The document contains a variety of GCSE mixed higher-level mathematics questions and answers covering topics such as estimation, indices, geometry, probability, transformations, and algebra. Each question involves different problem-solving techniques, including drawing graphs, calculating areas, and solving equations. It serves as a comprehensive resource for students preparing for GCSE mathematics exams, focusing on critical mathematical concepts and applications.

1. Mixed Higher GCSE Questions
and Answers
(Grades 4 to 7)

2. Estimation and Rules of Indices
1) Work out an estimate for
2) (a) Write as a power of 3
(b) Write down the value of 120
(c) Write down the value of 3–2

3. 3) (a) On the grid, draw a frequency
polygon for this information.
*(b) Nalini says that at least a quarter of these
teachers sent more than 30 emails.Is Nalini
correct? You must explain your answer.
Frequency Polygon
She is not correct as 15/51

4. 4) Work out
Give your answer in its simplest form.
4 Rules Fractions
5) (a) Work out
(b) Work out
Give your answer as a decimal 0.42

5. 6) ABC and DEF are parallel straight lines.
ABE is an isosceles triangle with AB = BE.
Angle CBE = 142°
Work out the size of angle x.
Give a reason for each stage in your working.
Angles between parallel lines and in a triangle

6. On the grid, draw the graph of
y = 2x – 3 for values of x from –2 to 2
Linear Graphs

7. Problem Solving: Area, perimeter and algebra
8) The area of rectangle A is equal
to the area of rectangle B.
Work out the perimeter of rectangle
B.
2x - 3
2.5
A
4x
7

8. 9) On the grid, draw an accurate plan of the
solid prism
Plans and Elevations

9. Plans and Elevations
10) On the grid, draw an accurate side
elevation of the solid prism from the
direction of the arrow.

10. Expand and Simplify Expressions /Solve Quadratic Equation
11) (a) Expand and
simplify (y + 2)(y + 5)
(b) Factorise e2
+ e – 12
(c) Solve 3x2
– x – 1 = 0
Give your solutions correct
to 2 decimal places.

11. Q12.
Fiza has 10 coins in a bag.
There are three £1 coins and
seven 50 pence coins.
Fiza takes at random, 3 coins
from the bag.
Work out the probability that
she takes exactly £2.50
Probability

12. Q13.
Tom and Amy set the alarms on
their phones to sound at 6.45 am.
Both alarms sound together at
6.45 am.
Tom's alarm then sounds every 9
minutes.
Amy's alarm then sounds every
12 minutes.
At what time will both alarms next
sound together?
LCM in Context

13. Q14.
Write 525 as a product of its prime factors.
Product of Prime Factors

14. Negative and Fractional Indices
Q15.
(a) Write down the value of
b) Find the value of

15. Q16.
The table gives some information about the
birds Paula sees in her garden one day.
Bird Frequency
Magpie 15
Thrush 10
Starling 20
Sparrow 27
Complete the accurate pie chart.
Pie Charts

16. Q17.
Work out
Give your answer as a mixed
number in its simplest form.
Q18.
Prove that the recurring decimal
has the value
Add Mixed Numbers / Recurring Decimals to Fractions

17. Q19. ABC, PQR and AQD are straight lines.
ABC is parallel to PQR.
Angle BAQ = 35°
Angle BQA = 90°
Work out the size of the angle marked x.
Give reasons for each stage of your working.
Angles between Parallel Lines and in a Triangle

18. Q20.
(a) On the grid, draw the graph of
y = 4x + 2 from x = –1 to x = 3
(b) (i) Write down the equation
of a straight line that is parallel
to y = 4x + 2
(ii) Write down the gradient of
a straight line that is
perpendicular to y = 4x + 2
Linear Graphs and Parallel and Perpendicular Lines
a) See grid
b) (i) y = 4x +c
(ii) y =

19. Q21. The diagram shows a
triangle inside a rectangle.
Show that the total area, in cm2
, of the
shaded regions is 18x – 30
Solving Area Problems with Algebra

20. Q22. Describe fully the single transformation
which maps triangle A onto triangle B.
Rotations

21. Q23. Describe fully the single
transformation that maps shape
P onto shape Q
Enlargement

22. Q24.
(a) (i) Factorise x2
– 12x + 27
(ii) Solve the equation x2
– 12x + 27 = 0
(b) Factorise y2
– 100
Factorise and Solve Quadratics

23. Q25.
Sandy has a 4-sided spinner. The
sides of the spinner are labelled A,
B, C and D. The spinner is biased.
The table shows the probability that
the spinner will land on A or on B or
on C.
Side A B C D
Probability 0.15 0.32 0.27
(a) Work out the probability that
the spinner will land on D.
Sandy spins the spinner 300 times
(b) Work out an estimate for the
number of times the spinner will land
on A.
Probability / Expected
Frequency

24. Q26. CALC
Work out
Give your answer in standard form.
Q27. NON CALC
Work out (2 × 107
) × (5.4 × 10–12
)
Give your answer as an ordinary number.
Standard Form
0.000108

25. Q28.
Gemma has the same number of sweets as
Betty. Gemma gives 24 of her sweets to Betty.
Betty now has 5 times as many sweets as
Gemma. Work out the total number of sweets
that Gemma and Betty have.
Q29.
Find the
perimeter given
angel ABC = ACB
Form and Solve Equations

26. Q30.
Chloe recorded the test marks of 20 students.
22 29 38 16 36 18 30 21 27 43
14 41 25 38 46 19 48 34 23 46
(a) Show this information in an ordered stem
and leaf diagram.
One of these students is going to be
chosen at random.
(b) Find the probability that this
student has a test mark less than 28
Stem and Leaf Diagrams

27. Q31.
Express the recurring decimal
as a fraction in its simplest form.
Q32.
Work out
Give your answer as a mixed
number in its simplest form.
Recurring decimal to fraction/ multiply mixed nos.

28. Q33. Only blue vans and white vans
are made in a factory.
The ratio of the number of blue vans
to the number of white vans is 4 : 3
(a) Write down the fraction of vans
that are blue.
For blue vans,
the number of small vans : the number of large vans
= 3 : 5
(b) Work out the fraction of the number of vans
made in the factory that are blue and large.
Ratio and Fractions

29. Q34. ABCDEFGHI is a regular 9-sided polygon.
The vertices B and E are joined with a
straight line.
Work out the size of angle BEF.
You must show how you get your answer.
Angles in Regular Polygons

30. Q35. At 9 am, Bradley began a journey on his bicycle.
From 9 am to 9.36 am, he cycled at an average speed of 15 km/h.
From 9.36 am to 10.45 am, he cycled a further 8 km.
(a) Draw a travel graph to show Bradley's journey.
From 10.45 am to 11 am, Bradley cycled at an average
speed of 18 km/h.
(b) Work out the distance Bradley cycled from 10.45 am
to 11 am.
Distance-Time Graphs

31. Q36. Change 2 m3
to cm3
.
Q37. (HINT: 1 litre = 1000cm3
)
Sally wants to fill the sand pit with sand.
A bag of sand costs £2.50 . There are 8
litres of sand in each bag. Sally says,
"The sand will cost less than £70"
Show that Sally is wrong.
Conversions and Problem Solving

32. Q38. Shape A is translated by the vector
to make Shape B.
Shape B is then translated by the vector
to make Shape C.
Describe the single transformation that
maps Shape A onto Shape C.
Translations and Vector Notation

33. Q39. On the grid, enlarge the triangle by
scale factor –1½, centre (0, 2)
Negative and Fractional Enlargement

34. Q40. 3 teas and 2 coffees
have a total cost of £7.80
5 teas and 4 coffees have a
total cost of £14.20
Work out the cost of one tea
and the cost of one coffee.
Q41. Solve the simultaneous equations
4x + y = 25
x − 3y = 16
Simultaneous Equations

35. Sameena has a round pencil case and a
square pencil case. There are 4 blue pens
and 3 red pens in the round pencil case.
There are 3 blue pens and 5 red pens in
the square pencil case. Sameena takes at
random one pen out of each pencil case.
(a) Complete the probability tree diagram
(b) Work out the probability that the pens
Sameena takes are both red.
4/7
3/7
3/8
3/8
5/8
5/8
3/7 x 5/8 = 15/56
Probability Tree Diagrams

36. Q43. Expand (1 + √2 )(3 − √2 )
Give your answer in the form a + b √2
where a and b are integers.
Q44. (a) Express in the form
where n is a positive integer.
(b) Rationalise the denominator of
Surds

37. Q45. Make d the subject of the formula
Q46. Make t the subject of
Rearranging Formulae

38. Jean records the maximum daily temperature each day for 10 days.
She also records the number of children going to a paddling pool for each of these days.
She draws this scatter graph for her information. Jean's information for one of these days is an outlier on
the scatter graph.
(a) Give a possible reason for this.
(b) What type of correlation does the scatter graph show?
On the 11th day, the maximum daily temperature was 19°C.
(c) Write down an estimate for the number of children
going to the paddling pool on the 11th day.
It would not be sensible to use the scatter graph to predict
the number of children going to the paddling pool on a day
when the maximum daily temperature was 13°C.
(d) Give a reason why.
Scatter Graphs

39. Q48. The ratio of the number of boys to
the number of girls in a school is 4:5
There are 95 girls in the school.
Work out the total number of students
in the school.
Q49. A supermarket car park has 200 spaces. 10% of
the spaces are for staff. The other spaces are for
disabled people, for parents and for other customers
in the ratio 1 : 2 : 7 Paul is going to paint a sign for
each of the spaces for staff, for disabled people and
for parents. He is not going to paint signs for the
spaces for other customers. Work out the total
number of spaces Paul is going to paint a sign for.
Ratio Problem Solving

40. Q50. Work out the size of angle KLM.
Give your answer correct to 3 significant
figures
Pythagoras’ and Trigonometry

41. Q51. All measurements are in centimetres.
The area of the triangle is 2.5 cm2
. Find the
perimeter of the triangle. Give your
answer correct to 3 significant figures.
You must show all of your working.
Pythagoras’ and Solve Quadratics –
Problem Solving

42. (a) Complete the table for the
values for y = 6 – x – x2
(b) On the grid, draw the graph of
y = 6 – x – x2
for values of x from
–4 to 3
(c) Find estimates for the solutions of
the equation 6 – x – x2
= 2
Approx -2.6 and 1.6
0
Quadratic Graphs
6 4 -6

43. Q53. The diagram shows a circle inside a square.
ABCD is a square of side 10 cm.
Each side of the square is a tangent to
the circle.
Work out the total area of the shaded
regions in terms of π.
Give your answer in its simplest form.
Area of Circle – Problem Solving

44. Q54. The diagram shows a sector of a circle of radius 4 cm.
Work out the length of the arc ABC.
Give your answer correct to 3 significant
figures.
Arc Length

45. Q55. (a) On the grid, construct the
graph of x2
+ y2
= 16
(b) Find estimates for the solutions of the
simultaneous equations
x2
+ y2
= 16
y = 2x + 1
Solving Simultaneous Equations Graphically

46. Q56. Here are the first five terms of
an arithmetic sequence.
4 9 14 19 24
(a) Find, in terms of n, an expression
for the nth term of this sequence.
Here are the first five terms of a different sequence.
2 2 0 −4 −10
An expression for the nth term of this sequence is 3n − n2
(b) Write down, in terms of n, an expression for the nth term
of a sequence whose first five terms are
4 4 0 −8 −20
Sequences – Nth Term Rule

47. Q57. Here are the first 5 terms of a
quadratic sequence.
1 3 7 13 21
Find an expression, in terms of n, for the
nth term of this quadratic sequence.
Quadratic Sequences

48. Q58. Dan, Harry and Regan sell cars.
Dan sells x cars.
Harry sells 5 more cars than Dan.
Regan sells twice as many cars as Dan.
Write an expression, in terms of x, for the
mean number of cars Dan, Harry and
Regan sell.
Form Expressions

49. Q59. Bob asked each of 40 friends how many
minutes they took to get to work.
The table shows some information about his results.
Work out an estimate for the mean time taken.
Estimating Mean for Group for Grouped Data
Time taken (m minutes) Frequency
0 < m ≤ 10 3
10 < m ≤ 20 8
20 < m ≤ 30 11
30 < m ≤ 40 9
40 < m ≤ 50 9

50. Q60. There are 15 children at a birthday
party.
The mean age of the 15 children is 7 years.
9 of the 15 children are boys.
The mean age of the boys is 5 years.
Work out the mean age of the girls.
Finding the Mean of a Sub-Group

51. Q61. In a company, the ratio of the number
of men to the number of women is 3 :2
40% of the men are under the age of 25
10% of the women are under the age of 25
What percentage of all the people in the
company are under the age of 25?
Ratio and Percentages – Problem Solving

52. Q62. There are some red counters and
some yellow counters in a bag.
There are 30 yellow counters in the bag.
The ratio of the number of red counters
to the number of yellow counters is 1:6
(a) Work out the number of red counters
in the bag.
Riza puts some more red counters into the bag.
The ratio of the number of red counters to the number of
yellow counters is now 1:2
(b) How many red counters does Riza put into the bag?
Ratio – Problem Solving

53. Q63. Kim, Laura and Molly share £385
The ratio of the amount of money Kim gets
to the amount of money Molly gets is 2 : 5
Kim gets £105 less than Molly gets.
What percentage of the £385 does Laura
get?
Ratio – Problem Solving

54. Ratio – Problem Solving
Q64. On a farm
the number of cows and the number of
sheep are in the ratio 6 : 5
the number of sheep and the number of
pigs are in the ratio 2 : 1
The total number of cows, sheep and pigs
on the farm is 189
How many sheep are there on the farm?

55. Trigonometry
Q65. Work out the value of

56. Derive a Formula for a Volume
Q66. The diagram shows a solid triangular prism.
All the measurements are in centimetres.
The volume of the prism is V cm3
.
Find a formula for V in terms of x.
Give your answer in simplified form.

57. Volume of a Frustum
Q67. A frustrum is made by removing a small cone from a
similar large cone.
The height of the small cone is 20 cm.
The height of the large cone is 40 cm.
The diameter of the base of the large cone is 30 cm.
Work out the volume of the frustrum.
Give your answer correct to 3 significant figures.
Hint:
Volume of a Cone =

58. Loci
Q68. Here is a scale drawing of an office.
The scale is 1 cm to 2 metres.
A photocopier is going to be put in the office.
The photocopier has to be closer to B than it is to A.
The photocopier also has to be less than 8 metres
from C.
Show, by shading, the region where the photocopier
can be put.