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Application of Derivatives

JEE Mains Maths. Try each one first, then open its answer.

Top 10 JEE Mains PYQs to practise first

The questions and kinds of question asked most often come first. Each one says how often and in which papers.

  1. Question 1

    This type asked 7 times: 2021 Paper 20, 2021 Paper 22, 14 Jun 2022, 27 Jun 2022, 28 Jun 2022, 6 Apr 2023, 29 Jan 2024. This one: 29 Jan 2024 See all 7 of this type

    84
    Let f(x)=2x−x2,x∈Rf(x) =2^{x}-x^{2},x \in R. If mm and nn are respectively the number of points at which the curves y=f(x)y = f(x) and y=f′(x)y =f^{'}(x) intersects the xx-axis, then the value of m+nm + n is

  2. Question 2

    This type asked 6 times: 2021 Paper 3, 2021 Paper 10, 2021 Paper 23, 25 Jul 2022, 7 Apr 2025, 2 Apr 2026 S2. This one: 2 Apr 2026 S2 See all 6 of this type

    68
    Let f(x)f(x) be a polynomial of degree 5 and have extrema at x=1x= 1 and x=−1x= - 1. If lim⁡x→0(f(x)x3)=−5\lim_{x\rightarrow 0} \left( \frac{f(x)}{x^{3}} \right)= - 5, then f(2)−f(−2)f(2) - f( - 2) is equal to :

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  3. Question 3

    This type asked 6 times: 2021 Paper 1, 2021 Paper 2, 2021 Paper 6, 28 Jul 2022, 29 Jan 2024, 1 Feb 2024. This one: 1 Feb 2024 See all 6 of this type

    79
    If 5f(x)+4f(1x)=x2−2,∀x≠05f(x) + 4f\left( \frac{1}{x} \right)=x^{2}- 2,\forall x \neq 0 and y=9x2f(x)y = 9x^{2}f(x), then yy is strictly increasing in:

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  4. Question 4

    This type asked 5 times: 2021 Paper 5, 2021 Paper 22, 14 Jun 2022, 26 Jul 2022, 9 Apr 2024. This one: 9 Apr 2024 See all 5 of this type

    172
    Let the set of all values of pp, for which f(x)=(p2−6p+8)(sin⁡22x−cos⁡22x)+2(2−p)x+7f(x) =\left( p^{2}- 6p + 8 \right)\left( \sin^{2}2x -\cos^{2}2x \right)+ 2(2 - p)x + 7 does not have any critical point, be the interval (a,b)(a,b). Then 16ab16ab is equal to

  5. Question 5

    This type asked 5 times: 14 Jun 2022, 26 Jun 2022, 25 Jul 2022, 31 Jan 2023, 1 Feb 2023. This one: 1 Feb 2023 See all 5 of this type

    160
    The sum of the absolute maximum and minimum values of the function f(x)=∣x2−5x+6∣−3x+2f(x) =\left| x^{2}- 5x + 6 \right|- 3x + 2 in the interval [−1,3]\lbrack - 1,3\rbrack is equal to:

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  6. Question 6

    This type asked 4 times: 27 Jun 2022, 31 Jan 2024, 22 Jan 2025, 28 Jan 2026 S2. This one: 28 Jan 2026 S2 See all 4 of this type

    73
    Let ff be a differentiable function satisfying
    f(x)=1−2x+∫0xe(x−t)f(t)dt,x∈R and let f(x) = 1 - 2x + \int_{0}^{x} e^{(x - t)}f(t)dt,x \in R\text{~and let~}
    g(x)=∫0x(f(t)+2)15(t−4)6(t+12)17dt,x∈Rg(x) = \int_{0}^{x} (f(t) + 2)^{15}(t - 4)^{6}(t + 12)^{17}dt,x \in R
    If pp and qq are respectively the points of local minima and local maxima of g , then the value of ∣p+q∣|p + q| is equal to ............ .

  7. Question 7

    This type asked 4 times: 2021 Paper 23, 29 Jun 2022, 28 Jul 2022, 30 Jan 2024. This one: 30 Jan 2024 See all 4 of this type

    67
    Let g:R→Rg:R \rightarrow R be a non constant twice differentiable such that g′(12)=g′(32)g^{'}\left( \frac{1}{2} \right)=g^{'}\left( \frac{3}{2} \right). If a real valued function ff is defined as f(x)=12[ g(x)+g(2−x)]f(x) =\frac{1}{2}\lbrack\text{ }g(x) + g(2 - x)\rbrack, then

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  8. Question 8

    This type asked 4 times: 2021 Paper 18, 2021 Paper 20, 29 Jun 2022, 31 Jan 2023. This one: 31 Jan 2023 See all 4 of this type

    65
    A wire of length 20 m20\text{ }m is to be cut into two pieces. A piece of length l1l_{1} is bent to make a square of area A1A_{1} and the other piece of length l2l_{2} is made into a circle of area A2A_{2}. If 2A1+3A22A_{1}+ 3A_{2} is minimum then (πl1):l2\left( \pi l_{1} \right):l_{2} is equal to:

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  9. Question 9

    This type asked 3 times: 30 Jan 2024, 22 Jan 2026 S1, 6 Apr 2026 S2. This one: 6 Apr 2026 S2 See all 3 of this type

    67
    Let A=∣13−121α01−1∣A =\left| \begin{matrix} 1 & 3 & - 1 \\ 2 & 1 & \alpha \\ 0 & 1 & - 1 \end{matrix} \right| be a singular matrix. Let f(x)=∫0x(t2+2t+3)dt,x∈[1,α]f(x) =\int_{0}^{x} \left( t^{2}+ 2t+ 3 \right)dt,x\in \lbrack 1,\alpha\rbrack. If MM and mm are respectively the maximum and the minimum values of ff in [1,α]\lbrack 1,\alpha\rbrack then 3(M−m)3(M-m) is equal to :

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  10. Question 10

    This type asked 3 times: 10 Apr 2023, 27 Jan 2024, 21 Jan 2026 S2. This one: 21 Jan 2026 S2 See all 3 of this type

    52
    Let f:R→Rf:R \rightarrow R be a twice differentiable function such that f′′(x)>0f^{''}(x) > 0 for all x∈Rx \in R and f′(a−1)=0f^{'}(a - 1) = 0, where a is real number. Let g(x)=f(tan⁡2x−2tan⁡x+a)g(x) = f\left( \tan^{2}x - 2\tan x + a \right), 0<x<π20 < x <\frac{\pi}{2}. Consider the following two statements : (I) g is increasing in (0,π4)\left( 0,\frac{\pi}{4} \right) (II) g is decreasing in (π4,π2)\left( \frac{\pi}{4},\frac{\pi}{2} \right) Then,

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