A bird flies through a distance in a straight line given by the vector i+2 j+ k. A man standing beside a straight metro rail track given by r=(3+λ) i+(2λ-1) j+3λ k is observing the bird. The projected length of its flight on the metro track is (A) (6)/(√(14)) units (B) (14)/(√(6)) units (C) (8)/(√(14)) units (D) (5)/(√(6)) units

MathsCBSE Class 12 · Sample paper, session 2025-26 · 1 mark · MCQ

CBSE Class 12 Mathematics sample paper 2025-26: question 13

From the official CBSE sample question paper for 2025-26, section A, question 13, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q13 · 1 mark · MCQCompetency-basedVector Algebra

A bird flies through a distance in a straight line given by the vector $\hat{i} + 2 \hat{j} + \hat{k}$ . A man standing beside a straight metro rail track given by $r = (3 + λ) \hat{i} + (2 λ - 1) \hat{j} + 3 λ \hat{k}$ is observing the bird. The projected length of its flight on the metro track is

(A) $\frac{6}{14}$ units
(B) $\frac{14}{6}$ units
(C) $\frac{8}{14}$ units
(D) $\frac{5}{6}$ units

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Collect the $λ$ terms to find the direction vector of the track.

Hint 2

Projection of $a$ on $b$ is $\frac{a \cdot b}{∣ b ∣}$ .

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.

The track is $r = (3 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} + 3 \hat{k})$ , so its direction is $b = \hat{i} + 2 \hat{j} + 3 \hat{k}$ .
Projection $= \frac{a \cdot b}{∣ b ∣} = \frac{1 + 4 + 3}{14} = \frac{8}{14}$ units. Option (C).
1 mark

Answer: $\frac{8}{14}$ units, option (C)

Where the marks are

Identifies the direction of the track and computes the projection $\frac{8}{14}$ (1 mark)

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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 paperSample paper 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 13

Hints

Step-by-step solution

Revise the chapter

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