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4024/21 Oct/Nov 2022, Question 6: Length of a Line Segment

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 9 marks

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The question, in short

Given position vectors for A and B, work with vector AB and a parallelogram ABCD, then use midpoint and gradient conditions for points P and Q to find r, t and u.

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

Part (ai) · 1 mark

Find the column vector from A to B using the given position vectors.

  1. 1→AB = (9 − 4, 2 − 7)
  2. 2= column vector (5, −5)

Answercolumn vector (5, −5)

Part (aii) · 2 marks

Calculate the magnitude of vector AB.

  1. 1→AB = (9 − 4, 2 − 7) = (5, −5)
  2. 2Use Pythagoras: |→AB| = √(5² + (−5)²)
  3. 3= √(25 + 25) = √50 = 7.071…

Answer7.071…

Part (aiii) · 2 marks

Using vector BC = (−4, 1) and parallelogram ABCD, find coordinates of C and D.

  1. 1C = B + →BC = (9 − 4, 2 + 1) = (5, 3)
  2. 2→AD = →BC, so D = (4 − 4, 7 + 1) = (0, 8)
  3. 3C = (5, 3), D = (0, 8)

AnswerC = (5, 3), D = (0, 8)

Part (b) · 4 marks

Given midpoint (1, 3) and gradient −1/4 of PQ, find r, t and u.

  1. 1Midpoint: (4 + u)/2 = 3, so u = 2
  2. 2(r + t)/2 = 1, so r + t = 2
  3. 3Gradient: (u − 4)/(t − r) = −2/(t − r) = −1/4, so t − r = 8
  4. 4Adding: 2t = 10, t = 5, r = −3
  5. 5r = −3, t = 5, u = 2

Answerr = −3, t = 5, u = 2

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Also searched as: 4024/21/O/N/22 · 4024_w22_qp_21 · 4024 w22 21 · question 6.
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