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WMA11/01 Oct 2020, Question 7: When a Line Meets, Touches or Misses a Curve

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 11 marks

The question, in short

Sketch the curve y = 1/(2 − x) showing axis intersections and asymptotes, then use the line y = 4x + k meeting the curve twice to derive and solve an inequality for k.

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

Part (a) · 3 marks

Sketch C, marking where it crosses the coordinate axes and stating the equations of its asymptotes.

  1. 1y = 1/(2 − x) = −1/(x − 2): the graph of y = 1/x reflected and moved 2 to the right
  2. 2Vertical asymptote x = 2; horizontal asymptote y = 0
  3. 3At x = 0, y = 1/2, so it crosses the y-axis at (0, 1/2); it never meets the x-axis
  4. 4For x < 2, y > 0 (top-left branch); for x > 2, y < 0 (bottom-right branch)

AnswerTwo-branch reciprocal curve with asymptotes x = 2 and y = 0: the left branch (x < 2) lies above the x-axis, crossing the y-axis at (0, 1/2) and rising towards x = 2; the right branch (x > 2) lies below the x-axis, coming up from −∞ near x = 2 towards y = 0

Part (b) · 4 marks

Using that line l intersects C at two distinct points, show that k² + 16k + 48 > 0.

  1. 14x + k = 1/(2 − x), so (4x + k)(2 − x) = 1
  2. 28x − 4x² + 2k − kx − 1 = 0, so 4x² + (k − 8)x + 1 − 2k = 0
  3. 3Two distinct points: b² − 4ac > 0, so (k − 8)² − 4(4)(1 − 2k) > 0
  4. 4k² − 16k + 64 − 16 + 32k > 0
  5. 5k² + 16k + 48 > 0

AnswerShown: 4x² + (k − 8)x + 1 − 2k = 0 has two roots, so (k − 8)² − 16(1 − 2k) > 0, which gives k² + 16k + 48 > 0

Part (c) · 4 marks

From the inequality in part b, determine the range of possible values of k.

  1. 1Solve k² + 16k + 48 = 0: (k + 12)(k + 4) = 0
  2. 2Critical values k = −12 and k = −4
  3. 3The quadratic is positive outside its roots
  4. 4k < −12 or k > −4

Answerk < −12 or k > −4

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Also searched as: WMA11 October 2020 · WMA11/01 · wma11 oct20 · IAS Maths Pure Mathematics 1 October 2020 · question 7.
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