Three person viz. Amber Bonzi and Comet are manufacturing cars

Class XII Case Study

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 : [CBSE 2025]

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

(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:- Let Amber manufacturing cars which run on petrol and on battery as well = A

Let Bonzi manufacturing cars which run on petrol and on battery as well = B

Let Comet manufacturing cars which run on petrol and on battery as well = C

(i) (a) Let selected car is an electric car = E

P(A) = 60/100, P(B) = 30/100, P(C) = 10/100

P(E/A) = 20/100, P(E/B) = 10/100, P(E/C) = 5/100

Probability of selected car is an electric car

P(E) = P(A)P(E/A) + P(B)P(E/B) + P(C)P(E/C)

⇒ $P(E) = \dfrac{60}{100}\times \dfrac{20}{100}+ \dfrac{30}{100}\times \dfrac{10}{100} + \dfrac{10}{100}\times \dfrac{5}{100}$

⇒ $P(E) = \dfrac{12}{100} + \dfrac{3}{100} + \dfrac{1}{200}$

⇒ $P(E) = \dfrac{24 + 6+ 1}{200}$

⇒ $P(E) = \dfrac{31}{200}= 0.155$

(i) (b) Let selected car is an petrol car = F

P(A) = 60/100, P(B) = 30/100, P(C) = 10/100

P(F/A) = 1- 20/100 = 80/100, P(F/B) = 1 – 10/100 = 90/100,

P(F/C) = 1 – 5/100 = 95/100

P(F) = P(A)P(F/A) + P(B)P(F/B) + P(C)P(F/C)

⇒ $P(F) = \dfrac{60}{100}\times \dfrac{80}{100}+ \dfrac{30}{100}\times \dfrac{90}{100} + \dfrac{10}{100}\times \dfrac{95}{100}$

⇒ $P(E) = \dfrac{48}{100} + \dfrac{27}{100} + \dfrac{9.5}{200}$

⇒ $P(E) = \dfrac{48 + 27 + 9.5}{200}$

⇒ $P(E) = \dfrac{84.5}{100}= 0.845$

(ii) A car is selected at random and is found to be electric.

The probability that it was manufactured by Comet = P(C/E)

$P(C/E) = \dfrac{P(C)P(E/C)}{P(A)P(E/A) + P(B)P(E/B) + P(C)P(E/C)}$

⇒ $P(C/E) = \dfrac{1/200}{31/200}$

⇒ $P(C/E) = \dfrac{1}{31}$

(iii) A car is selected at random and is found to be electric.

The probability that it was manufactured by Amber or Bonzi = 1 – 1/31 = 30/31

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

(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:- 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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