MathsCBSE Class 12 · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2024-25: question 28(b)

From the official CBSE sample question paper for 2024-25, section C, question 28(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(b) · 3 marks · Short answerThree Dimensional Geometry

Find the vector and the cartesian equation of the line that passes through $(- 1, 2, 7)$ and is perpendicular to the lines $r = 2 \hat{i} + \hat{j} - 3 \hat{k} + λ (\hat{i} + 2 \hat{j} + 5 \hat{k})$ and $r = 3 \hat{i} + 3 \hat{j} - 7 \hat{k} + μ (3 \hat{i} - 2 \hat{j} + 5 \hat{k})$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

A vector perpendicular to two given vectors is their cross product.

Hint 2

Use the point $(- 1, 2, 7)$ and that direction in $r = a + λ b$ .

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 required line is parallel to $b_{1} \times b_{2} = \hat{i} 13 \hat{j} 2 - 2 \hat{k} 55 = (10 + 10) \hat{i} - (5 - 15) \hat{j} + (- 2 - 6) \hat{k} = 20 \hat{i} + 10 \hat{j} - 8 \hat{k}$ , i.e. parallel to $10 \hat{i} + 5 \hat{j} - 4 \hat{k}$ .
1 mark
Vector equation through $(- 1, 2, 7)$ : $r = - \hat{i} + 2 \hat{j} + 7 \hat{k} + a (10 \hat{i} + 5 \hat{j} - 4 \hat{k})$ .
1 mark
Cartesian equation: $\frac{x + 1}{10} = \frac{y - 2}{5} = \frac{z - 7}{- 4}$ .
1 mark

Answer: $r = - \hat{i} + 2 \hat{j} + 7 \hat{k} + a (10 \hat{i} + 5 \hat{j} - 4 \hat{k})$ ; $\frac{x + 1}{10} = \frac{y - 2}{5} = \frac{z - 7}{- 4}$

Where the marks are

Direction $b_{1} \times b_{2} = 20 \hat{i} + 10 \hat{j} - 8 \hat{k}$ (1 mark)
Vector equation (1 mark)
Cartesian equation (1 mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Three Dimensional Geometry, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 28(b)

Hints

Step-by-step solution

Revise the chapter

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