The scalar product of the vector a = i - j + 2k with a unit vector along sum of vectors b = 2i - 4j + 5k and c = - 2j - 3k is equal to 1. Find the value of λ.

MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2025: question 31(a)

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, section C, question 31(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31(a) · 3 marks · Short answerVector Algebra

The scalar product of the vector $a = \hat{i} - \hat{j} + 2 \hat{k}$ with a unit vector along sum of vectors $b = 2 \hat{i} - 4 \hat{j} + 5 \hat{k}$ and $c = λ \hat{i} - 2 \hat{j} - 3 \hat{k}$ is equal to 1. Find the value of $λ$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/7/1.

Hints

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

Hint 1

First add $b$ and $c$ , then divide by the magnitude to get the unit vector.

Hint 2

Take the dot product with $a$ , set it equal to 1, and square both sides to solve for λ.

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.

$d = b + c = (2 + λ) \hat{i} - 6 \hat{j} + 2 \hat{k}$
1 mark
$\hat{d} = \frac{( 2 + λ ) i ^ - 6 j ^ + 2 k ^}{( 2 + λ ) ^{2} + 40}$
½ mark
$a \cdot \hat{d} = \frac{( 2 + λ ) + 6 + 4}{( 2 + λ ) ^{2} + 40} = 1$ , so $λ + 12 = (2 + λ)^{2} + 40$ .
1 mark
Squaring: $λ^{2} + 24 λ + 144 = λ^{2} + 4 λ + 44$ , so $20 λ = - 100$ and $λ = - 5$ . Check: $7 = 9 + 40$ ✓.
½ mark

Answer: λ = −5

Where the marks are

Finds $b + c$ (1 mark)
Writes the unit vector along it (½ mark)
Sets up $a \cdot \hat{d} = 1$ correctly (1 mark)
Solves to get λ = −5 (½ mark)

Revise the chapter

Chapter: Vector Algebra (notes, formulas, where marks are lost and every past question)
Topic: Scalar (Dot) Product of Vectors

Practise this chapterBoard-style questions from Vector Algebra, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/7/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 31(a)

Hints

Step-by-step solution

Revise the chapter

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