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Two quadratics share a root: find the unknown coefficient

This type of question, asked 3 times: CDS 2 2017, CDS 1 2019, CDS 1 2025. All 5 questions below, oldest first.

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  1. Question 1

    CDS 2 2017

    1
    What is the value of α\alpha (α≠0)(\alpha \ne 0) for which x2−5x+αx^2 - 5x + \alpha and x2−7x+2αx^2 - 7x + 2\alpha have a common factor ?

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  2. Question 2

    CDS 2 2017

    15
    If (x+2)(x + 2) is a common factor of x2+ax+bx^2 + ax + b and x2+bx+ax^2 + bx + a, then the ratio a:ba : b is equal to

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  3. Question 3

    CDS 1 2019

    19
    If the equations x2+5x+6=0x^2 + 5x + 6 = 0 and x2+kx+1=0x^2 + kx + 1 = 0 have a common root, then what is the value of kk ?

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  4. Question 4

    CDS 1 2025

    80
    The equations x2+px+q=0x^2 + px + q = 0 and x2+qx+p=0x^2 + qx + p = 0 (p≠q)(p \ne q) have a common root. What is the value of (p+q)(p + q) ?

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  5. Question 5

    CDS 1 2025

    82
    If two quadratic equations px2+px+4=0px^2 + px + 4 = 0 and x2+qx+q=0x^2 + qx + q = 0 have a common root 2, then what is p+qp + q equal to ?

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