AQA AS Level Mathematics Paper 1, June 2025: Question 15

1 mark · Easy difficulty · Short Answer

Identify which pair of given force vectors, if any, are parallel.

Practise this question

Question

Question 15 provides three forces in vector form: F1 = (-2i + 4j) N, F2 = (-3i + 6j) N, and F3 = (-6i + 9j) N. Students are asked to tick one of four boxes: 'F1 and F2 are parallel forces', 'F1 and F3 are parallel forces', 'F2 and F3 are parallel forces', or 'None of the forces F1, F2 and F3 are parallel'.
Question text

15 Three forces F1, F2 and F3 are given by:

F1 = (–2i + 4j) N

F2 = (–3i + 6j) N

F3 = (–6i + 9j) N

One of the following statements is true.

Identify the correct statement.

Tick ( ) one box.

[1 mark]

F1 and F2 are parallel forces.

F1 and F3 are parallel forces.

F2 and F3 are parallel forces.

None of the forces F1, F2 and F3 are parallel.

Mark scheme

Show the mark scheme Mark scheme for Question 15 shows 1 mark for AO 2.2a (R1) awarded for ticking the first box: 'F1 and F2 are parallel forces.'

Q Marking instructions AO Marks Typical solution

15 Ticks first box 2.2a R1 F1 and F2 are parallel forces.

Question 15 Total 1

How to answer it

Parallel Force Vectors in 2D

📌 What this question tests

This question assesses your understanding of vector multiples and parallelism in Mechanics. Specifically, it tests your ability to identify whether two force vectors given in component form ( ai + bj ) are scalar multiples of each other and therefore act in parallel directions.

Question 15 (1 Mark)

Identifying Parallel Force Vectors

✅ Correct Answer

First Box: F₁ and F₂ are parallel forces.

Mark Award: R1 (AO 2.2a - Mathematical Reasoning) for ticking the first box only.

📐 Step-by-Step Calculation

  1. State the condition for parallel vectors:
    Two vectors A and B are parallel if and only if there exists a scalar k such that B = kA (or their component ratios are identical: bj / bi = aj / ai ).
  2. Compare F₁ and F₂:
    F₁ = (-2i + 4j) N
    F₂ = (-3i + 6j) N
    Calculate the ratio of i and j components:
    Ratio of i: -3 / -2 = 1.5
    Ratio of j: 6 / 4 = 1.5
    Since F₂ = 1.5F₁, F₁ and F₂ are parallel.
  3. Verify other pairs (to ensure no errors):
    • For F₃ = (-6i + 9j) N:
      Ratio with F₁: i-ratio is -6 / -2 = 3; j-ratio is 9 / 4 = 2.25 (Not equal).
      Ratio with F₂: i-ratio is -6 / -3 = 2; j-ratio is 9 / 6 = 1.5 (Not equal).

💡 Key Knowledge

  • Scalar Multiples: If vector v = ku where k is a non-zero real scalar:
    • If k > 0, the vectors point in the same direction.
    • If k < 0, the vectors point in opposite directions (anti-parallel).
    • In both cases, they are parallel lines of action.
  • Gradient / Direction Angle: The direction angle θ with the positive x-axis satisfies:
    tan θ = j-component / i-component .
    For F₁: 4 / -2 = -2.
    For F₂: 6 / -3 = -2.
    Equal gradients confirm they are parallel.

🧠 Exam Technique

  • Quick Inspection Method: Simplify each vector by factoring out common factors to find unit direction ratios:
    F₁ = 2(-i + 2j)
    F₂ = 3(-i + 2j)
    F₃ = 3(-2i + 3j)
    It is instantly obvious that F₁ and F₂ share the exact base vector (-i + 2j) !
  • Multiple Choice Timing: This is a 1-mark question at the start of the mechanics section. Spend no more than 60–90 seconds on it.

❌ Common Errors

  • Difference Trap: Subtracting components instead of finding ratios (e.g., noticing the difference between -6 and -3 is -3, and assuming constant difference means parallel).
  • Sign Confusion: Forgetting that if a scalar multiplier is negative, the vectors are still geometrically parallel.
  • Selecting F₂ and F₃: Students frequently glance at the numbers 3, 6, 9 and assume an arithmetic pattern makes F₂ and F₃ parallel, overlooking that 6/(-3) = -2 while 9/(-6) = -1.5.

Topics

Mechanics · Pure Mathematics · J: Vectors · R: Forces and Newton’s laws

Question and mark scheme from the AQA AS Level Mathematics examination, Paper 1, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.