SPM 2022 Add Maths Paper 2, Question 8Vectors

Solved on video in English and Bahasa Melayu · 5 marks · Form 4, Vectors · 6:26 video

The question

(a)

Diagram 4 shows triangle OBQ and the straight line OAC. Point P lies on the straight line OQ and OC = 3OA. It is given that OA = a, OB = b, AP = (1/3)AB and OQ = k·OP, such that k is a constant.

(a) Express (i) OP in terms of a and b, (ii) BQ in terms of k, a and b.

(b)

Using OP = (2/3)a + (1/3)b and BQ = (2/3)k a + ((1/3)k − 1)b, and OC = 3OA:

(b) Hence, find the value of h and of k if BQ = h·BC, such that h is a constant.

[5 marks]

(c)

Using the result from part (b):

(c) State BQ : QC.

SPM 2022 Add Maths Paper 2, Question 8: the question as drawn in Eduly's video6:26

Part (a)

Step 1

AB = AO + OB = −a + b.

Why this step

We only know OA and OB, so we reach B from A by going through O. AO points opposite to OA, so it is −a. Watch the sign.

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

Final answer

OP = (2/3)a + (1/3)b; BQ = (2/3)k a + ((1/3)k − 1)b

Where students lose marks

Watch out: be careful when adding vectors. Reversing direction flips the sign, so AO = −a and BO = −b.

Part (b)

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

Final answer

h = 2/5, k = 9/5

Where students lose marks

Watch out: expand h(3a − b) carefully as 3h a − h b, then compare the coefficients of a and b separately.

Part (c)

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

Final answer

BQ : QC = 2 : 3

Where students lose marks

Watch out: the ratio is BQ : QC, so find QC = (3/5)BC first. Do not stop at BQ : BC = 2 : 5.

Watch every step of this question solved on video (6:26), with the reason behind each step and a box that checks your own answer.

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