CAT 2024 Question Paper | Quant Slot 2

CAT Questions

CAT 2024 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 21 If $x$ and $y$ are real numbers such that $4 x^2+4 y^2-4 x y-6 y+3=0$, then the value of $(4 x+5 y)$ is

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Video Explanation

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Explanatory Answer

$$4 x^2+4 y^2-4 x y-6 y+3=\left(4 x^2+y^2-4 x y\right)+\left(3 y^2-6 y+3\right)$$ We know that $4 x^2+y^2-4 x y=(2 x-y)^2$ $$3 y^2-6 y+3=3\left(y^2-2 y+1\right)=3(y-1)^2$$ Therefore, $4 x^2+4 y^2-4 x y-6 y+3=(2 x-y)^2+3(y-1)^2=0$.
The equation is satisfied only when $y=1$ and $x=1 / 2$.
The value of $(4 x+5 y)=4(1 / 2)+5(1)=2+5=7$.

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The question is "If $x$ and $y$ are real numbers such that $4 x^2+4 y^2-4 x y-6 y+3=0$, then the value of $(4 x+5 y)$ is"

Hence, the answer is '7'

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