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Case Study -1 A city's… (Class 12 Maths SQP 2025-26, Q36)

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MathsCBSE Class 12 · Sample paper, session 2025-26 · 4 marks · Case-based

CBSE Class 12 Mathematics sample paper 2025-26: question 36

From the official CBSE sample question paper for 2025-26, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedRelations and Functions

Case Study -1

A city's traffic management department is planning to optimize traffic flow by analyzing the connectivity between various traffic signals. The city has five major spots labelled and .

The department has collected the following data regarding one-way traffic flow between spots:
1. Traffic flows from to , to , and to .
2. Traffic flows from to and to .
3. Traffic flows from to .
4. Traffic flows from to and to .

The department wants to represent and analyze this data using relations and functions. Use the given data to answer the following questions:

  1. I

    Is the traffic flow reflexive? Justify.

    1 mark
  2. II

    Is the traffic flow transitive? Justify.

    1 mark
  3. III A

    Represent the relation describing the traffic flow as a set of ordered pairs. Also state the domain and range of the relation.

    2 marks

  4. OR

    III B

    Does the traffic flow represent a function? Justify your answer.

    2 marks

Figure in the paper: Illustration of a road intersection with traffic signals labelled A, C, B, D, E and white arrows painted on the road showing directions of traffic. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Write every one-way flow as an ordered pair (from, to).

Hint 2

For reflexive, check whether any pair like appears; for transitive, look for a chain without .

Hint 3

A function needs exactly one image for each element of the domain; count how many arrows leave .

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.

  1. I. The relation is = "traffic flows from first spot to second".

    It is not reflexive, because ; in fact no spot has traffic flowing to itself.

    1 mark

  2. II. It is not transitive, because and , but .

    1 mark

  3. III A. The relation as a set of ordered pairs is

    1 mark

  4. III A (contd.) The first elements give the domain .

    The second elements give the range .

    1 mark

  5. III B. No, the traffic flow does not represent a function.

    1 mark

  6. III B (contd.) Justification: in a function each element of the domain has exactly one image. Here is related to three spots , and , so has three images.

    (Also has no image. Similarly and each have two images.)

    1 mark

Answer: I. Not reflexive, since . II. Not transitive, since but . III A. ; domain , range . III B. Not a function, as has three images.

Where the marks are

  • States not reflexive with reason (1 mark)
  • States not transitive with the counterexample but (1 mark)
  • III A: writes correctly as ordered pairs (1 mark)
  • III A: domain (½ mark) and range (½ mark)
  • III B: says it is not a function (1 mark), justified by having more than one image (1 mark)

Revise the chapter

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 paperSample paper 2025-26: 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.