A small town is analyzing the pattern of a new street

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Class XII Case Study

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:- Given f(x) = e^x \sin x

⇒ \dfrac{d}{dx}f(x) = e^x\dfrac{d}{dx}\sin x + \sin x \dfrac{d}{dx}e^x

⇒ \dfrac{d}{dx}f(x) = e^x\cos x + e^x \sin x

⇒ \dfrac{d}{dx}f(x) = e^x(\sin x + \cos x) . . . (i)

Let \dfrac{d}{dx}f(x) = 0

e^x(\sin x + \cos x) = 0

Since, e^x>0 thus, \sin x + \cos x = 0

⇒ \sin x = -\cos x

⇒ \tan x = -1

⇒ x = \dfrac{3\pi}{4} \in [0, \pi]

(i)  there is two interval = [0, 3π/4] and [3π/4, π]

(a) at [0, 3π/4], let x = π/2

So, \dfrac{d}{dx}f(\pi/2) = e^{\pi/2}(\sin (\pi/2) + \cos (\pi/2))

⇒ \dfrac{d}{dx}f(\pi/2) =e^{\pi/2}(1 + 0) = e^{\pi/2}>0

Hence, f(x) is increasing function on [0, 3π/4]

(b) at [3π/4, π], let x= 5π/6

So, \dfrac{d}{dx}f(5\pi/6) = e^{5\pi/6}(\sin (5\pi/6) + \cos (5\pi/6)

⇒ \dfrac{d}{dx}f(\pi/2) =e^{\pi/2}(\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}) < 0

Hence, f(x) is decreasing function on [3π/4,  π]

(ii) From equation (i)

\dfrac{d}{dx}f(x) = e^x(\sin x + \cos x)

Let \dfrac{d}{dx}f(x) = 0

e^x(\sin x + \cos x) = 0

Since, e^x>0 thus, \sin x + \cos x = 0

⇒ \sin x = -\cos x

⇒ \tan x = -1

⇒ x = \dfrac{3\pi}{4} \in [0, \pi] is a critical point

Again differentiate with respect to x of equation (i)

\dfrac{d^2}{dx^2}f(x) = e^x \dfrac{d}{dx} (\sin x + \cos x) + (\sin x + \cos x)\dfrac{d}{dx}e^x

⇒ \dfrac{d^2}{dx^2}f(x) = e^x (\cos x - \sin x) + (\sin x + \cos x)e^x

⇒ \dfrac{d^2}{dx^2}f(x) = e^x( \cos x - \sin x + \sin x + \cos x)

⇒ \dfrac{d^2}{dx^2}f(x) = e^x(2 \cos x)

at x = \dfrac{3\pi}{4}

⇒ \dfrac{d^2}{dx^2}f(\dfrac{3\pi}{4}) = e^{\dfrac{3\pi}{4}}[2 \cos (\dfrac{3\pi}{4})]

⇒ \dfrac{d^2}{dx^2}f(\dfrac{3\pi}{4}) = e^{\dfrac{3\pi}{4}}[2\times \dfrac{1}{\sqrt{2}}]

⇒ \dfrac{d^2}{dx^2}f(\dfrac{3\pi}{4}) = \sqrt{2}.e^{\dfrac{3\pi}{4}}>0

Hence f(x) is minimum at critical point x = \dfrac{3\pi}{4}

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:- See full solution

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

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