PYQ Vault

Day 56: CBSE Class 12 Maths

5 questions (9 parts). Try each one first, then open its answer.

  1. Question 1 (case study, 4 parts)

    Application of Derivatives. This type asked 2 times: 2023, 2025. This one: 2025

    A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be xx metres and with parallel to the partition be y metres. Based on this information, answer the following questions :
    36 (i)
    Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of xx and y.
    36 (ii)
    Write the area of the solar panel as a function of xx.
    36 (iii) (a)
    Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.
    36 (iii) (b)
    Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.

  2. Question 2 (case study, 2 parts)

    Application of Derivatives. This type asked 2 times: 2023, 2025. This one: 2025

    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 xx metres from the start of the street can be modelled by f(x)=exsin⁡xf(x) = e^x \sin x, where xx is in metres. Based on the above, answer the following :
    38 (i)
    Find the intervals on which the f(x)f(x) is increasing or decreasing, x∈[0,π]x \in [0, \pi].
    38 (ii)
    Verify, whether each critical point when x∈[0,π]x \in [0, \pi] is a point of local maximum or local minimum or a point of inflexion.

  3. Question 3

    Application of Derivatives. This type asked 2 times: 2023, 2025. This one: 2025

    26
    The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm ?

  4. Question 4

    Continuity and Differentiability. This type asked 2 times: 2024, 2025. This one: 2025

    29 (b)
    If x1+y+y1+x=0x\sqrt{1 + y} + y\sqrt{1 + x} = 0, −1<x<1-1 < x < 1, x≠yx \ne y, then prove that dydx=−1(1+x)2\dfrac{dy}{dx} = \dfrac{-1}{(1 + x)^2}.

  5. Question 5

    Continuity and Differentiability. This type asked 2 times: 2024, 2025. This one: 2025

    23 (a)
    Differentiate (5xx5)\left(\dfrac{5^x}{x^5}\right) with respect to x.