PYQ Vault

Day 57: CBSE Class 12 Maths

5 questions (6 parts). Try each one first, then open its answer.

  1. Question 1

    Determinants. This type asked 2 times: 2024, 2025. This one: 2025

    1
    If A=[−100010001]A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, then A−1A^{-1} is

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  2. Question 2

    Determinants. This type asked 2 times: 2024, 2025. This one: 2025

    32
    If A is a 3×33 \times 3 invertible matrix, show that for any scalar k≠0k \ne 0, (kA)−1=1kA−1(kA)^{-1} = \frac{1}{k}A^{-1}. Hence calculate (3A)−1(3A)^{-1}, where A=[2−11−12−11−12]A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}.

  3. Question 3 (case study, 2 parts)

    Differential Equations. This type asked 2 times: 2024, 2025. This one: 2025

    During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature (25∘C)(25^\circ\text{C}). Initially the processor's temperature is 85∘C85^\circ\text{C}. The rate of cooling is defined by the equation ddt(T(t))=−k(T(t)−25)\frac{d}{dt}(T(t)) = -k(T(t) - 25), where T(t) represents the temperature of the processor at time t (in minutes) and k is a constant. Based on the above information, answer the following questions :
    38 (i)
    Find the expression for temperature of processor, T(t) given that T(0)=85∘CT(0) = 85^\circ\text{C}.
    38 (ii)
    How long will it take for the processor's temperature to reach 40∘C40^\circ\text{C} ? Given that k=0⋅03k = 0{\cdot}03, log⁡e4=1⋅3863\log_e 4 = 1{\cdot}3863.

  4. Question 4

    Integrals. This type asked 2 times: 2024, 2025. This one: 2025

    34 (b)
    Find : ∫dxsin⁡x+sin⁡2x\int \frac{dx}{\sin x + \sin 2x}

  5. Question 5

    Integrals. This type asked 2 times: 2024, 2025. This one: 2025

    11
    ∫1+sin⁡x dx\displaystyle\int \sqrt{1 + \sin x}\ dx is equal to :

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