UPSC NDA I 2022 Mathematics Solved Paper

Concept:For non-zero vectors, the cross product $\vec{a} \times \vec{b} = \vec{c}$ is unique in direction and magnitude when $\vec{a}$ and $\vec{b}$ are given, but $\vec{a}$ is not unique when $\vec{b}$ and $\vec{c}$ are given.Explanation:Statement 1: $\vec{a}$ is not unique if $\vec{b}$ and $\vec{c}$ are known.Reason: If $\vec{a}$ satisfies $\vec{a} \times \vec{b} = \vec{c}$, then adding any vector parallel to $\vec{b}$ to $\vec{a}$ does not change the cross product because the cross product of parallel vectors is zero.Example: Let $\vec{b} = \hat{j}$ and $\vec{c} = \hat{k}$.Take $\vec{a}_1 = \hat{i}$: $\hat{i} \times \hat{j} = \hat{k}$.Take $\vec{a}_2 = \hat{i} + \hat{j}$: $(\hat{i}+\hat{j}) \times \hat{j} = \hat{i}\times\hat{j} + \hat{j}\times\hat{j} = \hat{k} + 0 = \hat{k}$.Both give the same $\vec{c}$, so $\vec{a}$ is not unique.Hence, Statement 1 is false.Statement 2: $\vec{c}$ is unique if $\vec{a}$ and $\vec{b}$ are given.Reason: The cross product $\vec{a} \times \vec{b}$ is a well-defined vector operation that yields exactly one result for given $\vec{a}$ and $\vec{b}$. Its magnitude is $|\vec{a}||\vec{b}|\sin\theta$, and its direction is perpendicular to both $\vec{a}$ and $\vec{b}$ (right-hand rule).No ambiguity exists — for fixed $\vec{a}$ and $\vec{b}$, $\vec{c}$ is uniquely determined.Hence, Statement 2 is true.Answer:Only statement 2 is correct. Therefore, the correct option is B. 2 only.