UPSC NDA 2 2019 Mathematics Solved Paper

Concept:The magnitude of the cross product $|\vec{a} \times \vec{b}|$ gives the area of the parallelogram formed by vectors $\vec{a}$ and $\vec{b}$, not the area of a triangle.If the cross product of two non-zero vectors is zero, the vectors are parallel (collinear), so one is a scalar multiple of the other.Explanation:For statement 1: The area of a parallelogram with sides $\vec{a}$ and $\vec{b}$ is $|\vec{a} \times \vec{b}|$. A triangle with the same two sides has half that area, i.e., $\frac{1}{2} |\vec{a} \times \vec{b}|$. Therefore, the statement "the magnitude of $\vec{a} \times \vec{b}$ is same as the area of a triangle" is false.For statement 2: Given $\vec{a} \neq \vec{0}$, $\vec{b} \neq \vec{0}$, and $\vec{a} \times \vec{b} = \vec{0}$. This implies that $\vec{a}$ and $\vec{b}$ are parallel (or antiparallel). Hence there exists a scalar $\lambda$ such that $\vec{a} = \lambda \vec{b}$. So statement 2 is true.Thus only statement 2 is correct.Answer:B. 2 only