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 16 :The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm , and the sum of the lengths of all its edges is 144 cm . The volume, in cubic cm , of the sphere is

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Video Explanation

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Explanatory Answer

Surface Area of the closed rectangular box $=2 l b+2 l h+2 b h=846$
Sum of the length of all its edges $=4 l+4 b+4 h=144$ $$l+b+h=\frac{144}{4}=36 ---- (1)$$ Squaring equation (1) we get, $$\begin{array}{l} l^2+b^2+h^2+2 l b+2 l h+2 b h=36 \times 36=1296 \\ l^2+b^2+h^2=1296-846=450 \end{array}$$ Diameter of the sphere $=\sqrt{l^2+b^2+h^2}=15 \sqrt{2}=2 r$
Radius of the sphere $=r=\frac{15}{\sqrt{2}}$
Volume of the sphere $=\frac{4}{3} \pi r^3=\frac{4}{3} \pi \frac{15 \times 15 \times 15}{2 \sqrt{2}}=1125 \pi \sqrt{2}$

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The question is "The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm , and the sum of the lengths of all its edges is 144 cm . The volume, in cubic cm , of the sphere is"

Hence, the answer is '$1125 \pi \sqrt{2}$'

Choice B is the correct answer.

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