course textbook 1

St Andrew’s Academy

Department of Mathematics

Course Textbook

BOOK ONE

2

Book One

Section 1 - Solving Equations and Factorising Expressions 1.1 Working with Linear Equations and Inequalities……………………… 4

1.2 Algebraic Expressions with Brackets………………………………….. 9

1.3 Factorising Algebraic Expressions…………………………………. 11 1.4 Completing the square………………………………………………. 16

Section 2 – Trigonometry 2.1 The Tangent Ratio (lengths) …………………………………………… 17

2.2 The Tangent Ratio (angles)…………………………………………….. 18

2.3 Using the Tangent Ratio (mixed) ………………………………………. 19

2.4 Problem Solving Using the Tangent Ratio …………………….............. 20

2.5 Using the Sine and Cosine Ratios (lengths) ……………………………. 21

2.6 Using the Sine and Cosine Ratios (angles)……………………………... 22

2.7 Using the Sine and Cosine Ratios (mixed) …………………………….. 23

2.8 Choosing the Appropriate Ratio ……………………………………….. 24

2.9 Problem Solving Using Trigonometry …………………………………. 25

Section 3 - Circles 3.1 Applying Pythagoras’ Theorem ……………………………………….. 31

3.2 Applying Pythagoras’ Theorem - Exam Questions……………………. 35

3.3 Triangles, Chords and Perpendicular Bisectors in Circles……………… 42

3.4 Tangents and Angles……………………………………………………. 44

3.5 The Length of an Arc…………………………………………………… 45

3.6 The Area of a Sector……………………………………………………. 46

3.7 Properties of Circles - Exam Questions……………………………….. 47

3.8 Arc Length and Sector Area - Exam Questions……………………….. 52

Section 4 – Statistics 4.1 Quartiles and Interquartile Range………………………………………. 58

4.2 Standard Deviation……………………………………………………… 61

4.3 Scatter Graphs………………………………………………………….. 64

4.4 Mean and Standard Deviation - Exam Questions………………………. 68

4.5 Scatter Graphs – Exam Questions………………………………………. 70

Section 5 – Percentages and Fractions 5.1 Working with Percentages ………………………………………………73

5.2 Working with Percentages - Exam Questions………………………….. 74

5.3 Appreciation and Depreciation…………………………………………. 75

5.4 Appreciation and Depreciation - Exam Questions……………………… 76

5.5 Addition and Subtraction of Fractions………………………………….. 80

5.6 Multiplication and Division of Fractions……………………………….. 83

5.7 Mixed Questions on Fractions………………………………………….. 85

Section 6 – Similar Shapes 6.1 Linear Scale Factors…………………………………………………….. 88

6.2 Area Scale Factors……………………………………………………… 90

6.3 Volume Scale Factors………………………………………………….. 92

6.4 Similar Shapes - Exam Questions………………………………… 94

Section 7 – Equations of Straight Lines

7.1 Gradients of Straight Lines……………………………………………. 96

7.2 Equations of Straight Lines…………………………………………….. 99

7.3 Equations of Straight Lines - Exam Questions………………………… 103

Section 8 – Simultaneous Equations 8.1 Graphical Solutions…………………………………………………….. 106

8.2 Algebraic Solutions…………………………………………………….. 108

8.3 Simultaneous Equations in Context…………………………………….. 109

8.4 Simultaneous Equations - Exam Questions……………………………..114

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Section 9 – Functions and Formulae 9.1 Changing the Subject of a Formula…………………………………….. 117

9.2 Changing the Subject of a Formula - Exam Questions…………………120

9.3 Function Notation……………………………………………………… 121

Section 10 – Area and Volume 10.1 Properties of Shapes…………………………………………………… 123

10.2 Volumes of Cylinders…………………………………………………. 130

10.3 Volumes of Other Solids……………………………………………….. 132

10.4 Volumes of Solids - Exam Questions………………………………….. 134

4

Section 1 – Equations and Factors

1.1 Working with Linear Equations and Inequalities

1. Solve :

(a) x + 3 = 5 (b) x + 5 = 9 (c) x + 9 = 12 (d) x + 2 = 7

(e) a + 2 = 4 (f) y + 3 = 8 (g) p + 7 = 11 (h) c + 4 = 5

(i) b − 7 = 9 (j) q − 8 = 8 (k) d − 5 = 10 (l) x − 1 = 6

(m) c − 4 = 6 (n) p − 6 = 14 (o) a − 2 = 15 (p) y − 5 = 14

2. Solve:

(a) 2x = 6 (b) 5x = 20 (c) 8x = 16 (d) 3x = 27

(e) 4a = 16 (f) 7y = 28 (g) 6p = 18 (h) 5c = 25

(i) 9b = 36 (j) 2q = 18 (k) 7d = 70 (l) 4x = 32

(m) 8c = 56 (n) 3p = 15 (o) 5a = 35 (p) 6y = 42

3. Solve:

(a) 2a = −36 (b) −5m = −55 (c) −8q = 64 (d) −3y = −48

(e) 4x = −52 (f) −7c = −63 (g) −6d = 72 (h) −5a = −125

(i) 9p = −81 (j) −2q = −17 (k) −4x = 22 (l) −6q = −33

(m) 8c = −28 (n) −5x = −90 (o) −10a = 42 (p) −4y = −42

4. Solve :

(a) 2x − 3 = 5 (b) 4x + 5 = 9 (c) 3x + 3 = −12 (d) 5x + 2 = 7

(e) 2a − 2 = −14 (f) 5y + 3 = 18 (g) 2p + 7 = 21 (h) 3c − 4 = 17

(i) 6b + 7 = 49 (j) 8q − 8 = −8 (k) 2d − 5 = 35 (l) 3x + 5 = −25

(m) 8c + 4 = 36 (n) 7p + 6 = 55 (o) 12a + 2 = 26 (p) 9y + 5 = 50

5. Solve :

(a) 3x − 2 = 7 (b) 4x − 5 = 11 (c) 2x − 9 = 3 (d) 3x − 7 = 5

(e) 7a − 2 = 12 (f) 5y − 3 = 22 (g) 6p − 7 = 29 (h) 4c − 3 = 29

(i) 8b − 7 = 57 (j) 10q − 8 = 72 (k) 3d − 5 = 31 (l) 9x − 1 = 80

(m) 4c − 9 = 15 (n) 6p − 2 = 40 (o) 5a − 2 = 73 (p) 3y − 14 = 40

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6. Multiply out the brackets and solve :

(a) 2 (x + 5) = 12 (b) 5 (y + 7) = 45 (c) 3 (a + 6) = 36

(d) 6 (x + 4) = 54 (e) 4 (x + 9) = 48 (f) 3 (c + 8) = 30

(g) 7 (d + 3) = 56 (h) 5 (m + 5) = 55 (i) 2 (y + 14) = 50

(j) 8 (d − 6) = 24 (k) 3 (s − 8) = 9 (l) 4 (x − 15) = 20

(m) 10 (w − 2) = 50 (n) 5 (c − 5) = 35 (o) 3 (a − 10) = 33

7. Solve :

(a) 6y + 3 = y + 18 (b) 5a + 7 = a + 15

(c) 9c + 5 = c + 21 (d) 10x + 1 = 4x + 19

(e) 5b + 3 = 2b + 9 (f) 7n + 6 = 3n + 18

(g) 3x + 2 = x + 14 (h) 9c + 58 = 6c + 73

(i) 16 + 7y = 2y + 31 (j) 15a + 4 = 3a + 76

(k) 16 + 25x = 5x + 96 (l) 6n + 3·5 = 3n + 5

(m) 19b + 8 = 10b + 80 (n) 14x + 4 = 3x + 125

(o) 250 + 3x = 295 (p) 20y + 4 = 3y + 55

(q) 13a + 6 = a + 150 (r) 50x + 40 = 10x + 200

(s) 19y + 3 = 8y + 80 (t) 5b + 2 = 2b + 50

(u) 2 + 14x = 2x + 110 (v) 20x + 11 = 13x + 60

(w) 19x + 10 = 4x + 70 (x) 205a + 13 = 10a+ 403

8. Solve :

(a) 6y − 3 = 3y + 15 (b) 5a − 9 = a + 15

(c) 9c − 8 = 4c + 12 (d) 10x − 1 = 4x + 5

(e) 5b − 3 = 2b + 9 (f) 3n − 10 = n + 2

(g) 7x − 14 = 3x + 2 (h) 6c − 13 = 3c + 59

(i) 7y − 16 = 2y + 34 (j) 15a − 8 = 3a + 76

(k) 25x −16 = 5x + 84 (l) 6n − 3·5 = 3n + 4

6

(m) b + 13 = 9b − 7 (n) 3x + 12 = 4x − 4

(o) x + 25 = 3x − 5 (p) 5y + 4 = 20y − 26

(q) a + 6 = 13a − 18 (r) 10x + 40 = 50x − 120

(s) 8y + 3 = 19y − 74 (t) 2b + 2 = 5b − 16

(u) 2 + 2x = 10x − 14 (v) 13x + 11 = 20x − 38

(w) 4x + 10 = 9x − 50 (x) 10a + 13 = 20a − 387

9. Solve :

(a) x + 4 > 5 (b) x + 6 > 9 (c) x + 8 > 12 (d) x + 3 > 7

(e) a + 1 > 4 (f) y + 5 > 8 (g) p + 2 > 11 (h) c + 4 > 5

(i) b + 3 > 9 (j) q + 8 > 8 (k) d + 7 > 10 (l) x + 2 > 6

(m) c + 1 > 6 (n) p + 4 > 13 (o) a + 3 > 15 (p) y + 2 > 14

10. Solve :

(a) x + 5 < 7 (b) x + 1 < 8 (c) x + 3 < 13 (d) x + 5 < 9

(e) a + 3 < 6 (f) y + 5 < 11 (g) p + 2 < 10 (h) c + 1 < 5

(i) b + 8 < 13 (j) q + 3 < 20 (k) d + 7 < 7 (l) x + 10 < 15

(m) c + 3 < 9 (n) p + 2 < 16 (o) a + 4 < 15 (p) y + 9 < 10

11. Solve:

(a) 2x > 6 (b) 5x > 20 (c) 8x > 16 (d) 3x > 27

(e) 4a > 16 (f) 7y > 28 (g) 6p > 18 (h) 5c > 25

(i) 9b < 36 (j) 2q < 18 (k) 7d < 70 (l) 4x < 32

(m) 8c < 56 (n) 3p < 15 (o) 5a < 35 (p) 6y < 42

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12. Solve :

(a) x − 3 < 4 (b) x − 5 > 1 (c) x − 9 > 2 (d) x − 2 < 7

(e) a − 2 < 4 (f) y − 3 > 8 (g) p − 7 < 11 (h) c − 4 > 5

(i) b − 7 > 9 (j) q − 8 < 8 (k) d − 5 > 10 (l) x − 1 > 6

(m) c − 4 > 6 (n) p − 6 < 14 (o) a − 2 < 15 (p) y − 5 < 14

13. Solve :

(a) 2x + 1 < 5 (b) 4x + 1 > 9 (c) 3x + 3 > 12 (d) 5x + 2 > 12

(e) 2a + 2 < 8 (f) 5y + 3 < 13 (g) 2p + 5 > 21 (h) 3c + 1 < 16

(i) 6b + 13 > 49 (j) 8q + 8 < 8 (k) 3d + 5 < 35 (l) 4x + 5 > 21

(m) 8c + 12 < 36 (n) 7p + 6 < 55 (o) 12a + 2 > 26 (p) 9y + 23 < 50

14. Solve :

(a) 3x − 1 > 8 (b) 4x − 3 > 13 (c) 2x − 7 < 5 (d) 3x − 5 > 4

(e) 7a − 1 < 13 (f) 5y − 2 < 23 (g) 6p − 5 > 31 (h) 4c − 7 > 25

(i) 8b − 3 > 61 (j) 10q − 7 < 73 (k) 3d − 2 < 34 (l) 9x − 8 > 73

(m) 4c − 5 < 19 (n) 6p − 1 < 41 (o) 5a − 4 < 71 (p) 3y − 24 < 30

15. Solve each of the following inequalities where x can only take values from the set of numbers

{ }5,4,3,2,1,0,1,2 −− .

(a) 5326 +≤+ xx (b) 3137 +≥ xx

(c) 85)12(3 +≥+ xx (d) 128)56(2 +<+ xx

(e) 8)3(214 ≤−− x (f) xx 614)2(35 −≥−+

(g) 2)4(2 +<−− xxx (h) 17)2(6)2(43 −−>+− xx

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16. Solve each of the following inequalities.

(a) aa 21723 −≤+ (b) 278)32(7 +>+ xx

(c) 187)125(2 −≥− pp (d) kk −<+ 28340

(e) )7(2)2(53 +≤−− mm (f) )4(41)42(3 yy −>−−

(g) hh 1513)43(2 −<− (h) 5)1(2)2(32 −−>−− xx

17. Solve each of the following inequalities.

(a) aa 412182 +≤+ (b) 6314 +>− xx

(c) 105)2(3 −≥− pp (d) kk −<− 20316

(e) )12(2)2(7 −≤− dd (f) )1(108)12(2 yy +>−−

(g) hh +<− 12)43(4 (h) 7)31(2)2(3 −+>− yy

18. I think of a whole number, treble it and subtract 3. The answer must be less than or equal to 12.

Form an inequality and solve it to find the possible starting whole numbers.

19. I subtract a whole number from 8 and double the answer. The result must be greater than 10.

Form an inequality and solve it to find the possible starting whole numbers.

20. Fred and Jane are brother and sister. Fred is 3 years older than twice Jane's age.

The sum of their ages is less than 36 years.

Taking Jane's age to be x years form an inequality. What can you say about Jane's age?

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1.2 Algebraic Expressions with Brackets

1. Multiply out the brackets:

(a) 3 (x − 5) (b) 5 (y + 7) (c) 8 (a + 6) (d) 6 (3 + t)

(e) x (x + 9) (f) y (3 − y) (g) b (b − 4) (h) p (5 + p)

(i) a (b + c) (j) x (x − y) (k) p (q − r) (l) a (a + x)

2. Expand the brackets:

(a) 4 (2a + 5) (b) 7 (3y − 4) (c) 2 (12x + 11) (d) 9 (4c − 7)

(e) 2a (a + 3) (f) 5x (x − 8) (g) 10y (3 − y) (h) 3t (t + 6)

(i) 3x (2x − 9) (j) 2y (7 − 5y) (k) 4b (3b − 8) (l) 5x (5x + 4)

3. Expand and simplify:

(a) 3(3a − 1) + 2a (b) 2(5x + 3) − 3x (c) 8(b + 2) − 9

(d) 4(2h − 1) + 7 (e) 5(3 − 4x) + 11x (f) 3(2c + 1) − 8

(g) 2(4t + 3) − 10t (h) p(p + q) − 3pq (i) 7(1 − 3c) − 10

(j) 3 + 2(2x + 5) (k) 7a + 3(2a − 3) (l) 5 − 2(2x − 7)

(m) 6 + 5(3y − 2) (n) 9b − 2(4b −1) (o) 8 − 3(5x + 7)

(p) 12x − 4(4x − 5) (q) 3c + 5(1 − 2c) (r) 7 − 2(5a − 12)


(a) (x + 2)(x + 3) (b) (y +5)(y +2) (c) (a + 4)(a + 6)

(d) (b + 3)(b + 4) (e) (x + 9)(x +5) (f) (s + 3)(s + 8)

(g) (y + 7)(y + 4) (h) (b + 3)(b + 3) (i) (c + 6)(c + 7)

(j) (x − 6)(x − 5) (k) (b − 5)(b − 3) (l) (c − 10)(c − 4)

(m) (a − 3)(a − 9) (n) (y − 8)(y − 7) (o) (x − 12)(x − 3)

(p) (s − 4)(s − 7) (q) (d − 1)(d − 15) (r) (b − 10)(b − 1)

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(a) (x − 1)(x + 5) (b) (a + 3)(a − 7) (c) (t − 5)(t + 4)

(d) (y + 8)(y − 4) (e) (c + 2)(c − 7) (f) (x − 6)(x + 1)

(g) (b − 2)(b + 9) (h) (p − 10)(p + 2) (i) (y − 8)(y + 7)

(j) (z + 4)(z − 6) (k) (x + 1)(x − 1) (l) (a + 2)(a − 15)

(m) (c − 3)(c + 3) (n) (p − 7)(p + 1) (o) (b + 10)(b − 5)

6. Multiply out the brackets and simplify:

(a) (2x + 1)(x + 3) (b) (3x + 2)(x - 5) (c) (2t + 1)(2t + 3)

(d) (5y + 2)(y − 4) (e) (c - 3)(4c + 3) (f) (4x − 5)(2x + 1)

(g) (3b − 4)(4b - 7) (h) (8p + 3)(3p + 10) (i) (9y − 5)(2y - 5)

(j) 2(x + 2)(x + 5) (k) 3(a + 5)(a - 3) (l) 5(3x − 2)(x + 6)

(m) 6(3p + 1)(2p + 1) (n) t(t + 5)(t - 6) (o) x(4x − 3)(3x - 4)


(a) (x + 3)2 (b) (w − 2)

2 (c) (a − 5)

2 (d) (c + 8)

2

(e) (y − 4)2 (f) (a + 6)

2 (g) (b + 1)

2 (h) (s + 7)

2

(i) (b − 9)2 (j) (x − 10)

2 (k) (c − 1)

2 (l) (y − 3)

2

(m) (2x − 1)2

(n) (5y + 2)2

(o) (3x + 4)2

(p) (4b − 5)2


(a) (a + b)(c + d) (b) (2 + x)(3 + y) (c) (a + 4)(b + 5)

(d) (p − q)(r − s) (e) (1 − a)(7 − b) (f) (c − 6)(d + 8)


(a) x(x2 + x − 1) (b) 3(2x

2 −3x + 5) (c) x(3x

2 − 5x + 8)

(d) 2x(x2 + 2x + 3) (e) −5(x

2 − 8x + 2) (f) x(x

2 − 4x − 7)

11


(a) (x + 2)(x2 + 3x + 1) (b) (x + 5)(x

2 + 4x+ 2)

(c) (x + 1)(x2 + 5x + 4) (d) (x + 3)(x

2 + x + 5)

(e) (x + 8)(x2 + 2x + 3) (f) (x + 4)(x

2 + 7x + 6)

(g) (x + 12)(x2 + x + 7) (h) (x + 10)(x

2 + 3x +9)

(i) (x + 9)(x2 + 12x + 7) (j) (x + 7)(x

2 + 9x + 1)


(a) (x − 1)(x2 + x + 1) (b) (x − 7)(x

2 + 3x + 5)

(c) (x − 2)(x2 + 4x + 3) (d) (x − 4)(x

2 + 6x + 1)

(e) (x − 3)(x2 − 2x + 5) (f) (x − 6)(x

2 − 5x + 2)

(g) (x − 4)(x2 − x + 2) (h) (x − 1)(x

2 − 2x + 7)

(i) (x − 9)(x2 + 3x − 2) (j) (x − 5)(x

2 + 8x + 6)


(a) (x + 5)(2x2 + 4x + 9) (b) (x − 3)(5x

2 + x + 6)

(c) (x − 2)(6x2 − 5x + 7) (d) (x + 7)(3x

2 + 9x −2)

(e) (x − 4)(5x2 − x − 8) (f) (x + 1)(7x

2 − 2x + 11)

(g) (2x + 1)(3x2 + 4x + 1) (h) (3x + 4)(x

2 − 11x + 2)

(i) (5x − 2)(2x2 + 3x − 7) (j) (4x − 3)(3x

2 − 5x − 4)

13. Expand and simplify each of the following expressions:

(a) 2)2()4(3 ++− xx (b) )3(2)3)(12( −++− xxxx

(c) )1(4)32( 2 +−+ xx (d) xx 4)2( 2 ++−

(e) 22 12)12(3 xx +−− (f) 2)4()2)(3( +−+− xxx

(g) )4)(2()4(3 −+−− xxxx (h) )3()12()2( 22 +−−++ xxx

(i) )12)(3(4)32( 2 +−−− xxx (j) )3(2)3(3 2 −++ xxxx

(k) 22 )3()2(2 −++− xxxx (l) 22 )1()1( +−− xxx

12

14. Solve:

(a) x(x + 2) = x² + 6 (b) x(x + 4) = x² + 20

(c) x(2x + 3) = 2(x² + 6) (d) x(6x + 5) = 3(2x² + 5)

(e) 4x(x - 2) = 2(2x² - 8) (f) 3x(x - 4) = 3x² - 6

(g) (x + 2)(x + 4) = (x + 1)(x + 4) (h) (x + 3)(x + 5) = (x + 6)(x + 1)

(i) (x + 6)(x - 2) = (x - 1)(x + 3) (j) (x + 5)(x - 1) = (x + 2)(x - 1)

(k) (x - 5)(x + 5) + (x + 7)(x + 3) = 2x(x + 4) (l) (x - 2)(x - 3) + (x - 4)(x + 5) = x(2x + 3)

1.3 Factorising Algebraic Expressions

1. Factorise by first finding a common factor:

(a) 2x + 2y (b) 3c + 3d (c) 6s + 6t (d) 12x + 12y

(e) 9a + 9b (f) 8b + 8c (g) 5p + 5q (h) 7g + 7h

(i) 4m + 4n (j) 9e + 9f (k) 13j + 13k (l) 14v + 14w

2. Factorise by finding the common factor:

(a) 2x + 4 (b) 3d + 9 (c) 6s + 3 (d) 12x + 4

(e) 6 + 9a (f) 2b + 8 (g) 5y + 10 (h) 10 + 15c

(i) 12x + 16 (j) 18m + 24 (k) 30 + 36a (i) 14y + 21


(a) 3x − 6 (b) 4y − 8 (c) 16 − 8a (d) 10c − 15

(e) 9s − 12 (f) 2b − 14 (g) 12x − 20 (h) 22m − 33

(i) 15x − 10 (j) 18 − 12y (k) 25b − 20 (l) 18d − 30

13


(a) 2a + 4b (b) 10x − 12y (c) 18m + 24n (d) 10c + 15d

(e) 6a − 9x (f) 18s − 12t (g) 12x + 15y (h) 14a − 7b

(i) 25c + 10d (j) 9b − 15y (k) 18x + 24y (i) 6a + 28b

5. Factorise by finding the common factor

(a) ax + ay (b) xy2 + xa

2 (c) pqr + pst

(d) xay − bac (e) pq + p (f) y2 + y

(g) a2 − ab (h) ab − bc (i) n

2 − 3n

(j) xy + y2 (k) abc − abd (l) fgh − efg

6. Factorise by finding the highest common factor:

(a) 2ax + 6a (b) 3y + 9y2 (c) 24a − 16ab

(d) pq2 − pq (e) 12xy − 9xz (f) 6b

2 − 4b

(g) 3a2 + 27ah (h) 15abc + 20abd (i) 3s

3 − 9s

2

(j) 14x − 12xyz (k) 10b2c − 15bcd (l) 2πr

2 + 2πrh

7. Factorise by finding the highest common factor:

(a) ap + aq − ar (b) 2a + 2b + 2c (c) 6e − 2f + 4g

(d) p2 + pq + xp (e) 3ab − 6bc − 9bd (f) ½ ah + ½ bh + ½ ch

(g) 5x2 − 8xy + 5x (h) 4ac + 6ad − 10a

2 (i) 15p

2 + 10pq + 20ps

8. Factorise the following expressions, which contain a difference of squares:

(a) a2 − b

2 (b) x

2 − y

2 (c) p

2 − q

2 (d) s

2 − t

2

(e) a2 − 3

2 (f) x

2 − 2

2 (g) p

2 − 9

2 (h) c

2 − 5

2

(i) b2 − 1

(j) y

2 − 16 (k) m

2 − 25

(l) a

2 − 9

(m) 36 − d2 (n) 4 − q

2 (o) 49 − w

2 (p) x

2 − 64

14

9. Factorise the following expressions, which contain a difference of squares:

(a) a2 − 4b

2 (b) x

2 − 25y

2 (c) p

2 − 64q

2 (d) 16c

2 − d

2

(e) 81 − 4g2 (f) 36w

2 − y

2 (g) 4a

2 − 1 (h) g

2 − 81h

2

(i) 49x2 − y

2 (j) 9c

2 − 16d

2 (k) 4p

2 − 9q

2 (l) b

2 − 100c

2

(m) 25 −−−− 16a2 (n) 4d

2 − 121 (o) 225 − 49k

2 (p) 9x

2 − 0·25

10. Factorise the following expressions which contain a common factor and a difference of two

squares:

(a) 2a2 − 2b

2 (b) 5p

2 − 5 (c) 45 − 5x

2 (d) 4d

2 − 36

(e) 2y2 − 50 (f) 4b

2 − 100 (g) 3q

2 − 27 (h) 8a

2 − 32b

2

(i) ab2 − 64a (j) xy

2 − 25x (k) abc

2 − ab (l) 8p

2 − 50q

2

(m) 2x2 − 2·88 (n) ak

2 − 121a (o) 10s

2 − 2·5 (p) ½ y

2 − 450

11. Factorise the following quadratic expressions:

(a) x2+ 3x + 2 (b) a

2 + 2a + 1 (c) y

2 + 5y + 4

(d) x2 + 8x + 7 (e) x

2 + 6x + 9 (f) b

2 + 8b + 12

(g) a2 + 9a + 14 (h) w

2 + 10w + 9 (i) d

2 + 7d + 10

(j) x2 + 10x + 21 (k) p

2 + 9p + 20 (l) c

2 + 10c + 24

(m) s2 + 12s + 36 (n) x

2 +11x + 28 (o) y

2 + 10y + 25


(a) a2 − 8a + 15 (b) x

2 − 9x + 8 (c) c

2 − 9c + 18

(d) y2 − 4y + 4 (e) b

2 − 6b + 5 (f) x

2 − 15x + 14

(g) c2 − 10c + 16 (h) x

2 − 7x + 6 (i) y

2 − 12y + 32

(j) p2 − 11p + 24 (k) a

2 − 13a + 36 (l) x

2 − 15x + 36

(m) b2 − 4b + 3 (n) q

2 − 11q + 10 (o) a

2 − 7y + 12


(a) b2 + 3b − 10 (b) x

2 + 6x − 7 (c) y

2 − y − 6

(d) a2 − a − 20 (e) q

2 + 2q − 8 (f) x

2 − 8x − 20

(g) d2 + 4d − 21 (h) c

2 + 9c − 36 (i) p

2 − 5p − 24

(j) y2 − 7y − 8 (k) a

2 + 5a − 6 (l) x

2 − 5x − 36

(m) b2 − 4b − 5 (n) s

2 + 2s − 24 (o) d

2 + 6d − 16

15


(a) 3x2 + 7x + 2 (b) 2a

2 + 5a + 2 (c) 3c

2 + 8c + 5

(d) 2p2 + 11p + 9 (e) 2y

2 + 11y + 5 (f) 3d

2 + 11d + 6

(g) 5q2 + 9q + 4 (h) 4b

2 + 8b + 3 (i) 6x

2 + 13x + 6

(j) 3a2 + 14a + 15 (k) 10x

2 + 17x + 3 (l) 9c

2 + 6c + 1

(m) 6y2 + 11y + 3 (n) 3b

2 + 5b + 2 (o) 8x

2 + 14x + 3


(a) 2x2 − 7x + 3 (b) 2a

2 − 5a + 3 (c) 5p

2 − 17p + 6

(d) 5b2 − 7b + 2 (e) 6x

2 − 7x + 2 (f) 4y

2 − 11y + 6

(g) 7c2 − 29c + 4 (h) 4m

2 − 9m + 2 (i) 16a

2 − 10a + 1

(j) 8y2 − 22y + 5 (k) 3p

2 − 37p + 12 (l) 4x

2 − 25x + 6

(m) 15a2 − 16a + 4 (n) 24c

2 − 22c + 3 (o) 6b

2 − 35b + 36


(a) 3x2 − 2x − 1 (b) 2a

2 − a − 3 (c) 4p

2 − p − 3

(d) 2c2 + 7c − 4 (e) 6y

2 − 11y − 2 (f) 3w

2 + 10w − 8

(g) 3m2 + 2m − 5 (h) 4q

2 + 5q − 6 (i) 6b

2 + 7b − 20

(j) 4t2 − 4t − 3 (k) 12z

2 + 16z − 3 (l) 4d

2 − 4d − 15

(m) 7s2 − 27s − 4 (n) 15x

2 + 16x − 15 (o) 36v

2 + v − 2

(p) 7103 2 ++ vv (q) 5112 2 +− ll (r) 73112 2 +− mm

(s) 28193 2 +− vn (t) 25204 2 +− bb (u) 8189 2 ++ cc

(v) 5143 2 −+ qq (w) 126 2 −+ aa (x) 1528 2 −− bb

(y) 15812 2 −− mm (z) 282 2 −− nn

17. Fully factorise these expressions:

(a) 3x2 − 3 (b) 2p

2 + 12p + 10 (c) 9x

2 − 36

(d) 5x2 + 25x + 30 (e) ax

2 + 5ax + 6a (f) 3y

2 − 12y − 15

(g) 15c2 + 27c + 12 (h) 16b

2 + 28b + 6 (i) 9q

2 + 33q + 18

(j) 10s2 − 35s + 15 (k) 8m

2 −20m + 12 (l) 8a

2 −36a + 36

(m) 4t2 + 2t − 56 (n) 90d

2 −60d −80 (o) 400x

2 − 4

16


(a) 8w2 +16w -24 (b) 18x³ - 2x (c) 4ab – 6bc

(d) 2y + 2y² + 2y³ (e) r³ - 4r (f) 3x³ - 27x

(g) 11x2 + 22x + 12 (h) 6a²b – 8ab² (i) 8p² - 72

(j) 2n² - 2n – 144 (k) 3xy + 6b²y (l) 3b² - 12b

(m) 3f³ - 27f (n) 3u² - 36u + 105 (o) 30a² - 30a


(a) 180 – 5d² (b) 12n² - 33n + 18 (c) 2xyz – 8zyw

(d) 6v + 16v³ (e) 28q² - 14q - 14 (f) 16 – 400a²

(g) 17 – 34x + 17x² (h) m³ - 9mn² (i) 6x² - 17x + 12

(j) u²w² - 4w² (k) 28x² + 28x - 168 (l) 14x² + 20x + 6

(m) 9y – 16y³ (n) 36 + 72a – 108a² (o) abc² + a²bc

1.4 Completing the Square

1. Write the following in the form (x + a)² + b:

(a) x² + 6x (b) x² + 10x (c) x² - 4x (d) x² - 2x

(e) x² - 6x (f) x² - 12x (g) x² + 3x (h) x² + 5x

(i) x² - x (j) x² - 9x (k) x² + 20x (l) x² - ½x


(a) x² + 4x + 1 (b) x² + 2x + 5 (c) x² + 8x - 3 (d) x² - 2x - 9

(e) x² - 6x + 11 (f) x² - 8x + 7 (g) x² + 2x - 21 (h) x² - 14x + 13

(i) x² + 4x - 7 (j) x² - 10x + 9 (k) x² + 12x - 15 (l) x² - 20x + 11


(a) x² + x + 1 (b) x² + 3x - ¼ (c) x² - x + ½ (d) x² - x + ¾

(e) x² - x + ¾ (f) x² -5x - ½ (g) x² + 3x + ¼ (h) x² + 5x + 1

(i) x² + 3x - ¾ (j) x² - 3x + ½ (k) x² + x - 2 (l) x² - 7x + ¼

17

Calculate the length of the unknown side in each triangle. Give your answers to 1 decimal place.

2.1 The Tangent Ratio (lengths)

Section 2 – Trigonometry

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

10 cm

a

8 mm

9 cm

30o

40o

32o

b c

34 m

d

42o

15 mm

e

26o

7 km

f

68o

8 cm

25o

g

8.5 km

71o h

3.2 m

39o

j

8.7 mm

68o

k 65 cm

27o

m

12.4 km

60o

n

15.7 cm

17o

p 140 mm

47o

q

23.5 km

51o

r

14 cm

65o

h

46 m

44o

h

7.2 mm

53o

h

18

2.2 The Tangent Ratio (angles)

Calculate the size of the angle marked x in each triangle. Give your answer to 1 decimal place.

6.8 cm

7.5 cm

3 cm 8.7 cm

3.8 m 4.7 m

5.5 cm

4.9m

2.9 m

3 cm

1.3cm

2.5m

5.3 cm

3.7 m

7.8 m

2.8m 2.1cm

1.8 mm

18.6 m

12.7 m 6.2 mm

10.2 m

7.8 m

1.3cm

5.1 cm

7.6 cm

x

4 m

12.8 m

x 3.8m 2.9 m

x

x

x x

x x

x

x

x

x

x

x

x

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

19

Calculate the size of the unknown side or angle in each triangle.

Give your answers to 3 significant figures.

2.3 Using the Tangent Ratio (mixed)

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

5 cm

a

9 mm

14 cm

28o

10 mm

49o

b

c

6.3 m

d 2.1 m

15.3 mm

e

52o

8.6 km

f

5.9 km

7.81 cm

29.6o

g

34.8o

h

26.3 m

26.3o

j

29.2 mm

17.6 mm

k 102 cm

22.3o

m

62.3 km

54.1 km

n

31.7 cm

63.4o

p

91.7 mm

59.8o

q

386 km

17.3o

r

20 cm

70o

s 58 m

44 m

t

18 mm

h

20

2.4 Problem Solving Using the Tangent Ratio

21

Find the length of the unknown side in each triangle. Give your answer to 3 significant figures.

2.5a Using the Sine and Cosine Ratios (lengths)

21

a b

c

d e

f

g

h

j

k

m

n

p q

r

9 cm 14 m 23 mm

8.3 km

7.6 cm

16.2 mm

23.8 m

8.45 km

43.2 cm

3.81 mm

3.14 m 17.6 cm

29.4 km

6.48 m

12.8 cm

22

Replace each ? with a letter and find the length of the unknown side in each triangle.

2.5b Using the Sine and Cosine Ratios (lengths)

22

10cm

7 m

18 mm

11.4 km

34.4

8.65 mm

5.7 m

41.1 km

2.06 cm

4.22 mm

107 m

19.3 cm

269 km

68.48 m

439 cm

a b c

d

e

f

g

h

j

k

m

n

p q

r

23

Calculate the size of the angle in each triangle.

2.6 Using the Sine and Cosine Ratios (angles)

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

3 cm

a

5 cm 6 mm

c

11 m

d

15 m

13 mm

e

13 km

f

25 km

8 cm

9 cm

g

h 1.09 m

1.54 m

j

0.904 mm

k

58 cm

m

13.8 km

n

p

3.68 mm

6.4 mm

q

4.63 km 4.36 km

r

s 26.1 m

34.47 mm

8 cm

b

5.4 m

2.7 cm

21.4 cm

u

10 mm

8 cm

9 mm

t

2.74 mm

46 cm

15.3 km

5.1 cm

18.5 cm

22.1 m

28.35 mm

24



2.7 Using the Sine and Cosine Ratios (mixed)

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

a

7.6 cm

23 mm

63o

c

24 m

d

35 m

21.9 mm e

75.2o

13 km

f

48 km

48 cm

g

23.6o

h

3.84 m 5.36 m

j

12.8 mm

k

22.1 cm 54.4

o

m

2.09 km

n

57.2o

p

8.36 mm

10.4 mm

q

25.5 km

38.3 km

r

56.1o

s

3.75 m

t

1.064 mm

17 cm

40o

b

5.4 m

34.9o

65.7o

28.2 cm

56.1 cm

62.8o

42.7o

u

25



2.8 Choosing the Appropriate Ratio

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

a 32 cm 23 mm

50o

c

18 m

d

45 m

15.3 mm

e

78.6o

21.8 km

f

27.4 km

6.19 cm

g

61.8o

h

7.15 m

j

24.6 mm

k

7.28 cm

m

54.2 km

n

p

60.3 mm

q

52.6 km

r

64.1o

s

10.36 m

t

306 mm

17 cm

40o

b

20.3 m

73.7o

11.5 cm

3.45 cm

56.7o

46.5o

u

14 cm

25.7o

32.2o

45.8 mm

5.6 cm

27.8o

52.6o

15.1 cm

26

2.9a Problem Solving Using Trigonometry

27 27

28

Calculate all the values of x in each diagram

2.9b Problem Solving Using Trigonometry

29

2.9c Problem Solving Using Trigonometry

30

2.9d Problem Solving Using Trigonometry

31

Section 3 - Circles

3.1 Applying Pythagoras’ Theorem

1. Calculate the length of the side marked x in each triangle below

(a) (b) (c)

(d) (e) (f)

(g)

(h) (i)

2. Answer these questions about the framework opposite.

(a) Calculate the length of BD.

(b) Hence calculate the length of BC.

(c) Calculate the area of triangle ABC.

3. A rhombus has sides of 20cm and its longest diagonal measuring 34cm.

(a) Calculate the length of the shorter diagonal.

(b) Calculate the area of the rhombus

4. Calculate the distance between each pair of points below.

(a) A(2, 5) and B(7, 10) (b) P(1, 8) and Q(12, 2)

(c) E( )3,2− and F(2, 4− ) (d) R( 3,7 −− ) and F( 3 , 1− )

20cm

34cm

x x x

x

x

x

x x

x

8

5 14

16

5

12

15

11

2 71 ⋅

23 ⋅

6

4

13

25

10 70 ⋅

52 ⋅

A

B

C D

26

24 12

32

5. Answer the following about the cuboid opposite.

(a) Calculate the length of the face

diagonal AC.

(b) Hence calculate the length of the

space diagonal AG.

6. The pyramid opposite has a rectangular base.

(a) Calculate the length of the base diagonal PR.

(b) Given that edge TR = 18cm, calculate the

vertical height of the pyramid.

Start each of the following questions by drawing a diagram.

7. A ship sails 9km due North and then a further 17km due East.

How far is the ship from its starting point?

8. An aircraft flies 400km due West and then a further 150km due South.

How far is the aircraft from its starting point?

9. A ship sailed 428 ⋅ km due East followed by 74 ⋅ km due South.

How far would it have sailed if it had followed a direct course?

10. A ship sails 9km due North and then a further distance x km due West.

The ship is now 12km from its starting point. Calculate x.

11. How long is the diagonal of a square of side 11mm?

A

B C

D

E

F G

H

11cm

4cm

5cm

P

Q R

S

T

16cm

12cm

18cm

33

12. A rectangle measures 14cm by 9cm. Calculate the length of its diagonals.

13. A ladder of length 5 metres leans against a vertical wall with the foot of the ladder 2 metres

from the base of the wall. How high up the wall does the ladder reach?

14. A ladder is placed against a vertical wall. If the distance between the foot of the ladder and the

wall is 81⋅ metres, and the ladder reaches 4 metres up the wall, calculate the length of the

ladder.

15. A circle has a diameter of 20cm.

A chord is drawn which is 6cm from the centre of the circle.

Calculate the length of the chord.

16. A circle has a diameter of 12cm.

A chord is drawn which is 5cm from the centre of the circle.

Calculate the length of the chord.

17. The room shown opposite has two parallel sides.

Using the given dimensions calculate the perimeter of

the room.

18. Calculate the length of the banister rail shown in

the diagram if there are 6 stairs, and if each tread

measures 25cm and each riser 20cm.

Give your answer in metres.

6 m

64 ⋅ m

3 m

This diagram may

help you with

questions 15 and

16

chord

radius

1 m

1 m x

34

19. A solid cone has a base radius of 741⋅ cm and a vertical height of 322 ⋅ cm. Calculate the cone's

(i) slant height;

(ii) total surface area;

(iii) volume.

Give all answers correct to 3 significant figures. [ Surface Area = hrVrsr 2

3

12 ; πππ =+ ]

20. Use the converse of Pythagoras Theorem to prove that these triangles are right angled.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

cm322 ⋅

cm741⋅

6 cm

8 cm

8 m

15 m

24 mm 7 mm

5 cm

7∙9 cm

2 cm 9 cm 8∙3 m 12∙1 m

2∙1m

5∙8m

92mm

3∙2m

1∙9m

83mm

10cm 17cm

25mm

8∙1 cm

14∙7m

10∙3cm

6∙6m

124mm

2∙8m

35

21. Use the converse of Pythagoras Theorem to decide if these triangles are right angled or not.

(a) (b) (c)

(d) (e) (f)

3.2 Applying Pythagoras’ Theorem - EXAM QUESTIONS

1. In a switch mechanism lever AB rotates around A until it rests against the rod CD.

Point B touches rod CD at E.

AB = 11cm and AC = 8⋅4cm as shown.

For the switch to work the distance from C to E

must be more than 7cm.

Will this switch mechanism work?

Your answer must be accompanied by appropriate working and explanation.

2. The capital letter ‘M’ can be formed using straight lines as shown below.

Calculate the total length of the lines forming the letter.

A

B

C

D

E 11cm

8⋅4cm

8cm

10cm

8∙9m 15 m 1∙9 m

0∙6m 7cm

17 m

2∙0 m 16cm

15 cm

38 cm

2∙6 mm

18∙7 m

57 cm 3 mm

12∙9 m 68∙5 cm

22∙7 m

4 mm

36

3. A wall hanging is decorated with strips of sequences as shown in the diagram.

The sequences are represented by broken lines. ( ).

The shorter stripes are half the length of the longest strip.

Calculate the total length of sequences required to make this wall hanging.

4. The side view of a water trough is as shown in the diagram. The depth of it must by at least

11cm.

Is this container acceptable? Show working and give a reason for your answer.

5. EFGH is a rhombus. EG is 10cm and HF is 18cm.

Calculate the perimeter of the rhombus.

30cm

15cm

46cm

32cm

dcm 18cm

E

F

G

H

18cm

10cm

37

6. A special stage is being built for an outdoor concert. It has to be 20 metre wide, 2 metres high

and has a ramp on one side.

Special non-slip matting has to be laid along the stage and down the ramp.

The cost of the matting is £34 a metre and it is sold in complete metres.

Calculate the cost of the matting.

7. I have just built a new patio area in my garden.

The diagram shows the measurements of it.

I am going to put a low fence round its perimeter.

Calculate the length of fencing that I will require.

8. The opening of a tent is triangular in shape. The height of the tent is 2·5 metres and the sloping

edge of the tent is 3 metres long.

Calculate the length of the base of the tent.

20m

27∙5m

2m

6m

3⋅7m

5⋅5m

2∙5 m 3 m

b m

38

9. An isosceles triangle has its longest side 11 cm

and height 8 cm.

Find the perimeter of the triangle.

10. Two boys, Scott and Callum are playing football.

At one point Callum (C) is 25 m due east of Scott (S). The ball is at position B.

The positions of the 2 boys in relation to the ball are shown in the diagram.

Callum is 14 m away from the ball and angle SBC = 90

o.

Calculate how far Scott is away from the ball. Give your answer correct to 1 decimal place.

S C

B

25 m

14 m

S C

B

11 cm

8 c

m

39

11. The stones in an engagement ring are arranged as shown in the diagram.

The diameter of each stone is 4 mm and A, B and C are the centres of 3 of the stones.

(a) Write down the lengths of AB and BC.

(b) Calculate the length of AC.

12. The pattern below is formed from 5 congruent isosceles triangles.

The width of the pattern is 20cm and the equal sides of the triangles measure 8cm.

Calculate the height, h cm, of the pattern.

13. A ramp is being built to provide wheelchair access to a public building.

Find the length of the ramp, r to the nearest centimetre.

A

B

C

20 cm

h cm 8 cm

5 m

0⋅8 m

r

40

14. The Triangle Pizza Company uses this logo on their pizza boxes.

The logo is in the shape of the isosceles

triangle ABC as shown in the diagram below.

Calculate the height, h cm, of the logo.

Give your answer to the nearest centimetre.

15. The diagram shows the logo of the junior section of a sailing club called the ‘Windy Sails’ Club.

It consists of two identical right-angled triangles.

The dimensions of each right-angled triangle are shown in the diagram.

The outline of the logo is sewn on to the sailing suits.

Calculate the length of the stitching needed to sew the complete logo.

A

B C 20cm

18cm hcm

W

S

4cm

5⋅6cm

41

16. Answer the following questions about the diagram opposite. All lengths are centimetres.

(a) Calculate the length of AC.

(b) Calculate the length of ED.

(c) Prove that triangle ACD is right-angled at C.

(d) Hence calculate the length of BD and the area

of triangle ABD correct to the nearest whole number.

17. At a special school a sensory area was being created with a ‘log bridge’ as part of it.

The diagram shows the side view of the bridge.

The supports are 1·95 m high and the ramps at each side are 5 metres.

The overall length of the bridge is 25·74metres.

Use the converse of Pythagoras’ Theorem to check if the supports are vertical or not.

18. Mrs. Donaldson was wrapping up a parcel for her niece. For decoration, she put a cross of

ribbon across the top of it as shown in the diagram.

This diagram shows the top of the parcel.

Each length of ribbon was cut to a length 13·9cm.

Use the converse of Pythagoras Theorem to check if this is long enough. Give a reason for your

answer.

A

B

C

D

E 12

9

20

0418 ⋅

5m 5m

1∙95m 1∙95m

12cm

7cm

25∙74m

42

3.3 Triangles, Chords and Perpendicular Bisectors in Circles

1. In each of the diagrams below AB is a diameter. Find the missing angles in each diagram.

(a) (b) (c) (d)

2. Find the length of the diameter AB in each of the circles below, given the other 2 sides of

the triangle.

3. Use the symmetry properties of the circle to find the missing angles in the diagrams below.

In each diagram AB is a diameter.

(a) (b) (c)

4. Calculate the length of d in each diagram.

(a) (b) (c)

ao

bo 45

o

A B

35o

co

do

A

B

A

B

io

27o

j o

ko

lo

12o

A

A

B

47o

eo

f o

ho

g o

72o

ao

B

bo

50o c

o

f o

A

do

57o

eo

B

B

j o

A

io

28o g

o

ho

k o

lo

mo n

o

oo

d 9 cm

7 cm

B 7cm

A

7cm 8cm A

B

3cm

A

B

4cm

5cm

A

B 2cm

9cm

(a) (b) (c) (d)

d

12 cm

4 cm

d

7∙5 cm 6 cm

43

(d) (e) (f)

5. Find x in each of the triangles below.

(a) (b) (c)

(d) (e) (f)

6. A cylindrical pipe is used to transport

water underground.

The radius of the pipe is 30 cm and

the width of the water surface is 40 cm.

Calculate the height of the pipe above

the water.

d

12

cm

22 cm

40cm

30cm

x

15 cm

53o

x

56o 8 cm

57o

x 15 cm

x

20 cm

37o

xo

2∙4 cm

1∙8 cm x

32 cm

37o

d

20 cm

16 cm

d

2∙4 cm

1∙8 cm

44

3.4 Tangents and Angles

1. Calculate the sizes of the angles marked a, b, . . . . .r, in the diagrams below.

(a) (b)

(c) (d)

2. In each of the diagrams below, PQ is a tangent which touches the circle at R.

Calculate the lengths of the lines marked x.

3. In each of the diagrams below, AB is a tangent which touches the circle at C.

Calculate x for each diagram.

70o a

o

bo

co

120o

do

eo

f o

65o 65

o

go

ho

no

ko

mo

35o q

o

po

45o

r o

B

O

A

C

18 cm 30

o

x

Q

(a) (b) (c)

(a) (b) (c)

A B C

O

18 cm

27 cm

xo

O

A

B

C

10 cm

x 55o

x 25 cm

7 cm

x Q P R

O

x

P Q

O

5 cm

12 cm R P R

O

8 cm

10 cm

45

O O

C

D

E

F 14mm

2m

O

A

B

8cm

90o

140o 35

o

3.5 The Length of an Arc

1. Calculate the length of the arc in each diagram below, giving your answer correct to 1d.p.

(a) (b) (c)

2. Calculate the perimeter of each sector in Question 1. Giving your answers correct to 1 d.p.

3. Find the length of the minor arc AB in each of the following circles, giving your answers correct

to 1 d.p.

(a) (b) (c) (d)

(e) (f) (g) (h) h.

4. Calculate the length of the major arc in the circles shown in Question 3, giving your answers

correct to 1 d.p.

7 cm

A

A

A

A

A

A

A

B

B

B B

B

B

B

90o

90o

120o

80o

140o

150o 72

o

O O

O

O

O O

O

5 cm 3 cm

9 cm

2 cm

12 cm

8 cm

10 cm

A A B

30o

O

7cm

46

O O

C

D

E

F 14mm

2m

O

A

B

8cm

90o

140o 35

o

3.6 The Area of a Sector

1. Calculate the area of the sector in each diagram below, giving your answer correct to 3

significant figures.

(a) (b) (c)

2. Calculate the area of minor sector OAB in the circles shown below, giving your answers correct

to 3 significant figures.

(a) (b) (c) (d)

(e) (f) (g) (h)

3. Calculate the area of the major sector for the circles in Question 2, giving your answers correct

to 3 significant figures.

4. The length of minor arc CD is 7⋅33 cm.

Calculate the area of the circle.

C D

O

120o

7 cm

A

A

A

A

A

A

A

B

B

B B

B

B

B

90o

90o

120o

80o

140o

150o 72

o

O O

O

O

O O

O

5 cm 3 cm

9 cm

2 cm

12 cm

8 cm

10 cm

A A B

30o

O

7cm

47

3.7 Properties of Circles - EXAM QUESTIONS

1. The diagram shows a section of a cylindrical

drain whose diameter is 1 metre. The surface

of the water in the drain AB is 70 cm.

` (a) Write down the length of OA.

(b) Calculate the depth of water in the pipe, d.

(Give your answer to the nearest cm.)

2. The diagram shows a section of a disused

mineshaft whose diameter is 2 metres. The

surface of the water in the shaft, AB, is 140 cm.

(a) Write down the length of OB.

(b) Calculate the depth of water in the pipe, x.

(Give your answer to the nearest cm.)

3. A pool trophy is in the shape of a circular disc with

two pool cues as tangents to the circle.

O

A B

d

The radius of the circle is 6 cm and

the length of the tangent to the point

of contact (AB) is 9 cm.

The base of the trophy is 3 cm.

Calculate the total height of the

trophy, h, to the nearest centimetre.

h

6 cm

9 cm

3 cm

O

A

B

8Perfect Pool

2003

O

A

B B

x

A

48

4. A circular bathroom mirror,

diameter 48 cm, is suspended from

the ceiling by two equal wires from

the centre of the mirror, O.

The ceiling, AB, is a tangent to the

circle at C. AC is 45 cm.

Calculate the total length of wire used to hang the mirror.

5.

AC is a diameter and O is the centre

of the circle shown opposite. CD is a tangent to the

circle with C the point of contact.

If ∠BCD = 54o, find the size of ∠CAB.

6. A bowling trophy consists of a glass circle set into a rectangular wooden plinth as shown in the

diagram. The diameter of the circle, centre O, is 8cm and the height of the plinth is 3 cm.

The width of the glass at the plinth is 6cm.

Calculate the height, h cm, of the trophy.

7. In the diagram triangle ABC is

isosceles and BD is a diameter of the circle.

Calculate the size of angle ACD.

A B C

O 48 cm

3cm

hcm

6cm

O

46o

A

B C

D

A

O

C

B

54o

D

49

8. A and B are points on the circumference of a circle

centre O. BC is a tangent to the circle.

Angle ABC = 66o.

Calculate the size of angle AOB.

9. In the diagram shown, BD is a tangent to

the circle centre O.

Angle BAC = 28o.

Calculate the size of angle CBD.

10. The diagram shows a circle with centre O. ST is a tangent to the circle with point

of contact Q. ∠PQT = 56o.

(a) Calculate the size of ∠POQ.

(b) Hence calculate the length of the

major arc PQ given that the radius

of the circle is 14cm.

11. The sign outside a pet shop is formed from part of a circle.

The circle has centre O and radius 26cm.

66o

A

O

B

C

P

O

Q

T

56o

S

A

B

C O D

28o

50

Given that the line AB = 48cm, calculate the width, w cm, of the sign.

12. The Pot Black Snooker Club has this sign at its entrance.

It consists of 10 circles each with radius 8cm.

Calculate the height, h cm, of the sign.

13. The line DF is a tangent to the circle centre O

shown below. E is the point of contact of the tangent.

Given that angle CEF is 38˚, calculate the size

of angle EOC.

O

26cm

A

B

w cm

Pets'n

Us

P O T

L B A C K

hcm

O

38º

C

E D F

51

14. The circle in the diagram has centre

O and radius 6cm.

R is the point of contacT of the

tangent PQ.

Given that OQ = 10cm calculate

the length of RQ.

15. A child’s toy is in the shape of a sphere with a duck and some water inside.

As the ball rolls around the

water remains at the same level.

The diagram opposite shows the cross section when the

sphere has been halved.

Given that the radius of the sphere is 6 cm and that

the depth of the water is 4cm, calculate the width of

the water surface (w cm).

4cm

w cm

O

P

R

Q

52

3.8 Arc Length and Sector Area - EXAM QUESTIONS

Give your answers correct to 3 significant figures unless otherwise stated.

1. Calculate the area of the sector shown in the diagram, given that it has radius 6·8cm.

2. A table is in the shape of a sector of a circle with radius 1·6m.

The angle at the centre is 130o as shown in the diagram.

Calculate the

perimeter of the table.

3. The door into a restaurant kitchen swings backwards and forwards through 110o.

The width of the door is 90cm.

Calculate the area swept out by the door as it swings back and forth.

O

42o

135o

O

42o

130o

1∙6m

90cm

110o

53

P

Q R

S

4. The YUMMY ICE CREAM Co uses this logo.

It is made up from an isosceles triangle and a sector of a circle as shown in the diagram.

• The equal sides of the triangle are 6cm

• The radius of the sector is 3·3cm.

Calculate the perimeter of the logo.

5. A sensor on a security system covers a horizontal area in the shape of a sector

of a circle of radius 3·5m.

The sensor detects movement in an area with an angle of 105º.

Calculate the area covered by the sensor.

6. A biscuit is in the shape of a sector of a circle with

triangular part removed as shown in the diagram.

The radius of the circle, PQ, is 7 cm and PS = 1·5

cm.

Angle QPR = 80o.

Calculate the area of the biscuit.

100o

6cm

105º

54

7. Two congruent circles overlap to form the symmetrical shape shown below. Each circle has a

diameter of 12 cm and have centres at B and D.

Calculate the area of the shape.

8. A sector of a circle with radius 6cm is shown opposite.

Angle AOB �x=

If the exact area of the sector is 4π square centimetres,

calculate the size of the angle marked x.

9. A hand fan is made of wooden slats with material on the outer edge.

(a) Calculate the area of material needed for the hand fan.

(b) Calculate the perimeter of the shaded area in the diagram above.

A

O

B

6cm

�x

15cm 9cm

105˚

A

B

C

D

55

O 16cm

20cm 120

o

A

B

10.

The area of the shaded sector is 5Z024 cm2.

Calculate the area of the circle.

11. The area sector OPQ is 78.5 cm2.

Calculate the size of angle xo.

of the circle.

12. A school baseball field is in the shape of a

sector of a circle as shown.

Given that O is the centre of the circle, calculate:

(a) the perimeter of the playing field;

(b) the area of the playing field.

13. In the diagram opposite, O is the centre of two

concentric circles with radii 16cm and 20cm as shown.

Angle oAOB 120= .

Calculate: (a) The perimeter of the shaded shape.

(b) The shaded area.

O

P Q

72o

O

P Q

10 cm xo

O 80

o

80m

56

14. A Japanese paper fan is fully opened when angle oPQR 150= as shown.

(a) Using the dimensions shown in diagram 1, calculate

the approximate area of paper material in the fan.

(b) Decorative silk bands are placed along the edges

as shown in diagram 2, calculate the approximate total

length of this silk edging strip.

15. A grandfather clock has a pendulum which travels along an arc of a circle, centre O.

� The arm length of the pendulum is 60cm.

� The pendulum swings from position OA to OB.

� The length of the arc AB is 21cm.

Calculate the size of angle AOB to the nearest degree.

16. The shape opposite is the sector of a circle, centre P, radius 20m.

The area of the sector is 2251 ⋅ square metres.

Find the length of the arc QR.

36cm

19cm

150o

diagram 1

diagram 2

P

R Q

A B

O

P

Q

R

20m

20m

57

17. A metal strip has been moulded into an arc of a circle of radius 155 centimetres which subtends

an angle of 82o at the centre of the circle as shown in the diagram below.

The same strip of metal has now been remoulded to form an arc of a circle of radius

210 centimetres as shown.

Calculate the size of x, the angle now subtended by the metal strip.

18. Draw a diagram to help you answer these questions.

(a) A circle, centre O, has an arc PQ of length 40cm.

If the diameter of the circle is 80cm, calculate the size of angle POQ

correct to 1 d.p.

(b) A circle, centre O, has a sector EOF with an area of 50cm2.

If the radius of the circle is 8cm, calculate the size of angle EOF correct to 1 d.p.

(c) An arc AB on a circle, centre O, has a length of 16mm.

If angle AOB = 75o , calculate the radius of this circle.

(d) A sector of a circle has an area of 12cm2. If the angle at the centre is 60

o,

calculate the diameter of the circle correct to 2-decimal places.

155cm 210cm

82o x

o

Metal strip

58

Section 4 - Statistics

4.1 Five-figure summaries and Boxplots

1. For each of the data sets below make a five figure summary and calculate the

interquartile range (IQR).

2. For each of the data sets below draw a boxplot and calculate the semi-interquartile range

(SIQR).

(a) 2 4 4 6 7 8 10 14 15

(b) 29 30 32 33 34 37 40

(c) 17 19 20 22 23 25 26

(d) 0 0 0 1 1 2 2 2 3 3 4

(e) 1·8 1·8 2·8 2·9 4·0 4·0 4·0 4·7 5·1 5·2 5·3

(f) 0·13 0·18 0·18 0·19 0·25 0·26 0·29 0·29 0·30 0·31 0·33 0·39

(g) 133 136 136 138 140 141 143 145

(h) 371 375 376 379 380 384 385 387 389 390

(i) 57 58 58 60 63 67 67 69 82 85 86 90

(j) 11 11 11 12 13 14 15 15 16 18 20

.

(a) Ages of audience members at an opera:

47 56 58 48 60 65 50 52 61 53 63

(b) Pupil scores in a Maths test:

12 20 27 15 35 16 26 34 38 24 26

(c) Weekly earnings of a part-time shop assistant (£):

149 165 154 167 170 179 151 168 158

(d) Number of occupants in each flat in a new housing development:

1 8 3 1 2 5 3 1 4 3 2

(e) Number of newspapers sold daily in a shop:

108 114 132 95 144 120 116 125 172 188 155 160

(f) Cost of a can of Irn Bru in different shops (pence):

65 74 59 43 63 52 48 63 67 85 92 48

(g) Number of Smarties in a sample of giant Smartie tubes:

190 165 174 187 166 172 184 190 166 183 180

59

3. For each question, create a table of results (Median and IQR) and write two sentences

comparing the data.

4. Here are two sets of marks for a French test.

Draw a box plot for each class and compare the results.

5. A company that manufactures shoelaces spot checks the length (in cm) of the laces. Here are

the results for two different production lines.

(a) Draw a box plot for line A and line B.

(b) Which is the better production line? (Give a reason for your answer)

6. Two sixth year classes take part in a Sponsored Fast for Famine Relief. The number of hours

each pupil lasted is shown below.

Show each class on a box plot and comment on any differences.

Line A 26.8 27.2 26.5 27.0 27.3 27.5 26.1 26.4 27.9 27.3

Line B 26.8 26.7 27.1 27.0 26.9 27.0 27.3 26.9 27.0 27.3

6C1 20 22 21 20 22 20 22 20 20 24 21 22 23 22 22 23

6C2 15 20 24 23 22 24 18 24 22 23 24 17 20 24 24 20

73 95 80 72 85 90 91 88 91 93

83 76 93 75 88 94 88 91 91 75

Class 5B

98 94 92 78 88 78 82 98 68 66

100 96 84 86 84 94 86 92 82 100

Class 5A

(a) Ages of visitors to travel agent websites: (b) Test scores for S3 pupils in Maths and English:

(c) Resting heart rates of adults: (d) Hospital operation waiting times:

60

7. Mr. Khan timed how long, in seconds, it took each of his class to complete an exercise.

The results are shown below.

a) Calculate the mean and the median time for the pupils to complete the exercise.

Which average, in your opinion, best represents the data?

b) Illustrate the data in a boxplot.

8. The weights, in kilograms, of 20 new-born babies are shown below.

a) Find the median and modal weights and the range.

b) Draw a boxplot to illustrate the data.

9. The table shows the marks (out of 10) achieved by pupils in a class test.

a) Calculate the mean, median and modal test scores.

Which average, in your opinion, best represents the data?

b) Draw a boxplot of the test scores.

What conclusion might the teacher draw from the results about the overall performance

and progress of the class?

10. A passage was picked at random from a book and the number of letters in the first 100 words

was counted.

a) Calculate the mean, median and mode.

b) Repeat the exercise with a book of your choice and compare your results with the table

above in a comparative boxplot. Comment on the results.

300 480 216 311 419 333 281 295 308 276

402 343 398 290 364 378 399 294 401 300

mark 0 1 2 3 4 5 6 7 8 9 10 total

frequency 1 0 1 3 3 2 3 5 7 4 3 32

2.8 3.4 2.8 3.1 3.0 4.0 3.5 3.8 3.9 2.9

2.7 3.6 2.5 3.3 3.5 4.1 3.6 3.4 3.2 3.4

letters 1 2 3 4 5 6 7 8 9 10

frequency 4 12 30 24 17 5 2 3 3 1

61

4.2 Standard Deviation

1. Calculate the mean and standard deviation for the following sets of data.

2. A third year pupil conducting an experiment with a die got the following results

(a) Show these results in a frequency table

(b) Use your table to calculate the mean and standard deviation.

(a) 20 21 19 22 21 20 19 20 21 20

(b) 303 299 306 298 304 307 299 302 305 299 300

(c) 15·3 14·9 15·1 15·2 14·8 14·7 15·1 14·8 15·0 15·0

(d) 87 89 84 88 89 87 86 87 86 87

(e) 48 73 29 82 54 43 95 41 92 71

(f) 4·4 4·6 4·8 4·0 4·2 4·3 4·5 4·7 4·9 4·1

(g) 0·2 0·3 0·4 0·2 0·2 0·0 0·4 0·1 0·2 0·3

(h) 40 40 39 38 38 40 40 42 40 39

6 1 1 4 4 2 2 6 5 6

1 1 1 5 1 4 2 3 4 6

1 4 4 1 5 4 4 3 6 2

5 3 5 6 3 2 6 5 5 2

3 1 4 5 2 4 1 4 4 3

62

3. A company that manufactures shoelaces spot checks the length (in cm) of the laces.

Here are the results for two different production lines.

Calculate the mean and standard deviation and comment on any differences between line A and

line B.

4. The running times, in minutes, of films shown on television over a week are as follows.

Calculate the mean and standard deviation.

5. The temperatures, in oC, at a seaside resort were recorded at noon over a 10-day period.

Calculate the mean and standard deviation.

6. John James plays golf with his brother Joe each month. They keep a note of their scores.

Calculate the mean and standard deviation and comment on John’s and Joe’s performance over

the year.

110 95 135 70 100 125 140 105 95 105

95 95 110 90 110 100 125 105 90 120

19 20 19 17 21 18 19 24 25 28

John 74 73 74 73 71 73 72 75 73 73 72 73

Joe 68 74 70 67 80 81 69 68 79 67 70 71

Line A 26·8 27·2 26·5 27·0 27·3 27·5 26·1 26·4 27·9 27·3

Line B 26·8 26·7 27·1 27·0 26·9 27·0 27·3 26·9 27·0 27·3

63

7. The weekly takings in small store, to the nearest £, for a week in December and March are

shown below

Calculate the mean and standard deviation and comment on any differences.

8. Two sixth year classes take part in a Sponsored Fast for Famine Relief. The number of hours

each pupil lasted are shown below.

Calculate the mean and standard deviation for each class and comment on how well each class

did.

December 2131 2893 2429 3519 4096 4810

March 1727 2148 1825 2397 2901 3114

6C1 20 22 21 20 22 20 22 20 20 24 21 22 23 22 22 23

6C2 15 20 24 23 22 24 18 24 22 23 24 17 20 24 24 20

64

4.3 Scatter Graphs

1. Using the words positive, negative or no relation, describe the correlation in each of the

diagrams below.

(a) (b) (c)

2. What do the diagrams tell you about the correlation between the two variables

involved ?

a. b. c.

x

y

x

y

x

y

rainfall

umbrel

la

hair

pocket time

speed

65

3. A random survey of 20 pupils gave the following results

Draw a scatter diagram to find out if there is a correlation between

(a) age and height

(b) height and weight

(c) age and weight

(d) age and amount of cash carried.

4. Copy these graphs and use your ruler to draw what you think is the line of best fit.

Pupil 1 2 3 4 5 6 7 8 9 10

Age 16 17 14 17 14 12 12 16 18 15

Height(cm) 182 199 171 200 183 159 170 179 198 180

Weight (kg) 71 78 69 66 54 60 46 72 76 63

Cash carried (£) 4·23 10·90 25·50 1·43 2·98 6·24 3·18 0·72 1·98 0·25

Pupil 11 12 13 14 15 16 17 18 19 20

Age 18 18 17 16 11 11 13 12 14 14

Height (cm) 190 179 187 169 160 151 150 171 170 182

Weight (kg) 68 75 77 76 49 41 55 53 60 67

Cash carried 12∙06 4∙31 2∙38 12∙30 2∙15 4∙12 2∙71 0∙40 1∙80 3∙10

66

5. For the following sets of data, draw a scatter diagram and find the equation of the line of

best fit.

(a) (b)

(c) (d)

(e) (f)

6. The height of a plant measured over five days is shown below.

(a) Plot the points and draw the best fitting straight line through them

(b) Work out the equation of the line.

(c) Use your line to estimate the height after 1½ days.

7. The table shows the results of an experiment.

Plot the points, draw a best fitting straight line and find its equation.

x 1 2 3 4 5

y 5 7 8 10 12

x 1 2 3 4 5

y 2 2∙5 2∙5 3∙5 3

x 6 7 8 9 10

y 1 2 4 4∙5 6

x 1 2 3 4 5

y 8 6 5 4 2

x 1 2 3 4 5

y 8 10 8 5 3

x 5 6 7 8 9

y 6 5∙5 5∙4 5∙5 5

Days (D) 1 2 3 4 5

Height (H) 1∙6 1∙9 2∙5 3∙4 3∙5

x 1 2 3 4 5 6

y 9∙2 12∙0 18∙3 19∙0 25∙1 30∙2

67

Time, min (T) 1 3 5 7 9

Temperature (oC) 66 61 57 53 50

Number of meals 10 20 30 40 50 60

Cost in £ 188 192 220 216 232 248

8. The results below show the length of a spring when a force is applied.

(a) Plot the points and draw the best fitting straight line through them.

(b) Find the equation of the line.

(c) Use your graph to estimate the length when a force of 4·5 is applied.

9. The following table gives the temperature of a bottle of water as it cools.



(c) Use your graph to estimate the temperature after 2½ minutes.

10. The following table shows the speed of a car accelerating from rest.



(c) Use your graph to estimate the speed after 10 seconds.

11. A restaurant manager finds that the cost of running his restaurant depends on the

number of meals served.



(c) Use your equation to estimate the cost when 35 meals are served.

Force (F) 1 2 3 4 5 6

Length (l) 3∙0 3∙9 4∙8 5∙9 6∙9 8∙1

Time (secs) 0 2 6 8 12 16

Speed (mph) 0 14 44 56 82 110

68

12. The results of an experiment are shown in the table below.


(b) Find the equation of the line. (c) Use your graph to estimate R when V is 0.8.

4.4 Mean and Standard Deviation - EXAM QUESTIONS

1. The weights of 6 plums are

40·5g 37·8g 42·1g 35·9g 46·3g 41·6g

(a) Calculate the mean and standard deviation.

The weights of 6 apples are

140·5g 137·8g 142·1g 135·9g 146·3g 141·6g

(b) Write down the mean and standard deviation.

2. During a recent rowing competition the times, in minutes, recorded for a 2000 metre race were

27 ⋅ 37 ⋅ 37 ⋅ 57 ⋅ 67 ⋅ 48 ⋅

(a) Calculate the mean and standard deviation of these times. Give both answers correct to

2 decimal places.

(b) In the next race the mean time was 767 ⋅ and the standard deviation was 490 ⋅ .

Make two valid comments about this race compared to the one in part (a).

3. 6 friends joined "Super Slimmers", a weight loss class. Their weights were recorded and the

results are shown below.

65kg 72kg 74kg 81kg 90kg 98kg

(a) Calculate the mean and standard deviation of the weights.

After 6 weeks the mean weight was 74kg and the standard deviation was 8·6

(b) Compare the mean and standard deviation of the friend's weights.

V 0 0∙35 0∙6 0∙95 1∙2 1∙3

R 0∙60 0∙48 0∙33 0∙18 0∙11 0∙05

69

4. Stewart and Jenni complete a crossword puzzle every day. Here are the times (in minutes) that

Stewart took to complete it each day for a week.

63 71 68 59 69 75 57

(a) Calculate the mean and standard deviation for Stewart's times.

Every day Jenni took exactly 5 minutes longer than Stewart to complete the puzzle.

(b) Write down Jenni’s mean and standard deviation.

5. The number of hours spent studying by a group of 6 student nurses over a week were

20 23 14 21 27 24

(a) Calculate the mean and standard deviation of this data.

(b) A group of student teachers had a mean of 21·5 and a standard deviation of 6.

Make two valid comments to compare the study times of the 2 groups of students.

6. Barbara is looking for a new 'A-Pod' and searches for the best deal.

The costs of the 'A-Pod' are shown below.

£175 £185 £115 £87 £150 £230

(a) Calculate the mean and standard deviation of the above data.

(b) A leading competitor, the 'E-Pod', has a mean price of £170 and a standard deviation of

26·7. Make two valid comparisons between the 2 products.

7. In Bramley’s Toy Shop there are 6 styles of teddy bear. The price of each is shown below.

£19 £25 £17 £32 £20 £22

(a) Calculate the mean and standard deviation of these prices.

In the same shop the prices of the dolls have a mean of £22.50 and a standard

deviation of 2·3. .

(b) Compare the two sets of data making particular reference to the spread of the prices.

70

4.5 Scatter Graphs – EXAM QUESTIONS

1. A selection of the number of games won and the total points gained by teams in the

Scottish Premier League were plotted on this scattergraph and the line of best fit was

drawn.

(a) Find the equation of the line of best fit.

(b) Use your equation to calculate the points gained by a team who won 27 matches.

2. The graph below shows the temperature and sales of ice cream for one week during the summer.

(a) Make a copy of the graph and draw the line of best fit on it.

(b) Find the equation of the best-fit line.

Nu

mb

er

of

ice

cre

am

s so

ld.

10

20

30

40

50

0

0 5 10 15 20 25

Temperature (Celsius)

T

N

Wins

W

P

4 8 12 16 20

Po

ints

10

20

30

40

50

60

70

80

71

3. The scattergraph shows the marks gained in Physics and Maths by a group of college students.

Which of the following statements best describes the correlation between the 2 sets of marks?

A strong positive correlation

B strong negative correlation

C weak positive correlation

D weak negative correlation

4. A group of smokers were asked how many cigarettes they smoked in a day and how many chest

infections they had suffered in the last ten years. The results are shown in the scattergraph with

the line of best fit drawn.

(a) Comment on the correlation between the 2 sets of data.

Maths

Ph

ysi

cs

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 45

Number of cigarettes smoked in a day (C )

Nu

mb

er

of

che

st in

fect

ion

s in

la

st 1

0 y

ea

rs (I

)

72

(b) Find the equation of the line of best fit.

5. The graph below shows the relationship between the number of hours (h) a swimmer

trains per week and the number of races (R) they have won.

A best fitting straight line has been drawn.

(a) Use information from the graph to find the equation of this line of best fit.

(b) Use the equation to predict how many races should be won by a swimmer who trains 22

hours per week.

R

h

•

•

• •

•

• • •

•

•

•

• •

Number of

races won

(R)

Number of hours training per week

(h)

0 5 10 15

10

5

• •

73

Section 5 – Percentages and Fractions

5.1 Working with Percentages

1. These amounts have been reduced by 15%. What was the original amount?

(a) £85 (b) 212·5 mm (c) £63.75

(d) 25·5 litres (e) 357 miles (f) 435·2 m

(g) 1 275 km (h) £4

462.50 (i) 10

200 m

(j) 605·2 cm (k) £658.75 (l) 76·5 kg

2. These amounts have been increased by 22%. What was the original amount?

(a) £26.84 (b) £54.90 (c) £87.84

(d) 103·7 ml (e) £21.35 (f) 122 cm

(g) 3 111 m (h) 10

370 km (i) 68·32 m

(j) £13 664 (k) 118·95 litres (l) £7

564

3. A shop is having a sale. There is ‘20% OFF’. Calculate the original cost of these items.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

£32 £52 £20

£340

£5.60

£239.20

£1 640 £20.80

£21.20

74

4. A company gave their workers a 7% wage rise. Calculate how much each of these people were

earning each year before the increase.

(a) Irene £13 375 per year (b) Billy £19

324.20 per year

(c) Peter £26 322 per year (d) Isobel £40

060.80 per year

(e) Stewart £481.50 per week (f) Jackie £1 820 per month

(g) Alan £75 per week (h) Anne £1 200 per month

5.2 Working with Percentages - EXAM QUESTIONS

1. A gym’s membership has increased by 17% over the past year. It now has 585 members.

How many members did it have a year ago?

2. The number of school pupils not wearing school uniform has decreased by 72% since the start

of last year. There are now 42 pupils not wearing school uniform.

How many pupils were not wearing school uniform at the start of last year?

3. My house has increased in value by 15% in the last two years. It is now worth £230 000.

How much was it worth 2 years ago?

4. I bought a new car in September of last year. By this September the car had

depreciated by 20% and was now worth £9600.

How much did I pay for the car last September?

5. Jane bought a painting in an auction. Unfortunately the painting depreciated in value by 7% and

is now worth £4185.

How much was the painting worth when it was bought?

6. An antique chair has increased in value by 34% since it was bought.

It is now worth £3 484. What was it worth when it was bought?

75

5.3 Appreciation and Depreciation

1. For each of the investments below , calculate

(i) the amount due at the end of the term

(ii) the total interest

Bank/ Building Society Amount

Invested (£)

Rate of interest

(per year)

Number of

Years

(a) Hamilton Bank 2000 8 % 2

(b) Allied Friendly 5000 6 % 3

(c) Northern Hill 4800 7 % 2

(d) Highland Bank 3500 7·5 % 3

(e) Church National 1600 5·5 % 4

(f) Southern Rock 1750 11 % 3

(g) London Savings Bank 20 000 6% 3

(h) Bath & Eastern 18 000 8·5% 2

(i) Royal Bank of Britain 50 000 9% 3

(j) Bingford & Bradley 400 4·8% 2

2. At the beginning of the year, Mr. Bradford borrows £5000 from the bank. The rate of compound

interest is 8%. He agrees to pay back £108 per month.

Calculate how much he still owes at the end of the second year.

3. The Smiths buy a house for £60,000. If it appreciates in value at the rate of 9% per year, how

much will it be worth in 5 years time?

4. Amanda wins some money and decides to spend £200 on some jewellery. If it appreciates at the

rate of 2% per year, how much will the jewellery be worth 3 years from now?

5. In 1990 the world population was estimated to be 5300 million, and was increasing at the rate of

1·7% per annum.

What will the population be in the year 2000? (answer to 2 significant figures)

76

6. Peter buys a car for £3000. If it depreciates at the rate of 20% per annum, how much will he be

able to sell it for in 3 years time?

7. Brian buys a new car costing £12600. It depreciates in value by 30% in the first year

and by 20% each year after that.

How much will he be able to trade it in for in 3 years time?

8. Each year a factory’s machinery depreciates by 25% of its value at the beginning of the year.

The initial value of the machinery was £360 000.

(a) What was the value of the machinery after 1 year

(b) The machinery was to be scrapped at the end of the year when its value fell below half

its original value. After how many years should the machinery be scrapped?

5.4 Appreciation and Depreciation - EXAM QUESTIONS

1. Joseph invests £4500 in a bank that pays 6·4% interest per annum.

If Joseph does not touch the money in the bank, how much interest will he have gained after 3

years?

Give your answer to the nearest penny.

2. Jane bought a painting in an auction for £32 250.

Unfortunately the painting depreciated in value by 7% each year.

Calculate how much the painting was worth after 2 years.

Give your answer to 3 significant figures.

3. (Non calculator) Last year (2008) a company made a profit of £1 000

000. This year (2009) it

expects to increase its profit by 20% and by 2010 to have increased it by a further 25%.

Calculate the profit the company expects to make in 2010.

4. A patient in hospital is given 200mg of a drug at 0900. 12% of the amount of the drug at the

beginning of each hour is lost, through natural body processes, by the end of that hour.

How many mg of the drug will be lost by 1200?

77

5. Holly buys an antique watch costing £1200. The watch appreciates in value by 3·7% per annum.

How much will the watch be worth in 4 years time?

(Give your answer to the nearest pound.)

6. A local council recycles 28 000 tonnes of glass each year. After a publicity campaign they

expect to increase the amount of glass recycled by 12% each year.

(a) How much glass do they expect to recycle in 3 years time?

Give your answer correct to 3 significant figures.

(b) The council aim to double the amount of glass recycled in 6 years.

If this rate is maintained, will the council meet their target?

Give a reason for your answer.

7. (Non calculator) Arthur’s new car cost him £15 000. The value of it will depreciate by 20%

each year.

How much will Arthur’s car be worth when he trades it in for a new one in 2 years time?

8. Barry bought a house last year costing £115 000. This year it is valued at £110

400.

(a) Calculate the percentage decrease in the value of the house.

(b) If the value of the house continues to decrease at this rate what will the house

be worth in a further 3 years time?

Give your answer to 3 significant figures.

9. Marcus invested £3000 in a bank which paid 2·5% interest per year.

(a) Calculate how much money Marcus would have in his account after 3 years.

(b) How long would it take for Marcus’ money to increase by 12%?

10. In 2007 a company made a profit of £45 000. Over the next three years its profit dropped by 3%

each year due to increased manufacturing costs.

Calculate, correct to 3 significant figures, the company's profit in 2010.

78

11. The value of an industrial machine is expected to decrease each year by 14⋅2% of its value at the

beginning of the year.

If it was valued at £15500 at the beginning of 2011, what will its expected value be at the end

of 2013? Give your answer correct to the nearest pound.

12. The membership of the ‘Watch your Weight’ slimming club is 40 000 and is increasing at the

rate of 4% per month.

The membership of ‘World of Slimming’ is 70 000 but is decreasing at the rate of 9% per

month.

(a) Calculate the membership of the ‘Watch your Weight’ club after 3 months,

giving your answer correct to 4 significant figures.

(b) How many months will it take for the membership of the ‘Watch your Weight’ club to be

more than the ‘World of Slimming’?

13. A woman had a Body Mass Index (BMI) of 30. After following a healthy eating

plan she managed to reduce her BMI to 27·6 in 1 month.

(a) Calculate the percentage reduction in her BMI.

(b) If she managed to continue to reduce her BMI by the same percentage in each of the next

3 months, what was her BMI then? Give your answer correct to 3 significant figures.

14. The value of an antique chair increased in value by 12½ % each year.

The chair was bought for £4800. What was its value at the end of 3 years?

15. (Non Calculator) Charlene’s house is valued at £120 000 and is expected to appreciate at the

rate of 10% per annum for the next three years.

If this happens, what will the house be valued at in three years time?

79

16. Three years ago I bought a new car which cost £10 500. An offer from the garage at the time

stated:

“Keep the car for 3 years, return it to us and we will refund half the original cost”

The car depreciated in value by 20% during the first year and by 15% in subsequent years.

By calculating the value of the car after 3 years decide whether the garage’s offer, in this case,

was a good one or not. Give a reason for your answer.

17. A piece of jewellery was bought for £2580 two years ago. Its present value is 65% of its original

price.

(a) What is its present day value?

An expert estimates that it will increase in value at a rate of 12% per annum over the next few

years.

(b) How many years will it take for the jewellery to regain its original value?

18. Bill invested £10 000 in the Dodgy Building Society but his money lost 5% per annum over the

first 2 years.

At the end of this time he decided to move his money to the Goody Building Society which

guaranteed that his money would gain 6% per annum over the next 2 years.

How much did Bill gain or lose over the four years?

19. Chocolate fountains have become very popular at parties.

It takes a minimum of 900g of melted chocolate to operate a fountain properly.

On one occasion 2kg of melted chocolate was added to the fountain.

23% of the remaining chocolate was used every 20 minutes.

Was there still enough chocolate left to operate the fountain properly one hour later?

You must show all working and give a reason for your answer.

20. In 2008 the Portable Phone Company announced that their profits were £850 000. In the next 3

years their profits increased by 4⋅2% each year. How much profit did the company make in

2011? Give your answer to the nearest thousand.

80

5.5 Addition and Subtraction of Fractions

1. Express each sum as a fraction in its simplest form.

(a) 5

3

5

1+ (b)

10

1

5

2+ (c)

8

1

4

3+ (d)

3

2

6

1+

(e) 3

2

9

1+ (f)

4

1

3

1+ (g)

4

1

5

3+ (h)

6

1

4

1+

(i) 8

5

3

1+ (j)

5

2

2

1+ (k)

6

1

4

3+ (l)

7

3

2

1+

(m) 8

1

7

2+ (n)

8

3

5

1+ (o)

7

3

9

2+ (p)

5

3

6

1+


(a) 5

4

5

2+ (b)

10

9

5

2+ (c)

8

5

4

3+ (d)

3

2

6

5+

(e) 3

2

9

5+ (f)

4

3

3

2+ (g)

4

1

5

4+ (h)

6

5

4

3+

(i) 8

7

3

2+ (j)

5

4

2

1+ (k)

6

5

7

3+ (l)

9

5

7

5+

(m) 8

7

7

2+ (n)

8

3

5

4+ (o)

7

3

9

7+ (p)

5

4

8

5+

(q) 4

3

2

1

2

1++ (r)

10

1

5

3

2

1++ (s)

16

3

8

5

2

1++ (t)

4

1

2

1

3

2++

(u) 3

1

4

1

6

5++ (v)

4

3

3

1

12

7++ (w)

8

3

7

1

6

1++ (x)

5

2

3

2

4

3++

3. Express each difference as a fraction in its simplest form.

(a) 4

1

4

3− (b)

6

1

2

1− (c)

3

2

6

5− (d)

6

5

12

11−

(e) 3

2

12

11− (f)

16

1

2

1− (g)

4

1

3

2− (h)

5

2

2

1−

(i) 16

3

8

7− (j)

2

1

5

4− (k)

2

1

4

3− (l)

3

1

12

7−

(m) 5

2

8

5− (n)

5

3

6

5− (o)

7

3

9

7− (p)

3

2

8

7−

81


(a) 4

11

2

11 + (b)

4

31

2

11 + (c)

4

11

8

32 + (d)

6

51

2

13 +

(e) 4

12

8

53 + (f)

4

32

3

25 + (g)

5

31

5

31 + (h)

6

52

8

32 +

(i) 8

32

4

35 + (j)

12

72

3

16 + (k)

6

5

2

13 + (l)

16

3

8

14 +

(m) 5

2

10

72 + (n)

12

13

3

24 + (o)

8

32

16

111 + (p)

3

2

9

75 +

(q) 12

52

4

33 + (r)

2

12

3

25 + (s)

12

11

8

72 + (t)

8

58

16

95 +


(a) 3

23− (b)

12

74 − (c)

8

73− (d)

9

21−

(e) 2

11− (f)

4

33− (g)

4

12 − (h)

6

52 −


(a) 2

11

4

33 − (b)

3

14

8

76 − (c)

4

11

5

42 − (d)

3

11

12

74 −

(e) 4

31

5

45 − (f)

6

51

12

116 − (g)

7

11

3

24 − (h)

6

11

4

33 −

(i) 8

11

3

15 − (j)

2

12

8

55 − (k)

3

14

12

78 − (l)

5

42

10

94 −

(m) 3

14

3

19 − (n)

12

11

6

58 − (o)

10

31

5

28 − (p)

3

14

9

55 −


(a) 4

324 − (b)

7

437 − (c)

3

225 − (d)

3

2510 −

(e) 5

24

7

25 − (f)

6

53

12

77 − (g)

3

11

12

14 − (h)

4

31

5

35 −

82

(i) 6

51

4

13 − (j)

2

12

8

15 − (k)

3

14

4

18 − (l)

5

42

10

34 −

(m) 3

24

3

19 − (n)

4

31

5

22 − (o)

10

71

5

28 − (p)

3

24

9

15 −

8. Tom walked for 18

5kilometres and then walked another 2

5

3km.

How far did he walk in total?

9. A rectangle has length 7

53 cm and breadth

5

21 cm. Calculate its perimeter.

10. Siobhan likes to go to the gym. Last week she went on Monday, Tuesday, Thursday, Friday,

Saturday and Sunday. The table below shows the number of hours she trained on each of the six

days.

MON TUES THURS FRI SAT SUN

2

11

4

11

4

3

5

21

5

4

10

31

How many hours in total did she spend in the gym last week?

11. Billy is a long distance lorry driver. One day he drove for 2

12 hours, had a break and then drove

for another 3

23 hours.

How long did he drive in total?

12. Peter is walking to school. When he reaches half way he meets Mike. They walk a third of the

way together when they meet Anne. They walk the rest of the way together. What fraction of

the journey is this?

13. A group of friends went to a burger bar. 5

2 of them bought a burger,

3

1bought chips and the

rest bought cola/ What fraction of the group bought cola?

83

14. At a school 5

1 of the time is spent in Mathematics classes and

20

3 is spent in English.

(a) What fraction of the time is spent in Maths and English altogether?

(b) If 15

1 of the time is spent in PE, what fraction of the time is spent on all the other

subjects apart from the three subjects already mentioned?

5.6 Multiplication and Division of Fractions

1. Express each product as a fraction in its simplest form:

(a) 7

4

4

1× (b)

10

3

3

1× (c)

7

4

2

1× (d)

8

1

3

2×

(e) 16

1

5

4× (f)

3

2

7

6× (g)

21

10

5

3× (h)

21

4

8

3×

(i) 7

4

32

21× (j)

13

12

9

1× (k)

25

6

16

5× (l)

15

14

7

5×

(m) 35

12

9

7× (n)

48

39

13

12× (o)

9

5

3

2× (p)

8

3

9

2×

(q) 5

3

2

1× (r)

3

2

8

3× (s)

15

8

14

5× (t)

14

5

10

7×

2. Express each product as a fraction in its simplest form:

(a) 3

11

4

11 × (b)

3

21

4

11 × (c)

2

12

2

12 × (d)

3

21

4

31 ×

(e) 5

11

4

13 × (f)

3

22

3

11 × (g)

2

12

15

11 × (h)

5

11

4

33 ×

(i) 52

12 × (j) 4

2

17 × (k)

3

11

7

12 × (l)

7

23

8

52 ×

(m) 8

52

7

44 × (n)

3

13

5

33 × (o)

3

13

5

11 × (p)

2

13

4

12 ×

(q) 2

13

4

32 × (r)

3

23

9

41 × (s)

4

33

5

35 × (t)

5

42

7

11 ×

84

3. Express as a single fraction:

(a) 3

1

4

1÷ (b)

7

2

5

2÷ (c)

4

3

5

4÷ (d)

5

2

7

3÷

(e) 3

5

12

5÷ (f)

3

1

9

5÷ (g)

10

9

5

2÷ (h)

14

11

7

3÷

(i) 3

2

9

4÷ (j)

5

4

5

2÷ (k)

21

20

35

24÷ (l)

20

9

25

6÷

(m) 14

9

21

8÷ (n)

14

19

21

10÷ (o)

44

15

33

20÷ (p)

4

3

5

3÷

(q) 3

2

15

8÷ (r)

24

22

36

11÷ (s)

36

25

33

10÷ (t)

15

2

5

4÷


(a) 4

115 ÷ (b)

2

12

2

17 ÷ (c)

4

31

2

13 ÷ (d)

5

11

10

11 ÷

(e) 4

12

8

31 ÷ (f)

14

11

7

62 ÷ (g)

9

71

3

22 ÷ (h)

16

33

12

51 ÷

(i) 4

12

5

33 ÷ (j)

15

14

24

111 ÷ (k)

25

71

15

113 ÷ (l)

15

8

35

91 ÷

(m) 5

44

20

71 ÷ (n)

12

12

9

44 ÷ (o)

9

13

12

112 ÷ (p)

2

12

3

26 ÷

(q) 5

26

5

25 ÷ (r)

7

31

2

11 ÷ (s)

2

13

5

14 ÷ (t)

9

22

3

21 ÷


(a) 8

3

9

2

4

3×× (b)

6

1

2

15of

11

2× (c)

7

12

5

3

3

1××

(d) 3

18

8

11

3

13 ×× (e)

5

3

8

3416 ×× (f)

19

2

9

12

4

36 ××

(g) 2

17)

2

11

3

13( ÷× (h)

4

3)

5

42of

14

11( ÷ (i)

3

26)

2

15of

11

1( ÷

(j) )7

24of

10

9(18 ÷ (k) )

18

5of

5

17(

4

16 ÷ (l) )

7

26of

11

31(

13

4÷

85

(m) 12

1)

4

33

8

55( ×÷ (n)

21

8)

49

26

16

13( ×÷ (o) )

9

7of

8

3(

24

5÷

(p) )7

410of

2

117(

4

19 ÷ (q)

4

3)

2

11

3

18( ÷× (r)

9

8)

15

73

30

13( ×÷

6. A sack of potatoes weighs 11 kgs.

(a) How many bags each weighing 4

31 kgs can be filled from the bag?

(b) What weight of potatoes would be left over?

7. A twenty – one metre length of fabric is cut into 8

5metre pieces.

(a) How many pieces can be cut?

(b) What length of fabric would be left over?

8. A triangle has base 4

32 cm and height

5

23 cm. Calculate its area.

9. A rectangle measures 4

15 metres by

3

25 metres. Calculate its area.

86

5.7 Mixed Questions on Fractions

1. Anne mixed 3

21 kgs of flour with

4

11 kgs of sugar.

What is the total of flour and sugar?

2. Brendan ran 4

36 km of a 10 km run. How far did he still have to run?

3. At a Christmas party, David drank 4

31 litres of fruit punch. Simon drank

8

7of a litre and John

drank 3

11 litres.

(a) How much fruit punch did they drink altogether?

(b) If the bowl held 8 litres altogether, how much was left over?

4. A garden is rectangular in shape and measures 5

47 metres by

3

210 metres.

(a) Calculate the perimeter of the garden.

(b) Find its area.

5. Billy is a long distance lorry driver. One day he had to drive to Birmingham. He drove for 2

12

hours at a speed of 76km/h and then for 3

23 hours at a speed of 81km/h before arriving at his

destination.

(a) How far did he drive during the first part of his journey?

(b) How far did he drive during the second part?

(c) How far did he travel altogether?

(d) How many hours did it take him in total?

(e) Calculate Billy’s average speed for the whole journey.

87

6. Laura has applied to join the RAF and has to sit an ‘Entrance Test’. Part of it includes some

problems with fractions. Here are the answers that Laura worked out.

(a) 7

3

6

5+ (b)

16

5

8

33 − (c)

22

3

15

11× (d)

102

85

45

34÷

(e) A plank of wood 4

33 metres long is cut up into 5 equal pieces.

How long is each piece?

(f) Each cow in a herd of 25 produces 3

24 litres of milk.

How much milk is this in total?

(i) How many did Laura get correct out of the six questions?

(ii) Write down the correct answers for the ones that Laura got wrong.

13

8

16

13

10

1

27

17

4

3metre

3

2116 litres

88

Section 6 – Similar Shapes

6.1 Linear Scale Factors

1. Each diagram below shows a pair of similar shapes or objects. For each pair …….

(i) state the scale factor (from left to right) (ii) calculate the length marked x .

(a) (b)

(c) (d)

2. Calculate the length of the side marked x in each diagram below.

(a) (b)

3. In the diagram , AB = 28cm,

AC = 24cm and ED = 21cm.

(a) Explain why the triangles ABC and CDE are

similar.

(b) Calculate the length of CD.

(c) Given that the area of triangle ABC is 144 square

centimetres, calculate the area of triangle CDE.

x

x

x

x

4cm

6cm

9cm

30cm

21cm 14cm

18cm

20cm 50cm

280mm

160mm

96mm

A

B

C

D

E

28cm

21cm

24cm

CEDABC ∠=∠

36mm 20mm

54mm

x

x

24cm 30cm

26cm

89

4. Calculate the length of the side marked x in each diagram below.

5. The diagram opposite shows an aluminium pipe frame.

The cross members QS and PT are parallel.

RS = 48cm, QS = 24cm and PT = 32cm as shown.

Calculate the length of ST

6. In the diagram a ladder is laid against two walls as shown.

The higher wall is 61⋅ metres high, and the lower wall is 70 ⋅ metres.

The distance between the two left hand faces of the walls is 90 ⋅ metres.

Calculate the distance between the foot of the ladder

and the lower wall.

40mm

16mm

36mm

x

18cm

x

24cm

8cm

32cm

48cm

24cm

P

Q

R

S

T

0 9m .

0 7m .

1 6m .

90

6.2 Area Scale Factors

1. For each pair of pictures below (i) State the enlargement scale factor for the lengths

(ii) State the scale factor for the areas.

(iii) Calculate the area of the larger shape.

(a) (b)

(c) (d)

(e) (f)

2. For each pair of pictures below (i) State the reduction scale factor for the lengths.

(ii) State the scale factor for the areas.

(iii) Calculate the area of the smaller shape.

(a) (b)

10cm 5cm 70cm2

76mm

19mm

3648mm2

16cm2

5cm

10cm

12mm

36mm

12mm 8mm

14cm

33∙6cm

6cm 24cm

16cm

28∙8cm

96mm2

40mm2

50cm2

22cm2 120cm

2

91

(c) (d)

3. Each pair of shapes below is mathematically similar.

Calculate the area of each right-hand shape.

(a) (b)

(c) (d)

30cm

24cm

150cm2

24mm 18mm

400mm2

8cm 16cm

90mm 30mm

22cm2

1494mm2

20cm

14cm

18mm 12mm

100cm2

32mm2

92

6.3 Volume Scale Factors

1. For each pair of similar pictures below (i) State the enlargement scale factor for the

length.

(ii) State the scale factor for the volumes.

(iii) Calculate the volume of the larger solid.

8cm

4cm

12mm

36mm

12mm 8mm

8cm

19∙2cm

3cm

12cm

16cm

22∙4cm

48cm3

V = 216mm3

72mm3

V = 20cm3

9cm3

400cm3

(a)

(b)

(c) (d)

(e)

(f)

93

2. For each pair of similar pictures below (i) State the reduction scale factor for the

lengths.

(ii) State the scale factor for the volumes.

(iii) Calculate the volume of the smaller solid.

3. Each pair of containers below is mathematically similar.

Calculate the volume of each container with a question mark.

(a) (b) (c)

10cm 5cm

8mm 2mm

368cm3

144mm3

24cm

18cm

V = 912cm3

15mm

9mm

230mm3

(a) (b)

(c)

(d)

10cm 20cm

150ml

? 8cm 6cm

640ml ?

16cm

24cm

2 litres

?

94

6.4 Similar Shapes - EXAM QUESTIONS

1. The diagram below shows two candles. Each candle is in the shape of a cuboid with a square

base.

The length of time each candle will burn is proportional to its volume.

The small candle burns for 36 hours.

Nadia reckons that because the large candle’s measurements are double the small candle’s

measurements then the large candle should burn for 72 hours.

Is she correct? [You should show all working and give a reason for your answer]

2. An international perfume manufacturer prices their bottles of perfume by volume.

The two bottles below, although containing different volumes, are mathematically similar in

shape. Their heights and prices are shown.

The larger of the two bottles is for sale in France.

Assuming the smaller bottle to be priced correctly, determine whether or not the larger bottle

has the correct price tag given that the exchange rate is £1 = 1⋅10 euros.

20cm

12cm 6cm

10cm

£22.00 8cm 9⋅6cm

41.82 euros

95

3. John is looking to buy a new rug for his main room.

The two rugs below are mathematically similar in shape.

He is hoping that the length of the large rug will be enough to make the area of the large rug at

least 72 square feet.

Does the large rug have the required area?

You must show appropriate working with your answer.

4. In the diagram below triangles ABC and ADE are mathematically similar.

BC = 12 cm, DE = 9

cm and AE = 21

cm.

Find the length of CE.

6 ft 7⋅2 ft

8 ft

length

A

B

C

D

E 12 cm

9 cm 21cm

96


7.1 Gradients of Straight Lines

1. (a) Calculate the gradient of each line

in the diagram opposite.

(b) Copy and complete each statement below:

The gradient of any horizontal line is _______________ .

The gradient of any vertical line is _________________.

A line sloping upwards from left to right has a ___________________ gradient.

A line sloping upwards from right to left has a ___________________ gradient.

2. Find the gradients of the lines shown in each of the diagrams below:

(i)

(ii)

(iii)

(iv) (v)

a b

c

d

e

f

g

h

i j

k

l

97

3. Find the gradients of the lines below:

4. Calculate the gradient of the line joining each pair of points below:

(a) (2, 1) and ( 6, 3) (b) (1, 5) and (3, 1) (c) (2, 0) and (4, 6)

(d) (4, 3) and (8, 11) (e) (1, 9) and (3, 1) (f) (7, 3) and (5, 2)

(g) (−2, −3) and (2, 3) (h) (−1, 2) and (5, −1) (i) (−4, 2) and (4, −4)

(j) (−6, −2) and (−5, 3) (k) (4, −3) and (6, 5) (l) (−2, 3) and (0, −2)

5. Calculate the gradient of the line joining each pair of points below:

(a) A(−2, 6) and B(8, 8) (b) C(3, −3) and D(4, −1)

(c) E(5, −9) and F(8, −15) (d) G(0, 6) and H(5, 11)

(e) I(−1, −3) and J(7, −9) (f) K(−4, 0) and L(−1, 5)

(g) M(2, 2) and N(−3, 4) (h) P(5, −1) and Q(−2, 10)

(i) R(−3, −5) and S(8, −4) (j) T(4, −6) and U(7, −2)

(k) V(5, −6) and W(−2, 6) (l) X(−1, 7) and Y(−2, 6)

(m) J(6, 8) and K(−3, −5) (n) S(3, −5) and T(−2, 8)

(o) D(6, −3) and E(0, 4) (p) F(6, 9) and G(−5, −5)

x

y

O

2

4

−−−−2

−−−−4

2 4 −−−−2 −−−−4

d

e

f

b

c

x

y

O 2 4

2

4

−−−−2

−−−−2

−−−−4

−−−−4

a

98

6. Prove that the following sets of points are collinear:

(a) A(−6,−1), B(2, 3) and C(4, 4)

(b) P(1, −1), Q(−3, 5) and R(7, −10)

(c) E(5, −3), F(11, −2) and G(−7, −5)

(d) K(5, −4), L(−1, 4) and M(9½ , −10)

7. Given that each set of points are collinear, find the value of k in each case:

(a) P(−4, −2), Q(−1, −1) and R(8, k)

(b) A(1, 3), B(3, k) and C(4, −6)

(c) E(−4, −1), F(k, −1) and G(8, 7)

(d) S(k, 2), T(9, 1) and U(−3, 4)

8. The points E and F have coordinates (2, −5) and (−4, a) respectively.

Given that the gradient of the line EF is 32 , find the value of a .

9. If the points (3, 2) , (−1, 0) and (4, k) are collinear, find k

.

10. Given that the points (3, −2), (4, 5) and (−1, a) are collinear , find the value of a.

11. The line which passes through (1, 4) and (2, 5) is parallel to the line through (3, 7) and (k, 5).

Find the value of k.

12. The line which passes through (−2, 3) and (−5, −9) is parallel to the line through (4, k) and

(−1, −1). Find the value of k.

99

7.2 Equations of Straight Lines

1. For each line, write down the gradient and the coordinates of the point where it

crosses the y – axis.

(a) y = 3x + 1 (b) y = ½ x − 5 (c) y = −2x + 3

(d) y = −¼ x − 2 (e) y = 8x − ½ (f) y = −x + 4

2. Match these equations with the graphs shown below.

1. y = x + 1 2. y = −2x − 3 3. y = ½ x + 4

4. y = −¼ x +2 5. y = 6x − 2 6. y = 3x − 5

(a) (b) (c)

(d) (e) (f)

3. Sketch the graphs of lines with equations:

(a) y = ½ x − 2 (b) y = −2x − 1 (c) y = −3x + 2

(d) y = −x + 3 (e) y = 2x + 3 (f) y = 4x + 1

x

y

O

x

y

O x

y

O

x

y

O x

y

O

x

y

O

100

4. Write down the equation of the lines drawn in the diagrams below.

(a) (b)

(c) (d)

(e) (f)

y

(g) (h)

x

y

x O

y

x

x

y y

x

y

x

x

y

O

O

O

O

O

O

O

101

5. Identify the gradient and y − intercept of these lines.

(a) 3+= xy (b) 12 −−= xy (c) xy2

1=

(d) 22

1 +−= xy (e) 6=+ yx (f) 42 −= xy

(g) 123 += xy (h) 2054 =+ yx (i) 1223 =− yx

6. State the gradient and the y − intercept for each line below.

(a) 7−= xy (b) 35 +−= xy (c) 1035 −= xy

(d) xy 4−= (e) 112 =+ yx (f) 52 −= xy

(g) 183 =− xy (h) 02173 =−+ yx (i) 2054 =− yx

7. Write down the equation of the lines described below:

(a) with gradient 4, passing through the point (0, 5)

(b) with gradient −2, passing through the point (0, 1)

(c) with gradient 4

3 , passing through the point (0, −3)

(d) with gradient 4, passing through the point (3, 1)

(e) with gradient −5, passing through the point (−3, 1)

(f) with gradient 21 , passing through the point (−5, −2)

(g) with gradient 34 , passing through the point (2, 7)

(h) with gradient 43− , passing through the point (−2, −2)

(i) with gradient 23− , passing through the point (−5, 3)

102

8. Find the equation of the line joining each pair of points below.

(a) A(4, 3) and B(8, 11) (b) C(1, 9) and D(3, 1) (c) E(−2, 6) and F(8, 8)

(d) G(5, −9) and H(8, −15) (e) I(0, 6) and J(5, 11) (f) K(−1, −3) and L(7, −9)

(g) M(−4, 0) and N(−1, 5) (h) P(2, 2) and Q(−3, 4) (i) R(5, −1) and S(−2, 10)

9. Find the equations of the lines joining the following pairs of points:

(a) (2, 1) and ( 6, 3) (b) (1, 5) and (3, 1) (c) (2, 0) and (4, 6)

(d) (−2, −3) and (2, 3) (e) (−1, 2) and (5, −1) (f) (−4, 2) and (4, −4)

(g) (−6, −2) and (−5, 3) (h) (4, −3) and (6, 5) (i) (−2, 3) and (0, −2)

10. Establish the equation of the line passing through each pair of points below.

(a) A(2, 1) and B(6, 13) (b) C(3, 4) and D(5, −4) (c) E(−2, −1) and F(6, 3)

(d) G(4, −13) and H(−2, −1) (e) I(2, 8) and J(10, 12) (f) K(−3, 2) and L(9, −2)

103

7.3 Equations of Straight Lines - EXAM QUESTIONS

1. A straight line has the equation 3x − 2y = − 4.

Find the gradient and y-intercept of the line.

2. The line AB passes through the points (0, 6) and (8, 0) as shown in the diagram.

Find the equation of the line AB.

3. A straight line has equation 2y + 3x = 8.

Which line of these gives its gradient and y – intercept?

Show working to explain your answer.

A. 3 and (0, 8) B. – 3 and (0. 8)

C. 23 and (0, 4) D.

23− and (0, 4)

4. Find the gradient and y – intercept of the straight line with equation

1243 =− yx .

x

y

0

A

B

104

5. The diagram below shows the line with equation 123 += xy .

Find the coordinates of P, the point where the line cuts the y-axis.

6. Find the equation of the line shown in the diagram below.

7. A line has equation 2y + 6x = 9. Find its gradient and y - intercept.

8. A line has equation 3y + 4x = 15. Make a sketch of this line on plain paper

showing clearly where it crosses the y - axis.

−−−−1 −−−−2 −−−−3 −−−−4 −−−−5 −−−−1

−−−−2

−−−−3

−−−−4

−−−−5

0

1

2

3

4

5

6

7

1 2 3 4 5 x

y

o

123 += xy

P

x

y

105

9. The relationship between variables v and T produces a straight line graph as shown below.

The line passes through the point P(24,16) as shown.

(a) Find the gradient of the line.

(b) Hence, write down the equation of the line in terms of v and T.

10. A straight line has equation 3y − 2x = 6. Find the gradient and y-intercept of the line.

11. A straight line has equation 3x − 2y = 8. Find the gradient and y-intercept of the line.

12. Find the equation of the straight line which passes through the point A(3, −2) and is parallel to

the line 3y − 2x = 5

13. (a) A straight line has equation 4y – 3x = 6.

State the gradient and the y-intercept point for this line.

(b) Write down the equation of the line with gradient – ½ which has the same

y – intercept point as the line above.

14. (a) A straight line has equation 3y – 4x = 12.

State the gradient and the y-intercept point for this line.

(b) Write down the equation of the line with gradient – ¾ which has the same

y – intercept point as the line above.

T

v

0

10

P(24,16)

106

Section 8 – Simultaneous Equations

8.1 Graphical Solutions

1. (a) Copy and complete the tables below.

(b) Plot the points from table 1. Join them carefully with a straight line.

(c) Plot the points from table 2 on the same graph. Join them with a straight line.

(d) Write down the coordinates of the points where the lines cross.

2. (a) Copy and complete the tables below.

(b) Plot the points from table 1. Join them carefully with a straight line.

(c) Plot the points from table 2 on the same graph. Join them with a straight line.

(d) Write down the coordinates of the points where the lines cross.

3. Repeat the questions above for

(a) y = 7 − x and y = x − 1 (b) y = 14 − x and y = x − 8

(c) y = x − 3 and y = 15 − x (d) y = x − 7 and y = 17 − x

(e) y = 12 − x and y = x − 4 (f) y = 30 − x and y = x − 10

(g) y = 18 − x and y = x − 12 (h) y = 11 − x and y = x − 5

(i) x + y = 10 and x − y = 4 (j) x − y = 9 and x + y = 17

Table 1 : y = 9 −−−− x

x 0 3 7

y 6

Table 2 : y = x −−−− 1

x 2 5 7

y 1

Table 1 : y = 8 −−−− x

x 0 3 7

y 5

Table 2 : y = x −−−− 2

x 2 5 7

y 0

107

4. Solve the following simultaneous equations graphically.

(a) 82

6

=+

=+

yx

yx (b)

93

82

=+

=+

yx

yx (c)

2

63

=−

=+

yx

yx

Draw axes with x and y from 0 to 8 Draw axes with x and y from 0 to 9 Draw axes with x from 0 to 8 and y from –2 to 4

(d) 5

1232

=+

=+

yx

yx (e)

1823

2443

=+

=+

yx

yx (f)

4

105

−=−

=+

yx

yx

Draw axes with x and y from 0 to 7 Draw axes with x and y from 0 to 9 Draw axes with x from -4 to 4 and y from 0 to 10

5. Find the value of x and y by drawing the graphs of the following pairs of equations.

(a) 3y − x = 9 ( b) 2x − 3y = 6 (c) x + 2y = 10

x + y = 11 x + 2y = 10 2x + y = 8

(d) x − 2y = −2 (e) x − y = 7 (f) 3x + 2y = 6

2x − y = 2 3x − 2y = 24 x − 2y = 10

(g) 2y − x = 8 (h) x + y = 2 (i) x − 2y = 3

3y + x = 17 2x − y = 4 x + y = 0

(j) 2y − 3x = 0 (k) x − y = 2 (l) x + y = 0

x − y = −2 2x + 3y = 4 2x + 3y = 6

(m) 2x + 3y = 4 (n) 3x − 2y = 3 (o) 5x − y = 6

x − 2y = 9 x + y = −4 3x + 2y = 1

108

8.2 Algebraic Solutions

1. Solve each pair of equations below using the method of substitution.

(a) y = x and 3x − y = 10 (b) y = x and 5x − y = 4

(c) y = 2x and 5x + y = 14 (d) y = 2x and 2x + 3y = 24

(e) y = 3x + 1 and y = x + 7 (f) y = 5x − 4 and y = 2x + 11

2. Solve the following pairs of simultaneous equations:

(a) x + y = 4 (b) x + y = 9 (c) x + y = 7

x − y = 2 x − y = 5 x − y = 3

(d) x + y = 1 (e) x + y = 3 (f) x + y = −1

x − y = 3 x − y = 9 x − y = 9

(g) x + y = −5 (h) x + y = −14 (i) x + y = −18

x − y = −1 x − y = −8 x − y = 2


(a) 427

2442

=−

=+

yx

yx (b)

662

1834

−=+

=−

ba

ba (c)

3858

2672

=−

=+

fe

fe

(d) 323

25

=+

−=+

yx

yx (e)

1863

1032

=−

=−

yx

yx (f)

158

134

−=+

=+

qp

qp

(g) 2625

132

−=−

=+

hg

hg (h)

179

632

−=−

=+−

yx

yx (i)

1711

1642

−=−

−=+

vu

vu

(j) 1555

082

=−

=−

yx

yx (k)

1434

1123

−=+

−=+

qp

qp (l)

4086

46310

=−

=−

ba

ba

109


(a) 423

173

−=−

=+

yx

yx (b)

83

63

=+

=−

ba

ba (c)

2025

12

−=−

=+

fe

fe

(d) 04

735

=+

=+

yx

yx (e)

52

1452

−=−

−=−

yx

yx (f)

84

632

−=+

=+

qp

qp

(g) 9687

112

=−

=+

hg

hg (h)

35

2523

−=+

=−

yx

yx (i)

2229

104

=−

=−

vu

vu

(j) 95

532

=+

+=

yx

yx (k)

24

0723

−=+

=+−

qp

qp (l)

03856

0304

=−+

=−+

ba

ba


(a) 2x + y = 15 (b) 3x + 2y = 32 (c) 5x + 3y = 26

x − y = 6 x − 2y = 8 2x − 3y = 2

(d) 3x + y = 9 (e) 4x + y = 11 (f) 7x + 2y = 36

x + y = 5 2x + y = 5 2x + 2y = 16

(g) 2x − 5y = −21 (h) 3x + 8y = 23 (i) 3x + 4y = 10

3x + 10y = 56 x − 4y = 1 6x + 5y = 17

(j) 5x − 2y = 16 (k) 7x + 3y = −13 (l) 3x − 5y = 8

3x + 4y = 20 3x + y = −5 x − 7y = 8


(a) 5x + 2y = 9 (b) 4x + 5y = 7 (c) 5x + 2y = 14

2x − 3y = −4 7x − 3y = 24 4x − 5y = −2

(d) 3x + y = 16 (e) 8x − 3y = 19 (f) 5x + 3y = 19

2x + 3y = 13 3x − 2y = 1 7x − 4y = 43

(g) 2x − 5y = 21 (h) 2x − 3y = 17 (i) 8x + 2y = 23

3x + 2y = 3 7x − 4y = 40 5x + 6y = 31

(j) 2x + 3y = 7 (k) 7x + 2y = 11 (l) 7x − 5y = 35

4x + 5y = 12 6x −5y = −4 9x − 4y = 45

110

8.3 Simultaneous Equations in Context

1. Find two numbers whose sum is 56 and whose difference is 16.

2. Find two numbers whose sum is 22 and where twice the bigger one minus three times the

smaller one is 24.

3. Two numbers are such that twice the smaller plus the larger is equal to 18 and the difference

between twice the larger and the smaller is 11.

Find the two numbers.

4. Two numbers are such that three times the larger plus twice the smaller is equal to 31 and the

sum of twice the smaller plus the larger is 13.

Find the two numbers.

5. Four chocolate bars and six packets of crisps together cost £3.40.

Ten chocolate bars and three packets of crisps cost £4.90.

Form simultaneous equations and solve them to find the cost of

each packet of crisps and each bar of chocolate.

6. Four sandwiches and 3 hot-dogs cost £7.50.

Two sandwiches and 4 hot-dogs cost £6.

Form simultaneous equations and solve them

to find the cost of each sandwich and hot-dog.

111

7. At Smith’s Stationers, the cost of a ruler and a pencil together is 57p. The ruler costs 23p

more than the pencil.

Find the cost of each.

8. Blear’s new album First Sight is available on CD and as a download.

5 downloads and 4 CDs cost £97.

3 downloads and 3CDs cost £66

Calculate the cost of the download and of the CD.

9. A photographer produces 2 sizes of print, Standard and Jumbo.

A customer who orders 24 standard and 5 jumbo prints pays £7.79

Another customer pays £8.60 for 20 standard and 8 jumbo prints.

How much would I have to pay for 1 standard and 1 jumbo print ?

10. There are 2 types of ticket on sale for a football match – Side Stand and Centre Stand.

You are sent to buy tickets for various members of your family

and you pay £71.75 for 4 Side and 3 Centre tickets.

Your friend pays £75.25 for 2 Side and 5 Centre tickets.

What is the price for each type of ticket?

11. Two small glasses and five large glasses together contain 915 ml.

One small glass and three large glasses together

hold 530 ml.

How much does each glass hold?

First Sight

First Sight

\65

\60mm

112

12. On a camping holiday a group of 30 students take 3 frame tents and 2 ridge tents.

Another group of 25 students take 2 frame tents and 3

ridge tents.

How many people does each type of tent hold ?

13. A magazine pays different rates for Star Letters and Readers’ Letters.

In June the magazine editor paid out £195 for 3 Star Letters and 8 Readers’ Letters.

In July £215 was paid out for 2 Star Letters and 11 Readers’ Letters.

How much does the magazine pay for each type of letter?

14. Brian is a potter and is making 2 different sizes of vase.

Five small vases and four large ones require 17 kg of clay.

Three small vases and two large vases take 9·4 kg of clay.

How much clay is needed for each size of vase?

15. Karen is in charge of ordering the

lunches in the office where she works.

She keeps a note of what she orders

and the total costs.

She thinks she has been wrongly

charged on one of the days.

By forming and solving pairs of

equations, find out if she is correct

or not.

Burger

Meals

Chicken

Meals

Total

Cost(£)

Monday 7 8 29.70

Tuesday 3 12 30.30

Wednesday 8 3 21.35

Thursday 4 7 20.85

Friday 6 6 23.70

Saturday 5 10 30.00

113

16. Look at the two rectangles opposite.

The smaller one has a perimeter of 60cm.

The larger one has a perimeter of twice the smaller.

(a) Form two equations and solve them

simultaneously to find the values of x and y.

(b) Hence calculate the area of the smaller rectangle.

17. A van is carrying eight identical boxes and five identical parcels.

(a) If 3 boxes and 2 parcels weigh a total of 22kg and 4 boxes and 3 parcels

weigh 30kg, find the weight of an individual box and a single parcel.

(b) What is the total weight carried by the van?

18. 3 pounds of butter and 4 pints of milk costs £3.84.

5 pounds of butter and 7 pints of milk costs £6.48.

Find the cost of a pound of butter and a single pint of milk.

19. In a certain factory, the basic rate of pay is £4.50 per hour, with overtime at £6.40.

Paul’s total wage for a certain week was £215.80.

If he worked a total of 45 hours in all, how many hours did he work at the basis rate?

20. At a concert 500 tickets were sold. Cheap tickets cost £5 whereas more expensive ones cost £9.

If the total receipts were £3 220, how many cheap tickets were sold?

21. John saves money by putting every 50p and every 20p coin he receives in a box. After a while

he discovers that he has 54 coins amounting to £17.10.

How many of each coin does he have?

2x

2y

3x

8y

114

8.4 Simultaneous Equations - EXAM QUESTIONS

1. A small printing company sends out letters to customers every day.

On Monday they sent out 20 first class letters and 15 second class letters and the charge for

postage was £19·50.

On Tuesday they sent out 18 first class letters and 25 second class letters and the charge was

£23.30.

How much will it cost on Wednesday to send 10 first class letters and 30 second class?

2. A concert hall sells two types of tickets, stall tickets and balcony tickets. When all seats are sold

the concert hall holds a total of 640 people.

(a) Let s be the number of stall tickets and b the number of balcony tickets.

From the information above write down an equation connecting s and b.

(b) On a particular night a concert is sold out (all seats are taken) with stall tickets priced at

£8.50 and balcony tickets at £12.20. The total takings at the box office for that night was

£6143.

From this information write down a second equation connecting s and b.

(c) Hence find how many stall and balcony seats are in this concert hall.

3. In a fast food restaurant Ian buys 3 burgers and 4 portions of French fries and it costs £5.64.

Sarah buys 2 burgers and 3 portions of French fries and it costs £4.01.

Jack had a voucher to receive one burger and one portion of fries for free.

How much would it cost Jack for 5 burgers and 3 portions of French fries?

4. A hotel owner is buying some new duvets for his hotel.

One week he bought 7 double duvets and 12 single duvets which cost £168.

The next week he bought 4 double duvets and 9 singles for £111.

The hotel owner was given a 14% discount on his next order for 5 double duvets and 5 single

duvets.

How much did he pay for this third order?

115

5. Find the point of intersection of the lines with equations

1625 =− yx and 953 −=+ yx

6. Clare has baked 60 scones to sell at the school fayre. Some are fruit scones (f) and some are

treacle scones (t).

(a) Write down an equation using f and t to illustrate this information.

She sells the fruit scones for 25p and the treacle scones for 20p each.

She sells all the scones for a total of £13.25.

(b) Write down another equation using f and t to illustrate this information.

(c) Hence, find algebraically the number of treacle scones Clare sold.

7. At the funfair coloured tokens are awarded as prizes in some of the games. These tokens can be

saved up and exchanged for larger items.

3 green tokens and 4 red tokens have a total value of 26 points.

5 green tokens and 2 red tokens have a total value of 20 points.

Dave has 10 green tokens and 10 red tokens.

Does he have enough points to exchange for a large soft toy with a points value of 75?

8. In a week Peter downloads 5 tracks and 4 films and pays £21.23.

In the same week Frank downloads 7 tracks and 3 films and pays £18.49.

Calculate how much Richard would pay if he downloaded 3 tracks and 2 films.

9. Solve, algebraically, the equations

3x + 2y = 13

x = y + 1

10. Find the point of intersection of the lines with equations:

3x – 4y = 18

2y – 5x = –16

116

11. In the Garden Centre there are 2 types of plants on special offer.

Carly bought 3 Rose bushes and 2 Poppy plants which cost £15.23

Steph paid £26.71 for 4 Poppy plants and 5 Rose bushes.

How much would Sally pay for a Rose bush and 3 Poppy plants?

12. Peter is buying new furniture for his flat.

Two sofas and one chair will cost him £1145.

For £1310 he can buy one sofa and three chairs.

Find the cost of one sofa and the cost of one chair.

13. Eric orders goods from a mail-order company. 5 books and 2 CDs cost £40⋅80.

2 books and 3 CDs cost £37⋅78. Each order includes £2⋅95 post and packing

regardless of the size of the order.

How much would it cost Eric to have 3 books and 1CD and have them delivered?

14. Shereen goes shopping in the summer sales.

The store has an advert in the window, shown opposite.

Shereen buys 2 tops and 3 skirts and

pays £33·90. Her friend Nadia buys 3 tops and 4 skirts and pays £46·70.

Another friend Kay buys 2 tops and 2 skirts. How much does she pay?

15. Find the point of intersection of the straight lines with these equations.

4x + 3y = 7 and y = 2x + 9

This week’s

specials!

Rose bushes

and

Poppy plants

All tops one price!

SALE

All skirts one price!

117

Section 9 –Functions and Formulae

9.1 Changing the Subject of a Formula

1. Change the subject of each formula to x.

(a) y = x + 3 (b) y = x − 5 (c) y = x + a

(d) y = x − b (e) y = 3x (f) y = 10x

(g) y = kx (h) y = ax (i) y = 3p + x

(j) y = x − 5t (k) y = 2x + 1 (l) y = 3x − 7

(m) y = 7x + 4a (n) y = 3b + 4x (o) y = 8 + 10x

2. Make a the subject of each formula.

(a) b = 4 − a (b) d = 12 − a (c) y = 5x − a

(d) m = 2 − 2a (e) q = 7 − 5a (f) c = 20 − 3a

(g) r = s −2a (h) t = d − 4a (i) z = 4b −5a

(j) k = 2h − 7a (k) p = 6q − 11a (l) g = 2x − 9a

3. Make x the subject of each formula below.

(a) y = ax + b (b) y = mx + c (c) t = sx − r

(d) p = qx + 2r (e) m = fx − 3n (f) a = b + cx

(g) k = h − mx (h) d = 3b + cx (i) g = kc − hx

4. Change the subject of each formula to the letter shown in brackets.

(a) P = 4l (l) (b) V = IR (I) (c) S = DT (T)

(d) A = lb (b) (e) C = πd (d) (f) G = UT (U)

(g) v = u + at (t) (h) P = 2l + 2b (l) (i) H = xy + 5m (y)

118

5. Change the subject of each formula to c.

(a) b = 1/2 c (b) x =

1/5 c (c) y =

1/4 c

(d) m = 1/6 c (e) k =

1/9 c (f) d =

1/10 c

(g) a = 1/2 c + 2 (h) h =

1/3 c −5 (i) p =

1/4 c + q

(j) y = 1/10 c − x (k) t =

1/8 c + 2s (l) r =

1/5 c − 3q

6. Change the subject of each formula to x.

(a) y = x

3 (b) d =

x

c (c) m =

x

y

(d) s = x

a 2+ (e) w =

x

z 1− (f) a =

x

cb +

(g) a = 9

8+x (h) k =

2

5−x (i) p =

4

3 x−

(j) y = x

2 + 1 (k) z =

x

6 − 7 (l) h =

x

m + k

7. Change the subject of each formula to k.

(a) y = k (b) x = k (c) m = k

(d) a = b

k (e) c =

d

k (f) h =

g

k

(g) s = k

t (h) q =

k

p (i) w =

k

z

(j) r = k2 (k) ab = k

2 (l)

q

p = k

2

(m) y = x + k2 (n) c = k

2 − d (o) x = 3k

2 − 1

119

8. Change the subject of each formula to the letter shown in brackets.

(a) v2 = u

2 + 2as (s) (b) v

2 = u

2 + 2as (u)

(c) V = πr2h (h) (d) V = πr2

h (r)

(e) r = π

A (A) (f) L = 3 + a6 (a)

(g) 2k = )4( +p (p) (h) x2 =

t

yz4 (y)

(i) ar = b

x

2

1 (b) (j) st = A

2(x − 3y) (A)

(k) R = A2(x − 3y) (x) (l) na = )1( 2n− (n)

(m) d = n

nt )1( − (n) (n)

21

111

rrR+= (R)

(o) d = 4

)(2 bxa + (a)

120

9.2 Changing the Subject of a Formula - EXAM QUESTIONS

1. Change the subject of the formula to c.

ab = 22

1

c

x

2. The formula for the velocity that a body must have to escape the gravitational pull of Earth is

V = gR2

Change the subject of the formula to g.

3. For the formula given below, change the subject to x

A2 = x + 5

4. The formula for kinetic energy is

E = 2

21 mv

Change the subject of the formula to v.

5. Change the subject of the formula to a:

V = 3a2b

6. Change the subject of the formula to k.

k

mT π2=

7. A formula to convert temperature from degrees Celsius to degrees Farenheit is

F = 325

9+C

Change the subject of the formula to C.

121

8. The formula for finding the volume of a cone is given by

π3

1=V r

2h

Change the subject of the formula to r.

9. Change the subject of the formula ab

km

22 += to k.

10. Change the subject of the formula 72 += cba to b.

11. Change the subject of this formula to m 3

mnk =

12. The formula for finding the volume of a sphere is given by

π3

4=V r

3

Change the subject of the formula to r.

9.3 Function Notation

1. A function is given as 56)( −= xxf .

Find: (a) )3(f (b) )1(−f (c) )(2

1f (d) )(af

2. A function is given as 4)( 2 += xxf .

Find: (a) )2(f (b) )4(f (c) )3(−f (d) )( pf

3. A function is given as aah 212)( −= .

Find: (a) )4(h (b) )6(h (c) )2(−h (d) )(mh

4. A function is defined as xxxg 3)( 2 += .

Find: (a) )(ag (b) )2( pg (c) )1( +mg (d) )2( eg −

122

5. A function is defined as xxxf 4)( 2 −= .

Find: (a) )4(f (b) )3( af (c) )2( −af (d) )12( +pf

6. A function is given as 35)( += xxf . For what value of x is :

(a) 23)( =xf (b) 2)( −=xf (c) 5)( =xf ?

7. A function is given as tth 620)( −= . For what value of t is :

(a) 2)( =th (b) 16)( −=th (c) 32)( =th ?

8. A function is given as 16)( 2 −= aag . For what value(s) of a is :

(a) 9)( =ag (b) 15)( −=ag (c) 0)( =ag ?

9. A function is defined as xxxf 2)( 2 += .

(a) Evaluate: (i) )3(f (ii) )2(−f .

(b) Find )3( +af in its simplest form.

10. A function is defined as aah 633)( −= .

(a) Evaluate: (i) )4(h (ii) )1(−h .

(b) Given that 0)( =th , find the value of t.

(c) Express )2( −ph in its simplest form.

123

Section 10 – Area and Volume

10.1 Properties of Shapes

1. Copy and complete the blanks in these statements about QUADRILATERALS.

(a) A RHOMBUS that has 4 RIGHT ANGLES is a .

(b) A PARALLELOGRAM has lines of symmetry and

of order 2.

(c) The opposite angles of a PARALELLOGRAM are .

.

(d) The of a SQUARE

each other at angles.

(e) A is a QUADRILATERAL that has ONE pair

of parallel sides.

2. Pauline wanted to make a kite so she had to work out how much fabric she would need to cover

it. She drew a diagram to help her to do this. This is what she drew.

Calculate the area of her kite.

55 cm

70 cm

124

3. (a) Calculate the area of this parallelogram giving your answer in cm2.

(b) How many square metres is this?

4.

(a) Calculate the area of one rhombus in the above tessellation.

(b) Use the information given about one rhombus to calculate the area of the whole

rectangular pattern.

(c) What percentage of the whole rectangle is shaded?

5. Each of these shapes is a REGULAR POLYGON. For each one, write down what kind of

POLYGON, the size of the CENTRAL ANGLE, the INTERIOR ANGLE and the EXTERIOR

ANGLE.

(a) (b) (c)

2⋅3m

70cm

12cm

4cm

125

6. Choose words from the above to describe these triangles as fully as you can.

(a) (b) (c)

7. Draw a set of coordinate axes on a coordinate grid and plot the points:

A(2, 4), B(1, -3) and C(-5, 3).

Join the points up to form a triangle.

Write down a FULL description of triangle ABC.

Work out its area.

8. Pupils were asked to make up sets of three angles that could be used to draw a triangle.

30o, 60

o, 90

o 47

o, 34

o, 98

o 53

o, 47

o, 90

o 32

o, 36

o, 112

o

Which ones are correct? Why they are correct?

9. Calculate the values of a, b, c and d in these triangles:

53o ao

62o bo

co

2⋅5do 1⋅5do

0⋅5do

(a) (b) (c)

126

10. Pupils in the art department have been asked to design a logo for the school band. Here is one of

the entries.

It consists of 4 isosceles triangles each measuring 12cm on the base and 6cm high.

Calculate the area of fabric required to make up 40 of these.

11. The cost of advertising depends on the length of space that the advert takes up. Each column of

the paper measures 5cm in width and each cm costs £0·75.

Calculate the cost of these adverts:

(a) (b)

12. Calculate the cost of carpeting a room like this if the carpet costs £12⋅96 per m2.

13. Say whether the following statements are True or False

(a) If the diameter of a circle is 2⋅5m, then its radius is 5m.

(b) The perimeter of a circle is called its circumference.

(c) To find the circumference of a circle the radius is multiplied by .π

(d) The diameter divides a circle into two semi-circles.

12m

9m

4⋅5m

2⋅75m

127

14. Calculate the CIRCUMFERENCES of these circles:

15. The diameter of the ‘bell’ on the end of a trumpet measures

14 cm. Calculate its circumference.

16. Calculate the circumference of the circle drawn

with these compasses.

17. The radius of this lampshade is 95mm at the bottom. How much trim would by required to fit

round the bottom edge.

If the trim costs £2⋅75 a metre, how much would it cost to trim the

lamp if the trim is only sold in complete metres.

•

(a)

15m •

(b)

9cm

14cm

5⋅3cm

128

18. A florist is decorating her shop and wants to put pieces of coloured ribbon round white poles to

create a striped effect like this:

The pole has a radius of 12cm. Calculate how much ribbon she will

need to decorate the two poles.

Answer correct to the nearest metre.

19. Linzi’s Mum buys a frill of length 38cm to fit round her birthday cake.

Find out the biggest diameter that the cake can have

so that the frill fits.

20. Calculate the area of these circles:

21. The radius of a cymbal is 18cm. Calculate the area of one of them.

22. The diameter of the top of a pin is 7mm.

Calculate the total area of the tops of 5 of them.

7mm

• 30cm

(b)

•

12cm

(a)

129

23. Tea-light candles have to be packed into a box like this:

(a) What is the area of 1 tea light?

(b) Calculate the total area taken up by the

15 tea lights on the tray.

(c) What is the area of the top of the tray?

(d) What percentage of space on the tray is NOT taken up by the tea lights?

24. Mrs Ahmad has moved into a new house and has to sort out her garden. This is a plan of what

she wants to do. It consists of a circular flower-bed with diameter 3m and 4 quarter-circles with

radius 1⋅5m set in a rectangular lawn.

(a) Calculate the total area of the 5 flower

beds.

(b) What area is given over to the lawn?

(c) It costs £3⋅65 for each square metre of

lawn and a total of £197⋅50 for plants.

How much would it cost Mrs Ahmad

altogether for her new garden?

25. The weights at the end of these balloons each have an area of 20cm2.

Calculate their radius and then the circumference.

12cm

10m

6m

130

10.2 Volumes of Cylinders

1. Circular – based prism (cylinder)

Find the volume of a circular-based prism for the values of r and h given.

(a) r = 6 cm h = 15 cm

(b) r = 8 cm h = 24 cm

(c) r = 4 cm h = 12 cm

(d) r = 10 cm h = 8 cm

(e) r = 20 cm h = 60 cm

(f) r = 7 cm h = 20 cm

(g) r = 15 cm h = 40 cm

(h) r = 11 cm h = 35 cm

(i) r = 44 cm h = 125 cm

(j) r = 8.8 cm h = 30 cm

2. A milk dispenser is cylindrical in shape with diameter 30cm.

(a) If 14 litres of milk are poured into it, calculate the depth of the milk

in the cylinder.

(b) The height of the cylinder is 25cm.

How many more litres of milk are needed to completely fill it?

r

h

30cm

25cm

131

3.

Calculate the volume of a cylinder with diameter 12cm and height 8cm.

4. This paint tin has diameter 20 cm and height 30

cm as shown in the diagram.

It is claimed that it can hold 10 litres of paint. Is this claim correct?

You must show all working and give a reason for your answer.

10 litres

20 cm

30 cm

8cm

12cm

132

10.3 Volumes of Other Solids

1. Calculate the volume of each sphere described below, rounding your answer to 1 decimal place.

(a) r = 6cm

(b) r = 2m

(c) r = 9mm

(d) r = 3cm

2. Find the volume of a sphere for the following values of r and d.

(give your answers correct to 3 significant figures)

(a) r = 10cm (f) d = 18cm

(b) r = 25cm (g) r = 80mm

(c) d = 2m (h) d = 55cm

(d) r = 200mm (i) r = 3·5m

(e) d = 11cm (j) d = 48cm

3. A sphere has a diameter of 8cm.

Calculate its volume giving your answer correct to 3 significant figures.

4. Find the volume of a cone for the following values of r and h.


(a) r = 5cm h = 14cm

(b) r = 7cm h = 25cm

(c) r = 3cm h = 22cm

(d) r = 12cm h = 7cm

r

r

h

r

133

5. Find the volume of a cone for the following values of d and h.


(a) d = 15cm h = 40cm

(b) d = 11cm h = 37cm

(c) d = 22cm h = 125cm

(d) d = 8Z8cm h = 30cm

6. Calculate the volume of each cone described below, rounding your answers to 1 decimal place.

(a) r = 3cm and h = 6cm

(b) r = 8mm and h = 12mm

(c) r = 3cm and h = 5cm

(d) r = 2m and h = 6m

7. A cone has a base diameter of 8cm and a height of 5cm. Calculate the volume of this cone.

8. A cone has a base diameter of 10cm and a slant height of 13cm.

Calculate the volume of the cone.

9. A cone has a base radius of 9cm and a slant height of 15cm.

Calculate the volume of the cone.

10. A pyramid has a square base of side 4cm and a vertical height of 7cm.

Calculate the volume of the pyramid correct to 2 significant figures.

11. A pyramid has a rectangular base measuring 16mm by 12mm and a vertical height of 10mm.

Calculate the volume of the pyramid.

h

r

13 cm

5cm

134

10.4 Volumes of Solids - EXAM QUESTIONS

1. The Stockholm Globe Arena is the largest hemispherical building in the world.

The radius of the building is 110 m.

Calculate the volume of the building in cubic metres,

giving your answer in scientific notation correct to 3


2. A metal bottle stopper is made up from a cone topped with a sphere.

The sphere has diameter 1⋅5cm.

The cone has radius 0⋅9cm.

The overall length of the stopper is 6⋅5cm.

Calculate the volume of metal required to make the stopper.

Give your answer correct to 3 significant figures.

3. The volume of this sphere is 524cm

3.

Calculate the diameter, d cm.

4. Non Calculator!

Calculate the volume of this sphere which has radius 3m.

[Take 143 ⋅=π ]

6⋅5cm

Radius = 0⋅9cm

dcm

3m

135

5. Sherbet in a sweet shop is stored in a cylindrical container like the one shown

in diagram 1.

The volume of the cylinder, correct to the nearest 1000cm3, is 10

000 cm

3.

The sherbet is sold in conical containers with diameter 5 cm as

shown in diagram 2.

250 of these cones can be filled from the contents of the cylinder.

Calculate the depth, d cm, of a sherbet cone.

6. Non Calculator!

The diagram shows a cone with radius 10

centimetres and height 30 centimetres.

Taking π = 3·14, calculate the volume of the

cone.

7. A children’s wobbly toy is made from a cone, 21 cm

high, on top of a hemispherical base of diameter 20 cm.

The toy has to be filled with liquid foam.

Calculate the volume of foam which will be required.

32cm

20cm

Diagram 1

10 cm

30 cm

d cm

Diagram 2

5 cm

21 cm

20 cm

136

8. The lamp cover in a street lamp is in the

shape of a cone with the bottom cut off.

The height of the cone is 50cm and its

radius is 25cm. The height of the lamp is 30cm

and the base of the lamp has a radius of 18cm

Calculate the volume of the lamp cover. [Answer to 3 significant figures.]

9. A glass candle holder is in the shape of a cuboid with a cone

removed. The cuboid measures 4cm by 4cm by 6cm.

The cone has a diameter of 3cm and a height of 5cm.

Calculate the volume of glass in the candle holder.

10. For the Christmas market a confectioner has created a chocolate Santa.

It consists of a solid hemisphere topped by a solid cone.

Both have diameter 5cm and the height of the cone is 4cm as

shown in the diagram.

Calculate the volume of chocolate required to make one chocolate

Santa, giving your answer correct to 3 significant figures.

4 cm

4 cm

6 cm

5cm

4cm

137

11. The diameter of an ordinary snooker ball is 5⋅25cm.

Calculate the volume of a snooker ball giving your answer correct to 3


12. A dessert is in the shape of a truncated cone [a cone with

a ‘slice’ taken from the top].

The radius of the base is 4·1cm and is 1·6 cm at the top.

The other dimensions are shown in the diagram.

Calculate the volume of the dessert.

13. A young child was given a slab of moulding clay. It was a cuboid and measured 15·2cm by

4·8cm by 3·4cm.

(a) Calculate the volume of the cuboid rounding your answer to 2 significant figures.

The clay was made into 25 identical spheres.

(b) Using your answer from part (a), calculate the radius of one of the spheres.

5∙6cm

3∙7cm

138

14. An ice cream is shaped like the one in the diagram.

The overall height of is 11·7 cm.

The height of the cylinder is 3·2 cm.

The diameter of the cone and cylinder is 6·6 cm.

Calculate the volume of ice cream.

15. A company that produces bins uses the design of a cylindrical base with a hemispherical lid.

If the total height of the bin is 60cm and the

radius of the bin is 14cm, calculate the

total volume of the bin in litres correct to 3 significant

figures.

(Volume of cylinder = πr2h;

Volume of sphere = 4/3πr

3)

16. A Christmas bauble is made from a sphere of perspex with

a coloured cylinder in the middle.

The volume round the cylinder is filled with a thick liquid.

The sphere has a diameter of 8 cm.

The cylinder has a radius of 2·6 cm with a height of 6

cm.

Calculate the volume of liquid needed to fill the sphere, giving your answer correct to 2 significant

figures.

11∙7 cm

3∙2 cm

6∙6 cm

14cm

60cm

139

Answers

Section 1 – Equations and Factors

1.1 Working with Linear Equations and Inequations

1. (a) 2 (b) 4 (c) 3 (d) 5 (e) 2 (f) 5

(g) 4 (h) 1 (i) 16 (j) 16 (k) 15 (l) 7

(m) 10 (n) 20 (o) 17 (p) 19

2. (a) 3 (b) 4 (c) 2 (d) 9 (e) 4 (f) 4

(g) 2 (h) 5 (i) 4 (j) 9 (k) 10 (l) 8

(m) 7 (n) 5 (o) 7 (p) 7

3. (a) −18 (b) 11 (c) −8 (d) 16 (e) −13 (f) 9

(g) −12 (h) 25 (i) −9 (j) 8·5 (k) −5·5 (l) 5·5

(m) −3·5 (n) 18 (o) −4·2 (p) 10·5

4. (a) 4 (b) 1 (c) −5 (d) 1 (e) −6 (f) 3

(g) 7 (h) 7 (i) 7 (j) 0 (k) 20 (l) −10

(m) 4 (n) 7 (o) 2 (p) 5

5. (a) 3 (b) 4 (c) 6 (d) 4 (e) 2 (f) 5

(g) 6 (h) 8 (i) 8 (j) 8 (k) 12 (l) 9

(m) 6 (n) 7 (o) 15 (p) 18

6. (a) 1 (b) 2 (c) 6 (d) 5 (e) 3 (f) 2

(g) 5 (h) 6 (i) 11 (j) 9 (k) 11 (l) 20

(m) 7 (n) 12 (o) 21

7. (a) 3 (b) 2 (c) 2 (d) 3 (e) 2 (f) 3

(g) 6 (h) 5 (i) 3 (j) 6 (k) 4 (l) 0·5

(m) 8 (n) 11 (o) 15 (p) 3 (q) 12 (r) 4

(s) 7 (t) 16 (u) 9 (v) 7 (w) 4 (x) 2

8. (a) 6 (b) 6 (c) 4 (d) 1 (e) 4 (f) 6

(g) 4 (h) 24 (i) 10 (j) 7 (k) 5 (l) 2·5

(m) 2·5 (n) 16 (o) 15 (p) 2 (q) 2 (r) 4

(s) 7 (t) 6 (u) 2 (v) 7 (w) 12 (x) 40

9. (a) x > 1 (b) x > 3 (c) x > 4 (d) x > 4 (e) a > 3 (f) y > 3

(g) p > 9 (h) c > 1 (i) b > 6 (j) q > 0 (k) d > 3 (l) x > 4

(m) c > 5 (n) p > 9 (o) a > 12 (p) y > 12

10. (a) x < 2 (b) x < 7 (c) x < 10 (d) x < 4 (e) a < 3 (f) y < 6

(g) p < 8 (h) c < 4 (i) b < 5 (j) q < 17 (k) d < 0 (l) x < 5

(m) c < 6 (n) p < 14 (o) a < 11 (p) y < 1

140

11. (a) x > 3 (b) x > 4 (c) x > 4 (d) x > 9 (e) a > 4 (f) y > 4

(g) p > 3 (h) c > 5 (i) b < 4 (j) q < 9 (k) d < 10(l) x < 8

(m) c < 7 (n) p < 5 (o) a < 7 (p) y < 7

12. (a) x < 7 (b) x > 6 (c) x > 11 (d) x < 9 (e) a < 6 (f) y > 11

(g) p < 18 (h) c > 9 (i) b > 16 (j) q < 16 (k) d > 15 (l) x > 7

(m) c > 10 (n) p < 20 (o) a < 17 (p) y < 19

13. (a) x < 2 (b) x > 2 (c) x > 3 (d) x > 2 (e) a < 3 (f) y < 2

(g) p > 8 (h) c < 5 (i) b > 6 (j) q < 0 (k) d < 10 (l) x > 4

(m) c < 3 (n) p < 7 (o) a > 2 (p) y < 3

14. (a) x > 3 (b) x > 4 (c) x < 6 (d) x > 3 (e) a < 2 (f) y < 5

(g) p > 6 (h) c > 8 (i) b > 8 (j) q < 8 (k) d < 12 (l) x > 9

(m) c < 6 (n) p < 7 (o) a < 15(p) y < 18

15. (a) { }1,0,1,2 −− (b) { }1,2 −− (c) { }5 (d) { }1,2 −−

(e) { }0,1,2 −− (f) { }5,4,3,2,1 (g) { }2,1,0,1,2 −−

(h) { }5,4,3,2,1

16. (a) 3≤a (b) 1>x (c) 2≥p (d) 3−<k

(e) 7≤m (f) 92 ⋅>y (g) 1<h (h) 5

1>x

17. (a) 3≥a (b) 2<x (c) 2≤p (d) 2−>k

(e) 9

38≥d (f)

3

10−<y (g) 0>h (h)

9

11<y

18. {0, 1, 2, 3, 4, 5} 19. {0, 1, 2} 20. Jane must be younger than 11

1.2 Algebraic Expressions with Brackets

1. (a) 3x − 15 (b) 5y + 35 (c) 8a + 48 (d) 18 + 6t

(e) x² + 9x (f) 3y − y² (g) b² − 4b (h) 5p + p²

(i) ab + ac (j) x² − xy (k) pq − pr (l) a² + ax

2. (a) 8a + 20 (b) 21y − 28 (c) 24x + 22 (d) 36c −63

(e) 2a² + 6a (f) 5x² − 40 x (g) 30y − 10y² (h) 3t² + 18t

(i) 6x² − 27x (j) 14y − 10y² (k) 12b² − 32b (l) 25x² + 20x

3. (a) 11a − 3 (b) 7x + 6 (c) 8b + 7 (d) 8h + 3

(e) 15 − 9 x (f) 6c − 5 (g) − 2t + 6 (h) p² − 2pq

(i) − 3 − 21c (j) 13 + 4x (k) 13a − 9 (l) 19 − 4x

(m) − 4 + 15y (n) b + 2 (o) −13 −15x (p) − 4x + 20

(q) − 7c + 5 (r) 31 − 10a

141

4. (a) 652 ++ xx (b) 1072 ++ yy (c) 24102 ++ aa

(d) 1272 ++ bb (e) 45142 ++ xx (f) 24112 ++ ss

(g) 28112 ++ yy (h) 962 ++ bb (i) 42132 ++ cc

(j) 30112 +− xx (k) 1582 +− bb (l) 40142 +− cc

(m) 27122 +− aa (n) 56152 +− yy (o) 36152 +− xx

(p) 28112 +− ss (q) 15162 +− dd (r) 10112 +− bb

5. (a) 542 −+ xx (b) 2142 −− aa (c) 202 −− tt

(d) 3242 −+ yy (e) 1452 −− cc (f) 652 −− xx

(g) 1872 −+ bb (h) 2082 −− pp (i) 562 −− yy

(j) 2422 −− zz (k) 12 −x (l) 30132 −− aa

(m) 92 −c (n) 762 −− pp (o) 5052 −+ bb

6. (a) 2x² + 7x + 3 (b) 3x² - 13x - 10 (c) 4t² + 8t + 3

(d) 5y² - 18y - 8 (e) 4c² - 9c - 9 (f) 8x² - 6x - 5

(g) 12b² -37b + 28 (h) 24p² + 89p + 30 (i) 18y² -55y + 25

(j) 2x² + 14x + 20 (k) 3a² + 6a - 45 (l) 15x² - 80x - 60

(m) 36p² + 30p + 12 (n) t³ - t² - 30t (o) 12x³- 25x² + 12x

7. (a) x ² + 6x + 9 (b) w2 – 4w + 4 (c) a

2 – 10a + 25

(d) c² +16c + 64 (e) y2 – 8y + 16 (f) a ² + 12a + 36

(g) b² + 2b + 1 (h) s² + 14s + 49 (i) b2 – 18b + 81

(j) x2 – 20x + 100 (k) c

2 – 2c + 1 (l) y

2 – 6y + 9

(m) 4x2 – 4x + 1 (n) 25y² + 20y + 4 (o) 9x² + 24x + 16

(p) 16b² – 40b + 25

8. (a) ac + bc + ad + bd (b) 6 + 3x + 2y + xy (c) ab + 4b + 5a + 20

(d) pr − qr − ps + qs (e) 7 − 7a − b + ab (f) cd − 6d + 8c – 48

9. (a) x3+ x

2 − x (b) 6x

2 −9x + 15 (c) 3x

3 − 5x

2 + 8 x

(d) 2x3 + 4x

2 + 6x (e) −5x

2 + 40x − 10 (f) x

3 − 4x

2 − 7x

142

10. (a) x3 + 5x

2 + 7x + 2 (b) x

3 + 9x

2 + 22x + 10

(c) x3 + 6x

2 + 9x + 4 (d) x

3 + 4x

2 + 8x + 15

(e) x3 + 10x

2 + 19x + 24 (f) x

3 + 11x

2 + 34x +24

(g) x3 + 13x

2 + 19x + 84 (h) x

3 + 13x

2 + 39x + 90

(i) x3 + 21x

2 + 115x + 63 (j) x

3 + 16x

2 + 64x + 7

11. (a) x3 − 1 (b) x

3 − 4x

2 − 16x − 35

(c) x3

+ 2x2 − 5x − 6 (d) x

3 + 2x

2 − 23x − 4

(e) x3 − 5x

2 + 11x − 15 (f) x

3 − 11x

2 + 32x − 12

(g) x3 − 5x

2 + 6x − 8 (h) x

3 − 3x

2 + 9x − 7

(i) x3 − 6x

2 − 29x + 18 (j) x

3 + 3x

2 − 34x − 30

12. (a) 2x3 + 14x

2 + 29x + 45 (b) 5x

3 − 14x

2 + 3x − 18

(c) 6x3 − 17x

2 + 17x − 14 (d) 3x

3 + 30x

2 + 61x − 14

(e) 5x3 − 21x

2 − 4x + 32 (f) 7x

3 + 5x

2 + 9x + 11

(g) 6x3 + 11x

2 + 6x + 1 (h) 3x

3 − 29x

2 − 38x + 8

(i) 10x3 + 11x

2 − 41x + 14 (j) 12x

3 − 29x

2 − x + 12

13. (a) 872 −+ xx (b) 34 2 −− xx (c) 584 2 ++ xx

(d) 42 −− x (e) 312 −x (f) 229 −− x

(g) 8102 2 +− xx (h) 25 2 +− xx (i) 24821 xx −+

(j) xxx 21203 23 ++ (k) 922 23 +−− xxx (l) 3231 xxx −−−

14. (a) x = 3 (b) x = 5 (c) x = 4 (d) x = 3

(e) x = 2 (f) x = 21 (g) x = 4 (h) x = -9

(i) x = 29 (j) x = 1 (k) x = 2 (l) x = -2

1.3 Factorising Algebraic Expressions

1. (a) 2(x + y) (b) 3(c + d) (c) 6(s + t) (d) 12(x + y)

(e) 9(a + b) (f) 8(b + c) (g) 5(p + q) (h) 7(g + h)

(i) 4(m + n) (j) 9(e + f) (k) 13(j + k) (l) 14(v + w)

2. (a) 2(x + 2) (b) 3(d + 3) (c) 3(2s + 1) (d) 4(3x + 1)

(e) 3(2 + 3a) (f) 2(b + 4) (g) 5(y + 2) (h) 5(2 + 3c)

(i) 4(3x + 4) (j) 6(3m + 4) (k) 6(5 + 6a) (i) 7(2y + 3)

143

3. (a) 3(x − 2) (b) 4(y − 2) (c) 8(2 − a) (d) 5(2c − 3)

(e) 3(3s − 4) (f) 2(b − 7) (g) 4(3x − 5) (h) 11(2m − 3)

(i) 5(3x − 2) (j) 6(3 − 2y) (k) 5(5b − 4) (l) 6(3d − 5)

4. (a) 2(a + 2b) (b) 2(5x − 6y) (c) 6(3m + 4n) (d) 5(2c + 3d)

(e) 3(2a − 3x) (f) 6(3s − 2t) (g) 3(4x + 5y) (h) 7(2a − b)

(i) 5(5c + 2d) (j) 3(3b − 5y) (k) 6(3x + 4y) (i) 2(3a + 14b)

5. (a) a(x + y) (b) x(y2 + a

2) (c) p(qr + st)

(d) a(xy − bc) (e) p(q + 1) (f) y(y + 1)

(g) a(a − b) (h) b(a − c) (i) n(n − 3)

(j) y (x + y)

(k) ab(c − d) (l) fg(h − e)

6. (a) 2a(x + 3) (b) 3y(1 + 3y) (c) 8a(3 − 2b)

(d) pq(q − 1) (e) 3x(4y − 3z) (f) 2b(3b − 2)

(g) 3a(a + 9h) (h) 5ab(3c + 4d) (i) 3s2(s − 3)

(j) 2x(7 − 6yz) (k) 5bc(2b − 3d) (l) 2πr(r + h)

7. (a) a(p + q − r) (b) 2(a + b + c) (c) 2(3e − f + 2g)

(d) p(p + q + x) (e) 3b(a − 2c − 3d) (f) ½ h(a + b + c)

(g) x(5x − 8y + 5) (h) 2a(2c + 3d − 5a) (i) 5p(3p + 2q + 4s)

8. (a) (a − b)(a + b) (b) (x − y)(x + y)

(c) (p − q)(p + q)

(d) (s − t)(s + t)

(e) (a − 3)(a + 3) (f) (x − 2)(x + 2)

(g) (p − 9)(p + 9) (h) (c − 5)(c + 5) (i) (b − 1)(b + 1)

(j) (y − 4)(y + 4) (k) (m − 5)(m + 5)

(l) (a − 3)(a + 3)

(m) (6 − d)(6 + d) (n) (2 − q)(2 + q) (o) (7 − w)(7 + w)

(p) (x − 8)(x + 8)

9. (a) (a − 2b)(a + 2b)

(b) (x − 5y)(x + 5y) (c) (p − 8q)(p + 8q)

(d) (4c − d)(4c + d) (e) (9 − 2g)(9 + 2g) (f) (6w − y)(6w + y)

(g) (2a − 1)(2a + 1) (h) (g − 9h)(g + 9h) (i) (7x − y)(7x + y)

(j) (3c − 4d)(3c + 4d) (k) (2p − 3q)(2p + 3q) (l) (b − 10c)(b +10c)

(m) (5 − 4a)(5 + 4a) (n) (2d − 11)(2d + 11) (o) (15 − 7k)(15 + 7k)

(p) (3x − 0·5)(3x + 0·5)

10. (a) 2(a − b)(a + b) (b) 5(p − 1)(p + 1) (c) 5(3 – x)(3 + x)

(d) 4(d− 3)(d + 3) (e) 2(y − 5)(y + 5) (f) 4(b − 5)(b +5)

(g) 3(q − 3)(q + 3) (h) 8(a − 2b)(a + 2b) (i) a(b – 8)(b + 8)

(j) x(y – 5)(y + 5) (k) ab(c − 1)(c + 1) (l) 2(2p − 5q)(2p + 5q)

(m) 2(x −1·2)(x + 1·2) (n) a(k – 11)(k + 11) (o) 2·5(2s – 1(2s + 1)

(p) ½( y − 30)(y + 30)

144

11. (a) (x + 1)(x + 2) (b) (a + 1)(a + 1) (c) (y + 1)(y + 4)

(d) (x + 7)(x + 1) (e) (x + 3)(x + 3) (f) (b + 6)(b + 2)

(g) (a + 7)(a + 2) (h) (w + 1)(a + 9) (i) (d + 5)(d + 2)

(j) (x + 7)(x + 3) (k) (p + 4)(p + 5) (l) (c + 4)(c + 6)

(m) (s + 6)(s + 6) (n) (x + 7)(x + 4) (o) (y + 5)(y + 5)

12. (a) (a − 5)(a − 3) (b) (x − 1)(x − 8) (c) (a − 6)(a − 3)

(d) (y − 2)(y − 2) (e) (b − 5)(b − 1) (f) (x − 14)(x − 1)

(g) (c − 2)(c − 8) (h) (x − 6)(x − 1) (i) (y − 4)(y − 8)

(j) (p − 8)(p − 3) (k) (a − 9)(a − 4) (l) (x − 3)(x − 12)

(m) (b − 1)(b − 3) (n) (q − 10)(q − 1) (o) (a − 4)(a − 3)

13. (a) (b + 5)(b − 2) (b) (x + 7)(x − 1) (c) (y + 2)(y − 3)

(d) (a + 4)(a − 5) (e) (q + 4)(q − 2) (f) (x + 2)(x − 10)

(g) (d + 7)(d − 3) (h) (c + 12)(c − 3) (i) (p + 3)(p − 8)

(j) (y + 1)(y − 8) (k) (a + 6)(a − 1) (l) (x + 4)(x − 9)

(m) (b + 1)(b − 5) (n) (s + 6)(s − 4) (o) (d + 8)(d − 2)

14. (a) (3x + 1)(x + 2) (b) (2a + 1)(a + 2) (c) (3c + 5)(c + 1)

(d) (2p + 9)(p + 1) (e) (2y + 1)(y + 5) (f) (3d + 2)(d + 3)

(g) (5q + 4)(q + 1) (h) (2b + 3)(2b + 1) (i) (3x + 2)(2x + 3)

(j) (3a + 5)(a + 3) (k) (5x + 1)(2x + 3) (l) (3c + 1)(3c + 1)

(m) (3y + 1)(2y + 3) (n) (3b + 2)(b + 1) (o) (4x + 1)(2x + 3)

15. (a) (2x − 1)(x − 3) (b) (2a − 3)(a − 1) (c) (5p − 2)(p − 3)

(d) (5b − 2)(b − 1) (e) (3x − 2)(2x − 1) (f) (4y − 3)(y − 2)

(g) (7c − 1)(c − 4) (h) (4m − 1)(m − 2) (i) (8a − 1)(2a − 1)

(j) (4y − 1)(2y − 5) (k) (3p − 1)(p − 12) (l) (4x − 1)(x − 6)

(m) (5a − 2)(3a − 2) (n) (6c − 1)(4c − 3) (o) (3b − 4)(2b − 9)

16. (a) (3x + 1)(x − 1) (b) (a + 1)(2a − 3) (c) (4p + 3)(p − 1)

(d) (c + 4)(2c − 1) (e) (6y + 1)(y − 2) (f) (3w − 2)(w + 4)

(g) (3m + 5)(m − 1) (h) (q + 2)(4q − 3) (i) (2b + 5)(3b − 4)

(j) (2t + 1)(2t − 3) (k) (2z + 3)(6z − 1) (l) (2d + 3)(2d − 5)

(m) (7s + 1)(s − 4) (n) (3x + 5)(5x − 3) (o) (4v + 1)(9v − 2)

(p) )1)(73( ++ vv (q) )5)(12( −− ll (r) )14)(73( −− mm

(s) )4)(73( −− nn (t) )52)(52( −− bb (u) )23)(43( ++ cc

(v) )5)(13( +− qq (w) )43)(32( −+ aa (x) )32)(54( −+ bb

(y) )32)(56( −+ mm (z) )4)(72( −+ nn

145

17. (a) )1)(1(3 +− xx (b) )1)(5(2 ++ pp (c) )2)(2(9 +− xx

(d) )3)(2(5 ++ xx (e) )3)(2( ++ xxa (f) )1)(5(3 +− yy

(g) )1)(45(3 ++ cc (h) )32)(14(2 ++ bb (i) )3)(23(3 ++ qq

(j) )3)(12(5 −− ss (k) )1)(32(4 −− mm (l) )3)(32(4 −− aa

(m) )4)(72(2 +− tt (n) )43)(23(10 −+ dd (o) )110)(110(4 +− xx

18. (a) 8(w + 3)(w – 1 ) (b) 2x(3x – 1)(3x + 1) (c) 2b(2a – 3c)

(d) 2y(1 + y + y²) (e) r(r – 2)(r + 2) (f) 3x(x – 3)(x + 3)

(g) 11(x + 1)² (h) 2ab(3a – 4b) (i) 8(p – 3)(p + 3)

(j) 2(n – 9)(n + 8) (k) 3y(x + 2b²) (l) 3b(b – 4)

(m) 3f(f – 3)(f + 3) (n) 2(2w + 5)(w – 1) (o) 3(u – 5)(u – 7)

(p) 30a(a – 1)

19. (a) 5(6 – d)(6 + d) (b) 3(4n – 3)(n – 2) (c) 2yz(x – 4w)

(d) 2v(3 + 8v²) (e) 14(2q + 1)(q – 1) (f) 16(1 – 5a)(1 + 5a)

(g) 17(1 – x)² (h) m(m – 3n)(m + 3n) (i) (3x – 4)(2x – 3)

(j) w²(u – 2)(u + 2) (k) 28(x + 3)(x – 2) (l) 2(7x + 3)(x + 1)

(m) y(3 – 4y)(3 + 4y) (n) 36(1 + 3a)(1 – a) (o) abc(c + a)

1.4 Completing the Square

1. (a) (x + 3)² - 9 (b) (x + 5)² - 25 (c) (x - 2)² - 4 (d) (x - 1)² - 1

(e) (x - 3)² - 9 (f) (x - 6)² - 36 (g) (x + 23 )² -

49 (h) (x + 2

5 )² - 425

(i) (x - 21 )² - 4

1 (j) (x - 29 )² - 4

81 (k) (x + 10)² - 100 (l) (x - 41 )² - 16

1

2. (a) (x + 2)² - 3 (b) (x + 1)² + 4 (c) (x + 4)² - 19 (d) (x - 1)² - 10

(e) (x - 3)² + 2 (f) (x - 4)² - 9 (g) (x + 1)² - 22 (h) (x - 7)² - 36

(i) (x + 2)² - 11 (j) (x - 5)² - 16 (k) (x + 6)² - 51 (l) (x - 10)² - 81

3. (a) (x + 21 )² + 4

3 (b) (x + 23 )² - 2

5 (c) (x - 21 )² + 4

1 (d) (x - 21 )² + 2

1

(e) (x - 23 )² - 4

13 (f) (x - 25 )² - 4

27 (g) (x + 23 )² - 2 (h) (x + 2

5 )² - 421

(i) (x + 23 )² - 3 (j) (x -

23 )² - 4

7 (k) (x + 21 )² - 4

9 (l) (x - 27 )² - 12

Section 2 – Trigonometry

2.1 The Tangent Ratio (lengths)

1. 5.8cm 2. 6.7mm 3. 5.6cm 4. 30.6m 5. 7.3mm 6. 17.3km

7. 17.2cm 8. 2.9km 9. 4.0m 10. 3.5mm 11. 127.6cm 12. 7.2km

13. 4.8cm 14. 130.6mm 15. 19.0km 16. 15.0cm 17. 22.2m 18. 4.8mm

146

2.2 The Tangent Ratio (angles)

1. 33.9o 2. 72.1

o 3. 52.7

o 4. 52.0

o 5. 51.0

o 6. 21.8

o

7. 59.4o 8. 61.4

o 9. 45.0

o 10. 19.7

o 11. 34.0

o 12. 21.6

o

13. 38.8o 14. 67.2

o 15. 16.2

o

2.3 Using the Tangent Ratio (mixed)

1. 2.66cm 2. 48.0º 3. 12.2cm 4. 71.6º 5. 19.6m 6. 55.5º

7. 4.44cm 8. 55.2º 9. 13.0m 10. 58.9º 11. 249m 12. 49.0º

13. 63.3cm 14. 53.4º 15. 120km 16. 14.6cm 17. 56.6º 18. 15.6mm

2.4 Problem Solving Using the Tangent Ratio

1. 54.7m 2. 25.7º 3. 25.7m 4. 28.1º 5. 5.89m 6. 19.2m

7. 11.7m 8. 68.7º 9. 281m 10. 159m 11. 4.97m 12. 25.0m

13. 65.4º 14. 41.4m 15. 20.1cm 16. (a) 29.4m (b) 11.1º

17. (a) 36.5º (b) 34.8m 18. 40.9m 19. 31.8m 20. 185m 21. 58.1m

22. 67.4º and 112.6º

2.5a Using the Sine and Cosine Ratios (lengths)

1. 5.16cm 2. 10.4m 3. 14.2mm 4. 3.44km 5. 6.24cm 6. 11.9mm

7. 13.8m 8. 7.78km 9. 25.6cm 10. 3.53mm 11. 8.97m 12. 12.7cm

13. 8.74km 14. 6.39m 15. 1.03cm

2.5b Using the Sine and Cosine Ratios (lengths)

1. 13.7cm 2. 8.59m 3. 20.1mm 4. 14.1km 5. 36.9cm 6. 9.76mm

7. 7.12m 8. 46.3km 9. 5.01cm 10. 6.22mm 11. 707m 12. 24.3cm

13. 283km 14. 170m 15. 453cm

2.6 Using the Sine and Cosine Ratios (angles)

1. 37.0º 2. 57.0º 3. 38.7º 4. 47.2º 5. 30.0º 6. 45.6º

7. 27.3º 8. 44.9º 9. 35.3º 10. 70.7º 11. 37.5º 12. 64.4º

13. 32.0º 14. 54.9º 15. 70.3º 16. 59.8º 17. 32.1º 18. 55.3º

2.7 Using the Sine and Cosine Ratios (mixed)

1. 55.4º 2. 35.8mm 3. 16.7cm 4. 46.7º 5. 13.2mm 6. 8.42km

7. 15.7º 8. 13.5m 9. 44.2m 10. 15.6mm 11. 18.0cm 12. 2.29km

13. 33.7cm 14. 53.5º 15. 48.3º 16. 31.3cm 17. 1.71m 18. 0.722mm

2.8 Choosing the Appropriate Ratio

1. 50.5º 2. 14.8mm 3. 26.9cm 4. 66.4º 5. 75.9mm 6. 52.7º

7. 6.87cm 8. 23.0m 9. 3.81m 10. 57.5º 11. 37.6º 12. 15.2km

13. 49.6cm 14. 31.8º 15. 86.6km 16. 1.68cm 17. 5.69m 18. 222mm

147

2.9a Problem Solving Using Trigonometry

1. 8.82m 2. 12.5m 3. 7.1º 4. 1.03º 5. (a) 381.6m (b) 1.9º

6. 7.85m 7. 7.20m 8 (a) 8.76m (b) 6.02m (c) 6.56m (d) 5.09m

9. (a) 57.2m (b) 12.0m 10. 23.6º; 938.6m

11. (a) 8.96m (b) 38.5º (c) 6.72m

12. (a) (i) 373.4m (ii) 20.4º (b) 200.2m

2.9b Problem Solving Using Trigonometry

1. x1= 10.2cm; x2= 11.4cm 2. x1= 17.3cm; x2= 9.67cm

3. x1= 45.5cm; x2= 63.2º 4. x1= 11.3cm; x2= 3.72cm

5. x1= 21.5cm; x2= 43.1º 6. x1= 16.8cm; x2= 15.4cm

7. x1= 14.0cm; x2= 52.7º 8. x1= 65.2mm; x2= 76.7mm

9. x1= 49.8m; x2= 10.2º 10. x1= 45º; x2= 22.1cm; x3= 25.1cm

11. x1= 5.69cm; x2= 1.57cm 12. x1= 29.8m; x2= 32.0m

13. x1= 49º; x2= 8.41cm 14. x = 12.6cm;. 15. x= 54.5º;

2.9c Problem Solving Using Trigonometry

1. (a) 24.6º (b) 10.9cm (c) 3.9cm (d) 5.9cm (e) 122.7º

2. (a) 3.41cm (b) 17.3cm (c) 20.0cm

3. (a) 9.07cm (b) 127cm² (c) 49.3cm

4. (a) 22.3cm² (b) 15.1º

5. (a) 8.94cm (b) 2.01cm (c) 11.7cm (d) 30.4cm (e) 41.1cm²

2.9d Problem Solving Using Trigonometry

1. (a) 21.8º (b) 33.7º (c) 31.0º (d) 22.9cm (e) 29.2cm

(f) 30.9cm (g) 29.0º (h) 19.1º

2. 35.3º

3. (a) 6.50cm (b) 21.9cm (c) 63.4º (d) 23.3cm (e) 16.2º

4. (a) 22.6m (b) 34.8m (c) 41.5m (d) 65.3º

5. (a) 30.5cm (b) 29.3cm (c) 73.8º (d) 249cm²

Section 3 – Circles

3.1 Applying Pythagoras’ Theorem

1. (a) 9·43 (b) 21·3 (c) 13 (d) 10·2 (e) 1·05 (f) 5·07

(g) 12·4 (h) 26·9 (i) 2·4

2. (a) 10 (b) 15·6 (c) 180cm²

3. (a) 21 (b) 357cm²

4. (a) 7·07 (b) 12·5 (c) 8·06 (d) 10·2

5. (a) 11.7cm (b) 12·7cm

6. (a) 20cm (b) 15·0cm

7. 19·2km 8. 427·2km 9. 9·46km

148

10. 7·9km 11. 15·6mm 12. 16·6cm

13. 4·6m 14. 4·4m 15. 16cm

16. 6·6cm 17. 16·9m 18. 1·92m

19. (i) 2·9cm (ii) 25·4cm² (iii) 7·35cm³

20. proofs

21. (a) no (b) no (c) no

(d) yes (e) no (f) yes

3.2 Applying Pythagoras’ Theorem – EXAM QUESTIONS

1. Mechanism will work since 7·1 > 7 2. 34·8cm

3. 67cm 4. Suitable since 11·3 > 11 5. 41·2cm

6. £952 7. 21·5m 8. 3·3m 9. 37cm

10. 20·7m 11. (a) AB = 8mm; BC = 4mm (b) 6·9mm

12. 7·7cm 13. 5·06m or 506 cm 14. 15cm 15. 33cm

16. (a) 15cm (b) 16cm (c) proof (d) 30cm; 225cm²

17. supports are vertical 18. 13·9 > 13·89 so just long enough

3.3 Triangles, Chords and Perpendicular Bisectors in Circles

1. (a) 90o (b) 45

o (c) 90

o (d) 55

o (e) 90

o

(f) 43o (g) 90

o (h) 18

o (i) 90

o (j) 63

o

(k) 90o (l) 78

o

2. (a) 9·9 cm (b) 8·5 cm (c) 6·4 cm (d) 9·2 cm

3. (a) 40o (b) 40

o (c) 50

o (d) 33

o (e) 33

o

(f) 57o (g) 28

o (h) 62

o (i) 62

o (j)118

o

(k) 118o (l) 31

o (m) 31

o (n) 31

o (o) 31

o

4. (a) 4·5 cm (b) 5·7 cm (c) 7·2 cm (d) 3 cm (e) 8 cm

(f) 9·2 cm

5. (a) 36·9o (b) 24·1 cm (c) 9·0 cm (d) 12·6 cm

(e) 23·7 cm (f) 8 cm 6. 37·6 cm

3.4 Tangents and Angles

1. (a) 90o (b) 20

o (c) 110

o (d) 90

o (e) 60

o

(f) 30o (g) 35

o (h) 35

o

(k) 90

o (m) 65

o

(n) 90o (p) 55

o (q) 90

o (r) 45

o

2. (a) 6 cm (b) 13 cm (c) 24 cm

3. (a) 33·7o (b) 10·4 cm (c) 14·3 cm

149

3.5 The Length of an Arc

1. (a) 12.6cm (b) 34.2mm (c) 1.2m

2. (a) 28.6cm (b) 62.2mm (c) 5.2m

3. (a) 7.9cm (b) 4.7cm (c) 18.8cm (d) 4.4cm

(e) 4.9cm (f) 16.7cm (g) 20.9cm (h) 12.6cm

3. (a) 23.5cm (b) 14.1cm (c) 37.7cm (d) 40.3cm

(e) 7.0cm (f) 58.6cm (g) 29.3cm (h) 50.2cm

3.6 The Area of a Sector

1. (a) 50·2cm² (b) 239mm² (c) 1·22m²

2. (a) 19·6cm² (b) 7·07cm² (c) 84·8cm² (d) 12·8cm²

(e) 4·88cm² (f) 100cm² (g) 83·7cm² (h) 62·8cm²

3. (a) 58·9cm² (b) 21·2cm² (c) 170cm² (d) 141cm²

(e) 7·68cm² (f) 352cm² (g) 117cm² (h) 251cm²

4. 12·8cm²

3.7 Properties of Circles – EXAM QUESTIONS

1. (a) 50cm (b) 14cm 2. (a) 100cm (b) 171cm

3. 20cm 4. 102cm 5. 54o

6. 9·6cm 7. 23o 8. 132

o

9. 28o 10. (a) 112

o (b) 60·6cm 11. 36cm 12. 71·4cm

13. 76o 14. 8cm 15. 11·3cm

3.8 Arc Length and Sector Area - EXAM QUESTIONS

1. 54·4cm² 2. 6·83m 3. 7770cm² 4. 27cm 5. 11·2m²

6. 33·1cm² 7. 206 cm² 8. 40o

9. (a) 173cm² (b) 56·5cm 10. 25·1cm² 11. 90o

12. (a) 272m (b) 4470m² 13. (a) 83·4cm (b) 151cm²

14. (a) 1320cm² (b) 177cm 15. 20o 16. 25·1cm

17. 60·5o

18. (a) 57·3o (b) 89·6

o (c) 12·2mm (d) 9·58cm

150

2. L Q1 median Q3 H SIQR

(a) 47 50 56 61 65 5.5

(b) 12 16 26 34 38 9

(c) 149 152·5 165 169 179 8.25

(d) 1 1 3 4 8 1.5

(e) 95 115 128·5 157·5 188 21.25

(f) 43 50 63 70·5 92 10.25

(g) 165 166 180 187 190 10.5

Section 4 - Statistics

4.1 Quartiles and Interquartile Range

1. 2.

L Q1 Q2 Q3 H IQR

(a) 2 4 7 12 15 8

(b) 29 30 33 37 40 7

(c) 17 19 22 25 26 6

(d) 0 0 2 3 4 3

(e) 1.8 2·8 4·0 5·1 5.3 2·3

(f) 0.13 0·185 0·275 0·305 0.39 0·12

(g) 133 136 139 142 145 6

(h) 371 376 382 387 390 11

(i) 57 59 67 83·5 90 24·5

(j) 11 11 14 16 20 5

151

3.

4.

Class B has a higher median and a smaller range than class A, although IQRs are the same.

Although class A has a higher maximum mark there is a greater spread of ability.

5. (a)

(b) Line B is the better line as there is less variation in the length of the shoe-laces.

6.

Class 6C2 fasted longer on average (higher median) and class 6C1 were more consistent in their fasting

(smaller IRQ).

(a) Median IQR

Thomsons 50 15

First Choice 40 20

(b) Median IQR

Maths 75 30

English 70 15

(c) Median IQR

Teachers 75 30

Athletes 60 15

(d) Median IQR

Glasgow 40 25

Edinburgh 40 35

• On average, visitors to Thompsons website are

older (50 > 40)

• Ages of visitors to Thompsons website are

more consistent, as IRQ is smaller (15<20)

• On average, test score in Maths are higher (75 > 70)

• Pupils are more consistent in their English test scores

as IRQ is smaller (15<30)

• On average, teachers have a higher resting

heart rate (75 > 60)

• The athletes resting heart rates are more

consistent, as IRQ is smaller (15<30)

• On average, waiting times are the same in both

hospitals

• Waiting times for operations in Glasgow are

more consistent, as IRQ is smaller (25<35)

152

7. (a) Mean = 339.4s, mode = 322s (b)

8. (a) Median = 3.4kg, mode = 3.4kg, range = 1.6kg (b)

9. (a) Mean = 6.5 marks, median =7, mode = 8 (b)

10. (a) Mean = 3.96 letters, median = 4 letters, mode = 4 letters

(b) Check diagram

4.2 Standard Deviation

1.

2. 3·44, 1·72

3. line A 27, 0·55; line B 27,0·19;

line B more consistent

4. 106, 16·7

5. 21, 3·6

6. John 73, 1·04 ; Joe 72, 5·20

Joe has lower mean score but John has better

overall performance (lower standard deviation)

7. Dec 3313, 1025; Mar 2352, 565

December has higher mean takings

but March has less variation in takings

8. 6C1 21·5,1·26 ; 6C2 21·5, 2·88

Same average but 6C1 has lower SD so less spread out

Mean S.D.

(a) 20.3 0.95

(b) 302 3.19

(c) 14.99 0.19

(d) 87 1.49

(e) 62.8 22.9

(f) 4.45 0.30

(g) 0.23 0.13

(h) 39.6 1.17

153

4.3 Scatter Graphs

1. (a) no relation (b) positive (c) negative

2. (a) positive correlation (more rain – more people buy umbrellas)

(b) no relation

(c) negative correlation (the faster you go, the less time it takes)

3. (a) yes (b) yes, but not strong (c) yes (d) no

4. student’s best fit lines

5. Answers will vary depending on where line is drawn

(a) y = 1·67x + 3.3 (b) y = 0·4x + 1.5 (c) y = 1·2x − 6

(d) y = −1·5x + 9 (e) y = −1·5x + 12 (f) y = −0·25x + 7

6. H = 0·6D + 0·7, 1·6 7. y = 3·8x + 6

8. l = 0·9F + 2·2, 6·25 9. C = −2T + 67, 62oC

10. S = 7T, 70 mph 11. C = 1·1m + 177, £215.50

12. R = −0·35V + 0·61, 0·3

4.4 Mean and Standard Deviation - EXAM QUESTIONS

1. (a) 40·7g, 3·6 (b) 140·7g, 3·6

2. (a) 7⋅55; 0⋅44 (b) slightly higher mean so slower times on average in 2nd race

higher SD so times are less consistent than 1st race

3. (a) 80kg, 12·2 (b) on average weight is less and less spread out

4. (a) 66; 6·56 (b) 71; 6·56

5. (a) 21·5; 4·42 (b) On average study times same but teachers are more varied

6. (a) £157, 51·3 (b) on average E-Pod more expensive and less spread out

7. (a) £22·50, 5·4 (b) prices of dolls are less spread out than teddies

4.5 Scatter Graphs - EXAM QUESTIONS

1. (a) 53 += WP (b) 6 points

2. Answers depend on line drawn

3. A – strong positive correlation.

4. (a) strong positive correlation (b) I = 1/7C + 1

5. (a) R = ½h + 4 (b) 15

154

Section 5 – Percentages and Fractions

5.1 Working with Percentages

1. (a) £100 (b) 250mm (c) £75

(d) 30 litres (e) 420 miles (f) 512 m

(g) 1 500 km (h) £5

250 (i) 12

000m

(j) 712 cm (k) £775 (l) 90 kg

2. (a) £22 (b) £45 (c) £72

(d) 85 ml (e) £17.50 (f) 100 cm

(g) 2 550 m (h)

8

500 km (i) 56 m

(j) £11 200 (k) 97·5 litres (l) £6

200

3. (a) £40 (b) £65 (c) £25

(d) £425 (e) £7 (f) £299

(g) £2 050 (h) £26 (i) £26.50

4. (a) £12 500 (b) £18

060 (c) £24

600

(d) £37 440 (e) £23

400 (f) £20

411.22

(g) £3 644.86 (h) £13 457.94

5.2 Working with Percentages – EXAM QUESTIONS

1. 500 2. 150 3. £200 000

4. £12 000 5. £4

500 6. £2

600

5.3 Appreciation and Depreciation

1. (a) £2332.80, £332.80 (b) £5955.08, £955.08 (c) £5495.52, £695.52

(d) £4348.04, £848.04 (e) £1982.12, £382.12 (f) 2393.35, 643.35

(g) £23820.32, 3820.32 (h) £21190.05, 3190.05 (i) £64751.45, £14751.45

(j) £439.32, £39.32

2. £3240 3. £92317 4. £212.24

5. 6300 million 6. £1536 7. £5644.80

8. (a) £270 000 (b) after 3 years

155

5.4 Appreciation and Depreciation – EXAM QUESTIONS

1. £920.48 2. £27 900 3. £1

500

000

4. 136·3mg 5. £1 388

6. (a) 39 300 tonnes (b) just falls short of doubling

7. £9 600 8. (a) 4% (b) £97

700

9. (a) £3 230.67 (b) 5 years 10. £41

100

11. £9 790 12. (a) 44

990 (b) 5 months

13. (a) 8% (b) 21·5 14. £6 834.38 15. £159

720

16. Car is valued at more than half the original value.

17. (a) £1 677 (b) 4 years

18. Gained £140.49 19. Yes, since 913g > 900g 20. £962 000

5.5 Addition and Subtractions of Fractions

1. (a) 5

4 (b)

2

1 (c)

8

7 (d)

6

5 (e)

9

7 (f)

12

7 (g)

20

17 (h)

12

5 (i)

24

23 (j)

10

9 (k)

12

11 (l)

14

13

(m) 56

23 (n)

40

23 (o)

63

41 (p)

30

23

2. (a) 5

11 (b)

10

31 (c)

8

31 (d)

2

11 (e)

9

21 (f)

12

51 (g)

20

11 (h)

12

71 (i)

24

131 (j)

10

31 (k)

42

111 (l)

80

171

(m) 56

91 (n)

40

71 (o)

63

131 (p)

40

171 (q)

4

31 (r)

5

11

(s) 16

51 (t)

12

51 (u)

12

51 (v)

3

21 (w)

12

51 (x)

60

491

3. (a) 2

1 (b)

3

1 (c)

6

1 (d)

12

1 (e)

4

1 (f)

16

7

(g) 12

5 (h)

10

1 (i)

16

11 (j)

10

3 (k)

4

1 (l)

4

1

(m) 40

9 (n)

30

7 (o)

63

22 (p)

24

5

156

4. (a) 4

32 (b)

4

13 (c)

8

53 (d)

3

15 (e)

8

75 (f)

12

58

(g) 5

13 (h)

24

55 (i)

8

18 (j)

12

118 (k)

3

14 (l)

16

54

(m) 10

13 (n)

4

37 (o)

16

14 (p)

9

46 (q)

6

16 (r)

6

18

(s) 24

233 (t)

16

314

5. (a) 3

12 (b)

12

53 (c)

8

12 (d)

9

7

(e) 2

1 (f)

4

12 (g)

4

31 (h)

6

11

6. (a) 4

12 (b)

24

132 (c)

20

111 (d)

4

13 (e)

20

14 (f)

12

15

(g) 21

113 (h)

12

72 (i)

24

54 (j)

8

13 (k)

4

14 (l)

10

12

(m) 5 (n) 4

37 (o)

10

17 (p)

9

21

7. (a) 4

11 (b)

7

33 (c)

3

12 (d)

3

14 (e)

35

31 (f)

4

33

(g) 4

32 (h)

20

173 (i)

12

51 (j)

8

52 (k)

12

113 (l)

2

11

(m) 3

24 (n)

20

13 (o)

10

76 (p)

9

4

8. 40

94 km 9.

35

810 cm 10. 7 hours

11. 6

16 hours 12.

6

1 13.

15

4

14. (a) 20

7 (b)

12

7

157

5.6 Multiplication and Division of Fractions

1. (a) 7

1 (b)

10

1 (c)

7

2 (d)

12

1 (e)

20

1 (f)

7

4 (g)

7

2 (h)

14

1 (i)

8

3 (j)

39

4 (k)

40

3 (l)

3

2

(m) 15

4 (n)

4

3 (o)

27

10 (p)

12

1 (q)

10

3 (r)

4

1 (s)

21

4 (t)

4

1

2. (a) 3

21 (b)

12

12 (c)

4

16 (d)

12

112 (e)

10

93 (f)

9

53

(g) 3

22 (h)

2

14 (i)

2

112 (j) 30 (k)

7

62 (l)

8

58

(m) 12 (n) 14 (o) 4 (p) 8

77 (q)

8

59 (r)

27

85

(s) 21 (t) 5

13

3. (a) 4

3 (b)

5

21 (c)

15

11 (d)

14

11 (e)

4

1 (f)

3

21 (g)

9

4 (h)

11

6 (i)

3

2 (j)

2

1 (k)

25

18 (l)

15

8

(m) 27

16 (n)

57

20 (o)

9

71 (p)

5

4 (q)

5

4 (r)

3

1

(s) 55

24 (t) 6

4. (a) 4 (b) 3 (c) 2 (d) 12

11 (e)

18

11 (f)

3

22

(g) 2

11 (h)

9

4 (i)

5

31 (j)

16

91 (k)

12

112 (l)

14

52

(m) 32

9 (n)

15

22 (o)

16

15 (p)

3

11 (q)

32

27 (r)

20

9

(s) 5

11 (t)

4

3

158

5. (a) 16

1 (b)

6

1 (c)

7

3 (d)

4

131 (e) 42 (f)

2

11 (g)

10

7 (h)

15

142 (i)

40

3 (j)

3

24 (k)

8

13 (l)

26

1

(m) 8

1 (n)

12

7 (o)

7

5 (p)

20

1 (q)

3

216 (r)

9

1

6. (a) 6 (b) 2

1kg 7. (a) 33 (b)

8

3metre

8. 40

274 cm² 9.

4

329 cm²

5.7 Mixed Questions on Fractions

1. 12

112 kgs 2.

4

13 km 3. (a)

24

233 litres (b)

24

14 litres

4. (a) 15

1436 metres (b)

5

183 m²

5. (a) 190 km (b) 297 km (c) 487 km

(d) 6

16 hours (e)

37

3678 km/h

6. (i) 4 questions correct (ii) (a) 42

111 (d)

75

68

Section 6 – Similar Shapes

6.1 Linear Scale Factors

1.(a) (i) s.f. = 23 or 1·5 (ii) 13·5cm (b) (i) s.f. =

32 of 0·66... (ii) 20cm

(c) (i) s.f. = 25 of 2·5 (ii) 45cm (d) (i) s.f. =

53 or 0·6 (ii) 168mm

2. (a) x = 30mm (b) x = 532 ⋅ cm

3. (a) Because they are equiangular (b) CD = 18cm (c) 81 cm2

4. (a) x = 513 ⋅ cm (b) x = 414 ⋅ m 5. ST = 16cm

6. distance = 70 ⋅ m

159

6.2 Area Scale Factors

1. (a) s.f.(L) = 2; s.f. (A) = 4; A = 64cm2

(b) s.f.(L) = 3; s.f. (A) = 9; A = 864mm2

(c) s.f.(L) = 1⋅5; s.f. (A) = 2⋅25; A = 90mm2

(d) s.f.(L) = 2⋅4; s.f. (A) = 5⋅76; A = 288cm2

(e) s.f.(L) = 4; s.f. (A) = 16; A = 352cm2

(f) s.f.(L) = 1⋅8; s.f. (A) = 3⋅24; A = 388⋅8cm2

2. (a) s.f.(L) = 0⋅5; s.f. (A) = 0⋅25; A = 17⋅5cm2

(b) s.f.(L) = 0⋅25; s.f. (A) = 0⋅0625; A = 228mm2

(c) s.f.(L) = 0⋅8; s.f. (A) = 0⋅64; A = 96cm2

(d) s.f.(L) = 0⋅75; s.f. (A) = 0⋅5625; A = 225mm2

3. (a) 88 cm2 (b) 166 mm

2 (c) 49cm

2 (d) 72 mm

2

6.3 Volume Scale Factors

1. (a) s.f.(L) = 2; s.f.(V) = 8; V = 384cm3

(b) s.f.(L) = 3; s.f.(V) = 27; V = 5832mm3

(c) s.f.(L) = 1⋅5; s.f.(V) = 3⋅375; V = 243mm3

(d) s.f.(L) = 2⋅4; s.f.(V) = 13⋅824; V = 276⋅48cm3

(e) s.f.(L) = 4; s.f.(V) = 64; V = 576cm3

(f) s.f.(L) = 1⋅4; s.f.(V) = 2⋅744; V = 1097⋅6cm3

2. (a) s.f.(L) = 0⋅5; s.f.(V) = 0⋅125; V = 46cm3

(b) s.f.(L) = 0⋅25; s.f.(V) = 0⋅015625; V = 2⋅25mm3

(c) s.f.(L) = 0⋅75; s.f.(V) = 0⋅421875; V = 384⋅75cm3

(d) s.f.(L) = 0⋅6; s.f.(V) = 0⋅216; V = 49⋅68mm3

3. (a) 1200 ml (b) 270 ml (c) 756 ⋅ litres

6.4 Similar Shapes - EXAM QUESTIONS

1. no, will burn for 4 times the time 2. priced correctly

3. rug is too small since 69·1 < 72 4. 7cm

160


7.1 Gradients of Straight Lines

1. (a) (i) 2 (ii) 21− (iii)

34 (iv)

61− (v) 1

(b) 0; undefined; positive; negative

2. (a) 1 (b) 2 (c) 32 (d) 5 (e)

31 (f)

23

(g) 3− (h) 21− (i)

23− (j) 1− (k) 6− (l)

81−

3. (a) 3− (b) 21 (c) 1 (d)

21− (e)

52 (f) 4

4. (a) 21 (b) 2− (c) 3 (d) 2 (e) 4− (f)

21

(g) 23 (h)

21− (i)

43− (j) 5 (k) 4 (l)

25−

5. (a) 51 (b) 2 (c) 2− (d) 1 (e)

43− (f)

35

(g) 52− (h)

711− (i)

111 (j)

34 (k)

712− (l) 1

(m) 9

13 (n) 5

13− (o) 67− (p)

1114

6. (a) both gradients 21 (b) both gradients

23−

(c) both gradients 61 (d) both gradients

34−

7. (a) k = 2 (b) k = 3− (c) k = 4− (d) k = 5

8. a = 9− 9. k = 2·5 10. a = 30− 11. k = 1 12. k = 19

7.2 Equations of Straight Lines

1. (a) 3; (0, 1) (b) ½; (0, − 5) (c) −2; (0, 3)

(d) −¼; (0, − 2) (e) 8; (0, − ½) (f) −1; (0, 4)

2. 1 and (b) 2 and (d) 3 and (f) 4 and (c) 5 and (a) 6 and (e)

3. Lines sketched.

4. (a) 22 += xy (b) 12

1−= xy (c) 24 −= xy (d) 33 += xy

(e) 13

1−−= xy (f) 3

2

1+−= xy (g) 33 −−= xy (h) 52

4

1⋅−= xy

161

5. (a) 1; (0, 3) (b) )1,0(;2 −− (c) )0,0(;2

1

(d) )2,0(;2

1− (e) )6,0(;1− (f) )2,0(;

2

1−

(g) )4,0(;3

1 (h) )4,0(;

5

4− (i) )6,0(;

2

3−

6. (a) )7,0(;1 − (b) )3,0(;5− (c) )2,0(;5

3−

(d) )0,0(;4− (e) )11,0(;2− (f) )2

5,0(;

2

1−

(g) )6,0(;3

1 (h) )3,0(;

7

3− (i) )4,0(;

5

4−

7. (a) 54 += xy (b) 12 +−= xy (c) 34

3−= xy

(d) 114 −= xy (e) 145 −−= xy (f) 12 =− xy

(g) 1343 =− xy (h) 1443 −=+ yx (i) 932 −=+ yx

8. (a) 52 −= xy (b) 134 =+ xy (c) 325 += xy

(d) 12 −= xy (e) 6+= xy (f) 1534 −=+ yx

(g) 2053 =− xy (h) 1452 =+ yx (i) 48117 =+ xy

9. (a) 02 =− xy (b) 72 =+ xy (c) 63 −= xy

(d) 032 =− yx (e) 32 =+ xy (f) 434 −=+ xy

(g) 285 += xy (h) 194 −= xy (i) 452 −=+ xy

10. (a) 53 −= xy (b) 164 =+ xy (c) 02 =− xy

(d) 52 −=+ xy (e) 142 =− xy (f) 33 =+ xy

7.3 Equations of Straight Lines - EXAM QUESTIONS

1. )2,0(;2

3 2. 6

4

3+−= xy 3. D 4. )3,0(;

4

3

5. P(0, 4) 6. 22

3+−= xy 7. )54,0(;3 ⋅−

162

8. Line crossing at (0, 5) with gradient 3

4− 9. (a)

4

1 (b) 10

4

1+= vT

10. )2,0(;3

2 11. )4,0(;

2

3− 12. 1223 −=− xy

13. (a) )51,0(;4

3⋅ (b) 32 =+ xy 14. (a) )4,0(;

3

4 (b) 1634 =+ xy

Section 8 – Simultaneous Equations

8.1 Graphical Solutions

1. (a) Tables completed Table 1: 9, 6, 2 Table 2: 1, 4, 6

(b) and (c) Graphs drawn

(d) (5, 4)

2. (a) Tables completed Table 1: 0, 5, 1 Table 2: 0, 3, 5

(b) and (c) Graphs drawn

(d) (5, 3)

3. (a) (4, 3) (b) (11, 3) (c) (9, 6) (d) (12, 5)

(e) (8, 4) (f) (20, 10) (g) (15, 3) (h) (8, 3)

(i) (7, 3) (j) (13, 4)

4. (a) (2, 4) (b) (2, 3) (c) (3, 1) (d) (3, 2)

(e) (4, 3) (f) (1, 5)

5. (a) (6, 5) ( b) (6, 2) (c) (2, 4) (d) (2, 2)

(e) (10, 3) (f) (4, −3) (g) (2, 5) (h) (2, 0)

(i) (1, −1) (j) (4, 6) (k) (2, 0) (l) (−6, 6)

(m) (5, −2) (n) (−1, −3) (o) (1, −1)

8.2 Algebraic Solutions

1. (a) x = 5 and y = 5 (b) x = 1 and y = 1 (c) x = 2 and y = 4

(d) x = 3 and y = 6 (e) x = 3 and y = 10 (f) x = 5 and y = 21


(d) x = 2 and y = −1 (e) x = 6 and y = −3 (f) x = 4 and y = −5

(g) x = −3 and y = −2 (h) x = −11 and y = −3 (i) x = −8 and y = −10

163

3. (a) x = 2 and y = 5 (b) a = 3 and b = 2 (c) e = 6 and f = 2

(d) x = −1 and y = 3 (e) x = 2 and y = −2 (f) p = −2 and q = 3

(g) g = −4 and h = 3 (h) x = 3 and y = 4 (i) u = −2 and v = −3

(j) x = 4 and y = 1 (k) p = −5 and q = 2 (l) a = 4 and b = −2

4. (a) x = 2 and y = 5 (b) a = 3 and b = −1 (c) e = −2 and f = 5

(d) x = −1 and y = 4 (e) x = 3 and y = 4 (f) p = −3 and q = 4

(g) g = 8 and h = −5 (h) x = 7 and y = −2 (i) u = 2 and v = −2

(j) x = 4 and y = 1 (k) p = −1 and q = 2 (l) a = 8 and b = −2


(d) x = 2 and y = 3 (e) x = 3 and y = −1 (f) x = 4 and y = 4

(g) x = 2 and y = 5 (h) x = 5 and y = 1 (i) x = 2 and y = 1

(j) x = 4 and y = 2 (k) x = −1 and y = −2 (l) x = 1 and y = −1

6. (a) x = 1 and y = 2 (b) x = 3 and y = −1 (c) x = 2 and y = 2

(d) x = 5 and y = 1 (e) x = 5 and y = 7 (f) x = 5 and y = −2

(g) x = 3 and y = −3 (h) x = 4 and y = −3 (i) x = 2 and y = 3·5

(j) x = 0·5 and y = 2 (k) x = 1and y = 2 (l) x = 5 and y = 0

8.3 Simultaneous Equations in Context

1. 36 and 20 2. 18 and 4 3. 8 and 5 4. 9 and 2

5. Chocolate costs 40p and crisps cost 30p

6. Sandwich costs £1.20 and hotdog costs 90p.

7. Ruler costs 49p and pencil costs 26p.

8. Download is £9 and CD is £13

9. Standard print is 21p and Jumbo costs 55p.

10. Centre costs £11.25 and Side costs £9.50

11. Large glass holds 145ml and small holds 95ml.

12. Frame tent holds 8 and ridge tent holds 3.

13. Reader’s letter pays £15 and Star letter pays £25.

164

14. Small takes 1·8kg and the large takes 2kg.

15. Thursday should have been £21.95.

16. (a) 4x + 4y = 60; 6x + 16y = 120; x = 12 and y = 3 (b) 144cm²

17. (a) Box weighs 6kg and parcel weighs 2kg. (b) 58kg

18. Milk is 24p and butter is 96p.

19. 38 hours basic and 7 hours overtime

20. 320 cheaper tickets were sold

21. 33 × 20p coins and 21 × 50p coins

8.4 Simultaneous Equations - EXAM QUESTIONS

1. £21

2. (a) s + b = 640 (b) 8·5s + 12·2b = 6143

(c) 450 stalls tickets and 190 balcony tickets

3. £5.02 4. £81.70 5. (2, –3)

6. (a) f + t = 60 (b) 25f + 20t = 1325

(c) Clare sold 35 treacle scones

7. 75 points are needed – tokens give only 70 points so not enough.

8. £11.01. 9. x = 3 and y = 2 10. (2, –3)

11. £9.72 12. Sofa costs £425 and chair costs £295.

13. £23.87 14. £25.60 15. (–2, 5)

Section 9 – Functions and Formulae

9.1 Changing the Subject of a Formula

1. (a) x = y − 3 (b) x = y + 5 (c) x = y − a (d) x = y + b (e)

3

yx = (f)

10

yx = (g)

k

yx = (h)

a

yx =

(i) pyx 3−= (j) tyx 5+= (k) 2

1−=y

x (l) 3

7+=y

x

(m) 7

4ayx

−= (n)

4

38−=y

x (o) 10

8−=y

x

165

2. (a) a = 4 − b (b) a = 12 − d (c) a = 5x − y (d) 2

2 ma

−= (e)

5

7 qa

−= (f)

3

20 ca

−= (g)

2

rsa

−= (h)

4

tda

−= (i)

5

4 zba

−= (j)

7

2 kha

−= (k)

11

6 qpa

−= (l)

9

2 gxa

−=

3. (a) a

byx

−= (b)

m

cyx

−= (c)

s

rtx

+= (d)

q

rpx

2−= (e)

f

nmx

3+= (f)

c

bax

−= (g)

m

khx

−= (h)

c

bdx

3−=

(i) h

gkcx

−=

4. (a) 4

Pl = (b)

R

VI = (c)

D

ST = (d)

l

Ab =

(e) π

Cd = (f)

T

GU = (g)

a

uvt

−= (h)

2

2bPl

−= (i)

x

mHy

5−=

5. (a) bc 2= (b) xc 5= (c) yc 4= (d) mc 6= (e)

kc 9= (f) dc 10= (g) 42 −= ac (h) 153 += hc (i)

qpc 44 −= (j) xyc 1010 += (k) stc 168 −= (l) qrc 155 +=

6. (a) y

x3

= (b) d

cx = (c)

m

yx = (d)

s

ax

2+=

(e) w

zx

1−= (f)

a

cbx

+= (g) 89 −= ax (h) 52 += kx (i)

px 43−= (j) 1

2

−=y

x (k) 7

6

−=z

x (l) kh

mx

−=

7. (a) 2yk = (b) 2xk = (c) 2mk = (d) bak 2= (e)

dck 2= (f) 2ghk = (g) 2s

tk = (h)

2q

pk = (i)

2w

zk = (j) rk = (k) abk = (l)

q

pk =

(m) xyk −= (n) dck += (o) 3

1+=

xk

166

8. (a) a

uvs

2

22 −= (b) asvu 2

2−= (c)

2r

Vh

π= (d)

h

Vr

π=

(e) 2rA π= (f) 6

)3( 2−=

La (g) 44 2 −= kp (h)

z

txy

4

2

=

(i) 224 ra

xb = (j)

yx

stA

3−= (k)

2

23

A

yARx

+= (l)

1

12 +

=a

n

(m) dt

tn

−= (n)

12

21

rr

rrR

+= (o)

bx

da

+=

4

9.2 Changing the Subject of a Formula - EXAM QUESTIONS

1. 224 ba

xor

ab

x

2 2.

R

Vg

2

2

= 3. 22 )5( −= Ax

4. m

Ev

2= 5.

b

va

3= 6.

2

24

T

mk

π=

7. )32(9

5−= FC 8.

h

Vr

π

3= 9. 2−= mabk

10. c

ab

7−= 11.

n

km

3= 12. 3

4

3

π

Vr =

9.3 Function Notation

1. (a) 13 (b) 11− (c) 2− (d) 56 −a

2. (a) 8 (b) 20 (c) 13 (d) 42 +p

3. (a) 4 (b) 0 (c) 16 (d) m212 −

4. (a) aa 32 + (b) pp 64 2 + (c) 452 ++ mm (d) 1072 +− ee

5. (a) 0 (b) aa 129 2 − (c) 1282 +− aa (d) 344 2 −− pp

6. (a) 4 (b) 1− (c) 5

2

7. (a) 3 (b) 6 (c) 2−

8. (a) 5± (b) 1± (c) 4±

9. (a) (i) 15 (ii) 0 (b) 1582 ++ aa

10. (a) (i) 9 (ii) 39 (b) 5·5 (c) p645 −

167

Section 10 – Area and Volume

10.1 Properties of Shapes

1. (a) square (b) 0; rotational symmetry (c) equal

(d) diagonals; bisect; right (e) trapezium

2. 1925cm² 3. (a) 16 100cm² (b) 1·61m³

4. (a) 24cm² (b) 432cm² (c) 50%

5. (a) pentagon; 72o; 108

o, 72

o (b) octagon; 45

o; 135

o; 45

o

(c) hexagon; 60o; 120

o; 60

o

6. (a) right angled scalene (b) acute angled isosceles

(c) obtuse angled scalene

7. Acute angled isosceles; 23·8 units²

8. 30o, 60

o, 90

o and 32

o, 36

o, 112

o angles of a triangle add up to 180

o

9. (a) 37o (b) 62

o and 56

o (c) 40

o

10. 5760cm² 11. (a) £1.50 (b) £3

12. £792.18 13. (a) false (b) true (c) false (d) true

14. (a) 47·1m (b) 56·52cm 15. 43·96cm

16. 33·3cm 17. 596·6mm; £2.75 18. 9m 19. 12·1cm

20. (a) 452·16cm² (b) 706·5cm² 21. 1017·36cm² 22. 192mm²

23. (a) 12·56cm² (b) 188·4cm² (c) 240cm² (d) 21·5%

24. (a) 14·13m² (b) 45·87m² (c) £364.93

25. 2·5cm; 15·7cm

10.2 Volumes of Cylinders

1.(a) 1696·5 cm3 (b) 4825·5 cm

3 (c) 603·2 cm

3 (d) 2513·3 cm

3 (e) 75398·2 cm

3

(f) 3078·8 cm3 (g) 28274·3 cm

3 (h) 13304·6 cm

3 (i) 760265 cm

3 (j) 7298·5 cm

3

2. (a) 19·8cm (b) 3·7 litres 3. 904cm³

4. No; volume is 9·72 litres

168

10.3 Volumes of Other Solids

1.(a) 904·3cm³ (b) 33·5m³ (c) 3052·1mm³ (d) 113·0cm³

2.(a) 4190cm³ (b) 65400cm³ (c) 4·19m³ (d) 33500000mm³

(e) 697cm³ (f) 3050cm³ (g) 2140000mm³ (h) 87100cm³

(i) 180m³ (j) 57900cm³

3. 268cm³

4. (a) 366cm³ (b) 1280cm³ (c) 207cm³ (d) 1060cm³

5. (a) 2369cm³ (b) 1170cm³ (c) 15800cm³ (d) 608cm³

6. (a) 56·5cm³ (b) 803·8mm³ (c) 47·1cm³ (d) 25·1cm³

7. 83·7cm³ 8. 314cm³ 9. 1020cm³ 10. 37cm³

11. 640mm³

10.4 Volumes of Solids – EXAM QUESTIONS

1. 2·79 × 106 m³ 2. 6·01cm³ 3. 10cm 4. 113·04m³

5. 6·11cm 6. 3140cm³ 7. 4291cm³ 8. 25900cm³

9. 84·225cm³ 10. 58·9cm³ 11. 75·7cm³ 12. 93·4cm³

13. (a) 250cm³ (b) 1·3cm 14. 206cm³

15. 34·1 litres 16. 140 cm

3

course textbook 1

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