UPSC CDS 2 2023 Math Paper

Concept:Proportionality relationships can be manipulated using algebraic operations like squaring, composing, and dividing to deduce new proportionalities.Explanation:For statement 1: Given $(a + b) \propto (a - b)$.Let $(a + b) = k(a - b)$ where $k$ is a constant.Square both sides: $(a+b)^2 = k^2 (a-b)^2$.This gives $\frac{a^2 + b^2 + 2ab}{a^2 + b^2 - 2ab} = k^2$.Apply componendo and dividendo: $\frac{2(a^2 + b^2)}{4ab} = \frac{k^2+1}{k^2-1}$.Simplify: $a^2 + b^2 = \frac{k^2+1}{k^2-1} \cdot 2ab$.Since $\frac{k^2+1}{k^2-1} \cdot 2$ is constant, $a^2 + b^2 \propto ab$. So statement 1 is correct.For statement 2: Given $a \propto b$.Let $a = kb$ where $k$ is a constant. Then $\frac{a}{b}=k$.Using componendo and dividendo: $\frac{a+b}{a-b} = \frac{k+1}{k-1}$.Square both sides: $\frac{a^2+b^2+2ab}{a^2+b^2-2ab} = \frac{(k+1)^2}{(k-1)^2}$.Again apply componendo and dividendo: $\frac{2(a^2+b^2)}{4ab} = \frac{(k+1)^2+(k-1)^2}{(k+1)^2-(k-1)^2}$.Simplify: $a^2+b^2 = ab \cdot \frac{(k+1)^2+(k-1)^2}{(k+1)^2-(k-1)^2}$.The fraction is constant, so $a^2+b^2 \propto ab$, not $a^2-b^2$.Thus statement 2 is incorrect.Answer:Only statement 1 is correct. So the correct option is A. 1 only.