PYQ Vault

Day 147: CBSE Class 12 Maths

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

  1. Question 1 (case study, 4 parts)

    Application of Derivatives. Asked once: 2023 (same type asked 3 times)

    A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of 2 cm3/s2 \text{ cm}^3/\text{s}. The semi-vertical angle of the conical tank is 45∘45^\circ. On the basis of given information, answer the following questions :
    37 (i)
    Find the volume of water in the tank in terms of its radius r.
    37 (ii)
    Find rate of change of radius at an instant when r=22r = 2\sqrt{2} cm.
    37 (iii) (a)
    Find the rate at which the wet surface of the conical tank is decreasing at an instant when radius r=22r = 2\sqrt{2} cm.
    37 (iii) (b)
    Find the rate of change of height 'h' at an instant when slant height is 4 cm.

  2. Question 2

    Application of Integrals. Asked once: 2025 (same type asked 3 times)

    32
    In a rough sketch, mark the region bounded by y=1+∣x+1∣y = 1 + |x + 1|, x=−2x = -2, x=2x = 2 and y=0y = 0. Using integration, find the area of the marked region.

  3. Question 3

    Application of Integrals. Asked once: 2024 (same type asked 3 times)

    34
    Using integration, find the area of the region enclosed between the circle x2+y2=16x^2 + y^2 = 16 and the lines x=−2x = -2 and x=2x = 2.

  4. Question 4

    Continuity and Differentiability. Asked once: 2025 (same type asked 3 times)

    21 (a)
    Differentiate 2cos⁡2x2^{\cos^2 x} w.r.t cos⁡2x\cos^2 x.

  5. Question 5

    Continuity and Differentiability. Asked once: 2023 (same type asked 3 times)

    24
    If y=(x+x2−1)2y = \left(x + \sqrt{x^2 - 1}\right)^2, then show that (x2−1)(dydx)2=4y2(x^2 - 1)\left(\frac{dy}{dx}\right)^2 = 4y^2.