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0580/43 Oct/Nov 2022, Question 6: Finding Regions from Multiple Inequalities

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 16 marks

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The question, in short

Work with the formula P = 5k² − 7, solve and represent an inequality, and use four inequalities to identify region R and maximise 2x + y.

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

Part (ai) · 2 marks

Substitute k = 3 into P = 5k² − 7 and calculate the resulting value of P.

  1. 1P = 5 × 3² − 7 = 45 − 7
  2. 2= 38

Answer38

Part (aii) · 3 marks

Rearrange P = 5k² − 7 so that k is the subject, keeping the square root and signs correct.

  1. 1P + 7 = 5k²
  2. 2k² = (P + 7)/5
  3. 3k = ±√((P + 7)/5)

Answer±√((P + 7)/5)

Part (bi) · 2 marks

Solve the inequality x − 3 ≤ 5x + 7 for x, giving the range of values.

  1. 1x − 5x ≤ 7 + 3
  2. 2−4x ≤ 10
  3. 3x ≥ −2.5

Answerx ≥ −2.5

Part (bii) · 1 mark

Draw the solution set from part (b)(i) on the given number line from −6 to 6.

  1. 1Filled circle at −2.5 (≥ includes −2.5)
  2. 2Line from −2.5 going to the right

AnswerFilled circle at −2.5 with the line going to the right

Part (ci) · 6 marks

On the grid, draw the three remaining boundary lines, shade the unwanted regions, and label region R.

  1. 1x = 2: broken vertical line (strict inequality)
  2. 22x + 3y = 72: solid line through (0, 24) and (36, 0)
  3. 3y = 32 − x: solid line through (0, 32) and (32, 0)
  4. 4Shade the unwanted side of each line; label the unshaded region R

Answerx = 2 broken line, y = 32 − x solid line, 2x + 3y = 72 solid line, correct region R indicated

Part (cii) · 2 marks

Among integer coordinate pairs in region R, find the one giving the largest value of 2x + y.

  1. 12x + y is largest when x is as large as possible
  2. 2On y = 16 the largest x is 32 − 16 = 16, and 2 × 16 + 3 × 16 = 80 ≥ 72
  3. 32 × 16 + 16 = 48, the maximum
  4. 4(16, 16)

Answer(16, 16)

Stuck on a step? Read the free lesson: Finding Regions from Multiple Inequalities

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Also searched as: 0580/43/O/N/22 · 0580_w22_qp_43 · 0580 w22 43 · question 6.
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