Higher Tier - Year 10

Year 10 Paper 6: Full Year 10 Review

Covers number and calculation, fractions, decimals and percentages, indices, surds and standard form, ratio and proportion, linear algebra, sequences and graphs, geometry, and statistics and probability.

22 questions - 100 marks - calculator allowed

Year 10 here means a typical teaching order, not a syllabus rule. No exam board defines what belongs to Year 10, and schools sequence the course differently. Check it against your own scheme of work before using it to decide what a class has covered.

Download printable PDF

Questions

Question 1 [3 marks]

Angles, Polygons and Circle Theorems

A regular nonagon (9-sided polygon) is drawn.

Find the sum of its interior angles, and find the size of one interior angle.

Question 2 [3 marks]

Indices, Surds and Standard Form

A red blood cell has diameter 0.0000075 m. Write this number in standard form.

Question 3 [3 marks]

Ratio and Proportion

The cost of petrol is directly proportional to the number of litres bought.

35 litres costs 54.25 pounds. Find the cost of 52 litres.

Question 4 [4 marks]

Statistics and Probability

A bag contains 4 red counters and 6 blue counters.

A counter is drawn at random, its colour noted, and then replaced. A second counter is then drawn at random.

Find the probability that both counters are the same colour.

Question 5 [3 marks]

Indices, Surds and Standard Form

Simplify (x^7 y^3)/(x^3 y^-2).

Give your answer using positive indices.

Question 6 [4 marks]

Number and Calculation

The HCF of two numbers is 12. The LCM of the same two numbers is 360.

One of the numbers is 72. Find the other number.

Question 7 [4 marks]

Fractions, Decimals and Percentages

Ryan mixes 3.6 litres of orange squash.

He pours 40% of it into a jug for a party, then shares 1/4 of what remains between two bottles.

How many millilitres of squash are left in the original container?

Question 8 [3 marks]

Indices, Surds and Standard Form

Simplify sqrt(8) x sqrt(18), giving your answer as an integer.

Question 9 [5 marks]

Fractions, Decimals and Percentages

A roll of ribbon is 5.4 m long.

Priya cuts off 1/3 of the ribbon for a project, then cuts 2/5 of what remains for a second project.

How many centimetres of ribbon are left?

Question 10 [5 marks]

Linear Equations and Inequalities

n is an integer.

-3 < 2n - 5 <= 7.

Find all possible values of n.

Question 11 [5 marks]

Sequences and Graphs

A geometric sequence has first term 40 and common ratio 1/2.

Find the 5th term of the sequence, and find the sum to infinity of the sequence.

Question 12 [4 marks]

Angles, Polygons and Circle Theorems

In triangle ABC, D lies on AB and E lies on AC, such that DE is parallel to BC.

AD = 6 cm, DB = 4 cm and DE = 9 cm. Find the length of BC.

Question 13 [4 marks]

Fractions, Decimals and Percentages

A shop increases the price of a jacket by 12%, then later reduces the new price by 12%.

The final price is 98.56 pounds. Find the original price.

Question 14 [5 marks]

Sequences and Graphs

In a geometric sequence, the 2nd term is 12 and the 5th term is 324.

Find the first term and the common ratio.

Question 15 [5 marks]

Number and Calculation

The recurring decimal 0.458458458... means the block 458 repeats forever.

Convert 0.458458458... to a fraction in its simplest form.

Question 16 [5 marks]

Fractions, Decimals and Percentages

A bank account pays simple interest at a rate of 3.2% per year.

2400 pounds is invested for 4 years.

Find the total simple interest earned, and find the value of the investment at the end of the 4 years.

Question 17 [5 marks]

Indices, Surds and Standard Form

Solve 9^x = 1/3, giving x as a fraction.

Question 18 [6 marks]

Ratio and Proportion

y is directly proportional to x and inversely proportional to z.

When x = 8 and z = 4, y = 20. Find y when x = 15 and z = 6.

Question 19 [6 marks]

Fractions, Decimals and Percentages

Account A pays 4.8% simple interest per year. Account B pays 4.5% compound interest per year.

For an investment of 3000 pounds over 3 years, determine which account gives the higher return,

showing the value from each account.

Question 20 [6 marks]

Number and Calculation

A = 3.2 x 10^-3 and B = 8 x 10^5.

Work out A x B and B / A, giving both answers in standard form.

Question 21 [6 marks]

Fractions, Decimals and Percentages

An investment of 5000 pounds grows to 5788.13 pounds after 3 years of compound interest.

Find the annual rate of interest, to 1 decimal place.

Question 22 [6 marks]

Statistics and Probability

A box contains 10 tickets, of which 4 are winning tickets.

Two tickets are drawn at random without replacement.

Find the probability that at least one of the two tickets is a winning ticket.

Model solutions

Mark scheme for Question 1 [3 marks]
Question 1[3 marks]
Answer or working	Marks
using the sum of interior angles = (n - 2) x 180	M1
(9 - 2) x 180 = 1260 degrees	M1
one interior angle = 1260 / 9 = 140 degrees	A1
Final answer: sum = 1260 degrees, one interior angle = 140 degrees

Mark scheme for Question 2 [3 marks]
Question 2[3 marks]
Answer or working	Marks
identifying the first significant figures as 7.5	M1
counting the place value shift to give power -6	M1
7.5 x 10^-6	A1
Final answer: 7.5 x 10^-6 m

Mark scheme for Question 3 [3 marks]
Question 3[3 marks]
Answer or working	Marks
finding the cost per litre, 54.25 / 35 = 1.55	M1
multiplying by 52	M1
80.60 pounds	A1

Mark scheme for Question 4 [4 marks]
Question 4[4 marks]
Answer or working	Marks
P(red) = 4/10 and P(blue) = 6/10, unchanged after replacement	M1
P(both red) = 4/10 x 4/10 = 16/100	M1
P(both blue) = 6/10 x 6/10 = 36/100	M1
52/100, or 13/25	A1
Final answer: 13/25

Mark scheme for Question 5 [3 marks]
Question 5[3 marks]
Answer or working	Marks
subtracting powers of x to get x^4	M1
subtracting powers of y to get y^5	M1
x^4 y^5	A1

Mark scheme for Question 6 [4 marks]
Question 6[4 marks]
Answer or working	Marks
using HCF x LCM = product of the two numbers	M1
12 x 360 = 4320	M1
dividing 4320 by 72	M1
60	A1

Mark scheme for Question 7 [4 marks]
Question 7[4 marks]
Answer or working	Marks
60% of 3.6 = 2.16 litres remaining after the jug is filled	M1
finding 1/4 of 2.16	M1
subtracting to leave 3/4 of 2.16	M1
1620 ml	A1

Mark scheme for Question 8 [3 marks]
Question 8[3 marks]
Answer or working	Marks
multiplying to get sqrt(8 x 18) = sqrt(144)	M1
evaluating 8 x 18 = 144	M1
12	A1

Mark scheme for Question 9 [5 marks]
Question 9[5 marks]
Answer or working	Marks
finding 2/3 of 5.4 = 3.6 m remaining after the first cut	M1
finding 2/5 of 3.6 = 1.44 m	M1
subtracting to leave 3.6 - 1.44 = 2.16 m	M1
converting 2.16 m to centimetres	M1
216 cm	A1

Mark scheme for Question 10 [5 marks]
Question 10[5 marks]
Answer or working	Marks
adding 5 to all three parts of the inequality	M1
-3 + 5 < 2n <= 7 + 5, giving 2 < 2n <= 12	M1
dividing all three parts by 2	M1
1 < n <= 6	A1
n = 2, 3, 4, 5 or 6	A1

Mark scheme for Question 11 [5 marks]
Question 11[5 marks]
Answer or working	Marks
using the nth term formula a r^(n-1)	M1
5th term = 40 x (1/2)^4	M1
2.5	A1
using the sum to infinity formula a/(1 - r)	M1
sum to infinity = 80	A1
Final answer: 5th term = 2.5, sum to infinity = 80

Mark scheme for Question 12 [4 marks]
Question 12[4 marks]
Answer or working	Marks
recognising triangle ADE is similar to triangle ABC	M1
finding AB = AD + DB = 10 cm	M1
forming the ratio AD/AB = DE/BC, giving 6/10 = 9/BC	M1
BC = 15 cm	A1
Final answer: 15 cm

Mark scheme for Question 13 [4 marks]
Question 13[4 marks]
Answer or working	Marks
the multiplier for a 12% increase, 1.12	M1
the multiplier for a 12% decrease, 0.88	M1
the combined multiplier 1.12 x 0.88 = 0.9856	M1
original price = 100 pounds	A1
Final answer: 100 pounds

Mark scheme for Question 14 [5 marks]
Question 14[5 marks]
Answer or working	Marks
forming ar = 12 and ar^4 = 324	M1
dividing to eliminate a: r^3 = 27	M1
r = 3	A1
substituting back to find a = 12/3	M1
a = 4	A1
Final answer: first term = 4, common ratio = 3

Mark scheme for Question 15 [5 marks]
Question 15[5 marks]
Answer or working	Marks
setting x = 0.458458458...	M1
1000x = 458.458458...	M1
subtracting to get 999x = 458	M1
x = 458/999	A1
confirming 458/999 is already in its simplest form	A1
Final answer: 458/999

Mark scheme for Question 16 [5 marks]
Question 16[5 marks]
Answer or working	Marks
finding 3.2% of 2400	M1
3.2% of 2400 = 76.8	M1
multiplying by 4 years	M1
total interest = 307.20 pounds	A1
final value = 2707.20 pounds	A1
Final answer: interest = 307.20 pounds, final value = 2707.20 pounds

Mark scheme for Question 17 [5 marks]
Question 17[5 marks]
Answer or working	Marks
writing 9 as 3^2	M1
writing 1/3 as 3^-1	M1
forming the equation 3^(2x) = 3^-1	M1
equating powers to get 2x = -1	M1
x = -1/2	A1

Mark scheme for Question 18 [6 marks]
Question 18[6 marks]
Answer or working	Marks
y = kx/z	M1
20 = 8k/4, so 20 = 2k	M1
k = 10	A1
substituting x = 15 and z = 6 into y = 10x/z	M1
y = 10 x 15 / 6	M1
25	A1

Mark scheme for Question 19 [6 marks]
Question 19[6 marks]
Answer or working	Marks
simple interest per year = 3000 x 0.048 = 144	M1
total simple interest over 3 years = 432, giving Account A = 3432 pounds	M1
compound multiplier 1.045^3 = 1.141166...	M1
Account B = 3000 x 1.141166...	M1
Account B = 3423.50 pounds (nearest penny)	A1
Account A gives the higher return, by 8.50 pounds	A1
Final answer: Account A gives the higher return: 3432 pounds vs 3423.50 pounds

Mark scheme for Question 20 [6 marks]
Question 20[6 marks]
Answer or working	Marks
3.2 x 8 = 25.6	M1
10^-3 x 10^5 = 10^2	M1
A x B = 2.56 x 10^3	A1
8 / 3.2 = 2.5	M1
10^5 / 10^-3 = 10^8	M1
B / A = 2.5 x 10^8	A1
Final answer: A x B = 2.56 x 10^3, B / A = 2.5 x 10^8

Mark scheme for Question 21 [6 marks]
Question 21[6 marks]
Answer or working	Marks
forming 5000 x r^3 = 5788.13, where r is the multiplier	M1
r^3 = 5788.13 / 5000 = 1.157626	M1
taking the cube root of both sides	M1
r = 1.05 (to 3 s.f.)	M1
identifying the multiplier corresponds to a 5% increase	A1
5.0%	A1

Mark scheme for Question 22 [6 marks]
Question 22[6 marks]
Answer or working	Marks
recognising the complement: P(at least one winner) = 1 - P(no winners)	M1
P(first not a winner) = 6/10	M1
P(second not a winner given first not a winner) = 5/9	M1
P(no winners) = 6/10 x 5/9 = 1/3	M1
1 - 1/3	A1
2/3	A1