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Form the equation whose roots are changed versions of the given roots

This type of question, asked 2 times: CDS 2 2021, CDS 2 2023. All 2 questions below, oldest first.

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  1. Question 1

    CDS 2 2021

    10
    The sum and the product of the roots of a quadratic equation are 7 and 12 respectively. If the bigger root is halved and the smaller root is doubled, then what is the resulting quadratic equation ?

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  2. Question 2

    CDS 2 2023

    76
    Let α\alpha and β\beta be the roots of the equation 1x+a+b=1x+1a+1b\frac{1}{x + a + b} = \frac{1}{x} + \frac{1}{a} + \frac{1}{b}; a≠0a \ne 0, b≠0b \ne 0, x≠0x \ne 0. Which one of the following is a quadratic equation whose roots are α2\alpha^2 and β2\beta^2 ?

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