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0580/42 May/Jun 2021, Question 3: Exponential Equations

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 20 marks

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The question, in short

Work through index laws, HCF and LCM with prime factors, a cube-number multiplier, two factorisations, and an equation linking x and y.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (ai) · 1 mark

Write 7⁵ × 7⁶ as one power of 7.

  1. 17⁵ × 7⁶ = 7⁵⁺⁶ = 7¹¹

Answer7¹¹

Part (aii) · 1 mark

Write 7¹⁵ ÷ 7⁵ as one power of 7.

  1. 17¹⁵ ÷ 7⁵ = 7¹⁵⁻⁵ = 7¹⁰

Answer7¹⁰

Part (aiii) · 1 mark

Write 42 + 7 as one power of 7.

  1. 142 + 7 = 49 = 7²

Answer7²

Part (b) · 3 marks

Simplify the cube of 5x² × 2xy⁴, giving a single term.

  1. 15x² × 2xy⁴ = 10x³y⁴
  2. 2(10x³y⁴)³ = 1000x⁹y¹²

Answer1000x⁹y¹²

Part (ci) · 2 marks

Using P = 2⁵ × 3³ × 7 and Q = 540, find the HCF of P and Q.

  1. 1540 = 2² × 3³ × 5, P = 2⁵ × 3³ × 7
  2. 2HCF = 2² × 3³ = 108

Answer108

Part (cii) · 2 marks

Using the same P and Q, find their LCM.

  1. 1Write 540 in primes: 540 = 2² × 3³ × 5
  2. 2LCM takes the highest power of each prime: 2⁵ × 3³ × 5 × 7
  3. 3= 32 × 27 × 35 = 30 240

Answer30 240

Part (ciii) · 2 marks

Given P × R is a cube number, find the smallest integer R.

  1. 1In a cube number, every prime's power is a multiple of 3.
  2. 2P = 2⁵ × 3³ × 7¹
  3. 32⁵ needs one more 2 to make 2⁶. 3³ is already a cube. 7¹ needs two more 7s to make 7³.
  4. 4R = 2 × 7² = 98
  5. 5Check: P × R = 2⁶ × 3³ × 7³ = 84³

Answer98

Part (di) · 2 marks

Factorise x² − 3x − 28 fully.

  1. 1−7 × 4 = −28 and −7 + 4 = −3
  2. 2x² − 3x − 28 = (x − 7)(x + 4)

Answer(x − 7)(x + 4)

Part (dii) · 2 marks

Factorise 7(a + 2b)² + 4a(a + 2b) fully.

  1. 1Common factor (a + 2b): (a + 2b)[7(a + 2b) + 4a]
  2. 2= (a + 2b)(7a + 14b + 4a) = (a + 2b)(11a + 14b)

Answer(a + 2b)(11a + 14b)

Part (e) · 4 marks

From 3²ˣ⁻¹ = 1/9ˣ × 3²ʸ⁻ˣ, find y as an expression in x.

  1. 11/9ˣ = 3⁻²ˣ, so 3²ˣ⁻¹ = 3⁻²ˣ × 3²ʸ⁻ˣ
  2. 22x − 1 = −2x + 2y − x
  3. 32y = 5x − 1
  4. 4y = (5x − 1)/2

Answer(5x − 1)/2

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Also searched as: 0580/42/M/J/21 · 0580_s21_qp_42 · 0580 s21 42 · question 3.
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