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

CBSE Class 12 Mathematics 2024: question 36(a)

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

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

Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by $R = {(l_{1}, l_{2}) : l_{1} is parallel to l_{2}}$

On the basis of the above information, answer the following questions :

(i)
Find whether the relation R is symmetric or not.
1 mark
(ii)
Find whether the relation R is transitive or not.
1 mark
(iii)
If one of the rail lines on the railway track is represented by the equation $y = 3 x + 2$ , then find the set of rail lines in R related to it.
2 marks

Figure in the paper: Photograph of several parallel rail lines on a track near a railway station, with a train engine and hills in the background. See it in the official paper.

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

For symmetry, ask: if line 1 is parallel to line 2, is line 2 parallel to line 1?

Hint 2

For transitivity, think about two lines that are both parallel to a third line.

Hint 3

Parallel lines have equal slopes; what is the slope of $y = 3 x + 2$ ?

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.

(i) Let $(l_{1}, l_{2}) \in R$ . Then $l_{1} ∥ l_{2}$ , so $l_{2} ∥ l_{1}$ , and hence $(l_{2}, l_{1}) \in R$ . Therefore R is symmetric.
1 mark
(ii) Let $(l_{1}, l_{2}) \in R$ and $(l_{2}, l_{3}) \in R$ . Then $l_{1} ∥ l_{2}$ and $l_{2} ∥ l_{3}$ , so $l_{1} ∥ l_{3}$ [lines parallel to the same line are parallel to each other]. Hence $(l_{1}, l_{3}) \in R$ , and R is transitive.
1 mark
(iii) The lines related to $y = 3 x + 2$ are exactly the lines parallel to it, i.e. the lines with the same slope $3$ .
The required set is ${l : l is a line y = 3 x + c, c \in R}$ .
2 marks

Answer: (i) Symmetric; (ii) Transitive; (iii) ${l : y = 3 x + c, c \in R}$

Where the marks are

Shows $l_{1} ∥ l_{2} \Rightarrow l_{2} ∥ l_{1}$ , so R is symmetric (1 mark)
Shows $l_{1} ∥ l_{2}, l_{2} ∥ l_{3} \Rightarrow l_{1} ∥ l_{3}$ , so R is transitive (1 mark)
Set of lines $y = 3 x + c, c \in R$ (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(a)

Hints

Step-by-step solution

Revise the chapter

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