PYQ Vault

Day 94: CBSE Class 12 Maths

5 questions (8 parts). Try each one first, then open its answer.

  1. Question 1

    Continuity and Differentiability. Asked once: 2024 (same type asked 2 times)

    22 (b)
    If xy=ex−yx^{y} = e^{x-y}, prove that dydx=log⁡x(1+log⁡x)2\dfrac{dy}{dx} = \dfrac{\log x}{(1 + \log x)^{2}}.

  2. Question 2

    Continuity and Differentiability. Asked once: 2025 (same type asked 2 times)

    4
    If y=log⁡2x(2x)y = \log_{2x}(\sqrt{2x}), then dydx\frac{dy}{dx} is equal to :

    Tap an option to check your answer.

  3. Question 3

    Determinants. Asked once: 2024 (same type asked 2 times)

    18
    If A=[200030005]A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}, then A−1A^{-1} is :

    Tap an option to check your answer.

  4. Question 4

    Determinants. Asked once: 2024 (same type asked 2 times)

    32 (b)
    If A=[−1a212x311]A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix} and A−1=[1−11−87−5by3]A^{-1} = \begin{bmatrix} 1 & -1 & 1 \\ -8 & 7 & -5 \\ b & y & 3 \end{bmatrix}, find the value of (a+x)−(b+y)(a + x) - (b + y).

  5. Question 5 (case study, 4 parts)

    Differential Equations. Asked once: 2025 (same type asked 2 times)

    Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. (Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, dVdt=kS\dfrac{dV}{dt} = kS is the differential equation, where V is the volume, S is the surface area and t is the time in hours. Based upon the above information, answer the following questions :
    37 (i)
    Write the order and degree of the given differential equation.
    37 (ii)
    Substituting V=πr3V = \pi r^3 and S=2πr2S = 2\pi r^2, we get the differential equation drdt=23k\dfrac{dr}{dt} = \dfrac{2}{3}k. Solve it, given that r(0)=5r(0) = 5 mm.
    37 (iii) (a)
    If it is given that r=3r = 3 mm when t=1t = 1 hour, find the value of k. Hence, find t for r=0r = 0 mm.
    37 (iii) (b)
    If it is given that r=1r = 1 mm when t=1t = 1 hour, find the value of k. Hence, find t for r=0r = 0 mm.