Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are 55i - 2j, 5i + 8j and ai - 52j respectively, find the value of 'a'.

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2026: question 30(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section C, question 30(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(a) · 3 marks · Short answerCompetency-basedVector Algebra

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are $55 \hat{i} - 2 \hat{j}$ , $5 \hat{i} + 8 \hat{j}$ and $a \hat{i} - 52 \hat{j}$ respectively, find the value of 'a'.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Find two vectors joining the points, for example $A B$ and $B C$ .

Hint 2

If three points are collinear, these vectors are parallel, so their corresponding components are in the same ratio.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$A B = (5 - 55) \hat{i} + (8 + 2) \hat{j} = - 50 \hat{i} + 10 \hat{j}$
$B C = (a - 5) \hat{i} + (- 52 - 8) \hat{j} = (a - 5) \hat{i} - 60 \hat{j}$
1 mark
A, B and C lie on one line, so $A B$ and $B C$ are collinear and their components are proportional: $\frac{- 50}{a - 5} = \frac{10}{- 60}$
1 mark
$10 (a - 5) = 3000 \Rightarrow a - 5 = 300 \Rightarrow a = 305$
1 mark

Answer: $a = 305$

Where the marks are

Correct $A B$ and $B C$ (1 mark)
Uses the proportionality condition for collinear vectors (1 mark)
$a = 305$ (1 mark)

Revise the chapter

Chapter: Vector Algebra (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Vector Algebra, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 30(a)

Hints

Step-by-step solution

Revise the chapter

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