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 18 Three circles of equal radii touch (but not cross) each other externally. Two other circles, $X$ and $Y$, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of $X$ is more than that of $Y$, the ratio of the radii of $X$ and $Y$ is

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Video Explanation

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Explanatory Answer

$$\begin{array}{l} A B=B C=A C=a=2 r . \\ O B=2 r / \sqrt{ } 3 \end{array}$$ $x$ and $y$ are radii of bigger and smaller circle. $$\begin{array}{l} r+y=2 r / \sqrt{ } 3 \\ 2 r+y=x \end{array}$$ $\begin{array}{l}2 r+2 r / \sqrt{ } 3-r=x ; \\ r+(2 r / \sqrt{ } 3)=x ; \\ (2 r / \sqrt{ } 3)-r=y ; \\ x / y=r+(2 r / \sqrt{ } 3) /(2 r / \sqrt{ } 3)-r \\ =\{(2 / \sqrt{ } 3)+1\} /\{(2 / \sqrt{ } 3)-1\} \\ =(2+\sqrt{ } 3) /(2-\sqrt{ } 3) \\ =\{(2+\sqrt{ } 3)(2+\sqrt{ } 3)\} /\{(2-\sqrt{ } 3)(2+\sqrt{ } 3)\} \\ =(4+3+4 \sqrt{ } 3) /(4-3) \\ =(7+4 \sqrt{ } 3): 1\end{array}$

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The question is "Three circles of equal radii touch (but not cross) each other externally. Two other circles, $X$ and $Y$, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of $X$ is more than that of $Y$, the ratio of the radii of $X$ and $Y$ is "

Hence, the answer is '$7+4 \sqrt{3}: 1$'

Choice C is the correct answer.

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