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WMA11/01 Jan 2023, Question 7: When a Line Meets, Touches or Misses a Curve

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 10 marks

The question, in short

Sketch y = 6/x for x ≠ 0, describe the transformation to y = 6/(x − 2), then use the two intersections with y = kx + 7, where P has x = −4, to find k and Q's coordinates.

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

Part (a) · 2 marks

Sketch the curve y = 6/x, x ≠ 0, showing its shape and key features.

  1. 1y = 6/x is a reciprocal graph with k = 6 > 0
  2. 2For x > 0, y > 0 (1st quadrant); for x < 0, y < 0 (3rd quadrant)
  3. 3As x → 0, y → ±∞ and as x → ±∞, y → 0, so the axes are asymptotes
  4. 4Sketch two decreasing branches that approach but never touch the axes

AnswerTwo decreasing branches, one in the 1st quadrant and one in the 3rd; asymptotes x = 0 and y = 0; the curve does not meet either axis

Part (b) · 2 marks

Describe fully the transformation mapping y = 6/x onto y = 6/(x − 2), x ≠ 2.

  1. 1Replacing x by x − 2 shifts the graph 2 units right
  2. 2Translation by vector (2, 0)

AnswerTranslation by 2 units in the positive x-direction, vector (2, 0)

Part (c) · 2 marks

Given P has x-coordinate −4, find the constant k in y = kx + 7.

  1. 1At x = −4: y = 6/(−4 − 2) = −1
  2. 2−1 = −4k + 7, so 4k = 8
  3. 3k = 2

Answerk = 2

Part (d) · 4 marks

Using algebra, find the coordinates of the second intersection point Q.

  1. 16/(x − 2) = 2x + 7, so 6 = (2x + 7)(x − 2) = 2x² + 3x − 14
  2. 22x² + 3x − 20 = 0
  3. 3(2x − 5)(x + 4) = 0, so x = 5/2 (x = −4 is P)
  4. 4y = 2 × 5/2 + 7 = 12, so Q is (5/2, 12)

AnswerQ(5/2, 12)

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Also searched as: WMA11 January 2023 · WMA11/01 · wma11 jan23 · IAS Maths Pure Mathematics 1 January 2023 · question 7.
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