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WMA12/01 Jun 2022, Question 3: Circles Meeting Lines, and Tangents

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

Show a cubic difference is not always prime for natural n, and prove AB is a diameter of the circle through A(1, 0), B(3, −10), C(7, −6).

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

Part (i) · 2 marks

Find a natural number n for which (n + 1)³ − n³ is not prime, showing the statement fails.

  1. 1Try values: n = 1 gives 2³ − 1³ = 7 (prime), n = 2 gives 19 (prime), n = 3 gives 37 (prime), n = 4 gives 61 (prime)
  2. 2n = 5: 6³ − 5³ = 216 − 125 = 91
  3. 391 = 7 × 13, so it is not prime
  4. 4One counter-example is enough, so the statement is false

AnswerShown: n = 5 gives 6³ − 5³ = 91 = 7 × 13, which is not prime, so the statement is false.

Part (ii) · 5 marks

Using A(1, 0), B(3, −10), C(7, −6), prove AB is a diameter of their circle.

  1. 1Gradient AC = (−6 − 0)/(7 − 1) = −6/6 = −1
  2. 2Gradient BC = (−6 − (−10))/(7 − 3) = 4/4 = 1
  3. 3Product of gradients = −1 × 1 = −1, so AC is perpendicular to BC
  4. 4So angle ACB = 90°
  5. 5The angle in a semicircle is a right angle, so AB is a diameter

AnswerShown: gradient AC = −1 and gradient BC = 1, so AC × BC gradients = −1; angle ACB = 90°, so AB is a diameter (angle in a semicircle).

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Also searched as: WMA12 June 2022 · WMA12/01 · wma12 jun22 · IAS Maths Pure Mathematics 2 June 2022 · question 3.
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