PYQ Vault

Day 76: CBSE Class 12 Maths

5 questions (11 parts). Try each one first, then open its answer.

  1. Question 1 (case study, 4 parts)

    Application of Derivatives. Asked once: 2026 (same type asked 3 times)

    At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0⋅10{\cdot}1 cm3^3/s. On the basis of the above information, answer the following questions :
    36 (i)
    Establish a relation between the height h of the juice in the cup and radius r of the surface of the juice in the cup, if the semi-vertical angle of the cone is α\alpha.
    36 (ii)
    At what rate is the juice level in the cup rising when the juice is 6 cm deep ?
    36 (iii) (a)
    When the juice is 6 cm deep, then find at what rate is the upper surface area of juice increasing ?
    36 (iii) (b)
    When the juice is 6 cm deep, then find the rate at which the wetted surface area of the cup is increasing.

  2. Question 2

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

    33
    Draw a rough sketch for the curve y=2+∣x+1∣y = 2 + |x + 1|. Using integration, find the area of the region bounded by the curve y=2+∣x+1∣y = 2 + |x + 1|, x=−4x = -4, x=3x = 3 and y=0y = 0.

  3. Question 3 (case study, 4 parts)

    Application of Integrals. Asked once: 2026 (same type asked 3 times)

    A racing track is build around an elliptical ground whose equation is given by 9x2+16y2=1449x^2 + 16y^2 = 144. The width of the track is 3 m as shown below : Based on given information, answer the following questions :
    37 (i)
    Express y as a function of xx from the given equation of ellipse.
    37 (ii)
    Integrate the function obtained in (i) with respect to xx.
    37 (iii) (a)
    Find the area of the region enclosed within the elliptical ground excluding the track using integration.
    37 (iii) (b)
    Write the co-ordinates of the points P and Q where the outer edge of the track cuts xx axis and y axis in first quadrant and find the area of the triangle formed by points P, O, Q using integration.

  4. Question 4

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

    22 (a)
    Differentiate e2x\sqrt{e^{\sqrt{2x}}} with respect to e2xe^{\sqrt{2x}} for x>0x > 0.

  5. Question 5

    Continuity and Differentiability. Asked once: 2024 (same type asked 3 times)

    26 (b)
    Show that : ddx(∣x∣)=x∣x∣,x≠0\dfrac{d}{dx}\big(|x|\big) = \dfrac{x}{|x|}, x \ne 0