giving your answer as a surd in the form a√b, where a and b are integers and b is as small as possible, or if the answer is a whole number, just state the whole number.

 

CORRECT ANSWER
(√20)/4
EDDIE SAYS
4/80 reduces to 1/20.
Multiply top and bottom by √20.
Remember 20 = 4 × 5.
  • Question 10

Rearrange the following equation so that it is in the form y = ...

 

2y - 4x = 8

CORRECT ANSWER
y = 4 + 2x
EDDIE SAYS
2y - 4x = 8
2y = 8 + 4x
y = 4 + 2x
  • Question 11

Factorise the following quadratic expression.

 

4x2 + 39x - 10

CORRECT ANSWER
(4x - 1) (x + 10)
EDDIE SAYS
We look for factor pairs that multiply to give -40 and add to give +39.
These are 40 and -1.
Factorise 4x² + 40x - x - 10 in pairs.
  • Question 12

Use the quadratic formula to solve for x, giving your answers to 3 sig. figs.

 

17x + 21 = 3x2

CORRECT ANSWER
x = 6.71
x = -1.04
EDDIE SAYS
-3x2 + 17x + 21 = 0

a=-3 b=17 c=21
  • Question 13

Is this an equation with one solution, an equation with no solutions or an identity?

 

3x + 12 = 3(x + 4)

CORRECT ANSWER
identity
EDDIE SAYS
3(x + 4) simplifies to 3x + 12 so this is true for all values of x.
  • Question 14

Select all the possible equations for the curve labelled B below.

CORRECT ANSWER
y = (x + 2) (x + 3)
y = (x + 3) (x + 2)
y = x² + 5x + 6
EDDIE SAYS
The line passes through (-3, 0) and (-2, 0), so y = (x + 2)(x + 3).
This is y = x² + 5x + 6.
  • Question 15

Find all the x-intercepts of the graph with the following equation:

(i.e. where the line crosses the x-axis)

 

y = 8x2 - 10x - 3

CORRECT ANSWER
(1½, 0)
(-¼, 0)
EDDIE SAYS
When y = 0,
8x² - 10x - 3 = 0
(2x - 3)(4x + 1) = 0
x = -¼, 1½
  • Question 16

Select all the possible equations represented by the red line shown below.

 

CORRECT ANSWER
y = 4x - 5
y + 5 = 4x
EDDIE SAYS
It crosses the y-axis at -5 and has a gradient of 4 (1 to the right and 4 up).
So y = mx + c becomes y = 4x - 5.
y + 5 = 4x is a rearrangement of this.
  • Question 17

Find the gradient of a line perpendicular to the line with equation:

 

5x - 10y = 2

CORRECT ANSWER
-2
EDDIE SAYS
5x - 2 = 10y
y = ½x - 0.2 has a gradient of ½.
Product of gradients of perpendicular lines is always -1, so gradient of perpendicular line will be -2, since -2 × ½ = -1.
  • Question 18

Divide the following expression by 4x3.

 

40x7 + 36x6 - 16x4 - 20x3

CORRECT ANSWER
10x4 + 9x3 - 4x - 5
EDDIE SAYS
40x7 + 36x6 - 16x4 - 20x3
4x3 4x3 4x3 4x3
  • Question 19

Find the smallest integer that satisfies the inequality:

 

15 - 5x ≤ 2(1 - x)

CORRECT ANSWER
5
EDDIE SAYS
15 - 5x ≤ 2(1 - x)
15 - 5x ≤ 2 - 2x
15 - 2 ≤ 5x - 2x
13 ≤ 3x
3x ≥ 13
x ≥ 13/3 = 4.33333...
  • Question 20

Use the quadratic formula to solve for x, giving your answers to 3 sig. figs.

 

3x2 - 19x + 2 = 0

CORRECT ANSWER
x = 6.23
x = 0.107
EDDIE SAYS

a=3 b=-19 c=2
  • Question 21

Solve the following simultaneous equations:

 

18 - y2 = x2

y = 6 - x

CORRECT ANSWER
 solutions
x
y
EDDIE SAYS
Substitute for y to get:
18 - (6 - x)2 = x2
18 - (36 - 12x + x2) = x2
-18 + 12x - x2 = x2
2x2 - 12x + 18 = 0
x2 - 6x + 9 = 0
(x - 3)2 = 0
x = 3
y = 3
  • Question 22

You may use a calculator for this question.

 

William invests £50 000 at 3% compound interest p.a. 

He makes no further withdrawals or investments.

 

How much is his investment worth in pounds after five years?

(Just write the number with 2 decimal places.)

CORRECT ANSWER
57963.70
57 963.70
EDDIE SAYS
50 000 × 1.035
  • Question 23

You may use a calculator for this question.

 

Given that y is directly proportional to x and that x = 19 when y = 123.5,

find the value of x when y = 29.25.

CORRECT ANSWER
4.5
EDDIE SAYS
y = kx
123.5 = 19k
k = 123.5 ÷ 19 = 6.5
y = 6.5x
x = y ÷ 6.5 = 29.25 ÷ 6.5 = 4.5
  • Question 24

You may use a calculator for this question.

 

The population of a certain species is expected to fall by x% each year.

After 10 years the population is halved.

 

Calculate x to 1 decimal place.

CORRECT ANSWER
6.7
EDDIE SAYS
(1 - x/100)10 = 0.5
Work out the 10th root of 0.5 = 0.933.
x/100 = 1 - 0.933 = 0.066967...
x = 6.6967
  • Question 25

You may use a calculator for this question.

 

 

Calculate the volume (in cm3) of a cone with radius R equal to 2 cm and height H equal to 6 cm.

(Write your answer to 1 dp and just write the number.)

CORRECT ANSWER
25.1
EDDIE SAYS
Vol = 1 ÷ 3 × π × 2² × 6
  • Question 26

You may use a calculator for this question.

 

Find the length of the cuboid's diagonal AG of this 4 cm × 5 cm × 10 cm cuboid in cm to 3 sig. figs.

(Just write the number.)

 

CORRECT ANSWER
11.9
EDDIE SAYS
Use Pythagoras Theorem to get:
AG² = 4² + 5² + 10² = 16 + 25 + 100 = 141
  • Question 27

You may use a calculator for this question.

 

In triangle PQR, which is not drawn to scale,

PQ = 12 cm, PR = 22 cm, and ∠QPR = 35º.

 

Calculate the length QR in cm to 3 sig. figs.

(Just write the number.)

 

CORRECT ANSWER
14.0
EDDIE SAYS
Use the cosine rule to get QR² = 12² + 22² - 2 × 12 × 22 × cos 35°.
  • Question 28

You may use a calculator for this question.

 

Using trigonometry, calculate the angle θº to 3 sig. figs.

(Just write the number.)

 

 

CORRECT ANSWER
41.8
EDDIE SAYS
θ = sin-1(6/9)
  • Question 29

Without using a calculator, state the value of:

 

cos 30°

CORRECT ANSWER
√3/2
EDDIE SAYS

 


In the blue triangle, cos 30° = adj/hyp = √3/2
  • Question 30

The tangent SU touches the circle, with centre C, at T.

Calculate the angle TUP in degrees.

(Just write the number.)

 

CORRECT ANSWER
48
EDDIE SAYS
The tangent SU and the diameter PT meet at right angles,

so ∠PTU = 90°
∠TUP = 180° - 90° - 42° = 48°

  • Question 31

The diagram shows a circle, centre C and three points A, B and P on the circumference.

Calculate the angle ∠ABC.

(Just write the number.)

 

CORRECT ANSWER
53
EDDIE SAYS
∠ACB = 2∠APB so a = 74°
Triangle ACB is isosceles.
∠ABC = (180° - 74°) ÷ 2 = 106° ÷ 2 = 53°
  • Question 32

You may use a calculator for this question.

 

Fred's eye level is 1.6 metres above ground.

He is standing on a cliff which is h metres above sea level and measures the angle of depression to a boat 400 m away as 26°.

 

Calculate the height of the cliff, h, in metres.

(Give your answer as a number to 1 decimal place.)

 

CORRECT ANSWER
193.5
EDDIE SAYS
(h + 1.6)/400 = tan26°
h + 1.6 = 400tan26° = 195.093035
h = 193.493...
  • Question 33

In triangle ABC above,

a = 6.2 cm

b = 7 cm

∠BCA = 125º

 

Calculate the triangle's area in cm2 to 3 sig. figs.

(Just write the number.)

CORRECT ANSWER
17.8
EDDIE SAYS
Area = ½ × 6.2 × 7 × sin125°
  • Question 34

The probability that it will rain at the weekend is 6/24.

What is the probability that it will not rain at the weekend?

(Write your answer as a fraction in the form a/b in its lowest terms.)

CORRECT ANSWER
3/4
EDDIE SAYS
1 - 6/24 = 1 - 1/4 = 3/4
  • Question 35

A money bag contains 20 × £1 coins, 25 × 50p coins, n × 20p coins, 10 × 10p coins and 5 × 2p coins.

The probability that a 10p coin is selected at random is 2/19.

 

Calculate n, the number of 20p coins in the bag.

CORRECT ANSWER
35
EDDIE SAYS
10/(60 + n) = 2/19
190 = 2(60 + n)
190 = 120 + 2n
2n = 70
n = 35
  • Question 36

There are 16 red balls and 4 white balls in a bag.

Fatima randomly chooses 2 balls, removing each in turn from the bag.

 

What is the probability of choosing one of each colour?

(Give your answer as a reduced fraction in the form a/b.)

CORRECT ANSWER
32/95
EDDIE SAYS
16/20 × 4/19 + 4/20 × 16/19
  • Question 37

Look at these scatter graphs.

Match each situation to the most likely scatter graph.

 

A  

    

B

   

 

C 

CORRECT ANSWER

Column A

Column B

Outside Temperature and No. of Ic...
A
Shoe Size and Maths Test Marks
C
Distance Travelled and Petrol Lef...
B
EDDIE SAYS
A shows a positive correlation. The higher the temperature, the more ice creams are sold.
B shows a negative correlation. The further the distance, the less petrol there is remaining in the tank.
C shows no correlation. There is no connection between shoe size and marks gained in a test.
  • Question 38

A survey was carried out to record the marks some students scored in a test.

The marks were as follows:

 

25, 51, 20, 25, 18, 40, 30, 30, 40, 30, 19

 

Which box plot shows this information?

 

CORRECT ANSWER
B
EDDIE SAYS
First place the marks in ascending order.
Minimum mark = 18
Lower quartile = 20
Median = 30
Upper quartile = 40
Maximum mark = 51
  • Question 39

You may use a calculator for this question.

 

Over the course of five days, there were the following number of empty seats at a theatre.

 

12 seats, 14 seats, 10 seats, 20 seats, 24 seats

 

Calculate the standard deviation in the number of empty seats to 3 sig. figs.

CORRECT ANSWER
5.22
EDDIE SAYS

Step 1

Mean = (12 + 14 + 10 + 20 + 24) ÷ 5 = 80 ÷ 5 = 16


Step 2

(12 - 16)2 = (-4)2 = 16

(14 - 16)2 = (-2)2 = 4

(10 - 16)2 = (-6)2 = 36

(20 - 16)2 = (4)2 = 16

(24 - 16)2 = (8)2 = 64


Step 3

Mean of 16, 4, 36, 16 and 64 = (16 + 4 + 36 + 16 + 64) ÷ 5 = 136 ÷ 5 = 27.2


Variance = 27.2
Standard Deviation = √27.2 = 5.22
---- OR ----

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