Lemmara logoLemmaraWorked solution

4PM1/02 Jan 2022, Question 8: Solving Equations by Graphical Methods

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 11 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Complete a table for y = sin x − 3 cos 2x − 0.5 in radians to 1 d.p., plot the curve for 0 ≤ x ≤ 3, prove a double-angle identity, then use a line to estimate roots of f(x) = 2 sin x + 12 sin²x − x − 5.

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

Part (a) · 2 marks

Fill the missing y-values in the table for y = sin x − 3 cos 2x − 0.5, giving each to one decimal place.

  1. 1Substitute each x (in radians) into y = sin x − 3 cos 2x − 0.5
  2. 2x = 0: 0 − 3 − 0.5 = −3.5
  3. 3x = 0.5: −1.64… → −1.6
  4. 4x = 1.6: 3.49… → 3.5
  5. 5x = 2: 2.37… → 2.4

Answerx = 0: y = −3.5; x = 0.5: y = −1.6; x = 1.6: y = 3.5; x = 2: y = 2.4

Part (b) · 2 marks

Plot y = sin x − 3 cos 2x − 0.5 on the grid across the range 0 ≤ x ≤ 3.

  1. 1Plot the eight points from the table
  2. 2Join them with a smooth curve

AnswerPoints from the completed table plotted and joined with a smooth curve rising from (0, −3.5) to a maximum of about 3.5 near x = 1.6, then falling to (3, −3.2)

Part (c) · 2 marks

Starting from a formula on page 2, derive the result cos 2A = 1 − 2 sin²A.

  1. 1cos(A + B) = cos A cos B − sin A sin B
  2. 2Put B = A: cos 2A = cos²A − sin²A
  3. 3cos²A = 1 − sin²A
  4. 4cos 2A = 1 − sin²A − sin²A = 1 − 2 sin²A

Answercos 2A = 1 − 2 sin²A shown

Part (d) · 5 marks

Add a suitable straight line to the grid and use it to estimate, to 1 d.p., the roots of f(x) = 0 for 0 ≤ x ≤ 3.

  1. 112 sin²x = 6(1 − cos 2x)
  2. 2f(x) = 0: 2 sin x + 6 − 6 cos 2x − x − 5 = 0
  3. 3Divide by 2: sin x − 3 cos 2x − 0.5 = x/2 − 1
  4. 4Draw the line y = x/2 − 1 (through (0, −1) and (2, 0))
  5. 5Intersections with the curve: x ≈ 0.6 (or 0.7) and x ≈ 2.3 (or 2.4)

AnswerDraw y = x/2 − 1; x = 0.6 or 0.7 and x = 2.3 or 2.4

Stuck on a step? Read the free lesson: Solving Equations by Graphical Methods

Practise more solving equations by graphical methods questions

Real 4PM1 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on solving equations by graphical methods

Also searched as: 4PM1 2 January 2022 · 4PM1/02 · 4pm1 jan22 2 · question 8.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.