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Equal roots or a perfect square: set the discriminant to zero to find the parameter

This type of question, asked 5 times: CDS 2 2017, CDS 1 2021, CDS 1 2023, CDS 2 2024, CDS 2 2026. All 6 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 2 2017

    22
    If the roots of the equation a(b−c)x2+b(c−a)x+c(a−b)=0a (b - c) x^2 + b (c - a) x + c (a - b) = 0 are equal, then which one of the following is correct ?

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  2. Question 2

    CDS 1 2021

    23
    If the equation 4x2−2kx+3k=04x^2 - 2kx + 3k = 0 has equal roots, then what are the values of kk?

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  3. Question 3

    CDS 1 2021

    21
    For how many real values of kk is 6kx2+12kx−24x+166kx^2 + 12kx - 24x + 16 a perfect square for every integer xx?

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  4. Question 4

    CDS 1 2023

    44
    For what values of mm, is mx2+mx+8x+9mx^2 + mx + 8x + 9 a perfect square ?

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  5. Question 5

    CDS 2 2024

    38
    If the roots of the equation x2−(k−2)x+(k+1)=0x^2 - (k - 2)x + (k + 1) = 0 are equal, then what are the values of kk ?

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  6. Question 6

    CDS 2 2026

    13
    If 5 is a root of the equation x2+px−55=0x^2 + px - 55 = 0 and the quadratic equation x2+px+k=0x^2 + px + k = 0 has equal roots, then what is the value of k ?

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