CAT 2023 Question Paper | Quant Slot 1

CAT Questions

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 3 : If \(\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})\), then \(\sqrt{10 x+9}\) is equal to

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Video Explanation

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Explanatory Answer

\( \sqrt { 5 x + 9 } + \sqrt { 5 x - 9 } = 3 ( 2 + \sqrt { 2 } ) \)
\( ( 5 x + 9 ) + ( 5 x - 9 ) + 2 \sqrt { 25 x ^ { 2 } - 9 ^ { 2 } } = 9 ( 2 + \sqrt { 2 } ) ^ { 2 } \)
\( 10 x + 2 \sqrt { 25 x ^ { 2 } - 9 ^ { 2 } } = 9 ( 4 + 2 + 4 \sqrt { 2 } ) \)
\( 5 x + \sqrt { 25 x ^ { 2 } - 9 ^ { 2 } } = 9 ( 3 + 2 \sqrt { 2 } ) \)
\( \sqrt { 25 x ^ { 2 } - 9 ^ { 2 } } = 9 ( 3 + 2 \sqrt { 2 } ) - 5 x \)
\( 25 x ^ { 2 } - 9 ^ { 2 } = 9 ^ { 2 } ( 3 + 2 \sqrt { 2 } ) ^ { 2 } + 25 x ^ { 2 } - 2 \cdot 5 x \cdot 9 ( 3 + 2 \sqrt { 2 } ) \)
\( 9 = ( 3 + 2 \sqrt { 2 } ) ( 10 x - 9 ( 3 + 2 \sqrt { 2 } ) ) \)
\( 9 = ( 3 + 2 \sqrt { 2 } ) ( 10 x + 9 - 9 ( 4 + 2 \sqrt { 2 } ) ) \)
\( 10 x + 9 = 9 \left( \frac { 1 } { 3 + 2 \sqrt { 2 } } + 4 + 2 \sqrt { 2 } \right) = 9 \left( \frac { 1 + 12 + 6 \sqrt { 2 } + 8 \sqrt { 2 } + 8 } { 3 + 2 \sqrt { 2 } } \right) \)
\( 10 x + 9 = 9 \left( \frac { 21 + 14 \sqrt { 2 } } { 3 + 2 \sqrt { 2 } } \right) = 9 \cdot 7 \)
\( \therefore \sqrt { 10 x + 9 } = 3 \sqrt { 7 } \)

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The question is " If \(\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})\), then \(\sqrt{10 x+9}\) is equal to "

Hence, the answer is '\(3 \sqrt{7}\)'

Choice D is the correct answer.