CAT 2023 Question Paper | Quant Slot 3

CAT 2023 Quant was dominated by Arithmetic followed by Algebra. 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 4 : If \(x\) is a positive real number such that \(x^8+\left(\frac{1}{x}\right)^8=47\), then the value of \(x^9+\left(\frac{1}{x}\right)^9\) is

Video Explanation

Explanatory Answer

\( x ^ { 8 } + \frac { 1 } { x ^ { 8 } } = 47 \)
\( \left( x ^ { 4 } + \frac { 1 } { x ^ { 4 } } \right) ^ { 2 } = x ^ { 8 } + \frac { 1 } { x ^ { 8 } } + 2 = 47 + 2 = 49 \)
\( \left( x ^ { 4 } + \frac { 1 } { x ^ { 4 } } \right) = 7 \)
\( \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) ^ { 2 } = x ^ { 4 } + \frac { 1 } { x ^ { 4 } } + 2 = 7 + 2 = 9 \)
\( \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) = 3 \)
\( \left( x + \frac { 1 } { x } \right) ^ { 2 } = x ^ { 2 } + \frac { 1 } { x ^ { 2 } } + 2 = 3 + 2 = 5 \)
\( \left( x + \frac { 1 } { x } \right) = \sqrt { 5 } \)
\( \left( x + \frac { 1 } { x } \right) ^ { 3 } = x ^ { 3 } + \frac { 1 } { x ^ { 3 } } + 3 \times x \times \frac { 1 } { x } \left( x + \frac { 1 } { x } \right) \)
\( ( \sqrt { 5 } ) ^ { 3 } = x ^ { 3 } + \frac { 1 } { x ^ { 3 } } + 3 \sqrt { 5 } \)
\( x ^ { 3 } + \frac { 1 } { x ^ { 3 } } = 5 \sqrt { 5 } - 3 \sqrt { 5 } = 2 \sqrt { 5 } \)
\( \left( x ^ { 3 } + \frac { 1 } { x ^ { 3 } } \right) ^ { 3 } = x ^ { 9 } + \frac { 1 } { x ^ { 9 } } + 3 \times x ^ { 3 } \times \frac { 1 } { x ^ { 3 } } \left( x ^ { 3 } + \frac { 1 } { x ^ { 3 } } \right) \)
\( ( 2 \sqrt { 5 } ) ^ { 3 } = x ^ { 9 } + \frac { 1 } { x ^ { 9 } } + 3 ( 2 \sqrt { 5 } ) \)
\( x ^ { 9 } + \frac { 1 } { x ^ { 9 } } = 40 \sqrt { 5 } - 6 \sqrt { 5 } = 34 \sqrt { 5 } \)

The question is " If \(x\) is a positive real number such that \(x^8+\left(\frac{1}{x}\right)^8=47\), then the value of \(x^9+\left(\frac{1}{x}\right)^9\) is "

Hence, the answer is '\(34 \sqrt{5}\)'

Choice D is the correct answer.