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Number of real roots of an equation using its derivative

This type of question, asked 7 times: 2021 Paper 20, 2021 Paper 22, 14 Jun 2022, 27 Jun 2022, 28 Jun 2022, 6 Apr 2023, 29 Jan 2024. All 7 questions below, oldest first.

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  1. Question 1

    2021 Paper 20

    82
    The number of distinct real roots of the equation 3x4+4x3−12x2+4=03x^{4}+ 4x^{3}- 12x^{2}+ 4 = 0 is

  2. Question 2

    2021 Paper 22

    62
    The number of real roots of the equation e4x+2e3x−ex−6=0e^{4x}+ 2e^{3x}-e^{x}- 6 = 0 is :

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  3. Question 3

    14 Jun 2022

    161
    The number of distinct real roots of the equation x7−7x−2=0x^{7}- 7x - 2 = 0 is

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  4. Question 4

    27 Jun 2022

    66
    The number of distinct real roots of x4−4x+1=0x^{4}- 4x + 1 = 0 is:

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  5. Question 5

    28 Jun 2022

    72
    The number of real solutions of x7+5x3+3x+1=x^{7}+ 5x^{3}+ 3x + 1 = 0 is equal to

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  6. Question 6

    6 Apr 2023

    176
    The number of points, where the curve y=x5−20x3+50x+2y =x^{5}- 20x^{3}+ 50x + 2 crosses the xx-axis, is

  7. Question 7

    29 Jan 2024

    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