MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2023: question 23(b)

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, section B, question 23(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(b) · 2 marks · Very short answerThree Dimensional Geometry

Check whether the lines given by equations $x = 2 λ + 2$ , $y = 7 λ + 1$ , $z = - 3 λ - 3$ and $x = - μ - 2$ , $y = 2 μ + 8$ , $z = 4 μ + 5$ are perpendicular to each other or not.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

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

Hint 1

The coefficients of $λ$ and $μ$ are the direction ratios.

Hint 2

Two lines are perpendicular when $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$ .

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.

From the parametric forms, the coefficients of the parameters give the direction ratios: line 1 has $⟨ 2, 7, - 3 ⟩$ and line 2 has $⟨ - 1, 2, 4 ⟩$ .
1 mark
$a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 2 (- 1) + 7 (2) + (- 3) (4) = - 2 + 14 - 12 = 0$ .
Hence the given lines are perpendicular to each other.
1 mark

Answer: Yes, the lines are perpendicular.

Where the marks are

Correct direction ratios of both lines (1 mark)
Shows $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$ and concludes the lines are perpendicular (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 paper2023 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 23(b)

Hints

Step-by-step solution

Revise the chapter

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