UPSC CDS 1 2021 Math Paper

Concept:Solve a system of two linear equations in $x$ and $y$ using elimination.Explanation:Step 1: Write the first equation $\frac{x}{a}+\frac{y}{b}=a+b$ in linear form.Multiply by $ab$: $bx+ay=a^2b+ab^2=ab(a+b)$ …(1)Step 2: Write the second equation $\frac{x}{a^2}+\frac{y}{b^2}=2$ in linear form.Multiply by $a^2b^2$: $b^2x+a^2y=2a^2b^2$ …(2)Step 3: Eliminate $y$ by multiplying (1) by $a$: $abx+a^2y=a^3b+a^2b^2$ …(3)Subtract (2) from (3): $(ab-b^2)x = a^3b - a^2b^2 = a^2b(a-b)$.Factor the left side: $b(a-b)x = a^2b(a-b)$.Assuming $a\neq b$ and $b\neq 0$, we get $x=a^2$.Step 4: Substitute $x=a^2$ into (1): $b(a^2)+ay = a^2b+ab^2$.Simplify: $a^2b+ay = a^2b+ab^2$ ⇒ $ay=ab^2$ ⇒ $y=b^2$ (assuming $a\neq0$).Step 5: Compute the required expression:$\frac{x}{a^2}-\frac{y}{b^2} = \frac{a^2}{a^2}-\frac{b^2}{b^2}=1-1=0$.Answer:$0$ (Option C).