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WMA11/01 Jan 2022, Question 10: Lines Meeting Curves: the Discriminant

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Sketch curve C with y = 1/x² − 9, marking axis crossings and asymptotes, then find the range of k for which curve D, y = kx², meets C at four distinct points.

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

Part (a) · 4 marks

Sketch y = 1/x² − 9, showing where it crosses the coordinate axes and stating the equations of any asymptotes.

  1. 1y = 1/x² has two branches above the x-axis, symmetric about the y-axis
  2. 2Subtracting 9 moves it down 9: horizontal asymptote y = −9, vertical asymptote x = 0
  3. 3x-axis crossings: 1/x² = 9, so x = ±1/3, giving (−1/3, 0) and (1/3, 0)
  4. 4There is no y-intercept because x = 0 is an asymptote

AnswerTwo branches symmetric about the y-axis (the shape of y = 1/x² moved down 9): each rises to +∞ near the y-axis and levels off towards y = −9; crosses the x-axis at (−1/3, 0) and (1/3, 0); asymptotes x = 0 and y = −9

Part (b) · 5 marks

Given y = kx² intersects C at 4 distinct points, determine the range of possible values of k.

  1. 11/x² − 9 = kx², so 1 − 9x² = kx⁴
  2. 2kx⁴ + 9x² − 1 = 0 is a quadratic in x²
  3. 3Four distinct points need two distinct positive values of x²
  4. 4Distinct roots: 81 + 4k > 0, so k > −81/4
  5. 5Both roots positive: product −1/k > 0, so k < 0
  6. 6−81/4 < k < 0

Answer−81/4 < k < 0

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