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Question

# Assertion (A) If the volumes of two spheres are in the ratio 27 : 8, then their surface areas are in the ratio 3 : 2. Reason (R) Volume of a sphere = $\frac{4}{3}\mathrm{\pi }{R}^{3}$ Surface area of a sphere = $4\mathrm{\pi }{R}^{2}$ (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A). (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A). (c) Assertion (A) is true and Reason (R) is false. (d) Assertion (A) is false and Reason (R) is true.

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Solution

## (d) Assertion (A) is false and Reason (R) is true. Assertion (A): Let R and r be the radii of the two spheres. Then, ratio of their volumes$=\frac{\frac{4}{3}\mathrm{\pi }{R}^{3}}{\frac{4}{3}{\mathrm{\pi r}}^{3}}$ Therefore, $\frac{\frac{4}{3}\mathrm{\pi }{R}^{3}}{\frac{4}{3}{\mathrm{\pi r}}^{3}}=\frac{27}{8}\phantom{\rule{0ex}{0ex}}⇒\frac{{R}^{3}}{{r}^{3}}=\frac{27}{8}\phantom{\rule{0ex}{0ex}}⇒{\left(\frac{R}{r}\right)}^{3}={\left(\frac{3}{2}\right)}^{3}\phantom{\rule{0ex}{0ex}}⇒\frac{R}{r}=\frac{3}{2}$ Hence, the ratio of their surface areas$=\frac{4\mathrm{\pi }{R}^{2}}{4{\mathrm{\pi r}}^{2}}$ $=\frac{{R}^{2}}{{r}^{2}}\phantom{\rule{0ex}{0ex}}={\left(\frac{R}{r}\right)}^{2}\phantom{\rule{0ex}{0ex}}={\left(\frac{3}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}=\frac{9}{4}\phantom{\rule{0ex}{0ex}}=9:4$ Hence, Assertion (A) is false. Reason (R): The given statement is true.

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