cambridge algebra answers

1 Linear graphs

1. 2.

3.

4. a b (6, 1)

5. a 2 b 13 c 3 d 1 e 2 f 13 g 5 h 5 i

15j 34

6. a 1 b 1 They are perpendicular andsymmetrical about the axes.

7.

8. a

b (2, 7)

9. a y = 43x 2 or 3y = 4x 6 b y = x + 1c y = 2x 3 d 2y = x + 6 e y = xf y = 2x

10.a y = 2x + 1 b 2y = x c y = x + 1d 5y = 2x 5 e y = 32x 3

2 Patterns and sequences

1. a 21, 34: add previous 2 terms b 49, 64: nextsquare number c 47, 76: add previous 2 terms

2. 15, 21, 28, 36

3. 12, 35,

23, 57,

34

4. a 6, 10, 15, 21, 28 b It is the sums of the naturalnumbers, or the numbers in Pascals Triangle.

5. a 13, 15, 2n + 1 b 33, 38, 5n + 3c 20, 23, 3n + 2 d 21, 25, 4n 3e 42, 52, 10n 8

6. a 3n + 1, 151 b 5n 2, 248c 8n 6, 394 d 5n + 1, 251e 3n + 18, 168

7. a 64, 128, 256, 512, 1024b i 2n 1 ii 2n + 1 iii 3 2n

8. b 4n 3 c 97 d 50th diagram

9. b 2n + 1 c 121 d 49th set

10.a i 14 ii 3n + 2 iii 41 b 66

3 Substitution

1. a 13 b 3 c 5

2. a 2 b 8 c 10

3. a 6 b 3 c 2

4. a 4.8 b 48 c 32

5. a 13 b 74 c 17

6. a 75 b 22.5 c 135

7. a 2.5 b 20 c 2.5

4 Simplifying expressions

1. a 2 + x b x 6 c k + x d x te x + 3 f d + m g b y h p + t + w

i 8x j hj k x 4 or l 2 x or

m y t or n wt o a2 p g2

2. a x + 3 yr b x 4 yr

3. F = 2C + 30

4. a 3n b 3n + 3 c n + 1 d n 1

5. a $4 b $(10 x) c $(y x) d $2x

yt

2x

x4

8

6

4

20

2

4

6

8

6 4 2 2 4 6

iiiLinear (ii)Linear (i)

y = 3x + 1, y = 2x + 3

2

3

1

0

1

2

3

4

5

2 4 6 8 10 12 14

y = 2x2y = 1x3

1234567

8 6 4 2 0 2 4 6 8

y = + 4x3

2

56

43210123456

1 3 4 5 6

y = 2x 5

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5 6

y = 3x + 4

1

10

8

6

4

20

2

4

6

8

10

10 8 6 4 2 2 4 6 8 10

y = 2x + 6, y = x + 7, y = 14

x 3, y = x + 8

CAMBRIDGE IGCSE MATHEMATICS AnswersAlgebra

BANATTypewritten Textkamalmath.wordpress.com

26. a 75p b 15x p c 4A p d Ay p

7. $(A B)

8. a 6t b 8w c 2w2 d 6t2

9. a $t b $(4t + 3)

10.a 5a b 6c c 9e d 6f e 3gf 4i g 4j h 3q i 0 j wk 6x2 l 5y2 m 0

11.a 7x b 6y c 3t d 3t e 5xf 5k g 2m2 h 0 i f 2

12.a 7x + 5 b 5x + 6 c 5p d 5x + 6e 5p + t + 5 f 8w 5k g ch 8k 6y + 10

13.a 2c + 3d b 5d + 2e c f + 3g + 4hd 2i + 3k e 2k + 9p f 3k + 2m + 5pg 7m 7n h 6n 3p i 6u 3vj 2v k 2w 3y l 11x2 5ym y2 2z n x2 z2

5 Expanding and factorising

1. a 6 + 2m b 6 12f c t2 + 3t d k3 5k e 15a3 10ab

2. a 7t b 9d c 3e d 2t e 5t2

f 4y2 g 5ab h 3a2d

3. a 2 + 2h b 9g + 5c 17k + 16 d 6e + 20

4. a 9t2 + 13t b 13y2 + 5yc 10e2 6e d 14k2 3kp

5. a 6(m + 2t) b 3m(m p) c 2(2a2 + 3a + 4)d 3b(2a + 3c + d ) e 2ab(4b + 16 2a)

6. a x2 + 5x + 6 b m2 + 6m + 5c x2 + 2x 8 d f 2 f 6e x2 + x 12 f y2 + 3y 10g x2 9 h t2 25i m2 16

7. a 6x2 + 11x + 3 b 10m2 11m 6 c 6a2 7a 3 d 6 7t 10t2 e 4 + 10t 6t2

8. a x2 + 10x + 25 b m2 + 8m + 16 c t2 10t + 25d 9x2 + 6x + 1 e x2 + 2xy + y2

9. a (x + 2)(x + 3) b ( p + 2)( p + 12) c (a + 2)(a + 6)d (t 2)( t 3) e (c 2)(c 16) f ( p 3)( p 5) g (n + 3)(n 6) h (d + 1)2

10. a (x + 3)(x 3) b (t + 5)(t 5) c (m + 4)(m 4) d (k + 10)(k 10) e (x + y)(x y) f (3x + 1)(3x 1)

11. a (2x + 1)(x + 2) b (3t + 2)(8t + 1) c 3( y + 7)(2y 3) d (2t + 1)(3t + 5)

6 Solving equations

1. a 30 b 72 c 6 d 10 e 4

2. a 3 b 4 c 112 d 2

3. a x = 2 b p = 2 c d = 6 d y = 1 e b = 9

4. 55p

5. a 112 cm b 6.75 cm2

6. 17 sweets

7. 3 years old

8. 5

7 Rearranging formulae

1. k =

2. r =

3. m = gv

4. r =

5. p = m 2

6. d =

7. a t = u2 v b u = v + t

8. a w = K 5n2 b n =

9. a 8y b c

10. a b = b a =

11.a

d Same formula as in a

12.a Cannot factorise the expression.

b c Yes, 3

8 Functions

1. a f(1) = 2 + 1 = 3

b g(2) = (2)2 = 4

c fg(x) = f(x2) = x2 + 1

d f1(x) = x 1

e fg1(x) = f(x12 ) = x

12 + 1

f gf1(x) = g(x 1) = (x 1)2

3V5

3Vr2(2r + 3h)

2 + 2yy 1

Rbb R

Raa R

6 + st2 + s

a(q p)q + p

K w5

4A

C2

A 94

T3


2. a f : x 1/x is not defined when x = 0

b g : x (x 5) is not defined when x 5

c h : x 10/(x +1) is not defined when x = 1

3. The function f(x) is defined as f(x) = x(x 1)

a i f(3) = 3(3 1) = 6

ii f(3) = 3(3 1) = 12

b If f(x) = 6, then x(x 1) = 6

x2 x = 6

x2 x 6 = 0

(x + 2)(x 3) = 0

x = 2 or x = 3

4.

5. a i f(100) = 10012 = 10

ii g(1) = 3(1)2 + 4 = 7

iii fg(2) = f(3(2)2 + 4) = f(16) = 1612 = 4

b gf(x) = g(x12) = 3(x

12)2 + 4 = 3x + 4

6. The function f(x) is defined as f(x) = 2/(x + 2)

a If f(x) = 5 then 2/(x + 2) = 5 and x = 8/5

b Let y = 2/(x + 2)

y (x + 2) = 2

yx + 2y = 2

yx = 2 2y

x = 2 2y

y

So f1(x) = 2 2x

or 2/x 2x

7. f : x x3 and g : x 1/(x 1)

a i fg(2) = f(1) = 1

ii gf(1) = g(1) = 1/2

b i fg(x) = f(1/(x 1)) = (1/(x 1))3 =1/(x 1)3

ii gf(x) = g(x3) = 1/( x3 1)

iii gg(x) = g(1/(x 1))= 1(1/(x 1) 1)

= 1(1 (x 1))/(x 1)

= x 11 (x 1)

= x 1 1 x + 1

= x 1 2 x

c i fg(x) =1/(x 1)3 is not defined when x =1

ii gf(x) = 1/( x3 1) is not defined when x = 1

iii gg(x) = x 1 is not defined when x = 22 x

9 Algebraic indices

1. a b c d e

f g h i j

2. a 7x3 b 10p1 c 5t 2 d 8m5 e 3y1

3. a i 25 ii 1125 iii45 b i 64 ii

116 iii5256

c i 8 ii 132 iii 412 d 1 000 000 ii

11000 iii14

4. a a3 b a5 c a7 d a4 e a2 f a1

5. a 6a5 b 9a2 c 8a6 d 6a4 e 8a8

f 10a3

6. a 3a b 4a3 c 3a4 d 6a1

e 4a7 f 5a4

7. a 8a5b4 b 10a3b c 30a2b2

d 2ab3 e 8a5b7

8. a 3a3b2 b 3a2c4 c 8a2b2c3

9. a t b m c k d x

10 Linear programming

1. a x 3 b x 5 c x 6 d t 18

2. a x 6 b t 83 c y 4 d x 2e w 5.5 f x 145

3. a x 2 b x 38 c x 612 d x 7e t 10 f y 75

4. a x 1 b x 3 c x 2 d x 1e x 1 f x 1

32

25

34

23

78x5

45y3

34t4

12m

12x3

10y5

4q4

7m2

6t

5x3

3

f(x) = f1(x) =x + 2 x 2x 10 x + 10

2x 12xx/3 3x1/x 1/xx3 3x

sin x sin 1xcos x cos 1xtan x tan 1x

replace f(x) with y andchange the subject

replace y with x tocomplete the inverse


5. a

b

c

d

e

f

g

h

6. a x 4

b x 2

c x 312

d x 1

e x 112

f x 2

g x 50

h x 6

7. 8.

9.

10.

11. a

b i Yes ii Yes iii No

12. ad

e i No ii No iii Yes

13. a 45x + 25y 200 9x + 5y 40b y x + 2

14. a i Cost 30x + 40y 300 3x + 4y 30ii At least 2 apples, so x 2iii At least 3 pears, so y 3iv At least 7 fruits, so x + y 7

b Draw graph with inequalities and shading asquestion

c Three apples and five pears

4

6

2

02

4

6

8

10

0 2 4 6 8 10

22

2

4

6

4682

4

6

8

y

4 x0

8

6 8

11

1

2

2

3

5

y

2 x02

1

4y = 4

y = 1

1

1

1

2

1

2

23 2 x0

y

x = 1x = 2

1

11

1

2

4

y

2 x0

3

2

y = 3

1

11

1

2

2

x = 2

x0

y

7 6 5 4 3

20 30 40 50 60

3 2 1 0 1

0 1 2 3 4

2 1 0 1 2

0 1 2 3 4

3 2 1 0 1

1 2 3 4 5

3 2 1 0 1 2 3 4

1 0 1 2 3

1 2 3 4 5

2 1 0

1 2 3 4 5

1 0 1

2 1 0

0 1 2 3 4

4


15. a i Space 4x + 3y 48

b ii Cost 300x + 500y 6000 3x + 5y 60Draw graph with inequalities and shading asquestion

c Six sofas and eight beds

16. a i Number of seats required is 40x + 50y 300 4x + 5y 30

ii Number of 40-seaters x 6iii Number of 50-seaters y 5

b Draw graph with inequalities and shading asquestion

c Five 40-seater coaches and two 50-seatercoaches cost $740

11 Direct and inverse proportion

1. a 15 b 2

2. a 75 b 6

3. a 150 b 6

4. a 22.5 b 12

5. a 175 miles b 8 hours

6. a 100 b 10

7. a 27 b 5

8. a 56 b 1.69

9. a 192 b 2.25

10. a 25.6 b 5

11. a 3.2 C b 10 atm

12. a 388.8 g b 3 mm

13. Tm = 12 a 3 b 2.5

14. Wx = 60 a 20 b 6

15. Q(5 t) = 16 a 3.2 b 4

16. Mt2 = 36 a 4 b 5

17. WT = 24 a 4.8 b 100

18. gp = 1800 a $15 b 36

19. td = 24 a 3 C b 12 km

20. ds2 = 432 a 1.92 km b 8 m/s

21. WF = 0.5 a 5 t/h b 0.58 t/h

12 Simultaneous equations

1. a (4, 1) b (5, 5) c (2, 6) d (2, 6) e (712, 312)

2. a x = 4, y = 1 b x = 5, y = 2 c x = 214, y = 612

3. a x = 2, y = 3 b x = 2, y = 5 c x = 12, y = 34

4. a x = 5, y = 1 b x = 7, y = 3 c x = 3, y = 2

d x = 1, y = 212 e x = 12, y = 6

12

5. Amul $7.20, Kim $3.50

6. 84p

7. 4.40

8. $195

13 Quadratic equations

1. a 2, 5 b 3, 2 c 1, 2 d 3, 2

2. a 4, 1 b 3, 5 c 6, 2 d 2 e 2, 6

3. a 6, 4 b 6, 4

4. a (x + 2)2 4 b (x 2)2 4 c (x + 5)2 25d (x + 1)2 1

5. a (x + 2)2 5 b (x 2)2 5 c (x + 4)2 22d (x + 1)2 10

6. a 1.45, 3.45 b 5.32, 1.32 c 4.16, 2.16

7. a 1.77, 2.27 b 3.70, 2.70 c 0.19, 1.53 d 0.41, 1.84 e 2.18, 0.15 f 1.64, 0.61

8. 6, 8, 10

9. 15 m, 20 m

10.6.54, 0.46

11.48 km/h

12.5 h

14 Algebraic fractions

1. a b

c d

2. a b c d

3. a 3 b 5 c 3

4. a b c d

5. a b c 1 d

6. a x b 3 c d

7. a 3, 1.5 b 0, 1

8. a bx + 1x 1

x 12x + 1

2x2 6y2

93x2 5x 2

10

12x

2xy3

32

12x

6x2 + 5x + 18

2xy3

x2

6

8x + 710

x 14

3x 2y6

x6

12x 2310

7x 34

3x + 2y6

5x6

5


15 Quadratic and cubic graphs

1. a Values of y: 27, 12, 3, 0, 3, 12, 27b 6.8 c 1.8 or 1.8

2. a Values of y: 27, 18, 11, 6, 3, 2, 3, 6, 11, 18, 27b 8.3 c 3.5 or 3.5

3. a Values of y: 27, 16, 7, 0, 5, 8, 9, 8, 5, 0, 7b 8.8 c 3.4 or 1.4

4. a Values of y: 2, 1, 2, 1, 2, 7, 14 b 0.25c 0.7 or 2.7 e (1.1, 2.6) and (2.6, 0.7)

5. a Values of y: 15, 9, 4, 0, 3, 5, 6, 6, 5, 3, 0,4, 9

b 0.5 and 3

6. a Values of y: 5, 0, 3, 4, 3, 0, 5, 12b 4 and 0

7. a Values of y: 16, 7, 0, 5, 8, 9, 8, 5, 0, 7, 16b 0 and 6

8. a Values of y: 9, 4, 1, 0, 1, 4, 9 b +2c Only 1 root

9. a Values of y: 10, 5, 4, 2.5, 2, 1.33, 1, 0.67, 0.5b i 0.8 ii 1.6

10.a Values of y: 10, 5, 2.5, 2, 1, 0.5, 0.4, 0.25, 0.2 c 4.8 and 0.2

11.a Values of y: 25, 12.5, 10, 5, 2.5, 1, 0.5, 0.33,0.25

c 0.5 and 10.5

12.a Values of y: 24, 12.63, 5, 0.38, 2, 2.9, 3,3.13, 4, 6.38, 11, 18.63, 30 b 4.7

13.a Values of y: 27, 15.63, 8, 3.38, 1, 0.13, 0, 0.13,1, 3.38, 8, 15.63, 27 b 0.2

14.a Values of y: 16, 5.63, 1, 4.63, 6, 5.88, 5, 4.13,4, 5.38, 9, 15.63, 26 c 1.6, 0.4, 1.9

16 Gradients and tangents

1. Draw graph of y = x2 + 1 with tangents at thefollowing points

a x = 5 gradient = 10

b x = 1 gradient = 2

c x = 2 gradient = 4

d x = 5 gradient = 10

2. Draw graph of y = x (x 3) with tangents at thefollowing points




d x = 1.5 gradient = 0

3. Draw graph of y = x3 3 with tangents at thefollowing points




4. a Gradient when x = 2 is 9 NB graph is y = x3 3x 2

b (1, 0) and (1, 4)

5. Draw graph of the curve y = sinx for 0 < x < 360 with tangents at the following points

a x = 60 gradient = 0.5


c x = 240 gradient = 0.5

6. a Draw a graph of the curve y = x2

b

c Gradient = 2x

d They should be the same as the curves aretransformed vertically

6

x 3 2 1 0 1 2 3gradient at x 6 4 2 0 2 4 6


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cambridge algebra answers

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