A school is organizing a debate competition with participants

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Class XII Case Study 

Case Study 1:- A school is organizing a debate competition with participants as speakers S = \{S_1, S_2, S_3, S_4\} and these are judged by judges J = \{J_1, J_2, J_3\}. Each speaker can be assigned one judge. Let R be a relation from set S to J defined as R = {(x, y) : speaker x is judged by judege y, x ∈ S, y ∈ J}.     [CBSE 2025]

 

Based on the above, answer the following:

(i) How many relation can be there from S to J ?     (1)

(ii) A student identifies a function from S to J as f = \{(S_1, J_1), (S_2, J_2), (S_3,J_2), (S_4, J_3)\} Check if it is bijective.                  (1)

(iii) (a) How many one-one functions can be there from set S to set J ?                 2

             OR

(iii) (b) Another student considers a relation R_1 = \{(S_1, S_2), (S_2, S_4)\} in set S. Write minimum ordered pairs to be included in R_1 so that R_1 is reflexive but not symmetric.          (2)

Solution:- (i) S = \{S_1, S_2, S_3, S_4\} and J = \{J_1, J_2, J_3\}

n(S) = 4 and n(J) = 3

Number of relation = 2^{mn} = 2^{\{4 \times 3\}} = 2^{12}

(ii) f = \{(S_1, J_1), (S_2, J_2), (S_3,J_2), (S_4, J_3)\}

One-one :-f = \{(S_1, J_1), (S_2, J_2), (S_3,J_2), (S_4, J_3)\}

This relation is no one-one because in ordered pair (S_2, J_2), (S_3,J_2) J_2 related to two element of domain S_2, S_3.

Hence f is not one- one function

Onto:- Every element of codomain have preimage in domian , so it is onto function

Thus, f is not bijective function

(iii) If n(S)=m, n(J)=n then number of one-one functions from S to J = \begin{cases}m! & \text{if } m\leq n\\0&\text{if }m >n\end{cases}

n(S) = 4, n(J) = 3

n(S) > n(J)

then number of one- one function is 0

OR

(iii) R_1 = \{(S_1, S_2), (S_2, S_4)\} in set S

If we add minimum ordered pair (S_1, S_1), (S_2, S_2), (S_3, S_3), (S_4, S_4)

The relation R_1 will be reflexive but not symmetric

Case Study 2:- Three person viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60 %, 30% and 10% respectively. Of their respective production capacities, 20 %, 10 % and 5 % cars respectively are electric (or battery operated).

Based on the above, answer the following :

(i) (a) What is the probability that a randomly selected car is an electric car ?          (2)

                         OR

(i) (b) What is the probability that a randomly selected car is a petrol car ?                             (2)

(ii) A car is selected at random and is found to be electric. What is the probability that it was manufactured by Comet ?           (1)

(iii) A car is selected at random and is found to be electric. What is the probability that it was manufactured by Amber or Bonzi ?         (1)

Solution:- See full solution

Case Study 3:-  A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity of light at any point x meters from the start of the street can be modelled by f(x) = e^x \sin x, where x is in meters.

Based on the above, answer the following :

(i) Find the intervals on which the f(x) is increasing or decreasing, x ∈ [0, π].                   (2)

(ii) Verify, whether each critical point when x ∈ [0, π] is a point of local maximum or local minimum or a point of inflexion.                    (2)

Solution:- See full solution

 

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