MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2024: question 36(b)

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section E, question 36(b), for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36(b) · 4 marks · Case-basedCompetency-basedRelations and Functions

Let S be the relation defined by $S = {(l_{1}, l_{2}) : l_{1} is perpendicular to l_{2}}$ check whether the relation S is symmetric and transitive.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

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

Hint 1

Perpendicularity is a two-way condition between two lines; use this for symmetry.

Hint 2

For transitivity, sketch $l_{1} ⊥ l_{2}$ and $l_{2} ⊥ l_{3}$ and see how $l_{1}$ and $l_{3}$ are placed.

Hint 3

Try the lines $y = x$ , $y = - x$ and $y = x + 1$ .

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.

Symmetric: Let $(l_{1}, l_{2}) \in S$ . Then $l_{1} ⊥ l_{2}$ , which means $l_{2} ⊥ l_{1}$ , so $(l_{2}, l_{1}) \in S$ . Hence S is symmetric.
2 marks
Transitive: Let $(l_{1}, l_{2}) \in S$ and $(l_{2}, l_{3}) \in S$ , so $l_{1} ⊥ l_{2}$ and $l_{2} ⊥ l_{3}$ . Then $l_{1} ∥ l_{3}$ [lines perpendicular to the same line are parallel], so $l_{1}$ is not perpendicular to $l_{3}$ and $(l_{1}, l_{3}) \in / S$ .
Example: take $l_{1} : y = x$ , $l_{2} : y = - x$ and $l_{3} : y = x + 1$ . Here $l_{1} ⊥ l_{2}$ and $l_{2} ⊥ l_{3}$ , but $l_{1} ∥ l_{3}$ .
Hence S is not transitive.
2 marks

Answer: S is symmetric but not transitive

Where the marks are

Proves S is symmetric (2 marks)
Shows $l_{1} ⊥ l_{2}, l_{2} ⊥ l_{3} \Rightarrow l_{1} ∥ l_{3}$ , so S is not transitive (2 marks)

Revise the chapter

Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Relations and Functions, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/5/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2024: question 36(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Relations and Functions