MathsCBSE Class 12 · Board exam 2023, set 65/5/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2023: question 36

Set in the CBSE Class 12 board exam of 2023, paper 65/5/1, 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-I

An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.

Let $B = {b_{1}, b_{2}, b_{3}}$ and $G = {g_{1}, g_{2}}$ , where B represents the set of Boys selected and G the set of Girls selected for the final race.

Based on the above information, answer the following questions :

(I)
How many relations are possible from B to G ?
1 mark
(II)
Among all the possible relations from B to G, how many functions can be formed from B to G ?
1 mark
(III)
Let $R : B \to B$ be defined by $R = {(x, y) : x and y are students of the same sex}$ . Check if R is an equivalence relation.
2 marks
OR
(III)
A function $f : B \to G$ be defined by $f = {(b_{1}, g_{1}), (b_{2}, g_{2}), (b_{3}, g_{1})}$ . Check if f is bijective. Justify your answer.
2 marks

Figure in the paper: Photograph of two riders on racing bikes, one bike ahead tilted. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/5/1.

Hints

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

Hint 1

How many ordered pairs are in B × G? A relation is any subset of it.

Hint 2

For a function, count the choices of image for each boy.

Hint 3

In (III), notice that all members of B are boys. For the function, look for two boys with the same image.

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) We have $n (B \times G) = 3 \times 2 = 6$ .
Every subset of $B \times G$ is a relation from $B$ to $G$ . So the number of relations is $2^{6} = 64$ .
1 mark
(II) In a function, each of the 3 elements of $B$ must have exactly one image in $G$ . Each has 2 choices.
So the number of functions is $2 \times 2 \times 2 = 2^{3} = 8$ .
1 mark
(III)(a) All elements of $B$ are boys, so any two of them are of the same sex. Hence $R = B \times B$ .
- Reflexive: $(b_{i}, b_{i}) \in R$ for every $b_{i}$ .
- Symmetric: if $(x, y) \in R$ , then $(y, x) \in R$ , since they are of the same sex.
- Transitive: if $(x, y), (y, z) \in R$ , then $x$ and $z$ are of the same sex, so $(x, z) \in R$ .
Hence $R$ is an equivalence relation.
2 marks
(III)(b) We have $f (b_{1}) = g_{1}$ and $f (b_{3}) = g_{1}$ , but $b_{1} \neq = b_{3}$ .
So $f$ is not one-one. (It is onto, since both $g_{1}$ and $g_{2}$ have pre-images.)
Since $f$ is not one-one, $f$ is not bijective.
2 marks

Answer: (I) 64; (II) 8; (III)(a) R is an equivalence relation; (III)(b) f is not bijective (not one-one, since f(b₁) = f(b₃) = g₁)

Where the marks are

(I) Number of relations = 2^6 = 64 (1 mark)
(II) Number of functions = 2^3 = 8 (1 mark)
(III)(a) Shows R is reflexive, symmetric and transitive, hence equivalence (2 marks)
(III)(b) Shows f is not one-one as f(b₁) = f(b₃) (1 mark); concludes f is not bijective (1 mark)

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 paper2023 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 2023: question 36

Hints

Step-by-step solution

Revise the chapter

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