Maths: Full Practice Paper (Additional, Level 3) - Worksheets, Questions and Revision

22 original exam-style questions - 6 pages of questions with a full mark scheme - free printable PDF.

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13+ · Mathematics

CE.M14 Maths: Full Practice Paper (Additional, Level 3)

ISEB COMMON ENTRANCE CE.M14 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Time allowed: 60 minutes. Answer ALL 22 questions. Write your answers in the spaces provided. You ARE allowed to use a calculator. You must show all your working; marks are available for a correct method as well as for the final answer. The number of marks for each question, or part-question, is shown in brackets. This paper covers stretch and scholarship-track topics from the Additional (Level 3) syllabus: index laws, expanding pairs of brackets, factorising, solving simultaneous equations by substitution, Pythagoras' theorem, right-angled trigonometry, compound measures, and harder probability.

Write 5² * 5³ as a single power of 5. (1 mark)

(Total for Question 1 is 1 mark)

Work out the value of 7⁰ + 4². (1 mark)

(Total for Question 2 is 1 mark)

Simplify m⁷ / m³, giving your answer as a single power of m. (1 mark)

(Total for Question 3 is 1 mark)

Simplify (x³)² * x², giving your answer as a single power of x. (2 marks)

(Total for Question 4 is 2 marks)

Expand and simplify (x + 3)(x - 5). (2 marks)

(Total for Question 5 is 2 marks)

Expand and simplify (2x - 1)(3x + 4). (3 marks)

(Total for Question 6 is 3 marks)

Factorise x² + 7x + 10. (2 marks)

(Total for Question 7 is 2 marks)

Factorise x² - 9. (2 marks)

(Total for Question 8 is 2 marks)

Solve the simultaneous equations using substitution. y = 2x + 1 and x + y = 10 (3 marks)

(Total for Question 9 is 3 marks)

Solve the simultaneous equations using substitution. y = 3x + 2 and 2x + y = 17 (4 marks)

(Total for Question 10 is 4 marks)

A right-angled triangle has two shorter sides of length 5 cm and 12 cm. Calculate the length of the hypotenuse. (3 marks)

(Total for Question 11 is 3 marks)

A right-angled triangle has a hypotenuse of length 17 cm and one shorter side of length 8 cm. Calculate the length of the other shorter side. (3 marks)

(Total for Question 12 is 3 marks)

Jamal leans a ladder of length 6.5 m against a vertical wall. The foot of the ladder is 2.5 m from the base of the wall on level ground. Calculate how far up the wall the ladder reaches. (4 marks)

(Total for Question 13 is 4 marks)

In a right-angled triangle, one of the angles is 40 degrees. The side adjacent to this angle has length 6 cm. Calculate the length of the side opposite this angle, correct to 1 decimal place. (3 marks)

(Total for Question 14 is 3 marks)

In a right-angled triangle, the side opposite an angle x has length 7 cm and the side adjacent to x has length 4 cm. Calculate the size of angle x, correct to the nearest degree. (3 marks)

(Total for Question 15 is 3 marks)

Aisha cycles 18 km in 45 minutes at a constant speed. Work out her average speed in km/h. (3 marks)

(Total for Question 16 is 3 marks)

A metal cube has a mass of 540 g and a volume of 60 cm³. Work out the density of the metal, in g/cm³. (3 marks)

(Total for Question 17 is 3 marks)

A bag contains 5 red counters, 3 blue counters and 2 green counters. A counter is picked at random from the bag. Work out the probability that the counter is blue. Give your answer as a fraction in its simplest form. (2 marks)

(Total for Question 18 is 2 marks)

A box contains 4 yellow pens and 6 black pens. Two pens are taken from the box at random, one after the other, without replacement. Work out the probability that both pens are the same colour. (4 marks)

(Total for Question 19 is 4 marks)

A vertical flagpole stands on horizontal ground. From a point P on the ground, 15 m from the base of the flagpole, the angle of elevation to the top of the flagpole is 35 degrees.

(a)Calculate the height of the flagpole, correct to 1 decimal place. (3 marks)(3)

(b)A supporting wire runs in a straight line from the top of the flagpole to a peg in the ground, 8 m from the base of the flagpole, on the opposite side of the flagpole from P. Using a height of 10.5 m for the flagpole, calculate the length of the wire, correct to 1 decimal place. (2 marks)(2)

(Total for Question 20 is 5 marks)

Two adult tickets and three child tickets to a theme park cost 65 pounds in total. One adult ticket costs 5 pounds more than one child ticket.

(a)Using c for the cost of one child ticket, in pounds, write down an expression for the cost of one adult ticket, in pounds. (1 mark)(1)

(b)Form and solve a pair of simultaneous equations to find the cost of one child ticket and the cost of one adult ticket. (4 marks)(4)

(Total for Question 21 is 5 marks)

The three sides of a right-angled triangle have lengths x cm, (x + 7) cm and (x + 8) cm, where (x + 8) cm is the hypotenuse.

(a)Use Pythagoras' theorem to form an equation in x. (1 mark)(1)

(b)Expand and simplify your equation to show that it can be written as x² - 2x - 15 = 0. (2 marks)(2)

(c)Factorise your equation and hence find the value of x, giving a reason for rejecting the other solution. (2 marks)(2)

(Total for Question 22 is 5 marks)

Mark scheme · CE.M14 Maths: Full Practice Paper (Additional, Level 3)

Question 1

B1 5⁵ oe (accept 3125) cao
Answer: 5⁵

Question 2

B1 17 cao
Answer: 17

Question 3

B1 m⁴ cao
Answer: m⁴

Question 4

M1 (x³)² = x⁶ seen
A1 x⁸ cao
Answer: x⁸

Question 5

M1 x² - 5x + 3x - 15, or three correct terms of the expansion, oe
A1 x² - 2x - 15 cao
Answer: x² - 2x - 15

Question 6

M1 6x² + 8x - 3x - 4, or a correct partial expansion, oe
M1 collects the two middle terms: 8x - 3x = 5x
A1 6x² + 5x - 4 cao
Answer: 6x² + 5x - 4

Question 7

M1 identifies a factor pair of 10 that sums to 7 (2 and 5)
A1 (x + 2)(x + 5) cao
Answer: (x + 2)(x + 5)

Question 8

M1 recognises the difference of two squares, x² - 3²
A1 (x - 3)(x + 3) cao
Answer: (x - 3)(x + 3)

Question 9

M1 substitutes y = 2x + 1 into the second equation: x + (2x+1) = 10
M1 simplifies to 3x + 1 = 10 and solves to find x = 3
A1 x = 3, y = 7 cao
Answer: x = 3, y = 7

Question 10

M1 substitutes y = 3x + 2 into the second equation: 2x + (3x+2) = 17
M1 collects terms to give 5x + 2 = 17
A1 x = 3
A1 y = 11 (follow through 3 x their x + 2)
Answer: x = 3, y = 11

Question 11

M1 5² + 12² seen
M1 25 + 144 = 169
A1 √169 = 13 cm cao
Answer: 13 cm

Question 12

M1 17² - 8² seen
M1 289 - 64 = 225
A1 √225 = 15 cm cao
Answer: 15 cm

Question 13

M1 recognises Pythagoras applies with the ladder as the hypotenuse
M1 6.5² - 2.5² seen
M1 42.25 - 6.25 = 36
A1 √36 = 6 m cao
Answer: 6 m

Question 14

M1 tan(40) = opposite / 6 oe
M1 opposite = 6 * tan(40)
A1 5.0 cm (1 d.p.) cao
Answer: 5.0 cm

Question 15

M1 tan(x) = 7 / 4 oe
M1 x = arctan(1.75)
A1 60 degrees (nearest degree) cao
Answer: 60 degrees

Question 16

M1 converts 45 minutes to 0.75 hours (or 3/4 hour), oe
M1 speed = 18 / 0.75
A1 24 km/h cao
Answer: 24 km/h

Question 17

M1 quotes density = mass / volume
M1 540 / 60
A1 9 g/cm³ cao
Answer: 9 g/cm³

Question 18

M1 total number of counters = 10
A1 3/10 cao
Answer: 3/10

Question 19

M1 P(both yellow) = 4/10 * 3/9
M1 P(both black) = 6/10 * 5/9
M1 adds the two probabilities: 2/15 + 1/3 oe
A1 7/15 cao
Answer: 7/15

Question 20

(a) M1 tan(35) = height / 15 oe
(a) M1 height = 15 * tan(35)
(a) A1 10.5 m (1 d.p.) cao
(a) Answer: 10.5 m
(b) M1 wire² = 10.5² + 8² (follow through their part (a) height)
(b) A1 13.2 m (1 d.p.) cao
(b) Answer: 13.2 m

Question 21

(a) B1 c + 5 oe
(a) Answer: c + 5
(b) M1 forms 2(c+5) + 3c = 65, using their part (a)
(b) M1 expands and simplifies to 5c + 10 = 65
(b) A1 c = 11 (child ticket costs 11 pounds)
(b) A1 adult ticket costs 16 pounds (follow through their c + 5)
(b) Answer: Child ticket = 11 pounds, adult ticket = 16 pounds

Question 22

(a) B1 x² + (x+7)² = (x+8)² oe
(a) Answer: x² + (x+7)² = (x+8)²
(b) M1 expands both brackets correctly: x² + x² + 14x + 49 = x² + 16x + 64
(b) A1 correctly rearranges to x² - 2x - 15 = 0
(b) Answer: x² - 2x - 15 = 0
(c) M1 factorises to (x-5)(x+3) = 0
(c) A1 x = 5, rejecting x = -3 since a length cannot be negative
(c) Answer: x = 5

Question 22

5 marks

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