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4024/21 Oct/Nov 2024, Question 10: Constructing Equations

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 10 marks

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The question, in short

Given bag A has x balls with green exceeding red by 6, work through fractions for bags A and B, derive and solve a quadratic, then use the valid root to count green balls in A.

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

Part (a) · 2 marks

Express the share of red balls in bag A as (x − 6)/(2x), showing the reasoning.

  1. 1Let r = number of red balls, so green = r + 6
  2. 2r + r + 6 = x, so 2r = x − 6, r = (x − 6)/2
  3. 3Fraction red = r/x = (x − 6)/2 ÷ x = (x − 6)/(2x)

Answerr + r + 6 = x, so r = (x − 6)/2, leading to fraction red = (x − 6)/(2x)

Part (b) · 2 marks

Bag B has x red balls and four times bag A's green count; show its red fraction equals x/(3x + 12).

  1. 1Green in A = x − (x − 6)/2 = x/2 + 3
  2. 2Green in B = 4(x/2 + 3) = 2x + 12
  3. 3Total in B = x + 2x + 12 = 3x + 12
  4. 4Fraction red in B = x/(3x + 12)

Answergreen in A = x/2 + 3; total in B = x + 4(x/2 + 3), leading to fraction red = x/(3x + 12)

Part (c) · 3 marks

Starting from the two fractions set equal, rearrange to obtain x² − 6x − 72 = 0.

  1. 1(x − 6)(3x + 12) = 2x × x
  2. 23x² + 12x − 18x − 72 = 2x²
  3. 33x² − 6x − 72 − 2x² = 0
  4. 4x² − 6x − 72 = 0

Answer(x − 6)(3x + 12) = 2x², 3x² − 18x + 12x − 72 = 2x², correct rearrangement to x² − 6x − 72 = 0

Part (d) · 2 marks

Find the roots of x² − 6x − 72 = 0 using factorisation.

  1. 1x² − 6x − 72 = (x − 12)(x + 6) = 0
  2. 2x = 12 or x = −6

Answer12, −6

Part (e) · 1 mark

Given roots x = 12 and x = −6, choose the sensible value and state how many green balls bag A holds.

  1. 1Part (d): x² − 6x − 72 = 0 factorises to (x − 12)(x + 6) = 0
  2. 2x cannot be negative, so x = 12
  3. 3Red = (12 − 6)/2 = 3, so green = 3 + 6 = 9

Answer9

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Also searched as: 4024/21/O/N/24 · 4024_w24_qp_21 · 4024 w24 21 · question 10.
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