# GCSE Non-Calculator Questions 2 (2013/Higher Tier)

This worksheet contains a sample of practice questions based on the 2013 GCSE Higher Level syllabus. Calculators may not be used.

Key stage:  KS 4

Curriculum topic:  GCSE Practice Papers

Curriculum subtopic:  Selection of Topics for Non-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

Find the area of this trapezium in cm2.

All measurements are given in cm.

(Just write the number)

not drawn accurately

Work out:

 3 + 1 4 6

Simplify

p4 x p6

p46

2p10

p10

p24

Simplify

(p4)6

p10

p46

p24

p4096

Factorise

5a2b - 15a

Expand and simplify

(1 + √2)2 - (1 - √2)2

2√2

4

(2 + 2√2)²

4√2

The diagram shows a circle of radius 5 cm drawn inside a square.

(not drawn accurately)

Calculate the area of the part shaded in red.

¼(100 - 25π) cm²

25 - 25π cm²

75π cm²

¼(75π) cm²

An irregular hexagon is shown below.

Calculate the value of a.

(just write the number)

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

The marks were as follows:

26, 39, 18, 28, 51, 31, 31, 24, 31

Which box plot shows this information?

A

B

C

none of them

Solve for x:

15(x - 4) = 3(2x - 2)

(just write the value of x as a number)

Simplify

3p4q3 x 4p6q2

12p24q6

12p10q32

7p10q6

12p10q5

Evaluate as a decimal:

64-⅓

The following histogram shows the distribution of ages of people staying at a hotel.

Work out the frequency for the 40 ≤ a < 55 class interval.

Solve simultaneously:

4x - 3y = 10

2y + x = -14

x =4, y = 3

x = -2, y = -6

x = 2, y = -6

Write the following numbers in order of size.

Place the smallest at position 1 and the largest at position 6.

3.2 x 102

3.2 x 10-2

0.32 x 102

3.2 x 103

3.2

3.2 x 10-3

## Column B

1
3.2 × 103
2
3.2 × 102
3
3.2
4
0.32 × 102
5
3.2 × 10-2
6
3.2 × 10-3

Find the length of the line segment whose endpoints are at:

(15, 20) and (8, -4)

(Give your answer to 3 sig. figs. if it is not an integer)

P and Q are points on the circumference of a circle with centre C.

PB and QB are tangents to the circle.

∠ABC = 48°.

Calculate the size of ∠PCQ.

(just write the number)

Last year, the price of a printer was £120.

This year, the price of the same model of printer has decreased by 15%.

Calculate the price of the printer this year in £s.

(Just write the number)

Solve the quadratic equation by selecting all the correct solutions below.

10x2 = 12 - 7x

x = -1½

x = 1½

x = 4/5

x = 1¼

The first five terms of a number sequence are:

10      16      22      28      34 ...

What is the 99th term of this sequence?

Estimate the value of the following to 1 significant figure:

 1.993 - √25.1) 2.982

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

## Column B

A
y = x² + 5x + 6
B
y = x² + 1
C
y = x²

Calculate the volume of the cone shown below in cm3 leaving π in your answer

(not drawn accurately)

108π

12π

36π

Simplify the following and select the correct answer:

 3a2 + 5a - 2 a2 - 4

 1. 3a + 1 2. 3a - 1 3. 3a - 1 a + 2 a + 2 a - 2

1.

2.

3.

y is inversely proportional to the square of x.

When x = 4, y = 9.

Work out the positive value of x, when y = 4.

• Question 1

Find the area of this trapezium in cm2.

All measurements are given in cm.

(Just write the number)

not drawn accurately

44
EDDIE SAYS
Average of parallel sides = ½(3 + 8) = 5.5
Area = 8 × 5.5 = 44 cm²
• Question 2

Work out:

 3 + 1 4 6

11/12
EDDIE SAYS
Change to 12ths to get:
 9 + 2 12 12
• Question 3

Simplify

p4 x p6

p10
EDDIE SAYS
Add the indices to get p10
• Question 4

Simplify

(p4)6

p24
EDDIE SAYS
This is p4 × p4 × p4 × p4 × p4 × p4
• Question 5

Factorise

5a2b - 15a

5a(ab-3)
EDDIE SAYS
The common factor is 5a, so this will be the number outside the brackets. Then divide by 5a to get the terms inside the brackets.
• Question 6

Expand and simplify

(1 + √2)2 - (1 - √2)2

4√2
EDDIE SAYS
(1 + √2)² = (1 + √2)(1 + √2)
= 1 + √2 + √2 + 2 = 3 + 2√2
Similarly (1 - √2)² = 3 - 2√2
• Question 7

The diagram shows a circle of radius 5 cm drawn inside a square.

(not drawn accurately)

Calculate the area of the part shaded in red.

¼(100 - 25π) cm²
EDDIE SAYS
Square has side 10 cm and so its area is 100 cm²
Circle has area = πr² = 25π cm²
Area shaded = ¼(100 - 25π) cm²
• Question 8

An irregular hexagon is shown below.

Calculate the value of a.

(just write the number)

99
EDDIE SAYS
The interior angle sum = 4 × 180° = 720°
Five of the interior angles are 101°, 141°, 90°, (360 -200)°, (180-51)°
Sum of these five angles = 101° + 141° + 90° + 160° + 129° = 621°
a° = 720° - 621° = 99°
• Question 9

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

The marks were as follows:

26, 39, 18, 28, 51, 31, 31, 24, 31

Which box plot shows this information?

C
EDDIE SAYS
First place the marks in ascending order
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 = 35
Maximum mark = 51
• Question 10

Solve for x:

15(x - 4) = 3(2x - 2)

(just write the value of x as a number)

6
EDDIE SAYS
15(x - 4) = 3(2x - 2)
Multiply out brackets to get:
15x - 60 = 6x - 6
Add 60 to both sides to get:
15x = 6x + 54
Subtract 6x from both sides to get:
9x = 54
Divide both sides by 9 to get:
x = 6
• Question 11

Simplify

3p4q3 x 4p6q2

12p10q5
EDDIE SAYS
Add the indices of matching letters. Multiply the coefficients.
• Question 12

Evaluate as a decimal:

64-⅓

0.25
EDDIE SAYS
64 = cube root of 64 = 4
The negative index means that we must find the reciprocal of 4, which is 1 ÷ 4 = ¼ = 0.25
• Question 13

The following histogram shows the distribution of ages of people staying at a hotel.

Work out the frequency for the 40 ≤ a < 55 class interval.

18
EDDIE SAYS
Class width = 55 - 40 = 15
Frequency = Frequency Density × Class Width = 1.2 × 15 = 18
• Question 14

Solve simultaneously:

4x - 3y = 10

2y + x = -14

x = -2, y = -6
EDDIE SAYS
Rearrange the second equation to read:
x = -14 - 2y
Substitute this into the first equation:
4(-14 - 2y) - 3y = 10
Simplify:
-56 - 8y - 3y = 10
-11y = 66
y = -6
x = -14 - 2y = -14 + 12 = -2
• Question 15

Write the following numbers in order of size.

Place the smallest at position 1 and the largest at position 6.

3.2 x 102

3.2 x 10-2

0.32 x 102

3.2 x 103

3.2

3.2 x 10-3

## Column B

1
3.2 × 10-3
2
3.2 × 10-2
3
3.2
4
0.32 × 102
5
3.2 × 102
6
3.2 × 103
EDDIE SAYS
3.2 × 10-3 = 0.0032
3.2 × 10-2 = 0.032
0.32 × 102 = 32
3.2 × 102 = 320
3.2 × 103 = 3200
• Question 16

Find the length of the line segment whose endpoints are at:

(15, 20) and (8, -4)

(Give your answer to 3 sig. figs. if it is not an integer)

25
EDDIE SAYS
The x-part of the right-angled triangle is 15 - 8 = 7
The y-part of the right-angled triangle is 20 - -4 = 24
Pythagoras' Equation is d² = 7² + 24² = 625 (7,24,25 triad)
• Question 17

P and Q are points on the circumference of a circle with centre C.

PB and QB are tangents to the circle.

∠ABC = 48°.

Calculate the size of ∠PCQ.

(just write the number)

132
EDDIE SAYS
PCQB is a quadrilateral and so the sum of its interior angles is 360°
A tangent and a radius always meet at 90°
So ∠PCQ = 360° - 90° - 90° - 48° = 132°
• Question 18

Last year, the price of a printer was £120.

This year, the price of the same model of printer has decreased by 15%.

Calculate the price of the printer this year in £s.

(Just write the number)

102
EDDIE SAYS
120 × 0.85 = 102
• Question 19

Solve the quadratic equation by selecting all the correct solutions below.

10x2 = 12 - 7x

x = -1½
x = 4/5
EDDIE SAYS
(5x - 4)(2x + 3) = 0
• Question 20

The first five terms of a number sequence are:

10      16      22      28      34 ...

What is the 99th term of this sequence?

598
EDDIE SAYS
The gap between each term is 6, so the expression will have a 6n in it.
6n would give the sequence 6, 12, 18, 24, 30 etc.
The given sequence is four more than this, so the nth term is 6n + 4
With n = 99, we get 6 × 99 + 4 = 600 - 6 + 4 = 598
• Question 21

Estimate the value of the following to 1 significant figure:

 1.993 - √25.1) 2.982

0.3
EDDIE SAYS
1.99³ ≅ 2³ = 8
√25.1 ≅ √25 = 5
2.98² ≅ 3² = 9
So we have (8 - 5) ÷ 9 = 3 ÷ 9 = 0.33333....≅ 0.3
• Question 22

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

## Column B

A
y = x²
B
y = x² + 5x + 6
C
y = x² + 1
EDDIE SAYS
A is simply y = x²
B is y = x² + 5x + 6.
The line passes through (-3, 0) and (-2, 0), so y = (x + 2)(x + 3)
C is Graph A shifted up 1 unit, so y = x² + 1
• Question 23

Calculate the volume of the cone shown below in cm3 leaving π in your answer

(not drawn accurately)

12π
EDDIE SAYS
Volume = ⅓πr²h
Volume = ⅓ × π × 2² × 9 = (36 ÷ 3) π = 12π
• Question 24

Simplify the following and select the correct answer:

 3a2 + 5a - 2 a2 - 4

 1. 3a + 1 2. 3a - 1 3. 3a - 1 a + 2 a + 2 a - 2

3.
EDDIE SAYS
3a² + 5a - 2 factorises to (a + 2)(3a - 1)
a² - 4 factorises as the difference of two squares to (a + 2)(a - 2)
The (a + 2) terms cancel on the top and on the bottom.
• Question 25

y is inversely proportional to the square of x.

When x = 4, y = 9.

Work out the positive value of x, when y = 4.

6
EDDIE SAYS
y = k/x²
9 = k/4² so k = 9 × 16 = 144
So the formula is y = 144/x²
When y = 4, x² = 144/4 = 36
x = √36 = ±6
So the positive value is 6.
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