NinetynineCAT 2023 Slot 1 QA Q20 Progressions & Series
Question 20 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.
For some positive and distinct real numbers \(x, y\) and \(z\), if \(\frac{1}{\sqrt{y}+\sqrt{z}}\) is the arithmetic mean of \(\frac{1}{\sqrt{x}+\sqrt{z}}\) and \(\frac{1}{\sqrt{x}+\sqrt{y}}\), then the relationship which will always hold true, is
- A\(\sqrt{x}, \sqrt{y}\) and \(\sqrt{z}\) are in arithmetic progression
- B\(\sqrt{x}, \sqrt{z}\) and \(\sqrt{y}\) are in arithmetic progression
- C\(y, x\) and \(z\) are in arithmetic progression
- D\(x, y\) and \(z\) are in arithmetic progression
Answer: (C) y, x and z are in arithmetic progression Progressions & Series MCQ
Solution
- Let \(a=\sqrt x, b=\sqrt y, c=\sqrt z\): \(\frac{2}{b+c} = \frac{1}{a+c} + \frac{1}{a+b} = \frac{2a+b+c}{(a+c)(a+b)}\).
- Cross-multiply: \(2(a^2+ab+ac+bc) = 2ab+2ac+b^2+2bc+c^2\).
- This simplifies to \(2a^2 = b^2 + c^2\), i.e. \(2x = y + z\).
- So \(y, x, z\) are in AP.