Q30 / 58 · KVPY 2016 (SA) · Mathematics
Let $S_{1}$ be the sum of areas of the squares whose sides are parallel to coordinate axes. Let $\mathrm{S}_{2}$ be the sum of areas of the slanted squares as shown in the figure. Then $\mathrm{S}_{1} / \mathrm{S}_{2}$ is
- Correct answerA2
- B$\sqrt{2}$
- C1
- D$\frac{1}{\sqrt{2}}$
Solution
$\mathrm{S}_{1}=\mathrm{a}^{2}+\frac{\mathrm{a}^{2}}{4}+\frac{\mathrm{a}^{2}}{16}+\ldots \ldots \infty=\frac{\mathrm{a}^{2}}{1-\frac{1}{4}}=\frac{4 \mathrm{a}^{2}}{3}$
$\mathrm{~S}_{2}=\frac{\mathrm{a}^{2}}{2}+\frac{\mathrm{a}^{2}}{8}+\frac{\mathrm{a}^{2}}{32}+\ldots \ldots \infty=\frac{\frac{\mathrm{a}^{2}}{2}}{1-\frac{1}{4}}=\frac{4 \mathrm{a}^{2}}{6}$
$\therefore \frac{\mathrm{S}_{1}}{\mathrm{~S}_{2}}=2$
Hence, the correct answer is option: 'A'.