PYQ Vault

Day 27: 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. This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026

    Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length l1l_1 and l2l_2 are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
    36 (i)
    Express p(x)=l1+l2p(x) = l_1 + l_2 in terms of x.
    36 (ii)
    Find p′(x)p'(x).
    36 (iii) (a)
    Find the value of x for which l12+l22l_1^2 + l_2^2 is minimum.
    36 (iii) (b)
    If the 22 m long pole is also replaced by a 16 m long pole, at what distance from either pole should the ladders be kept so that the sum of squares of lengths of ladders needed to reach the top of the pole is minimum ?

  2. Question 2

    Application of Derivatives. This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026

    22
    Find the sub-interval(s) of (0,π2)\left(0, \dfrac{\pi}{2}\right) in which f(x)=tan⁡x−4xf(x) = \tan x - 4x is increasing.

  3. Question 3

    Continuity and Differentiability. This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026

    22 (a)
    Show that the function f(x)={cos⁡x−x+π2,x≠π21,x=π2f(x) = \begin{cases} \dfrac{\cos x}{-x + \dfrac{\pi}{2}}, & x \neq \dfrac{\pi}{2} \\[2ex] 1, & x = \dfrac{\pi}{2} \end{cases} is continuous at x=π2x = \dfrac{\pi}{2}.

  4. Question 4

    Continuity and Differentiability. This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026

    21 (a)
    Check whether function f(x) defined as f(x)={∣x−3∣2(x−3),x<3x−66,x≥3f(x) = \begin{cases} \frac{\left| x - 3 \right|}{2(x - 3)}, & x < 3 \\ \frac{x - 6}{6}, & x \geq 3 \end{cases} is continuous at x=3x = 3 or not ?

  5. Question 5

    Continuity and Differentiability. This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026

    22 (b)
    Find whether the function f(x)={x−1,x<22x−3,x≥2f(x) = \begin{cases} x - 1, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases} at x=2x = 2 is differentiable or not.