 GCSE Sample Test
Paper 2
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1 marks

Which of these numbers is one more than a multiple of 6 ?

15
19
26
30
[1 mark] ?
19 = 3 × 6 + 1
1 marks

Which of these numbers has exactly three factors?

23
24
25
26
[1 mark] ?
Factors are 1, 5, 25
1 marks

Which of these numbers is 6 less than −5.7?

−10.7
−11.7
−0.3
0.3
[1 mark] ?
−5.7 − 6 = −11.7
1 marks

Which shape is congruent to shape X ? [1 mark] ?
Rotate shape X by 90° clockwise to get shape D
2 marks

The map shows the positions of two ships, A and B. The two ships are 12 cm apart on the map.

Work out the actual distance between the ships.

km [2 marks] ?
12 cm on map becomes 12 × 1.5 km = 18 km actual distance
3 marks

A gym has 375 members.

40% have bronze membership.

28% have silver membership.

The rest have gold membership.

Work out the number with gold membership.

members have gold membership. [3 marks] ?
Percentage gold = 100 – (40 + 28) = 32% and 32% of 375 = × 375 = × 375 = 8 × 375 = 120
2 marks

(a)

Alan is looking at number machines. He says,

“If I know y then I can work out x.
I subtract 4 and then I add 6.”

Does this method work?

[1 mark] ?
𝑥 + 4 − 6 = 𝑦 ⇒ 𝑥 = 𝑦 − 4 + 6 hence method is correct
(b) Alan says,

“If I know d then I can work out c.
I divide by 2 and then subtract 3.”

Does this method work?

[1 mark] ?
𝑐 × 2 + 3 = 𝑑 ⇒ 𝑐 × 2 = 𝑑 − 3 ⇒ 𝑐 = (𝑑 − 3) ÷ 2 or “subtract 3 and divide by 2”. This is the reverse of the “Divide by 2 and subtract 3” so the method is incorrect. (Note: you could give an example to show it does not work, that is if 𝑐 = 4, then 𝑑 = 11 and the method gives 𝑑 = 11 then 𝑐 = 2.5 which is wrong!)
4 marks

(a)

Solve 7w − 4 = 24

w = [2 marks] ?
7𝑤 = 24 + 4 = 28 so 𝑤 = 28 ÷ 7 = 4
(b)

Write an expression for the total cost, in pounds, of

x shirts at £15 each

and

y belts at £8 each.

[1 mark] ?
This is 15 × 𝑥 + 8 × 𝑦 = 15𝑥 + 8𝑦
(c)

Simplify a + b × b + a + a

[1 mark] ?
Note that 𝑏 × 𝑏 = 𝑏2 so rearranging gives 𝑎 + 𝑎 + 𝑎 + 𝑏2 = 3𝑎 + 𝑏2
1 marks

Lucy says,

“3 is odd and 5 is odd,
so when you add a multiple of 3 to a multiple of 5 the answer is always even.”

Is she correct?

[1 mark] ?
Just give example to show this is incorrect; e.g. 3 × 7 + 5 × 4 = 21 + 20 = 41 and this is odd
3 marks

Tom earns £8.80 per hour.

He works for

22 hours each week

48 weeks each year.

He pays tax if he earns more then £10 000 per year.

(a)

How much does Tom earn in a year?

£ [2 marks] ?
22 × 48 × 8.80 = 9292.80
(b)

Does Tom pay tax?

[1 mark] ?
9292.80 is less than 10 000 so no tax to pay
3 marks

Three boxes contain counters. There are 72 counters in total.

The total number of counters in box A and box B is 33 and A contains more counters.

The difference between the number of counters in box A and box B is 9.

How many counters are in each box?

(i)

Box A:

(ii)

Box B:

(iii)

Box C:

[3 marks]
3 marks

The pie chart shows information about the sales of 1000 tickets.

There were four times as many adult ticket sales as senior ticket sales. How many senior ticket sales were there?

[3 marks] ?
Number of “Child” = x 1000 = 450 so number of “Senior + Adult” = 550 and of these are “Senior” so × 550 = 110
3 marks

Alice makes cards.

Each card used 32 cm of ribbon.

She has 1000 cm of ribbon.

(a)

Work out the maximum number of cards she can make.

[2 marks] ?
1000 ÷ 32 = 31 remainder 8 so 31 cards can be made
(b)

How much ribbon will be left over?

cm [1 mark] ?
Remainder is 8 cm
4 marks

Luke saves 1p coins and 2p coins.

He has

five times as many 1p coins as 2p coins a total of £14

How many 1p coins does he have?

[4 marks] ?
If x = no. of 1p coins and y = no. of 2p coins, then x = 5y and 14000 = x + 2y = 7y and hence 7y = 14000 and y = 200 and x = 5y = 5 × 200 = 1000
7 marks

A company has bikes for hire.

The cost, £C, to hire a bike for n days is given by the formula

C = 15 + (n − 1)

(a)

Write down the cost to hire a bike for 1 day.

£ [1 mark] ?
𝐶 = 15 + × (1 − 1) = 15 + 0 = 15
(b)

Special offer!

Hire a bike for £9 per day
(i)

How much does it cost to hire a bike for 7 days with this special offer.

£
(ii)

How much does it cost using the first formula?

£
(iii)

Which is cheaper?

?
7 × £9 = £63; 𝐶 = 15 + × (7 − 1) = 15 + × 6 = 15 + = 15 + 40.5 = £55.50; £55.50 < £63
[3 marks]
(c)

The graph shows the cost to hire a bike for one to five days at a different company. The cost, £C, to hire a bike for n days from this company is given by the formula

𝐶 = 𝑎 + 𝑏(𝑛 − 1)

Work out the values of a and b.

(i)
a = [1 marks]
(ii)
b = [2 marks] ?
For a graph of a straight line: 𝑦 = mx + c, m is the gradient and c is the intercept on the y-axis; so here,
𝐶 = 𝑎 + 𝑏(𝑛 − 1) = (𝑎 − 𝑏) + 𝑏𝑛 and the gradient, b = 5 and the intercept, (𝑎 − 𝑏) = 12; hence 𝑏 = 5 and 𝑎 = 17
2 marks

Here is the outline of a company’s logo. (a)

How many lines of symmetry does this have?

[1 mark] ?
The vertical line through the top vertex is the one line of symmetry
(b)

What is the name of this shape?

[1 mark] ?
This is a pentagon as it has 5 sides
5 marks

Mr Jones works for five days each week.

If he uses his car to travel to work, each day he drives a total distance of 27 miles his car travels 32 miles per gallon of petrol petrol costs £1.31 per litre.

If he uses the bus to travel to work, he can buy a weekly ticket for £22.00.

Use 1 gallon ≈ 4.5 litres.

(a)

What is the weekly cost if he uses car?

£ (to 2 d.p.) [4 marks]
(b)

Is it cheaper if he uses his car or the bus to travel to work?

[1 marks] ?
For car, miles = 5 × 27 = 135 each week so gallons used = or × 4.5 litres and costs £ × 4.5 × 1.31 = £24.87 to two decimal places or £25 to the nearest £; so bus at £22 per week is the cheaper option
4 marks

Here are two number machines, A and B. Both machines have the same input.

Work out the input that makes the output of A two times the output of B.

[4 marks] ?
For input x the output of A = 4𝑥 + 4 and for B is 6𝑥 − 2 ; hence 4𝑥 + 4 = 2(6𝑥 − 2) = 12𝑥 − 4 This gives 8 = 8𝑥 so that 𝑥 = 1
2 marks

Josef runs 400 metres in 1 minute.

He says,

“I would run 20 000 metres in 50 minutes.”

Choose a box to show whether his time to run 20 000 metres is likely to be accurate.

No, the time will be longer
Yes, the time will be 50 minutes
No, the time will be shorter
[2 marks] ?
The time will be longer as Josef could not keep the same pace (over 400 m) to 20 000 metres
1 marks

Which sequence is a geometric progression?

1 2 3 4
1 8 27 64
1 2 4 8
1 4 9 16
[1 mark] ?
1, 2, 4, 8 is a geometric progression as the common ratio is 2, i.e. 1 × 2 = 2, 2 × 2 = 4, 4 × 2 = 8
3 marks

This pyramid has a square base. Volume of a pyramid = × area of base × perpendicular height.

Work out the volume of the pyramid.

cm3 (to the nearest cm3) [3 marks] ?
Volume = × 11.5 × 11.5 × 16.8 = 740.6 cm3 to one decimal place, giving 741 as nearest cm3
4 marks

ξ = {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}

S = square numbers

E = even numbers

(a)

Complete a Venn diagram, as shown below. (i)

What numbers are in S ?

{}
(ii)

What numbers are in E ?

{}
(iii)

What numbers are in S ∩ E ?

{} ?
S is the set of ODD numbers, that is 16 = 42 and 25 = 52;
E is the set of EVEN numbers, that is 16, 18, 20, 22, 24 and the intersection of S and E gives only one member, that is 16
[3 marks]
(b)

One of the numbers is chosen at random.

Write down P(S ∩ E)

[1 mark] ?
As there is only one member of S∩E and 11 members of the universal set 𝜉, the probability =
4 marks

A coin is rolled onto a grid of squares.
It lands randomly on the grid.
To win, the coin must land completely within one of the squares.
Meera and John each roll the coin a number of times and record their results.

 Number of wins Number of losses Meera 7 45 John 27 73
(a)
(i)

What is Meera’s estimate of the probability that she will win? Give your answer correct to 2 d.p.

(ii)

What is John’s estimate of the probability that he will win? Give  your answer correct to 2 d.p.

?
Meera’s estimate is = 0.13 to 2 decimal places and John’s is = 0.27
[2 marks]
(b)

Combine the data to give an improved estimate of winning.

[2 marks] ?
A better estimate can be obtained by combining data to give = = 0.22 to 2 decimal places
3 marks

In a sale, the original price of a pair of shoes was reduced by .
The sale price of the shoes is £21.00.
What was the original price?

£ [3 marks] ?
If 𝑝 = price then 𝑝 × = £21 and 𝑝 = £21 × = £28 (Check: discount = = = )
1 marks

Which of these is not used to prove that triangles are congruent? SSS
SAS
SSA
ASA
[1 mark] ?
SSA (side, side, angle) is the only one that does not uniquely define a triangle - see attached diagram
marks

E is the centre of rectangle ABCD. Work out the length DE.

cm (to 1 d.p.) [3 marks] ?
DB2 = 152 + 252 = 225 + 625 = 850 giving DB ≈ 29.155 and DE = DB ≈ 14.578 = 14.6 to 1 decimal place
marks

Which of these is the equation of a line parallel to y = 3x + 2?

𝑦 = 2𝑥 − 5
𝑦 = 5𝑥 + 2
𝑦 = 3𝑥 − 2
𝑦 = 𝑥 − 2
[1 mark] ?
Lines are PARALLEL if they have the same gradient; the gradient is the coefficient of 𝑥 so only 𝑦 = 3𝑥 − 2 has 3 as its gradient
marks

At a school

number of boys : number of girls = 8 : 7

There are 96 more boys than girls.

Work out the total number of students at the school.

[3 marks] ?
are boys and are girls so the difference is = 96; hence total = 15 x 96 = 1440
marks

Which of these equations has roots 2 and −3?

(𝑥 + 2)(𝑥 + 3) = 0
(𝑥 − 2)(𝑥 − 3) = 0
(𝑥 − 2)(𝑥 + 3) = 0
(𝑥 + 2)(𝑥 − 3) = 0
[1 mark] ?
Roots of (𝑥 − 2)(𝑥 + 3) = 0 are either (𝑥 − 2) = 0 or (𝑥 + 3) = 0; that is 𝑥 = 2 or 𝑥 = −3
marks

A pattern is made from two similar trapeziums. Find the area of the shaded part.

cm2 [1 mark] ?
Area of large trapezium = = 260 cm2 and area of smaller trapezium is this, as both length and height are halved. So this is 65 cm2 so area of shaded part = 260 – 65 = 195 cm2
Test check
Total: - of marks