Kerala KEAM 5 June 2024 Solved Paper

We are given that: $$\sin^{-1} x + \cos^{-1} y = 0.$$This implies that: $$\cos^{-1} y = -\sin^{-1} x.$$Now, recall that: $$\cos^{-1} y = \sin^{-1} (\sqrt{1 - y^2}),$$and since $\cos^{-1} y$ and $\sin^{-1} x$ are inverses of each other, we conclude that $x = \sqrt{1 - y^2}$.Step 1: Now, we compute $x^2 + y^2$: $$x^2 = 1 - y^2 \quad \Rightarrow \quad x^2 + y^2 = (1 - y^2) + y^2 = 1.$$Thus, the value of $x^2 + y^2$ is $1$.Therefore, the correct answer is option (C).Quick Tip: When solving equations involving inverse trigonometric functions, use the relationships between sine and cosine functions, such as $\sin^{-1} x + \cos^{-1} y = 0$, to express one variable in terms of the other. This will help simplify the expression.