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4PM1/01 Jun 2024, Question 6: Length of a Line Segment

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 16 marks

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The question, in short

Line l through A(−2, 2) and B(3, 12); C on l with AC : CB = 3 : 2. Find C, perpendicular line k through C, length CD where k meets x-axis, and X on l with triangle DXC area 80 and m > 0.

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

Part (a) · 2 marks

Find p and q for point C(p, q) on l dividing AB in ratio AC : CB = 3 : 2.

  1. 1C = A + (3/5)(B − A)
  2. 2p = −2 + (3/5)(5) = 1
  3. 3q = 2 + (3/5)(10) = 8

Answerp = 1, q = 8

Part (b) · 4 marks

Show that the perpendicular line k through C has equation 2y + x − 17 = 0.

  1. 1Gradient of l = (12 − 2)/(3 − (−2)) = 2
  2. 2Gradient of k = −1/2
  3. 3y − 8 = −(1/2)(x − 1)
  4. 42y − 16 = −x + 1, so 2y + x − 17 = 0

Answer2y + x − 17 = 0 shown

Part (c) · 3 marks

Find the exact length of CD, where k crosses the x-axis at D.

  1. 1On k, y = 0 gives x = 17, so D = (17, 0)
  2. 2CD = √((17 − 1)² + (0 − 8)²)
  3. 3= √320 = 8√5

AnswerCD = 8√5

Part (d) · 7 marks

For X(m, n) on l with m > 0 and area of triangle DXC = 80 units², find m and n.

  1. 1Angle DCX = 90°, so area = ½ × CD × CX = 80
  2. 2CX = 160/(8√5) = 4√5
  3. 3X lies on l: n = 2m + 6
  4. 4(m − 1)² + (2m + 6 − 8)² = 80
  5. 55m² − 10m − 75 = 0, so (m + 3)(m − 5) = 0
  6. 6m > 0 gives m = 5, n = 16

Answerm = 5, n = 16

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