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WMA11/01 Oct 2022, Question 6: Sketching Cubics and Reciprocal Curves

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 6 marks

The question, in short

Sketch two graphs for 0 < k < 4, giving axis intercepts in terms of k, then state how many roots the equation formed by equating the two expressions has, with a reason.

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

Part (ai) · 3 marks

Sketch y = (2x − k)(x + 4)² for 0 < k < 4, marking where it crosses or meets the axes, with coordinates in terms of k.

  1. 1Roots: 2x − k = 0 ⇒ x = k/2 (crosses); (x + 4)² = 0 ⇒ x = −4 (repeated, touches)
  2. 2x = 0: y = (−k)(16) = −16k
  3. 3Leading term 2x³ is positive, so the curve rises from bottom left to top right
  4. 4Touches at (−4, 0) as a maximum, minimum between −4 and k/2, crosses at (k/2, 0)

AnswerPositive cubic: touches the x-axis at (−4, 0), which is a maximum; dips to a minimum below the x-axis; crosses the y-axis at (0, −16k); crosses the x-axis at (k/2, 0) and rises to the right

Part (aii) · 2 marks

Sketch y = k/x² for 0 < k < 4, showing any points where it meets or crosses the coordinate axes.

  1. 1k > 0 and x² > 0, so y > 0 for all x ≠ 0
  2. 2y → ∞ as x → 0 and y → 0 as x → ±∞
  3. 3Branches in quadrants 1 and 2, symmetric about the y-axis, never meeting the axes

AnswerTwo branches, one in the first quadrant and one in the second, symmetric about the y-axis; y > 0 everywhere; the axes are asymptotes; no points on the axes

Part (b) · 1 mark

State the number of roots of (2x − k)(x + 4)² = k/x² and give a reason, using the sketches.

  1. 1The cubic is only above the x-axis for x > k/2, where it rises steeply
  2. 2y = k/x² is always positive and falling for x > 0, so the graphs cross once: 1 root

Answer1 root, because the two graphs intersect exactly once

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Also searched as: WMA11 October 2022 · WMA11/01 · wma11 oct22 · IAS Maths Pure Mathematics 1 October 2022 · question 6.
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