CAT 2023 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like **Speed-time-distance**, **Mixture and Alligations**. This year, there was a surprise. The questions from **Geometry** were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 20 : 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

- \(\sqrt{x}, \sqrt{y}\) and \(\sqrt{z}\) are in arithmetic progression
- \(\sqrt{x}, \sqrt{z}\) and \(\sqrt{y}\) are in arithmetic progression
- \(y, x\) and \(z\) are in arithmetic progression
- \(x, y\) and \(z\) are in arithmetic progression

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Given: \( \frac { 1 } { \sqrt { y } + \sqrt { z } } = \frac { \frac { 1 } { \sqrt { x } + \sqrt { z } } + \frac { 1 } { \sqrt { x } + \sqrt { y } } } { 2 } \)

\( \frac { 2 } { \sqrt { y } + \sqrt { z } } = \frac { 2 \sqrt { x } + \sqrt { z } + \sqrt { y } } { ( \sqrt { x } + \sqrt { z } ) ( \sqrt { x } + \sqrt { y } ) } \)

\( \frac { 2 ( \sqrt { x } + \sqrt { 2 } ) ( \sqrt { x } + \sqrt { y } ) } { ( \sqrt { y } + \sqrt { 2 } ) } = 2 \sqrt { x } + \sqrt { 2 } + \sqrt { y } \)

\( \frac { 2 [ x + \sqrt { x } ( \sqrt { y } + \sqrt { z } ) + \sqrt { z } \sqrt { y } ] } { \sqrt { y } + \sqrt { z } } = 2 \sqrt { x } + \sqrt { z } + \sqrt { y } \)

\( \frac { 2 [ x + \sqrt { z } \sqrt { y } ] } { \sqrt { y } + \sqrt { z } } = \sqrt { z } + \sqrt { y } \)

\( 2 x + 2 \sqrt { z } \sqrt { y } = ( \sqrt { z } + \sqrt { y } ) ^ { 2 } \)

\( 2 x + 2 \sqrt { z } \sqrt { y } = z + y + 2 \sqrt { z } \sqrt { y } \)

\( 2 x = z + y \)

\( x = \frac { z + y } { 2 } \)

x is the Arithmetic Mean of z & y, therefore, z, x, y form an A.P.

*It goes without saying that y, x, z also forms an A.P.*

Choice C is the correct answer.

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