UPSC CDS 1 2021 Math Paper

Given: 1. If $x$ is directly proportional to $z$ and $y$ is directly proportional to $z$, then $(x^{2}-y^{2})$ is directly proportional to $z^{2}$. 2. If $x$ is inversely proportional to $z$ and $y$ is inversely proportional to $z$, then ( $x y$ ) is inversely proportional to $z^{2}$ Calculation Case I. According to the question $x \propto z$ and $y \propto z$ $\Rightarrow x=z k$ and $y=r z$ where $k$ and $r$ are constants $\Rightarrow x^{2}=z^{2} k^{2}$ and $y^{2}=r^{2} z^{2}$ $\Rightarrow x^{2}-y^{2}=z^{2}(k^{2}-r^{2})$ $\Rightarrow x^{2}-y^{2} \propto z^{2}$ Hence, Case I is correct. Case II. $x \propto \frac{1}{z}$ and $y \propto \frac{1}{z}$ $\Rightarrow x z=k$ and $y z=r$ where $k$ and $r$ are constants $\Rightarrow x z \cdot y z=k r$ $\Rightarrow x y z^{2}=k r$ $\Rightarrow x y=\frac{k r}{z^{2}}$ $\Rightarrow x y \propto \frac{1}{z^{2}}$ Hence, Case II is correct.