MathsCBSE Class 12 · Sample paper, session 2024-25 · 4 marks · Case-based

CBSE Class 12 Mathematics sample paper 2024-25: question 37

From the official CBSE sample question paper for 2024-25, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedRelations and Functions

An organization conducted bike race under 2 different categories-boys and girls. In all, there were 250 participants. 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 $B = {b_{1}, b_{2}, b_{3}}$ , $G = {g_{1}, g_{2}}$ where $B$ represents the set of boys selected and $G$ the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
On the basis of the above information, answer the following questions:

(i)
Ravi wishes to form all the relations possible from $B$ to $G$ . How many such relations are possible?
1 mark
(ii)
Write the smallest equivalence relation on G.
1 mark
(iii)
(a) Ravi defines a relation from B to B as $R_{1} = {(b_{1}, b_{2}), (b_{2}, b_{1})}$ . Write the minimum ordered pairs to be added in $R_{1}$ so that it becomes (A) reflexive but not symmetric, (B) reflexive and symmetric but not transitive.
2 marks
OR
(iii)
(b) If the track of the final race (for the biker $b_{1}$ ) follows the curve $x^{2} = 4 y$ ; (where $0 \leq x \leq 202$ & $0 \leq y \leq 200$ ), then state whether the track represents a one-one and onto function or not. (Justify).
2 marks

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

A relation from $B$ to $G$ is just a subset of $B \times G$ . Count the elements of $B \times G$ first.

Hint 2

A reflexive relation on $B$ must contain every $(b_{i}, b_{i})$ . To spoil symmetry or transitivity, add a pair that is missing its partner.

Hint 3

For $f (x) = x^{2} /4$ on $[0, 202]$ , use $x \geq 0$ to prove one-one. For onto, solve $y = x^{2} /4$ for $x$ and check that $x$ stays in the domain.

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) A relation from $B$ to $G$ is any subset of $B \times G$ . Here $n (B \times G) = 3 \times 2 = 6$ , so the number of relations is $2^{6} = 64$ .
1 mark
(ii) An equivalence relation must at least be reflexive. So the smallest equivalence relation on $G$ is ${(g_{1}, g_{1}), (g_{2}, g_{2})}$ , which is also symmetric and transitive.
1 mark
(iii)(a)(A) For reflexivity, add $(b_{1}, b_{1}), (b_{2}, b_{2}), (b_{3}, b_{3})$ . To break symmetry, add one pair such as $(b_{2}, b_{3})$ without its reverse. So the minimum pairs to add are $(b_{1}, b_{1}), (b_{2}, b_{2}), (b_{3}, b_{3}), (b_{2}, b_{3})$ . Any one of $(b_{3}, b_{2}), (b_{1}, b_{3}), (b_{3}, b_{1})$ may be used instead of $(b_{2}, b_{3})$ .
1 mark
(iii)(a)(B) Add $(b_{1}, b_{1}), (b_{2}, b_{2}), (b_{3}, b_{3}), (b_{2}, b_{3}), (b_{3}, b_{2})$ . The relation is now reflexive and symmetric. It is not transitive, because $(b_{1}, b_{2}) \in R$ and $(b_{2}, b_{3}) \in R$ but $(b_{1}, b_{3}) \in / R$ .
1 mark
(iii)(b) One-one: $y = f (x) = \frac{x ^{2}}{4}$ , $f : [0, 202] \to [0, 200]$ . If $f (x_{1}) = f (x_{2})$ , then $x_{1}^{2} = x_{2}^{2}$ , so $(x_{1} - x_{2}) (x_{1} + x_{2}) = 0$ . Since $x_{1}, x_{2} \geq 0$ , this gives $x_{1} = x_{2}$ . So $f$ is one-one.
1 mark
(iii)(b) Onto: for any $y \in [0, 200]$ , take $x = 2 y$ . Then $0 \leq x \leq 2200 = 202$ and $f (2 y) = \frac{4 y}{4} = y$ . Every $y$ has a pre-image, so $f$ is onto. Hence the track represents a one-one and onto function.
1 mark

Answer: (i) $2^{6} = 64$ ; (ii) ${(g_{1}, g_{1}), (g_{2}, g_{2})}$ ; (iii)(a) (A) add $(b_{1}, b_{1}), (b_{2}, b_{2}), (b_{3}, b_{3}), (b_{2}, b_{3})$ ; (B) add $(b_{1}, b_{1}), (b_{2}, b_{2}), (b_{3}, b_{3}), (b_{2}, b_{3}), (b_{3}, b_{2})$ ; (iii)(b) Yes, it is one-one and onto

Where the marks are

(i) Gets $2^{3 \times 2} = 64$ relations (1 mark)
(ii) Writes ${(g_{1}, g_{1}), (g_{2}, g_{2})}$ (1 mark)
(iii)(a)(A) Adds the three diagonal pairs plus one non-symmetric pair (1 mark)
(iii)(a)(B) Adds the diagonal pairs plus $(b_{2}, b_{3}), (b_{3}, b_{2})$ (1 mark)
(iii)(b) Proves one-one using $x \geq 0$ (1 mark)
(iii)(b) Proves onto using the pre-image $2 y$ (1 mark)

Revise the chapter

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

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 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 37

Hints

Step-by-step solution

Revise the chapter

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