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WMA11/01 Jan 2023, Question 9: Trig Graphs: Shifts and Counting Solutions

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 4 marks

The question, in short

Using the sketch of y = tan x and the line y = πx on −π < x < π, state the period of tan x and count roots of related equations over given intervals.

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

Part (a) · 1 mark

State the period of the tangent function.

  1. 1tan x repeats every π radians
  2. 2Period = π

Answerπ (180°)

Part (bi) · 1 mark

Find how many solutions tan x = (π + 2)x has for −π < x < π.

  1. 1y = (π + 2)x is steeper than y = tan x at O, so like y = πx it meets the central branch at O and once either side
  2. 2On the outer branches tan x and the line have opposite signs, so no more roots: 3 roots

Answer3

Part (bii) · 1 mark

Find how many solutions tan x = πx has for −2π < x < 2π.

  1. 1In −π < x < π there are 3 roots (from Figure 3)
  2. 2The branches on π/2 < x < 3π/2 and −3π/2 < x < −π/2 each cross the line once, both inside −2π < x < 2π
  3. 3Total = 3 + 2 = 5

Answer5

Part (biii) · 1 mark

Find how many solutions tan x = πx has for −100π < x < 100π.

  1. 1The central branch gives 3 roots (O and one either side)
  2. 2Each branch centred at nπ crosses the line once, close to its right-hand asymptote x = (n + ½)π
  3. 3For x > 0 this gives the branches n = 1, 2, …, 99 (the n = 100 crossing is beyond 100π): 99 roots
  4. 4By symmetry 99 more for x < 0, so total = 3 + 2 × 99 = 201

Answer201

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