Mathematics (9-1) - iGCSE 2018-20 Year 09 Unit 17 – Answers

17 - Prior knowledge check Page 699 84 a. 2 55 b. 43 36 or 1 7 36 c. 3 8 d. 25 27 a. 5 2 b. 4 5 a. 14 – 35x b. 2x2 + 5x – 12 c. 4x2 – 4x + 1 a. n = 40 b. p = 5 c. d = 9

17 - Prior knowledge check Page 699 a. x = 𝑦 −7 4 b. x = 𝑊 −ℎ 3ℎ c. x = 𝑦 4 - 1 d. x = 3𝑃 −1 6 a. y3 b. 28y6 c. y7 d. 2y5 5 a. (x + 1)(x + 5) b. (x - 10)(x + 3) c. (x - 2)(x - 3) d. (x - 6)(x + 6)

17 - Prior knowledge check Page 699 a. x = -5, -6 b. x = 1, 11 c. x = -1, - 7 2 x = −5 ± 17 2 x = -4 ± 6

17 - Prior knowledge check Page 700 a. Students' own answers, e.g. 1, 2, 3, 4 and 5. Students' own answers, e.g. 15. Students' own answers, e.g. 2 + 3 + 4 + 5 + 6 = 20, 3 + 4+ 5 + 6 + 7 = 25, 4 + 5 + 6 + 7 + 8 = 30, 5 + 6 + 7 + 8 + 9 = 35. The sum of 5 consecutive numbers is a multiple of 5. n + (n + 1) + (n + 2) + (n + 3) + (n + 4) = 5n + 10

17.1 – Rearranging Formulae Page 700 a. a = v − u t b. r = 𝐶 2π c. h = 2𝑎 𝑏 d. r = A π a. y(x + 2) b. q(p - 1) c. k(a - 4) v = 2𝐸 m x = H2 + y

17.1 – Rearranging Formulae Page 700 a. x = ( T2k 4p2 ) b. x = 16 𝑦 c. x = 𝑧𝑃2 𝑦 d. x = -1 a. r = 3 3V 4𝑝 b. x = 3 V 4 c. x - y3 5 d. y = x 𝑧3 y - ℎ 3 + 𝑥

17.1 – Rearranging Formulae Page 700 d = ( 𝐻 + 𝑎𝑐 𝑎 −𝑏 ) a. y = 𝑤 − 2 2𝑥 b. x = 𝑤 − 2 2𝑦 a. There cannot be an x on both sides of the formula. b. Zoe should have factorised the x first. c. x = 2𝑡 𝑡 + 1 x = 1 𝑉 − 7 12. k = 2𝑡 𝑡 − 1

17.2 – Algebraic Fractions a. 29 36 b. 85 99 c. - 1 20 Page 700 a. 29 36 b. 85 99 c. - 1 20 a. 10 77 b. 54 35 c. 5 16 a. 𝑥2 6 b. 6xy 20 = 3xy 10 c. 12 45𝑦2 = 4 15𝑦2 a. 3x 2𝑦2 b. 5y4 3x2 c. 3𝑥4 2𝑦2 a. 4 3 b. 𝑥4𝑦2 c. 5 4𝑥2𝑦4 d. 5𝑦 𝑦 − 7

17.2 – Algebraic Fractions Page 700 a. 8x 10 = 4x 5 b. 13x 12 c. 5x 14 a. 10x b. 6x c. 28x d. 12x a. 3 12x and 4 12x b. 7 12x a. 11 18x b. 1 20x c. 13 18x

17.2 – Algebraic Fractions Page 700 a. x −4 2 = 5(𝑥 − 4) 5 x 2 = 5𝑥 −20 10 b. x + 7 5 = 2(𝑥 +7) 2 x 5 = 2𝑥 + 14 10 c. 7x −6 10 a. 5x + 8 6 b. 5x + 41 14 c. x + 67 36

9𝑥 + 24 10 a = 𝑏 𝑏 − 1 u = vf(v – f) 17.2 – Algebraic Fractions Page 700 9𝑥 + 24 10 a = 𝑏 𝑏 − 1 u = vf(v – f)

17.3 – Simplifying Algebraic Fractions Page 700 a. 1 𝑥2 b. 5x2 c. 5x2 a. (x - 6)(x - 3) b. (x - 9)(x + 9) c. (5x + 1)(x + 4) a. 1 𝑦 b. 1 3 c. 1(x - 7) d. x + 2 𝑥 − 5 e. x − 3 𝑥 f. x 𝑥 − 1 a. x(x - 6) b. x a. x + 8. b. 3x c. 5 2𝑥

17.3 – Simplifying Algebraic Fractions Page 700 a. Sally is incorrect because the two terms on the denominator have nothing in common. The denominator cannot be factorised, The expression cannot be simplified because the denominator cannot be factorised. a. 2 𝑥 + 5 b. x − 3 5 a. x + 3 𝑥 − 3 b. x − 5 𝑥 +7 c. x − 5 𝑥 + 5

17.3 – Simplifying Algebraic Fractions Page 700 x + 7 x − 7 a. 2x − 3 3x −2 b. 5x −1 6x + 5 c. 5x −1 5x + 1 x + 4 2x − 2 a. (6 - x) = -(x - 6) b. i. -1 c. 2x 2x −3

17.3 – Simplifying Algebraic Fractions Page 700 a. - 4 − x x b. x − 6 2(x +6) c. 2x 2x −3 Numerator: (x + 4)(x - 3)(x + 3)(x - 1)(2x)(5x + 6) Denominator: (3 - x)(3 + x)(5x + 6)(x + 4)(7)(x - 1) x − 3 3 − x = -1; other factors cancel to leave - 2x 7

17.4 – More Algebraic Fractions Page 700 a. 15x y b. 75 4 c. 3x x − 2 a. 13x 15 b. 5 24x c. 7x +11 12 a. (x + 3)(x - 4) b. x + 2 x + 5 c. 1 9 a. (x - 3)(x + 3) b. (x + 2)(x + 3) c. 2x −6 x +2

17.4 – More Algebraic Fractions Page 700 a. (x − 2)(x − 3) (x + 1)(x + 4) b. 7(x + 7) (x − 3)(x − 7) a. x(x + 2) b. (x + 2)(x + 3) c. (x + 4)(x + 5) d. (x + 1)(x - 1) e. (7x - 3)(2x - 4) a. 2x + 9 (x + 4)(x + 5) b. 7x + 1 (x + 1)(x − 1) c. 6x + 26 (x − 5)(x + 3) d. 7 (2x − 3)(2x + 4)

17.4 – More Algebraic Fractions Page 700 (x + 10) (x − 4)(x + 3) a. i. 3(x + 3) ii. 4(x + 3) b. 12(x + 3) c. 7 12(x + 3) a. (x - 4)(x + 4) b. x − 3 (x + 4)(x − 4)

17.4 – More Algebraic Fractions Page 700 a. −x (3x + 5)(x +1) b. 1 − x 2(x + 1)(x + 6) c. 4x − 1 (x + 2)(x +4)(x − 7) d. −11 − 3x (5 − x)(5 + x) x2 + 3x − 2 (x + 2)(x + 4)(x − 7) Students own answer. A = 5

17.5 – Surds Page 700 a. 5 b. 3 3 c. 8 2 a. 3 b. 30 c. 5 7 a. 50 = 25 x 2 = 5 2 , k = 5 b. k = 2 3 c. k = 4 a. 10 10 b. 15 5 c. 2 a. i. 3 5 ii. 2 5 b. 2

a. 12 + 2 = 2 3 + 2 = 2( 3 +1) b. 3(3 + 6 ) c. 3(6 - 5 ) d. 5( 3 - 2 ) 17.5 – Surds Page 701 a. 3 b. 22 2 c. 15 2 a. 12 + 2 = 2 3 + 2 = 2( 3 +1) b. 3(3 + 6 ) c. 3(6 - 5 ) d. 5( 3 - 2 ) a. 4 5 + 5 b. 11 + 5 7 c. 22 + 2 2 d. 6 - 4 2 e. 26 - 8 10 f. 52 + 14 3

17.5 – Surds Page 701 30 -10 5 , a = 30, b = -10, c = 5 a. i. 53 - 6 2 ii. 12 + 4 8 = 12 + 8 2 The perimeter of the first shape would be 32 units, wh ich is rational. The perimeter of the second shape would be 8 + 4 8 or 8 + 8 2 , which is irrational.

17.5 – Surds Page 701 a. (3 2 + 2) 2 b. 2 3 - 1 c. (19 7 −7) 7 d. 5 + 1 a = -3, b = 4

17.5 – Surds Page 701 a. (1 − 2 ) −1 b. 5 + 3 22 c. 7(4 + 5 ) −11 d. 4(1 − 6 ) −5 e. 5 + 5 −4 f. 25 + 7 2 ) 31 a. x = 3 ± 2 2 b. x = 5 ± 2 3 c. x = 8 ± 2 14

17.6 – Solving Algebraic Fraction Equations Page 701 a. 2(x + 3) b. 4(x + 4) a. 5 x b. 4 2x = 2 x c. 10 x −6 a. x = -2, x = -4 b. x = 12 2 , x = 1 c. x = 4, x = 1 x = 1.40, x = -4.06 a. x = 5 4 b. x = 11 7 c. x = - 11 2

17.6 – Solving Algebraic Fraction Equations Page 701 a. x = - 5 3 , x = 4 b. x = 3 2 , x = -2 c. x = - 7 5 , x = 2 d. x = - 5 2 , x = -4 6x - 9 + 2x + 2 = 2x2 + 2x - 3x = 3 8x - 7 = 2x2 - x - 3 -2x2 + 9x - 4 = 0 (2x - 1)(-x + 4) x = 1 2 , x = 4

17.6 – Solving Algebraic Fraction Equations Page 701 a. Students’ own answer b. x = 1, x = 9 a. x = 3 b. x = 0, x = 8 c. x = 1, x = 2 d. x = -4, x = 1 a. x = 0.29, x = -10.29 b. x = 1.21, x = -1.81 c. x = 6.37, x = 0.63 d. x = 5.70, x = -0.70 x = 6 ± 31

17.7 – Functions Page 701 a. x → x 2 → + 5 → y b. x → ÷ 2 → - 6 → y c. x → + 1 → x 3 → y a. x = 7 5 b. x = 16 7 a. H = 12t b. P = y 6 a. 2 b. -5 c. 20 d. - 1 2 a. Alice first multiplied 5 by 2 to get 10. Then she worked out 10 squared, which is 100. b. 20

17.7 – Functions Page 701 a. 54 b. -2 c. 1 4 d. -250 a. 5 b. 56 c. 480 d. 2.5 e. 600 f. -33 a. a = 3 b. a = 1 5 c. - 4 5 a. a = ± 5 b. a = ± 2 c. a = ± 2 2 d. a = ± 2 5 a. a = 0, a = -3 b. a = 1, a = -5 c. a = -1, a = -2 d. a = -1, a = -3

17.7 – Functions Page 701 a. 5x +1 b. 5x - 13 c. 10x - 8 d. 35x – 28 e. 10x – 4 f. 20x – 4 a. 3x2 + 3 b. 6x2 – 8 c. 12x2 – 4 d. 3x2 –4 a. 11 b. 71 c. -40 d. -58 a. -4x + 13 b. 37 – 4x c. 4x2 + 25 d. 16x2 – 24x + 16 e. –x2 + 3 f. 107 – 20x + x2

a. x → x − 9 4 b. x → 3(x + 4) c. x → x 2 - 6 d. x → x + 1 7 + 4 17.7 – Functions Page 701 a. x → x − 9 4 b. x → 3(x + 4) c. x → x 2 - 6 d. x → x + 1 7 + 4 a. x → x 4 + 1 b. x → x 4 - 1 c. x → x 2 d. a = 2

17.8 – Proof Page 701 a. Students' own answer b. i. 9999 (use 1002 - 1) ii. 39 999 (use 2002 - 1) a. (x + 5)(x + 2)= x2 + 7x + 10 b. x(x + 1) = x2 + x c. x2 + 7x + 10 - (x2 + x) = 6x + 10 x(3x + 4) - 5x = 70 3x2 + 4x - 5x - 70 = 0 3x2 - x - 70 = 0

17.8 – Proof Page 701 a. 2 is a prime. Any number less than 1 gives a cube that is less than its square. For example -5 - -2 = -3, -5 + -2 = -7 For example 16 - 4 = 12. 2n + 1 + 2n = 4n + 1 = odd. a. The next even number will be two more (because the next number, which is one more, will be odd), (2n)(2n + 2) = 4n2 + 4n = 4(n2 + n). This is divisible by 4.

(2n + 1)(2n - 1) = 4n2 - 1.4n2 must be even, so 4n2 -1 must be odd. 17.8 – Proof Page 701 (2n + 1)(2n - 1) = 4n2 - 1.4n2 must be even, so 4n2 -1 must be odd. 2x - 2a = x + 5 x = 2a + 5 2a is even, even + odd = odd

17.8 – Proof Page 701 a. i. 1 30 ii. 1 12 iii. 1 56 b. 1 90 It will be 1 divided by 99 times 100. i. 1 x(x +1) This shows that the difference between two fractions with 1 on the numerators and consecutive numbers on the denominator will be 1 divided by the denominators multiplied together.

17.8 – Proof Page 701 A = 4 n2 + n = n(n +1) When n is even, even x odd = even When n is odd, odd x even = even b. n3 - n - (n -1) n (n + 1) = even x odd x even = even Or = odd x even x odd = even (n + 1)2 – n2 = n2 + 2n +1 – n2 = 2n + 1 (n + 1) + n = 2n + 1

17 – Problem-Solving Page 702 6370 km a. 3390 km b. 3.73 m/s2 11.29m/s2 7320 km (3 s.f.)

17 – Check-Up Page 702 a. 20 2 b. 23 = 8 7 a. 3 5 − 10 5 b. 6 + 3 3 a. x → 2x + 5 b. x → 𝑥 − 4 3 a. 26 b. -16 c. 96 d. -20 y = 𝑥 + 1 z3

17 – Check-Up y = 9 − 3𝑥 5x + 2 k = 2𝑝2𝑥 𝑇4 a. 29 b. a = ± 7 2 Page 702 y = 9 − 3𝑥 5x + 2 k = 2𝑝2𝑥 𝑇4 a. 29 b. a = ± 7 2 a. 𝑥 − 2 3 b. 𝑥 − 4 𝑥 + 1 a. 1 6𝑥 b. 4𝑥 − 11 (x + 4)(x − 5) c. 16 −2𝑥 (x − 6)(x −1)

17 – Check-Up a. 8𝑥2 9𝑦5 b. 4(𝑥 + 10) 4(𝑥 + 1) x = -1 ± 2 Page 702 a. 8𝑥2 9𝑦5 b. 4(𝑥 + 10) 4(𝑥 + 1) x = -1 ± 2 Students' own answer. Students' own answer, e.g. 13 + 33 = 28 or 23 + 44 = 72 a. (2n - 1) + (2n + 3) = 4n + 4 2n + (2n + 2) + (2n + 4) = 6n + 6 (2n + 1) + (2n + 3) + (2n + 5) + (2n + 7) = 8n + 16

17 – Strengthen Surds a. 3 b. 7 c. 4 d. 6 5 - 5 a. 4 3 Page 702 Surds a. 3 b. 7 c. 4 d. 6 5 - 5 a. 4 3 4 + 11 11 = 4 + 11 11 x 11 11 = 4 x 11 + 11 x 11 11 x 11 = 4 11 + 11 11 8 5 − 5 5

17 – Strengthen Page 702 a. 1 + 2 7 b. 27 - 10 7 c. 4 d. -7 e. 9 f. Students’ own answer g. i. 6 - 8 ii. 3 + 11 a. 40 + 8 2 23 b. 14 - 7 3 c. 42 +6 10 39 = 14 +2 10 13

17 – Strengthen Formulae and Functions Page 702 Formulae and Functions a. i. y2 = 3 ii. y2 = x iii. y2 = 3x + 1 b. x = 𝑦2 + 1 3 Rewrite the formula so there is no fraction. xy = 7 + y Get all the terms containing y on the left-hand side and all other terms on the right-hand side. xy - y = 7 Factorise so that y appears only once. y(x - 1) = 7 Get y on its own on the left hand side, y = 7 𝑥 − 1

a. y = 1 b. f(2) = 1 c. i. f(5) = 16 ii. f(-3) = -24 iii. f(0) = -9 17 – Strengthen Page 702 y = 1 𝐹 +5 a. y = 1 b. f(2) = 1 c. i. f(5) = 16 ii. f(-3) = -24 iii. f(0) = -9 a. i. x = 1 8 ii. x = 2 7 b. i. a = 1 8 ii. a = 2 7 c. a = 4 9

a. y = 𝑥 + 9 2 b. y = 𝑥 3 + 5 c. 𝑥 − 8 𝑥 + 7 d. 𝑥 + 5 𝑥 − 2 17 – Strengthen Page 702 a. 2 b. 35 c. 70 d. 8 e. 70 f. 4x2 – 1 g. i. 8 ii. 24 iii. 5 iv. 0 y = 𝑥 + 4 5 a. y = 𝑥 + 9 2 b. y = 𝑥 3 + 5 c. 𝑥 − 8 𝑥 + 7 d. 𝑥 + 5 𝑥 − 2

17 – Strengthen Algebraic Fractions Page 702 Algebraic Fractions a. 1 2 b. x c. 𝑥 − 8 𝑥 + 7 d. 𝑥 + 5 𝑥 − 2 e. (𝑥 + 4)(𝑥 −2) (𝑥 + 8)(𝑥 − 4) f. 5(𝑥 + 1) 9(𝑥 −1) a. i. 3 4 ii. 3 2 iii. 1 𝑥2 iv. 𝑦4 1 b. 9𝑦4 8𝑥2 a. 4𝑥2 9𝑦 b. 8𝑥 9𝑦3

17 – Strengthen a. i. 3(x + 6) ii. x(x + 6) b. 3 x c. x − 5 2 Page 702 a. i. 3(x + 6) ii. x(x + 6) b. 3 x c. x − 5 2 a. 8x + 32 x2 + 12x + 32 = 8(x + 4) (x + 4)(x + 8) = 8 (x + 8) b. x + 8 x − 9 c. x − 8 x + 3 a. i. 3(x - 3) ii. (x + 6)(x + 3) iii. (x + 3)(x + 5) iv. 2(x + 5) b. i. 3(x + 3) 2(x + 6) ii. 2(x + 6) 3(x + 3)

a. x = 8, x = -7 b. x = 9, x = -7 b. x = 3, x = -2 d. x = -5, x = 2 17 – Strengthen Page 702 a. x = 8, x = -7 b. x = 9, x = -7 b. x = 3, x = -2 d. x = -5, x = 2 a. x(x – 1) b. i. 3(x − 1) x(x − 1) ii. 2x x − 1 x c. (5x − 3) x(x − 1) d. x = 3 ± 6 5 − 2𝑥 x2 − 5𝑥 + 4

17 – Strengthen Proof a. x2 – 8x + 16 b. x2 - 8x - 7 c. x - 8x + 7 Page 702 Proof a. x2 – 8x + 16 b. x2 - 8x - 7 c. x - 8x + 7 Students show that 1 side of the identity is the same as the other side of the identity. Students show work to prove that the right- hand side is equal to the left hand side. a. 1, 8, 27, 64, 125 b. Students' own answer, e.g. 64 - 8 = 56

17 – Extend a. Students should show that both answers are correct, Page 702 a. Students should show that both answers are correct, Ruth's answer is considered the better answer because the denominator is positive. x = 𝑃 −3𝑑 2 R2 = 𝑅𝑅1 (𝑅1 −𝑅) a. 1 𝑑 = 1 𝑏 + 1 𝑐 - 1 𝑎 b. Students should show their own work to make d the subject.

17 – Extend Page 702 a. x = - 5 2 , x = 4 b. x = 10 ± 2 5 5 a. Students’ own answer b. Using the equivalent expression, it is 9 10 a. -21 b. 15 – 16x i. f-1(x) = x −3 −4 = x −4 + 3 4 ii. g-1(x) = x −3 −4 = x −4 - 3 4 When the two functions are added together, the Jr’s and the number term cancel, leaving zero. Students should show this.

17 – Extend a. i. x2 + 13 ii. x2 + 14x + 55 b. x = -3 a. i. x ii. x Page 702 a. i. x2 + 13 ii. x2 + 14x + 55 b. x = -3 a. i. x ii. x Yes, because fg(x) = gf(x) = x, the functions are inverses, These two functions are inverses. Students should show that fg(x) = gf(x) = x. Students show their own work. The factors cancel to leave -1 as (7 - x) = -(x - 7).

17 – Extend Page 703 Students should show that the expression simplifies to 12n, which is a multiple of 12. A = - 4 and B = 1 a. r = 𝐺𝑚1𝑚2 𝐹 b. 1.5 x 1011nm

17 – Unit Test Sample student answer Page 703 Sample student answer The student has made an error in multiplying both sides by (b - 5): a x (b - 5) = ab - 5a, not ab - 5. The student could avoid this mistake by putting the terms in a bracket before multiplying them up, which would remind them to multiply BOTH terms in the brackets by a.