Three honey bees were found flying along the vectors a = 2i - 3j + k, b = 4j - 2k and c = 3i + 2k respectively. Find the value of λ such that the path for a + λb is perpendicular to c.

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2026: question 24(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section B, question 24(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24(a) · 2 marks · Very short answerCompetency-basedVector Algebra

Three honey bees were found flying along the vectors $a = 2 \hat{i} - 3 \hat{j} + \hat{k}$ , $b = 4 \hat{j} - 2 \hat{k}$ and $c = 3 \hat{i} + 2 \hat{k}$ respectively. Find the value of $λ$ such that the path for $a + λ b$ is perpendicular to $c$ .

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

Two vectors are perpendicular when their dot product is zero.

Hint 2

Expand $(a + λ b) \cdot c$ as $a \cdot c + λ b \cdot c$ and compute each dot product.

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.

$a + λ b ⊥ c \Rightarrow (a + λ b) \cdot c = 0$ , i.e. $a \cdot c + λ b \cdot c = 0$
½ mark
$a \cdot c = 2 (3) + (- 3) (0) + 1 (2) = 8$ and $b \cdot c = 0 (3) + 4 (0) + (- 2) (2) = - 4$ , so $8 - 4 λ = 0$
1 mark
$λ = 2$
½ mark

Answer: $λ = 2$

Where the marks are

Uses the perpendicularity condition $(a + λ b) \cdot c = 0$ (½ mark)
Gets $8 - 4 λ = 0$ (1 mark)
$λ = 2$ (½ 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 24(a)

Hints

Step-by-step solution

Revise the chapter

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