AQA A-Level Mathematics Paper 2, June 2025: Question 17

7 marks · Medium difficulty · Multi-step Problem

Find the displacement vector AB, calculate its magnitude, and show that triangle ABC is scalene using 3D position vectors.

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Question

Question text

17 A triangle has vertices A, B and C with position vectors given by

–3 5 1

OA = 5 , OB = 4 and OC = –1

1 –2 0

17 (a) (i) Find AB

[1 mark]

17 (a) (ii) Find the magnitude of AB

[1 mark]

17 (b) Hence show that triangle ABC is a scalene triangle.

[5 marks]

… 27

(26) …

Mark scheme

Show the mark scheme

Q Marking instructions AO Marks Typical solution

17(a)(i) 8

−1

Obtains AB = −

1.1b B1 1

−3

ACF

Subtotal 1

17(a)(ii) Finds the magnitude of

their AB AB = 64 +1+ 9 = 74

1.1b B1F

– A-LEVEL MATHEMATICS – 7357/2 –

AWRT 8.6

Subtotal 1

Uses vector geometry to work

out OC - OA or OC - OB OE

17(b) 4 −4 1 -3 4

AC = -1 - 5 = -6

PI −6 or −5

−1 2 1.1a M1 0 1 -1

Condone missing arrows 1 5 -4

Condone missing or incorrect BC = -1 - 4 = -5

labels

Obtain the correct vectors for 0 -2 2

AC or CA

and

AB = 74

BC or CB 1.1b A1

The vectors must be correctly BC = 16+25+ 4 = 45

labelled

AC = 53

Condone missing arrows

Find the magnitude of

All sides are different in length.

their AC or BC OE

Therefore, triangle ABC is a scalene

PI 53 or 3 5 OE triangle.

AWRT 7.3 or 6.7 3.1a M1

If 3 5 is obtained from OB

score M0

Obtains 53 and 3 5 OE

from correct vectors. 1.1b A1

AWRT 7.3 and 6.7

Completes reasoned argument

to obtain the three lengths:

BC = 45

AC = 53 OE

AB = 74

and

Concludes that ABC is a 2.1 R1

scalene triangle with no

incorrect reasoning.

Must have scored M1A1M1A1

Must not equate a vector to a

length.

Subtotal 5

Question 17 Total 7

How to answer it

3D Vectors & Triangle Classification

📌 What this question tests

This question evaluates your core competency in 3D vector geometry: finding displacement vectors between given position vectors ( AB = OB − OA ), calculating the magnitude (length) of 3D column vectors using Pythagoras’ theorem, and applying these lengths in a geometric context to formally prove that a triangle is scalene.

Question 17 (a)(i)

Finding the displacement vector AB [1 Mark]

📐 Calculation

Displacement vector is given by final position minus initial position:

AB = OB − OA

AB = 5 4 −2 − −3 5 1 = 5 − (−3) 4 − 5 −2 − 1 = 8 −1 −3

✅ Correct Answer

AB = 8 −1 −3 or 8i − j − 3k

Mark scheme: [B1] for correct vector in any acceptable form (ACF).

❌ Common Errors

Reversing order: Calculating OA − OB to give −8 1 3 (this is BA, not AB).
Sign slip with negatives: Doing 5 − 3 = 2 instead of 5 − (−3) = 8.

🧠 Exam Technique

Always write out the explicit vector subtraction rule: AB = OB − OA . Never add coordinates or vectors when moving between two points.

Question 17 (a)(ii)

Magnitude of vector AB [1 Mark]

📐 Step-by-Step Calculation

Apply 3D magnitude formula: |v| = √(x² + y² + z²)
Substitute components of AB:
|AB| = √(8² + (−1)² + (−3)²)
Square each term carefully:
|AB| = √(64 + 1 + 9) = √74
Decimal equivalent: √74 ≈ 8.6023... (AWRT 8.6)

✅ Correct Answer

|AB| = √74 (or awrt 8.6)

Mark scheme: [B1F] follow-through for correctly calculating the magnitude of their vector from (a)(i).

❌ Common Errors

Squaring negative numbers into negatives: Writing (−1)² = −1 or (−3)² = −9, giving √(64 − 1 − 9) = √54. Squares of real components are always positive!
Forgetting the square root: Leaving the answer as 74 instead of √74.

💡 Key Knowledge

Exact surd form (√74) is preferred in Pure Mathematics unless a decimal approximation or rounding requirement is explicitly asked for.

Question 17 (b)

Show that Triangle ABC is a Scalene Triangle [5 Marks]

💡 Definition of a Scalene Triangle

A triangle is scalene if and only if all three of its side lengths are different. To provide a rigorous mathematical proof, you must find the lengths of all three sides: |AB|, |BC|, and |AC|, and explicitly state that they are unequal.

📐 Step-by-Step Proof

Side 1: Length of AB
From part (a)(ii): |AB| = √74 (≈ 8.60)
Side 2: Vector AC and its length
AC = OC − OA = 1 −1 0 − −3 5 1 = 4 −6 −1
|AC| = √(4² + (−6)² + (−1)²) = √(16 + 36 + 1) = √53 (≈ 7.28)
Side 3: Vector BC and its length
BC = OC − OB = 1 −1 0 − 5 4 −2 = −4 −5 2
|BC| = √((−4)² + (−5)² + 2²) = √(16 + 25 + 4) = √45 = 3√5 (≈ 6.71)
Conclusion / Reasoning
|AB| = √74, |AC| = √53, |BC| = √45
Since √74 ≠ √53 ≠ √45 (all three side lengths are different), triangle ABC is a scalene triangle.

✅ Mark Breakdown (5 Marks)

M1 (1.1a): Method to find displacement vector: calculates OC − OA or OC − OB.
A1 (1.1b): Correct vector for both AC (or CA) AND BC (or CB) with correct labels.
M1 (3.1a): Finds magnitude of their AC or BC.
A1 (1.1b): Obtains both correct lengths: √53 and 3√5 (or √45).
R1 (2.1): Complete reasoned argument stating all 3 lengths (|AB| = √74, |AC| = √53, |BC| = √45) and concluding that ABC is scalene.

❌ Critical Pitfalls where Marks are Lost

Equating a vector to a scalar: Writing AC = √53 instead of |AC| = √53 or Length AC = √53 . The mark scheme explicitly states: "Must not equate a vector to a length."
Finding magnitude of position vector instead: Calculating |OB| = √(5² + 4² + (−2)²) = √45 = 3√5 instead of finding displacement BC. Note the mark scheme specifies: "If 3√5 is obtained from OB, score M0."
Missing conclusion: Calculating the three surds but failing to write a concluding sentence explicitly stating they are unequal, hence scalene.

Topics

Pure Mathematics · J: Vectors

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.