UPSC NDA I 2021 Mathematics Solved Paper

Concept:Two vectors are perpendicular if their dot product is zero.Explanation:Given $\vec{c} = \vec{a} + \vec{b}$ and $|\vec{a}| = |\vec{b}| \neq 0$.Statement 1: $\vec{c}$ is perpendicular to $(\vec{a} - \vec{b})$.Compute $\vec{c} \cdot (\vec{a} - \vec{b}) = (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b})$.Using dot product properties, this simplifies to $|\vec{a}|^2 - |\vec{b}|^2$.Since $|\vec{a}| = |\vec{b}|$, the result is $0$.Hence $\vec{c} \cdot (\vec{a} - \vec{b}) = 0$, so statement 1 is true.Statement 2: $\vec{c}$ is perpendicular to $\vec{a} \times \vec{b}$.Compute $\vec{c} \cdot (\vec{a} \times \vec{b}) = (\vec{a} + \vec{b}) \cdot (\vec{a} \times \vec{b})$.This expands to $\vec{a} \cdot (\vec{a} \times \vec{b}) + \vec{b} \cdot (\vec{a} \times \vec{b})$.In a scalar triple product, if two vectors are equal, the product is zero.Both terms are zero because $\vec{a}$ repeats in the first and $\vec{b}$ in the second.Thus $\vec{c} \cdot (\vec{a} \times \vec{b}) = 0$, so statement 2 is also true.Answer:Both statements 1 and 2 are correct.