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Mathematics · 2026 · 4 marks
CBSE 2026 · Region 5 · Set 2 · Q38
A school wants the students of class XII to do a project on 'Sustainability' keeping the world environment in mind. They select the student…
A school wants the students of class XII to do a project on 'Sustainability' keeping the world environment in mind. They select the student participants on the basis of an essay writing competition. $\displaystyle 7$ students out of $\displaystyle 80$ are selected for the project and are categorized into two sets such that : Girl students belong to Set $\displaystyle \mathrm{A}=\left\{\mathrm{G}_{1}, \mathrm{G}_{2}, \mathrm{G}_{3}, \mathrm{G}_{4}\right\}$, Boy students belong to Set $\displaystyle \mathrm{B}=\left\{\mathrm{B}_{1}, \mathrm{~B}_{2}, \mathrm{~B}_{3}\right\}$. Based on the above information, answer the following questions :
(i)
How many relations are possible from Set A → Set B ?
(ii)
Let R be a relation from $\displaystyle \mathrm{A} \rightarrow \mathrm{B}$ such that \[\mathrm{R}=\left\{\left(\mathrm{G}_{1}, \mathrm{~B}_{1}\right),\left(\mathrm{G}_{2}, \mathrm{~B}_{2}\right),\left(\mathrm{G}_{3}, \mathrm{~B}_{2}\right),\left(\mathrm{G}_{4}, \mathrm{~B}_{3}\right),\left(\mathrm{G}_{1}, \mathrm{~B}_{2}\right)\right\} . \] Is R an injective function ? Justify your answer.
(iii)
(a)
Let the relation R from $\displaystyle \mathrm{A} \rightarrow \mathrm{A}$ be such that $\displaystyle \mathrm{R}=\{(\mathrm{x}, \mathrm{y}), \mathrm{x}, \mathrm{y} \in \mathrm{A}, \mathrm{x}$ and y are students from the same colony in the city} Verify if R is an equivalence relation.
OR
(b)
Verify if any function f : B → A is bijective. Give reason to support your answer.
Marking-scheme solution
(i)
Number of relations from A to B is $\displaystyle 2^{12}$ or $\displaystyle 4096$
(ii)
No, R is not injective.
It is not a function as $\displaystyle G_{1}$ has two images $\displaystyle B_{1}$ and $\displaystyle B_{2}$
(iii)
R is reflexive as $\displaystyle (x, x) \in R\ \forall x \in A$ because x and x are students from same colony
Let $\displaystyle (x, y) \in R$ so x and y are students from same colony
Hence y and x are students from same colony. Thus, $\displaystyle (y, x) \in R$
$\displaystyle \therefore$ R is symmetric
Let $\displaystyle (x, y) \in R$ and $\displaystyle (y, z) \in R$ so x and y are students from same colony, y and z are students from same colony
Thus, x and z are students from same colony $\displaystyle \therefore(x, z) \in R$
Thus, R is transitive
As R is reflexive, symmetric and transitive, hence R is an equivalence relation
Any function f from B to A will not be bijective
As f from B to A cannot be surjective because $\displaystyle n(B)<n(A)$
or A and B are finite sets and $\displaystyle n(A) \neq n(B)$
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CBSE Class 12 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.