Cambridge Assessment International Education O Level 4024

Cambridge Assessment International Education

O Level 4024

MATHEMATICS TOPICAL

Paper 1 / Paper 2 With Answers

2009-2019

Prepared By

TASLEEM KHALID LGS JT Girls, BSS Crescent

Mob: 0333-4254312 / Email: [email protected]

1st Edition | 2nd Impression

………………..Formula Sheet …………..

Topic 1: Numbers …………………………………………………………………………………… 10

Topic 2: Set language and notation……………………………………………………………….17 Topic 3: Squares, square roots, cubes and cube roots…………………………………………29 Topic 4: Directed numbers…………………………………………………………………………….33 Topic 5: Vulgar and decimal fractions and percentages……………………………………..37 Topic 6: Ordering…………………………………………………………………………………………42 Topic 7: Standard form………………………………………………………………………………..47 Topic 9: Estimation………………………………………………………………………………………54 Topic 10: Limits of accuracy…………………………………………………………………………..59 Topic 11: Ratio, proportion, rate……………………………………………………………………….65 Topic 12: Percentages…………………………………………………………………………………….72 Topic 14,15: Time, Money…………………………………………………………………………....77 Topic 16: Personal and small business finance………………………………………………….84 Topic 17: Algebraic representation and formulae……………………………………………....87 Topic 18: Algebraic manipulation……………………………………………………………………91 Topic 19: Indices…………………………………………………………………………………………..97 Topic 20: Solutions of equations and inequalities……………………………………………102 Topic 21: Graphical representation of inequalities…………………………………………….119 Topic 22: Sequences……………………………………………………………………………………129 Topic 23: Variation……………………………………………………………………………………..143 Topic 24: Graphs in practical situations………………………………………………………..148 Topic 25: Graphs of Functions ………….………………………………………………………..161 Topic 26: Function notation…………………………………………………………………………186 Topic 27: Coordinate geometry…………………………………………………………………….193 Topic 30: Similarity and congruence……………………………………………………………200 Topic 31: Symmetry…………………………………………………………………………………..206 Topic 32.: Angles( Circles) ………………………………………………………………………..222 Topic 33: Loci…………………………………………………………………………………………..230 Topic 35: Mensuration………………………………………………………………………………..247 Topic 36: Trigonometry……………………………………………………………………………...265 Topic 37: Vectors in two dimensions……………………………………………………………..285 Topic 38: Matrices………………………………………………………………………………………298 Topic 39: Transformations…………………………………………………………………………..303 Topic 40: Probability…………………………………………………………………………………..321 Topic 41: Categorical, numerical and grouped data………………………………………….343 Topic 42: Statistical diagrams……………………………………………………………………..348

Content

O level Mathematics Topical Topic 1: Numbers

Topic 1: Numbers

Subject Contents:

Candidates should be able to identify and use: Natural numbers, Integers (positive, negative and zero), Prime numbers, Square numbers, Cube numbers, Common factors and common multiples, Rational and irrational numbers (e.g. , 2 ), real numbers.

Notes & examples: Includes expressing numbers as a product of prime factors, Finding the Lowest Common Multiple (LCM) and Highest Common Factor

(HCF) of two or more numbers.

_____________________________________________________________________________________________________________________________10 Airport Road | Bahria Town | Johar Town 03214567519 03134567519 10

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O level Mathematics Topical Topic 1: Numbers 1. M/J/19/12/15 (a) Write 168 as a product of its prime factors. (b) The highest common factor of 168 and N is 42. Given that 200 N 300, find two possible values of N. 2. O/N/18/11/8 (a) Write down an irrational number which has a value between 4 and 5. (b) Kofi is using number cards to form a 5-digit number. His number is a multiple of 8. Complete the final digit of his number.

3. O/N/18/11/17 120 = 3 5 (a) Express 1200 as the product of its prime factors. (b) Find the smallest value of n, such that 120n is a square number.

4. M/J/18/21/1C (i) Write 540 as the product of its prime factors. (ii) p is the smallest possible integer such that is a square number. Find , giving your answer as the product of its prime factors.


O level Mathematics Topical Topic 1: Numbers 5. M/J/18/22/4b (b) Find the lowest common multiple (LCM) of 140 and 770.

(c) A rectangular field measures 450 m by 306 m. The whole field is divided into identical square plots with no land remaining. Find the largest possible side length for the squares.

6. S/P/18/1 (a) Find the Highest Common Factor (HCF) of 36 and 54.

7. O/N/02/1/7 Written as product of prime factors, 198= × ×11 & Use these results to find highest common factor of 198 and 90.

8. M/J/10/12/9 Written as product of prime factors, 168= ×3×7. (a) Express 140 as a product of its prime factors. (b) Find highest common factor of 168 and 140.

(c) Find the smallest positive integer, n, such that 168n is a square number.

9. M/J/07/1/6 Find lowest common multiple of 154 & 49.


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O level Mathematics Topical Topic 2 : Set language and notation

Topic 2: Set Language & notation

Subject Contents:

Use language, notation and Venn diagrams to describe sets and represent relationships between sets

Definition of sets: e.g. A = {x : x is a natural number} B = {(x, y): y = mx + c} C = {x : a x b} D = {a, b, c…} Notes & examples: Includes using Venn diagrams to solve problems. Notation: Number of elements in set A n(A) “… is an element of …” ∈ “… is not an element of …” Complement of set A A’ The empty set Universal set A is a subset of B A ⊆ B A is a proper subset of B A ⊂ B A is not a subset of B A B A is not a proper subset of B A B Union of A and B A ∪ B Intersection of A and B A ∩ B



1. O/N/18/21/4 (a) = {x : x is an integer 1 x 10} , A = {x : x is a factor of 20} , B = {x : x is a multiple of 4} (i) Complete the Venn diagram. [2] (ii) State n (A B). (iii) Describe in words the set A . 2. M/J/19/12/10b

= {x : x is a positive number} , A = { x : 9 x 10} , B = {x : x is an irrational number} Write down an element of A B.

3. O/N/18/12/22 = {0,1, 2, 3, 4, 5, 6}, P = { x : x = 0, 1, 2}, Q = { y : y = 0, 2} (a) List the members of P Q. (b) Find n( . (c) R = { z : z = 2x + y, x P, y Q } List the members of R.

4. M/J/12/22/ = { x : x is an integer, 2 x 14}, A = { x: x is a prime number } , B = { x: x is a multiple of 3 } List the members of

5. M/J/10/22/ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, A = {x :x is a multiple of 3}, B = {x : x is a factor of 24}

C = {x :x is an odd number} (i) Find n(B) (ii)


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6. O/N/13/22/5 = {x : x is an integer and 2 x 12}, M = {x :x is a multiple of 3} , P = {x :x is a prime number}

(i) a Find a. (ii) ( M U P )/ 7. O/N/12/22/ = { x : x is an integer, 2 x 14}, A = { x: x is a prime number }, B = { x: x is a multiple of 3 } (i) List the members of (ii) Find n (A B).

8. M/J/13/11 = {natural numbers} , P = {factors of 8}, Q = {factors of 2}, List the elements of the set P Q.

9. O/N/17/11/23

P = { 1, 4 }, Q = { -1, 1, 2 } , R = (i) Find n( (ii) List the members of R.

10. M/J/16/11/23 = { x: x is an integer and 22 x 33 }, E = { x: x is an even number }, T = { x: x is a multiple of 3 }, F = { x: x is a multiple of 4 }

(i) List the members of T F (ii) Find n(E T ) .

11. O/N/16/11/12 = {84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96} , P = {x : x is a even number}, Q = {x : x is a multiple of 3}

Find the value of n (P Q) . (ii) Given that y and that y is a prime number, write down the value of y.

12. M/J/15/11/18(a) = { x : x is an integer, 40 x 50 }, P = { x: x is a prime number }, Q = { x: x is a multiple of 6 } Find (i) n(P). (ii) List the members of Q.


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O level Mathematics Topical Topic 3: Squares, square roots, cubes and cube roots

Topic 3: Squares, square roots, cubes and cube roots

Subject Contents:

calculate squares, square roots, cubes and cube roots of numbers Notes & examples:

Includes recall of squares and their corresponding roots from 1 to 15 and cubes and their corresponding roots from 1 to 10.



1. M/J/18/12/20 N = 2 (b) Find the smallest positive integer M, given that MN is a cube number.

2. O/N/17/11/14 The table shows the square roots, given correct to 4 significant figures, of some numbers from 31.0 to 32.9. For example, the square root of 32.5 is 5.701. Use the table to find (a) The difference between the square root of 32.5 and the square root of 31.3, (b) An estimate of the square root of 32.25, (c) The number which has a square root of 56.39. 3. Without using a calculator, find square root of (a) 784 (b) 1 936

4. M/J/04/1/25 Evaluate

5. O/N/1/1/2 Find the value of


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6. Find the square root of 1296.

7. O/N/2/1/2 Find the value of

8. Without using a calculator, find the cube root of (a) 3 375 (b) 5 832

9. M/J/07/1/6 (i) Write 54 as the product of its prime factors. (ii) Find the smallest possible integer m such that 54m is a cube number.

10. a) Express 60 as a product of its prime factors.

(b) Find the smallest possible integer m such that 60m is a square number.

11. O/N/03/1/4 Write down the square root of 6


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O level Mathematics Topical Topic 4, 8 : Directed numbers, Four Operations

Topic 4, 8 : Directed numbers, Four Operations

Subject Contents:

Use directed numbers in practical situations Notes & examples:

e.g. temperature changes or flood levels



1. Evaluate i. M/J/19/11/1b 1+ 0.6 0.02.

ii. M/J/19/12/1b iii. M/J/19/11/2 In the grids below, the number in each square is the sum of the two numbers immediately below it.

This grid has been completed for you.

Complete the two grids below.

iv. O/N/18/11/2b v. M/J/18/11/1b 9 6 3 + 4 vi. 12 - 6 3 + 4 vii. viii.

ix. 12 + 6 ÷ 3 + 1 × 5 x. 16 – 8 ÷ 2 xi. 3 + 25 ÷ 2


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xii. 17 − 5 × 3 + 1 xiii. 10 – 8 ÷ 2 + 3 xiv. 20 − 8 ÷ 2 + 1

xv. 12 + 6 ÷ 2 – 8 xvi. 24 6 + 2 9 xvii. 3 + 2 (4 – 5) xviii. 3 + 5(3 – 1.4) xix. Add brackets to the expression in the answer space to make it correct. Ans.: 1 + 72 ÷ 4 × 2 = 10 xx. Add brackets to the equation in the answer space to make it correct. Ans.: 4 + 6 × 7 – 5 = 16 xxi. The temperature at the bottom of a mountain was 8°C. The temperature at the top was –26°C. Find the difference between the two temperatures, 2. i. 0.4 0.02 ii. 0.9 0.011 iii. 1.5 – 0.2 × 4 iv. 52.3 × 10 − 3.76 × 100


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O level Mathematics Topical Topic 5 : Vulgar fraction & Percentages

Topic 5 : Vulgar fraction & Percentages

Subject Contents:

Use the language and notation of simple vulgar and decimal fractions and Percentages in appropriate contexts Recognize equivalence and convert between these forms Notes & examples:



i. O/N/18/11/2a Write 17 % as a fraction in its simplest form. ii. Express as a fraction in its lowest terms.

iii. (a) Write 0.09 as a percentage.

iv. Express 0.03 as a fraction.

v. Express 0.45 as a fraction, giving your ans. in its lowest terms.

vi. Express as a decimal.

vii. Express as a decimal.

viii. Evaluate as a decimal.

ix. Express 22 % as a fraction in its lowest terms.

x. O/N/16/11/3a Express 32 % as a fraction in its simplest form.


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xi. M/J/12/12/1a Express 72% as a fraction in its lowest terms.

xii. M/J/11/11/2a Express 18% as a fraction in its lowest terms.

xiii. Express 7 % (a) as a decimal, (b) as a fraction in its simplest form. xiv. M/J/11/12/2b Express as a percentage.

xv. Express 0.08 as a percentage.

xvi. Express as a percentage.

xvii. Express 0.08 as a percentage.

xviii. Express 0.527 as a percentage.

xix. Express 0.82 as a percentage.

xx. Express as a percentage.

xxi. Express as a percentage.


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Topic 1 Numbers Marking Scheme

1. (a) (b) 210, 294

2. (b)

3. (a) 3 (b) 30

4. (i) (ii) 2 5

5. (b)1540 (c) 18

6. (a) 18

7. 28

8. (a) (b) 28

(c) 42

9. 107

10. 660 (b) 6.11am

11. (a) 12 (b) 27

12. (a) × (b) ×

13. ×

14. (a) 12 (b) 27

15. (a) 700 (b)

16. 11:20

17. 14 30

18. (b) 13, 2

19. (i) (ii)

20.

21.

22. 31,37

23. 24. (i) × (ii)

25. (a) (b) 8

26. (a) 2 (b) 8 (c) 27. (a) 619 (b) 196 (c) 169,196, 961

28. (b) 13 (c) ,

29. (a) 0 (b)

Topic 2 : Set language and notation

1.

(i)

(ii) 6 (iii) Factors of 10

2. Any irrational number in range 9 < x < 10

3. (a) { 0, 2} (b) 6 (c)

4. {4,8,10,14 }

5. (i) 7 (ii)

6. (i) 3 (ii)

7. (i) (ii) 1

8.

9. (i) 3 (ii)

10. (i) 24 (ii) 8

11. (i) 9 (ii) 89

12. (i) 3 (ii) 42, 48

Marking Scheme

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13.

14. 3 15.

16. (i) 6 (ii)

17. 7

18.

(i)

(ii) 4 (iii) 1 (iv) B

19. (i) Complete the Venn diagram (ii) 36 (iii) 13 (iv) 1,4,6,9,12,18

[[

20. (P∪ R)∩Q'

21.

22. 23. 24. 25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37. (P∪ Q)′ or P′∩Q′

38.

39.

40. ∪

41. Express in set not ∪

Marking Scheme


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42. ∪

43. ∪

44. (b) C

45. P ∩ Q R'

46.

47. (i)10 (ii)

48. (i)

(ii) 17

49. (i)

(ii) 55 (b) (i) 40 (ii) 39

50. 9

51. 12

52. 16

53. 21 54. 12

55. 16

56. 66

57. (a) p = 6 and q = 4 (b) 33 (c) 34

58.

59.

60. i. 3 ii.11 iii. 18 61. i. 3 ii. 51

Topic 3: Squares, square roots, cubes

and cube roots 1.

2. (a) (b) (c) 3180

3. (a) 28 (b) 44

4. 84

5.

6. 36

7. 30

8. (a) 15 (b) 18

9. (ii) 4

10. a) (b)15

11.

12. 18

13. 28

14.

15.

16. 22

17.

Marking Scheme


18. 27, 64

19. 42

20. (a) 225 (b) 216

21. 64

22.

Topic 4, 8 : Directed numbers, Four Operations

1. Evaluate i. 31 ii. 3

iii. iv. 3.6 v. 11 vi. 14 vii. 0.69 viii. 3.2 ix. 19 x. 12 xi. 15.5 xii. 3 xiii. 9 xiv. 17 xv. 7 xvi. 22 xvii. 1 xviii. 11 xix. 1 + 72 ÷ (4 × 2) = 10 xx. (4 + 6) × (7 – 5) = 16 xxi. 34

2.

i. 0.008 ii. 0.0099 iii. 0.7 iv. 147 v. 1.9 vi. 2.44 vii. 6.7 viii. 0.051 ix. 0.0035 x. 0.09 xi. 0.168 xii. 0.021 xiii. 0.53 xiv. 0.018 xv. 0.018 xvi. 0.054 xvii. 0.012 xviii. 300 xix. 60

Topic 5 : Vulgar fraction & Percentages

i.

ii.

iii. (a)

iv.

v.

vi. vii. viii. ix.

x.

xi.

xii.

xiii. (a) 0.075 (b) as a fraction in its simplest form.

xiv. 80 xv. 8% xvi. 80% xvii. 8%

xviii. 52.7 xix. 82% xx. 42.5% xxi. 32.5 xxii.

2.

i.

ii. 1.015 iii. 4.85

iv. 5/14

v. 13/18 vi. 0.225 vii. d = 12, n =26

3. Giving the ans. as simply as possible,

Calculate

i.

ii. 6/77 iii. 5/12 iv. 1/14 v. 9/20. vi. 8/9 vii. 8/15 viii. 12/25

ix. 8/45

x. xi. 24/35

Marking Scheme


Cambridge Assessment International Education O Level 4024

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