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

1 mark · Easy difficulty · Short Answer

Identify the correct sketch of the graph of y = cosec x° for 0 ≤ x ≤ 360 from four given options.

Practise this question

Question

Question 2 asks to identify the graph of y = cosec x° for 0 ≤ x ≤ 360 by ticking one of four boxes. Four candidate graphs are presented on axes from x = 0 to 360. The top-left graph has asymptotes at x = 180 and x = 360, with a U-shaped branch with minimum at y = 1 between 0 and 180, and an inverted U-shaped branch with maximum at y = -1 between 180 and 360. The top-right graph reverses these branches. The bottom two graphs have asymptotes marked at x = 90 and x = 270.
Question text

2 One of the diagrams below shows the graph of y = cosec x° for 0 ≤ x ≤ 360

Identify the correct graph.

Tick ( ) one box.

[1 mark]

y y

O O

180 360 x 180 360 x

–1 –1

y y

O O

90 270 x 90 270 x

–1 –1

Mark scheme

Show the mark scheme Mark scheme for Question 2 indicates 1 mark (B1, AO 1.2) awarded for selecting 'Ticks the top left box', showing the sketch of y = cosec x° with asymptotes at 180 and 360, a minimum at (90, 1) and a maximum at (270, -1).

Q Marking instructions AO Marks Typical solution

2 Ticks the top left 1.2 B1

Question 2 Total 1

How to answer it

Graph of the Reciprocal Trigonometric Function cosec(x)

📋 What this question tests

This question assesses your knowledge of reciprocal trigonometric functions, specifically:

  • Recognising the definition: cosec x = 1 / sin x.
  • Deducing the shape, vertical asymptotes, and turning points of the cosecant graph from the sine wave.
  • Distinguishing between the graphs of cosec x and sec x in the interval 0° ≤ x ≤ 360°.
Question 2 • 1 Mark

Identifying the Graph of y = cosec x° (0 ≤ x ≤ 360)

AQA A-Level Mathematics • Pure Core

✅ Correct Answer

Tick the Top-Left Box.

Mark Scheme Breakdown:
• B1 (AO 1.2): Correct box ticked (top-left diagram). No working required.

💡 Key Knowledge

  • Reciprocal Definition: cosec x = 1 / sin x
  • Vertical Asymptotes: Occur where denominator equals zero. Since sin x = 0 at x = 0°, 180°, and 360°, the asymptotes are at x = 0 , x = 180 , and x = 360 .
  • Key Coordinates:
    • At x = 90°: sin(90°) = 1 ⇒ cosec(90°) = 1 (local minimum)
    • At x = 270°: sin(270°) = -1 ⇒ cosec(270°) = -1 (local maximum)
  • Range: y ≥ 1 or y ≤ -1. The graph never enters the strip -1 < y < 1.

📐 Step-by-Step Graphical Deduction

  1. Step 1: Identify the vertical asymptotes
    Since cosec x = 1 / sin x, asymptotes occur when sin x = 0. In [0°, 360°], this happens at x = 0°, 180°, and 360°.
    • Eliminates the bottom two graphs, which show asymptotes at x = 90° and x = 270° (those are the asymptotes of sec x).
  2. Step 2: Check signs and turning points
    For 0° < x < 180°, sin x > 0, so cosec x must be positive (≥ 1).
    At x = 90°, cosec(90°) = +1, curving upwards towards +∞ as x approaches 0° and 180°.
    For 180° < x < 360°, sin x < 0, so cosec x must be negative (≤ -1).
    At x = 270°, cosec(270°) = -1, curving downwards towards -∞ as x approaches 180° and 360°.
  3. Step 3: Match with the diagrams
    The top-left graph has a positive U-shape above y = 1 for 0 < x < 180 and an inverted U-shape below y = -1 for 180 < x < 360. This matches cosec x exactly.

🧠 Exam Technique

  • Third Letter Rule: Remember cosec x = 1/sin x, and sec x = 1/cos x.
  • Quick Test Value: If ever unsure in the exam, calculate one value using your calculator in degree mode:
    cosec(90°) = 1 / sin(90°) = 1 / 1 = 1 .
    Look at x = 90°: the curve must pass through y = +1. The top-right graph has y = -1 at x = 90°, instantly ruling it out!

❌ Common Errors

  • Confusing cosec with sec: Selecting the bottom-left option (which represents y = sec x, with asymptotes at 90° and 270°).
  • Sign Inversion: Selecting the top-right option, which shows y = -cosec x.
  • Ticking multiple boxes: This is a 1-mark multiple-choice question. Marking more than one box scores 0 marks.

Topics

Pure Mathematics · E: Trigonometry

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 2, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.