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 2 : If \(x\) and \(y\) are real numbers such that \(x^2+(x-2 y-1)^2=-4 y(x+y)\), then the value \(x-2 y\) is

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Video Explanation

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Explanatory Answer

\( x ^ { 2 } + ( x - 2 y - 1 ) ^ { 2 } = - 4 y ( x + y ) \)
\( x ^ { 2 } + ( 2 y ) ^ { 2 } + 2 \cdot x \cdot 2 y + ( x - 2 y - 1 ) ^ { 2 } = 0 \)
\( ( x + 2 y ) ^ { 2 } + ( x - 2 y - 1 ) ^ { 2 } = 0 \)
Since \( x \& y \) are real numbers, \( ( x + 2 y ) \& ( x - 2 y - 1 ) \) are both real.
A square of a real number is always non-negative.
For this reason, for the equation: \( ( x + 2 y ) ^ { 2 } + ( x - 2 y - 1 ) ^ { 2 } = 0 \) to be true, both \( ( x + 2 y ) \& ( x - 2 y - 1 ) \) must be equal to 0.
\( x - 2 y - 1 = 0 \)
\( x - 2 y = 1 \)
We solve the linear equations \( ( x + 2 y = 0 ) \& ( x - 2 y - 1 = 0 ) \) to get the exact values of x & y, but that is not required to solve the question at hand.

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The question is " If \(x\) and \(y\) are real numbers such that \(x^2+(x-2 y-1)^2=-4 y(x+y)\), then the value \(x-2 y\) is "

Hence, the answer is '1'

Choice B is the correct answer.