CAT 2024 Question Paper | Quant Slot 1

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 4 :In the $X Y$-plane, the area, in sq. units, of the region defined by the inequalities $y \geq x+4$ and $-4 \leq x^2+y^2+4(x-y) \leq 0$ is

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Video Explanation

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Explanatory Answer

$y \geq x+4$ represents the portion above the line $y=x+4$ whose intercepts are $(-4,0)$ and $(0,4)$ $$\begin{array}{c} -4 \leq x^2+y^2+4(x-y) \leq 0 \\ -4 \leq x^2+y^2+4 x-4 y \leq 0 \\ -4 \leq x^2+4 x+y^2-4 y \leq 0 \\ 4 \leq x^2+4 x+4+y^2-4 y+4 \leq 8 \\ 4 \leq(x+2)^2+(y-2)^2 \leq 8 \end{array}$$ is the region between the two concentric circles with centre $(-2,2)$ and radii $2 \sqrt{ } 2$ and 2

Thus, the common region is shaded in the diagram Hence, the area of the shaded region $$\begin{array}{l} =\frac{1}{2}\left[\pi(2 \sqrt{2})^2-\pi(2)^2\right] \\ =\frac{1}{2}[8 \pi-4 \pi] \\ =\frac{1}{2}[4 \pi] \\ =2 \pi \end{array}$$

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The question is "In the $X Y$-plane, the area, in sq. units, of the region defined by the inequalities $y \geq x+4$ and $-4 \leq x^2+y^2+4(x-y) \leq 0$ is"

Hence, the answer is '$2 \pi$'

Choice B is the correct answer.

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