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Componendo and dividendo on (sqrt A + sqrt B)/(sqrt A - sqrt B) = k

This type of question, asked 4 times: CDS 1 2017, CDS 1 2021, CDS 2 2022, CDS 2 2025. All 5 questions below, oldest first.

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  1. Question 1

    CDS 1 2017

    9
    If x=a+b−a−ba+b+a−bx = \dfrac{\sqrt{a + b} - \sqrt{a - b}}{\sqrt{a + b} + \sqrt{a - b}}, then what is bx2−2ax+bbx^2 - 2ax + b equal to (b≠0)(b \ne 0) ?

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  2. Question 2

    CDS 1 2021

    8
    If p=3q+2+3q−23q+2−3q−2p = \frac{\sqrt{3q + 2} + \sqrt{3q - 2}}{\sqrt{3q + 2} - \sqrt{3q - 2}} then what is the value of p2−3pq+2p^2 - 3pq + 2?

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  3. Question 3

    CDS 1 2021

    10
    What is the value of xx, if b+b2−2bxb−b2−2bx=a\frac{b + \sqrt{b^2 - 2bx}}{b - \sqrt{b^2 - 2bx}} = a?

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  4. Question 4

    CDS 2 2022

    4
    If x+20+x−1x+20−x−1=73\frac{\sqrt{x + 20} + \sqrt{x - 1}}{\sqrt{x + 20} - \sqrt{x - 1}} = \frac{7}{3}, then what is the value of (x+20)(x−1)\sqrt{(x + 20)(x - 1)} ?

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  5. Question 5

    CDS 2 2025

    24
    If p+x+p−xp+x−p−x=p\frac{\sqrt{p+x}+\sqrt{p-x}}{\sqrt{p+x}-\sqrt{p-x}}=p, then what is xx equal to?

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