CAT 2025 Question Paper | Quant Slot 3

CAT Questions

CAT 2025 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 4 : If \( \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) = 25 \) and \( x > 0 \), then the value of \( \left( x ^ { 7 } + \frac { 1 } { x ^ { 7 } } \right) \) is

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Video Explanation

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Explanatory Answer

Given:
\[ x^2+\frac{1}{x^2}=25,\quad x>0 \]

Find \(x+\frac{1}{x}\):
\[ \left(x+\frac{1}{x}\right)^2 =x^2+\frac{1}{x^2}+2 =25+2=27 \]
Since \(x>0\):
\[ x+\frac{1}{x}=3\sqrt3 \]

Find \(x^3+\frac{1}{x^3}\):
\[ x^3+\frac{1}{x^3} =\left(x+\frac{1}{x}\right)^3-3\left(x+\frac{1}{x}\right) \]
\[ =(3\sqrt3)^3-3(3\sqrt3) =81\sqrt3-9\sqrt3 =72\sqrt3 \]

Find \(x^4+\frac{1}{x^4}\):
\[ \left(x^2+\frac{1}{x^2}\right)^2-2 =25^2-2=623 \]

Multiply \(x^4+\frac{1}{x^4}\) and \(x^3+\frac{1}{x^3}\):
\[ 623(72\sqrt3) =x^7+\frac{1}{x^7}+x+\frac{1}{x} \]
\[ x^7+\frac{1}{x^7} =623(72\sqrt3)-3\sqrt3 =44853\sqrt3 \]

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The question is " If \( \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) = 25 \) and \( x > 0 \), then the value of \( \left( x ^ { 7 } + \frac { 1 } { x ^ { 7 } } \right) \) is "

Hence, the answer is '\( 44853 \sqrt { 3 } \)'

Choice 3 is the correct answer.

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