SPM 2024 Add Maths Paper 1, Question 13Vectors

Solved on video in English and Bahasa Melayu · Form 4, Vectors · 5:11 video

The question

(a)(i)

Diagram 10 shows points P(4h, 2h), Q(0, -h) and R(-8, -31/5) on a Cartesian plane. (a)(i) Find vector PQ in terms of h.

(a)(ii)

Using the same diagram and points as part (a)(i), (a)(ii) given that PQ is a unit vector, find the value of h.

(b)

Using the same points P(4h, 2h), Q(0, -h) and R(-8, -31/5), (b) hence, by using vectors, show that the points P, Q and R are collinear.

Part (a)(i)

Step 1

Using the triangle law: PQ = PO + OQ = -OP + OQ.

Why this step

Travel P → O → Q. Since PO = −OP, we get PQ = OQ − OP: end point minus start point. Reversing this is the classic slip.

2 more steps, each with the reason for it, in the full solution

Final answer

PQ = (−4h, −3h)

Where students lose marks

Watch out: PQ = OQ − OP (end minus start). Doing OP − OQ flips every sign and gives (4h, 3h).

Part (a)(ii)

5 more steps, each with the reason for it, in the full solution

Final answer

h = 1/5

Where students lose marks

Watch out: square each component first, then add: (−4h)² + (−3h)² = 16h² + 9h² = 25h², so |PQ| = 5|h|.

Part (b)

7 more steps, each with the reason for it, in the full solution

Final answer

Shown: PQ = (1/10)QR, Q common point

Where students lose marks

Watch out: after showing PQ = (1/10)QR, also state that Q is a common point. Parallel alone does not prove collinear.

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