GCSE Calculator Questions 3 (2013/Higher Tier)

This worksheet contains a sample of practice questions based on the 2013 GCSE Higher Level syllabus. Calculators may be used.

Key stage:  KS 4

Curriculum topic:  GCSE Practice Papers

Curriculum subtopic:  Selection of Topics for Calculator Practice

Difficulty level:

QUESTION 1 of 10

Formula Sheet

Higher Tier

Area of trapezium = ½ (a + b) h

Volume of prism = area of cross-section x length

In any triangle ABC:

Area of triangle = ½ab sinC

Sine rule

 a = b = c sin A sin B sin C

Cosine rule

a2 = b2 + c2 - 2bc cos A

The solutions of ax2 + bx + c = 0, where a ≠ 0, are given by

Sam is decorating a room.

3 tins of white emulsion paint at £12.50 per tin.

1 roller at £4.50.

4 paintbrushes at £3.99 each.

2 tins of blue gloss paint at £8.99 per tin.

VAT at 20% must be added to all prices.

Calculate the total cost of Sam's purchases.

What is the smallest integer that satisfies this inequality:

17 - 9x < 6(2 - x)

Calculate the surface area of this cylinder in cm2 to 3 sig. figs.

(Just write the number to 3 sig. figs)

Write the following fraction as a percentage:

 11 40

Write the following as an ordinary number:

4.23 x 10-3

Calculate the value of the following to 2 significant figures:

 √15.1 + 5.14 0.15

y is directly proportional to x.

When x = 300, y = 24.

Calculate the value of y when x = 250.

Factorise fully:

64x2 - 96x - 64

(64x + 32)(x - 2)

(2x + 1)(32x - 64)

32(2x + 1)(x - 2)

4(16x + 1)(x - 2)

Solve simultaneously:

4x + 3y = 22

3x - 4y = 4

x = -4, y = 2

x = 4, y = -2

x = -4, y = -2

x = 4, y = 2

Solve for a:

 a + 4 = a - 3 4 8

(Just write down the numerical value of a)

The diagram shows two fair spinners.

Both spinners are spun and the product of the scores is found.

Calculate the probability that the product of the scores is odd.

Here is a formula:

v = u + at

Calculate t when u = 6.2 x 102, v = 1255 and a = 9.8

Use trial and improvement to find a solution to the equation:

2x3 - 7x = 83

The first step is shown below.

Give the solution to 1 decimal place

x  2x3 - 7x  Comment
4  2 × 43 - 7×4 = 128 - 28 = 100  too large

The front face of a V-shaped tent is shown in the diagram below.

Find the height from the ground to the top of the tent in metres to 3 s.f.

(Just write the number)

A hardware shop sells cement mix is sold in three sizes.

Which size of box represents the best value for money?

2 kg

3 kg

7 kg

The mean height of 15 girls in a class is 155 cm and the mean height of 12 boys is 152 cm.

Calculate the mean height in cm of all the children in the class to 1 decimal place.

(Just write the number)

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

The marks were as follows:

26, 41, 18, 28, 51, 31, 31, 24, 31

Which box plot shows this information?

A

B

C

none of them

A string of lights is to be suspended from the tops of two trees as shown in the diagram below.

Calculate the least length the string should be in metres to the nearest cm.

(Just write the number)

Find the positive solution to:

3x2 + 4x = 8

Calculate the area of this triangle to 3 sig. figs.

(Diagram not drawn accurately)

In △ABC above,

c = 12 cm

a = 9.5 cm

∠ABC = 82º

Calculate the value of b to 3 s.f.

(Just write the number)

A semi-circle is shown in the diagram below.

Calculate its perimeter in cm to 1 decimal place.

(Just write the number)

Look at this number pattern:

92 = 81

992 = 9801

9992 = 998001

99992 = 99980001

999992 = 9999800001

Write down the value of:

99999992

Triangle ABC has been cut by the line DE to form a smaller triangle and a trapezium.

Calculate the area of the trapezium EDBC in cm2 to 3 significant figures.

(Just write the number)

A table is 142.6 cm long.

What is the greatest possible error?

5 mm

0.5 mm

0.5 cm

5 cm

50 mm

• Question 1

Sam is decorating a room.

3 tins of white emulsion paint at £12.50 per tin.

1 roller at £4.50.

4 paintbrushes at £3.99 each.

2 tins of blue gloss paint at £8.99 per tin.

VAT at 20% must be added to all prices.

Calculate the total cost of Sam's purchases.

£91.13
£91.13
EDDIE SAYS
Total = 3 × £12.50 + £4.50 + 4 × £3.99 + 2 × £8.99 = £75.94
Add 20% VAT to get 1.2 × £75.94 = £91.128 which rounds up to £91.13
• Question 2

What is the smallest integer that satisfies this inequality:

17 - 9x < 6(2 - x)

2
EDDIE SAYS
17 - 9x < 12 - 6x
-3x + 17 < 12
-3x < -5
3x > 5
x > 5/3 = 1.6666...
Smallest integer greater than 1.6666..... is therefore 2
• Question 3

Calculate the surface area of this cylinder in cm2 to 3 sig. figs.

(Just write the number to 3 sig. figs)

363
EDDIE SAYS
Diameter d = 11cm
Circumference = π × d = π × 11 = 34.5575 cm
Area of side = 34.5575 × 5 = 172.7875..cm²
Area of circular top and bottom = 2 × π × r² = 2 × π × 5.5² = 190.066.. cm²
Total surface area = 190.066 cm² + 172.7875 = 362.85.. cm²
• Question 4

Write the following fraction as a percentage:

 11 40

27.5%
EDDIE SAYS
11 ÷ 40 = 0.275 = 27.5%
• Question 5

Write the following as an ordinary number:

4.23 x 10-3

0.00423
EDDIE SAYS
4.23 ÷ 1000 = 0.00423
• Question 6

Calculate the value of the following to 2 significant figures:

 √15.1 + 5.14 0.15
4500
EDDIE SAYS
On the calculator, type:
(√15.1 + 5.1^4) ÷ 0.15 =
4536.0398... rounds to 4500 to 2 significant figures.
• Question 7

y is directly proportional to x.

When x = 300, y = 24.

Calculate the value of y when x = 250.

20
EDDIE SAYS
y = kx
24 = 300k so k = 24 ÷ 300 = 0.08
Formula is therefore y = 0.08x
When x = 250, y = 0.08 × 250 = 20
• Question 8

Factorise fully:

64x2 - 96x - 64

32(2x + 1)(x - 2)
EDDIE SAYS
This divides by 32 to get 2x² - 3x - 2
Full factorisation is 32(2x + 1)(x - 2)
• Question 9

Solve simultaneously:

4x + 3y = 22

3x - 4y = 4

x = 4, y = 2
EDDIE SAYS
Multiply first equation by 3 and second equation by 4 to get:
12x + 9y = 66
12x - 16y = 16
Subtract to get:
25y = 50
y = 2
Substitute back into either equation to find x.
• Question 10

Solve for a:

 a + 4 = a - 3 4 8

(Just write down the numerical value of a)

-11
EDDIE SAYS
Multiply both fractions by 8 to get:
2a + 8 = a - 3
a = -11
• Question 11

The diagram shows two fair spinners.

Both spinners are spun and the product of the scores is found.

Calculate the probability that the product of the scores is odd.

1/4
EDDIE SAYS
For the product to be odd, we must have two odd numbers to start with. If one or both is even, then the product must be even.
Prob of getting an odd number on first spinner is ½
Prob of getting an odd number on second spinner is also ½
Probability required = ½ × ½ = ¼
• Question 12

Here is a formula:

v = u + at

Calculate t when u = 6.2 x 102, v = 1255 and a = 9.8

64.8
EDDIE SAYS
t = (v - u)/a = (1255 - 620) ÷ 9.8 = 64.7959...
• Question 13

Use trial and improvement to find a solution to the equation:

2x3 - 7x = 83

The first step is shown below.

Give the solution to 1 decimal place

x  2x3 - 7x  Comment
4  2 × 43 - 7×4 = 128 - 28 = 100  too large

3.8
EDDIE SAYS
2 × 3.5³ - 7×3.5 = 61.25 too small
2 × 3.8³ - 7×3.8 = 83.144 too large
2 × 3.7³ - 7×3.7 = 75.406 too small
2 × 3.75³ - 7×3.75 = 79.21875 too small
So answer lies between 3.75 and 3.8, which will give a solution of 3.8 to 1 decimal place.
• Question 14

The front face of a V-shaped tent is shown in the diagram below.

Find the height from the ground to the top of the tent in metres to 3 s.f.

(Just write the number)

1.98
EDDIE SAYS
Divide the triangle into two as shown, to form a right-angled triangle with top angle 29° and opposite side of 1.1 m.
Height, h = 1.1/tan29º
• Question 15

A hardware shop sells cement mix is sold in three sizes.

Which size of box represents the best value for money?

7 kg
EDDIE SAYS
The 2 kg box costs £3.50 ÷ 2 = £1.75/kg
The 3 kg box costs £4.50 ÷ 3 = £1.50/kg
The 7 kg box costs £10.00 ÷ 7 = £1.43/kg
• Question 16

The mean height of 15 girls in a class is 155 cm and the mean height of 12 boys is 152 cm.

Calculate the mean height in cm of all the children in the class to 1 decimal place.

(Just write the number)

153.7
EDDIE SAYS
Total of girls' heights = 15 × 155 = 2325 cm.
Total of boys' heights = 12 × 152 = 1824 cm.
Total of all heights = 2325 + 1824 = 4149 cm.
Mean height of 27 children = 4149 ÷ 27 = 153.6666...cm.
• Question 17

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

The marks were as follows:

26, 41, 18, 28, 51, 31, 31, 24, 31

Which box plot shows this information?

none of them
EDDIE SAYS
First place the marks in ascending order
18, 24, 26, 28, 31, 31, 31, 41, 51.

Remember that to find the median of an even number of numbers, we must find the mean of the two middle numbers.
Minimum mark = 18.
Lower quartile = 25
Median = 31
Upper quartile = 36
Maximum mark = 51
No box plot shows an upper quartile of 36
• Question 18

A string of lights is to be suspended from the tops of two trees as shown in the diagram below.

Calculate the least length the string should be in metres to the nearest cm.

(Just write the number)

20.22
EDDIE SAYS
We can form a right-angled triangle with the string of lights as the hypotenuse, h. The triangle has a base of 20 m and a height of (10 - 7) = 3 m.
Using Pythagoras' Theorem h² = 3² + 20² = 9 + 400 = 409
h = √409 = 20.2237... m
• Question 19

Find the positive solution to:

3x2 + 4x = 8

1.10
EDDIE SAYS
Rearrange to read 3x² + 4x - 7 = 0
Use the quadratic formula with a = 3, b = 4, c = -8 to get:
x = (-4 ± √(16 + 96)) ÷ 6
Positive solution will be
x = (-4 + √(16 + 96)) ÷ 6 = 1.097....
• Question 20

Calculate the area of this triangle to 3 sig. figs.

(Diagram not drawn accurately)

26.7
EDDIE SAYS
x = 6 ÷ tan34° = 8.89536...
Area = ½ 8.89536 × 6 = 26.686... units²
• Question 21

In △ABC above,

c = 12 cm

a = 9.5 cm

∠ABC = 82º

Calculate the value of b to 3 s.f.

(Just write the number)

14.2
EDDIE SAYS
Using the cos rule, we get:
b² = 9.5² + 12² - 2 × 9.5 × 12cos82° = 202.5185...
b = √202.5185 = 14.2309.... cm
• Question 22

A semi-circle is shown in the diagram below.

Calculate its perimeter in cm to 1 decimal place.

(Just write the number)

38.6
EDDIE SAYS
Cicumference of whole circle = π ×15 = 47.12 cm
Perimeter of curved part = 47.12 ÷ 2 = 23.56 cm
Perimeter including diameter = 23.56 + 15 = 38.56 cm
• Question 23

Look at this number pattern:

92 = 81

992 = 9801

9992 = 998001

99992 = 99980001

999992 = 9999800001

Write down the value of:

99999992

99999980000001
EDDIE SAYS
Following the pattern there will be six 9s, an 8, six 0s and a 1.
• Question 24

Triangle ABC has been cut by the line DE to form a smaller triangle and a trapezium.

Calculate the area of the trapezium EDBC in cm2 to 3 significant figures.

(Just write the number)

43.7
EDDIE SAYS
So AB = 12 × 10 ÷ 7 = 17.1428...cm
So DB = 17.1428 - 12 = 5.1428 cm
Area of trapezium = ½(7 + 10) × 5.1428 = 43.714 cm²
• Question 25

A table is 142.6 cm long.

What is the greatest possible error?