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Differentiation

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 6 times: 2021 Paper 3, 2021 Paper 22, 29 Jul 2022, 31 Jan 2024, 1 Feb 2024, 4 Apr 2026 S2. This one: 4 Apr 2026 S2 See all 6 of this type

    72
    Let f(x)={ex−1,x<0x2−5x+6,x≥0f(x) = \left\{ \begin{matrix} e^{x - 1} & ,x < 0 \\ x^{2} - 5x + 6 & ,x \geq 0 \end{matrix} \right. and g(x)=f(∣x∣)+∣f(x)∣g(x) = f(|x|) + |f(x)|. If the number of points where g is not continuous and is not differentiable are α\alpha and β\beta respectively, then α+β\alpha + \beta is equal to ____\_\_\_\_

  2. Question 2

    This type asked 6 times: 2021 Paper 4, 26 Jun 2022, 15 Apr 2023, 28 Jan 2025, 4 Apr 2025, 4 Apr 2026 S1. This one: 4 Apr 2026 S1 See all 6 of this type

    75
    The number of points, at which the function f(x)=max{6x,2+3x2}+∣x−1∣cos⁡∣x2−14∣,x∈(−π,π)f(x) = max\left\{ 6x,2 + 3x^{2} \right\} + |x - 1|\cos\left| x^{2} - \frac{1}{4} \right|,x \in ( - \pi,\pi), is not differentiable, is ____\_\_\_\_ .

  3. Question 3

    This type asked 5 times: 2021 Paper 24, 25 Jul 2022, 29 Jul 2022, 6 Apr 2023, 24 Jan 2025. This one: 24 Jan 2025 See all 5 of this type

    132
    Let [x]\lbrack x\rbrack denote the greatest integer function, and let mm and nn respectively be the numbers of the points, where the function f(x)=[x]+∣x−2∣f(x) = \lbrack x\rbrack + |x - 2|, −2<x<3- 2 < x < 3, is not continuous and not differentiable. Then m+nm + n is equal to:

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  4. Question 4

    This type asked 4 times: 2021 Paper 7, 2021 Paper 21, 14 Jun 2022, 4 Apr 2026 S1. This one: 4 Apr 2026 S1 See all 4 of this type

    66
    If y=tan⁡−1(3cos⁡x−4sin⁡x4cos⁡x+3sin⁡x)+2tan⁡−1(x1+1−x2)y=\tan^{- 1}\left( \frac{3\cos x- 4\sin x}{4\cos x+ 3\sin x} \right)+ 2\tan^{- 1}\left( \frac{x}{1 +\sqrt{1 -x^{2}}} \right), then dydx\frac{dy}{dx} at x=32x =\frac{\sqrt{3}}{2} is equal to :

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  5. Question 5

    This type asked 4 times: 2021 Paper 9, 8 Apr 2023, 30 Jan 2024, 24 Jan 2026 S1. This one: 24 Jan 2026 S1 See all 4 of this type

    66
    Let α,β∈R\alpha,\beta \in R be such that the function f(x)={2α(x2−2)+2βx,x<1(α+3)x+(α−β),x≥1f\left( x \right)=\left\{ \begin{matrix} 2\alpha\left( x^{2}- 2 \right)+ 2\beta x & ,x < 1 \\ \left( \alpha + 3 \right)x +\left( \alpha - \beta \right) & ,x \geq 1 \end{matrix} \right. be differentiable at all x∈Rx \in R. Then 34(α+β)34(\alpha + \beta) is equal to

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  6. Question 6

    This type asked 4 times: 24 Jan 2023, 1 Feb 2023, 27 Jan 2024, 21 Jan 2026 S2. This one: 21 Jan 2026 S2 See all 4 of this type

    53
    Let f(x)=x3+x2f′(1)+2xf′′(2)+f′′′(3)f(x) =x^{3}+x^{2}f^{'}(1) + 2xf^{''}(2) +f^{'''}(3), x∈Rx \in R. Then the value of f′(5)f^{'}(5) is :

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  7. Question 7

    This type asked 4 times: 2021 Paper 10, 26 Jun 2022, 29 Jan 2023, 22 Jan 2025. This one: 22 Jan 2025 See all 4 of this type

    66
    Let f(x)f(x) be a real differentiable function such that f(0)=1f(0) = 1 and f(x+y)=f(x)f′(y)+f′(x)f(y)f(x + y) = f(x)f^{'}(y) +f^{'}(x)f(y) for all x,y∈Rx,y \in R. Then ∑n=1100log⁡ef(n)\sum_{n = 1}^{100} \log_{e}f(n) is equal to :

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  8. Question 8

    This type asked 3 times: 25 Jun 2022, 4 Apr 2024, 5 Apr 2024. This one: 5 Apr 2024 See all 3 of this type

    77
    Let f(x)=x5+2x3+3x+1,x∈Rf(x) =x^{5}+ 2x^{3}+ 3x + 1,x \in R, and g(x)g(x) be a function such that g(f(x))=xg(f(x)) = x for all x∈Rx \in R. Then g(7)g′(7)\frac{g(7)}{g^{'}(7)} is equal to:

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  9. Question 9

    This type asked 3 times: 2021 Paper 17, 29 Jun 2022, 27 Jan 2024. This one: 27 Jan 2024 See all 3 of this type

    152
    Consider the function f:(0,2)→Rf:(0,2) \rightarrow R defined by f(x)=x2+2xf(x) =\frac{x}{2}+\frac{2}{x} and the function g(x)g(x) defined by g(x)={min⁡{f(t):0<t≤x},0<x≤132+x,1<x<2g(x) =\left\{ \begin{matrix} \min\{ f(t) : 0 < t \leq x\}, & 0 < x \leq 1 \\ \frac{3}{2}+ x, & 1 < x < 2 \end{matrix} \right.. Then

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  10. Question 10

    This type asked 3 times: 2021 Paper 5, 26 Jun 2022, 11 Apr 2023. This one: 11 Apr 2023 See all 3 of this type

    170
    Let ff and gg be two functions defined by f(x)={x+1,x<0∣x−1∣,x≥0f(x) =\left\{ \begin{matrix} x + 1, & x < 0 \\ |x - 1|, & x \geq 0 \end{matrix} \right. and g(x)={x+1,x<01,x≥0g(x) =\left\{ \begin{matrix} x + 1, & x < 0 \\ 1, & x \geq 0 \end{matrix} \right.. Then (gof) (x) is

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