The lines r= i+ j- k+λ(2 i+3 j-6 k) and r=2 i- j- k+μ(6 i+9 j-18 k); (where λ & μ are scalars) are (A) coincident (B) skew (C) intersecting (D) parallel

MathsCBSE Class 12 · Sample paper, session 2023-24 · 1 mark · MCQ

CBSE Class 12 Mathematics sample paper 2023-24: question 5

From the official CBSE sample question paper for 2023-24, section A, question 5, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q5 · 1 mark · MCQThree Dimensional Geometry

The lines $r = \hat{i} + \hat{j} - \hat{k} + λ (2 \hat{i} + 3 \hat{j} - 6 \hat{k})$ and $r = 2 \hat{i} - \hat{j} - \hat{k} + μ (6 \hat{i} + 9 \hat{j} - 18 \hat{k})$ ; (where $λ$ & $μ$ are scalars) are

(A)coincident
(B)skew
(C)intersecting
(D)parallel

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

Compare the two direction vectors.

Hint 2

Check whether a point of the first line lies on the second line.

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.

$6 \hat{i} + 9 \hat{j} - 18 \hat{k} = 3 (2 \hat{i} + 3 \hat{j} - 6 \hat{k})$ , so the two lines have parallel directions.
Check whether the point $(1, 1, - 1)$ lies on the second line: $1 = 2 + 6 μ$ gives $μ = - \frac{1}{6}$ , but $1 = - 1 + 9 μ$ gives $μ = \frac{2}{9}$ . The values do not agree, so the point does not lie on it. The lines are therefore parallel, not coincident: option (D).
1 mark

Answer: Parallel, option (D)

Where the marks are

Shows the directions are proportional and the lines are not coincident (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 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 5

Hints

Step-by-step solution

Revise the chapter

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