find the area of the region bounded by the line y = 5x + 2

Question 1:- Using integration, find the area of the region bounded by the line y = 5x + 2, the x- axis and the ordinates x = -2 and x = 2. [CBSE 2025]

Solution:- Given line, y = 5x + 2

(i) Let x = 0, then y = 2

(o, 2)

(ii) Let y = 0, then x = -2/5 = -0.4

(-2/5, 0)

find the area of the region bounded by the line y = 5x + 2

Area of bounded region from x = -2 to x = 2

$= -\displaystyle \int_{-2}^{-2/5} y dx + \int_{-2/5}^2 y dx$

$= -\displaystyle \int_{-2}^{-2/5} 5x + 2 dx + \int_{-2/5}^2 5x + 2 dx$

$= -[5\dfrac{x^2}{2} + 2x]_{-2}^{-2/5} +[5\dfrac{x^2}{2} + 2x]_{-2/5}^2$

$= -\left[\{5\dfrac{(-2/5)^2}{2} + 2(-2/5)\}-\{5\dfrac{(-2)^2}{2} + 2(-2)\}\right] +\left[\{5\dfrac{(-2)^2}{2} + 2(-2)\}-\{5\dfrac{(-2/5)^2}{2} + 2(-2/5)\}\right]$

$= -\left[\{\dfrac{2}{5}-\dfrac{4}{5}\} - \{10 - 4\}\right] + \left[\{10 - 4\} - \{\dfrac{2}{5}-\dfrac{4}{5}\}\right]$

$= -\left[-\dfrac{2}{5} -6\right] + \left[6 + \dfrac{2}{25}\right]$

$= \dfrac{32}{5} + \dfrac{32}{5}$

$= \dfrac{64}{5}$

Area of bounded region = 64/5

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:- See full solution

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