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Real or no real roots: discriminant sign gives the range of the parameter

This type of question, asked 4 times: CDS 2 2020, CDS 2 2021, CDS 2 2022, CDS 1 2024. All 5 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 2 2020

    3
    If the roots of the quadratic equation x2−4x−log⁡10N=0x^2 - 4x - \log_{10} N = 0 are real, then what is the minimum value of NN ?

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  2. Question 2

    CDS 2 2021

    19
    If the roots of the equation x2−4x−log⁡10N=0x^2-4x-\log_{10}N=0 are real, then what is the minimum value of NN ?

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  3. Question 3

    CDS 2 2021

    11
    For which values of kk, does the equation x2−kx+2=0x^2-kx+2=0 have real and distinct solutions ?

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  4. Question 4

    CDS 2 2022

    32
    What are the values of k for which the polynomial (k−3)x2−kx−1(k - 3)x^2 - kx - 1 has no real linear factors ?

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  5. Question 5

    CDS 1 2024

    14
    What is the greatest value of k for which 2x2−4x+k=02x^2 - 4x + k = 0 has real roots ?

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