AQA A-Level Mathematics Paper 3, June 2025: Question 2

1 mark Β· Easy difficulty Β· Short Answer

Identify the value of the horizontal asymptote parameter $a$ for the graph of $y = \arctan x$.

Practise this question

Question

A diagram shows the graph of y = arctan x on Cartesian axes, passing through the origin (0, 0). The curve is bounded by two dashed horizontal asymptotes at y = a and y = -a. Below the graph, the text states 'State the value of a. Circle your answer.' with four multiple-choice options: pi/4, 1, pi/2, and pi.

Mark scheme

Show the mark scheme Mark scheme table for Question 2 with columns Q, Marking instructions, AO, Marks, and Typical solution. It indicates 1 mark (B1) for AO 1.2 for circling the third answer, with a typical solution of pi/2.

How to answer it

Horizontal Asymptotes of y = arctan x

πŸ“‹ What this question tests

This question assesses your recall and understanding of inverse trigonometric functions, specifically the properties of the graph of y = arctan x (or tan⁻¹ x ), its range, and its corresponding horizontal asymptotes.

Question 2 (Multiple Choice) • [1 Mark]

Identifying the Value of the Asymptote a

AQA A-Level Mathematics • Pure Core

βœ… Correct Answer

Circle the third option: Ο€/2

Therefore, the value of a = Ο€/2 , meaning the two horizontal asymptotes are at y = Ο€/2 and y = -Ο€/2 .

Mark Scheme Breakdown:
β€’ B1 (AO 1.2): Circles the third answer ( Ο€/2 ).

πŸ’‘ Key Knowledge

  • Relationship to y = tan x: To make y = tan x a one-to-one function so that an inverse exists, its domain is restricted to (-Ο€/2, Ο€/2) .
  • Domain and Range Swap:
    β€’ For y = tan x : Domain is (-Ο€/2, Ο€/2) , Range is (-∞, ∞) .
    β€’ For y = arctan x : Domain is (-∞, ∞) , Range is (-Ο€/2, Ο€/2) .
  • Asymptotes: The vertical asymptotes of tangent at x = Β±Ο€/2 become the horizontal asymptotes of arctangent at y = Β±Ο€/2 .

πŸ“ Step-by-Step Explanation

  1. Recall the definition: If y = arctan x , then tan y = x for -Ο€/2 < y < Ο€/2 .
  2. Consider the limit as x approaches infinity:
    As x → +∞ , we look for an angle whose tangent approaches positive infinity.
    tan y → +∞ &implies; y → Ο€/2
  3. Identify the constant a:
    Since the upper asymptote is given by y = a , we have a = Ο€/2 .

❌ Common Misconceptions

  • Selecting Ο€/4: Confusing the asymptote with the known special value arctan(1) = Ο€/4 . The curve passes through (1, Ο€/4) , but it does not level off there.
  • Selecting 1: Confusing radians with gradient values, or mistaking the asymptote for a unity bound like the maximum value of sine/cosine.
  • Selecting Ο€: Thinking of the period of tan x (which is Ο€ ) rather than the distance from the origin to the asymptote.

🧠 Exam Technique & Examiner Commentary

This is a standard 1-mark recall question designed to test direct familiarity with the core inverse trigonometric graphs required by the specification (arcsin, arccos, and arctan).

  • Quick Check via Calculator: If you ever freeze in the exam, evaluate arctan(999999) in radian mode on your calculator. You will get approximately 1.57079... , which is immediately recognizable as Ο€/2 .
  • Ensure only one option is circled: Multiple circled options without a clear cancellation will score 0 marks.

Topics

Pure Mathematics Β· E: Trigonometry

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 3, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.