UPSC NDA I 2021 Mathematics Solved Paper

Concept:Unit vectors have magnitude 1. Their cross product, dot product, and sum/difference magnitudes all depend on the angle between them.Explanation:Statement 1: For two unit vectors $\vec{a}$ and $\vec{b}$, $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta = \sin\theta$.Since $\sin\theta$ ranges from $-1$ to $1$, the cross product magnitude is not always $1$.Thus, statement 1 is false.Statement 2: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = \cos\theta$.$\cos\theta$ ranges from $-1$ to $1$, so the dot product is not always unity.Thus, statement 2 is false.Statement 3: Choose $\vec{a} = \hat{i}$ and $\vec{b} = \hat{j}$. Then $|\vec{a}+\vec{b}| = \sqrt{2}$ and $|\vec{a}-\vec{b}| = \sqrt{2}$.These magnitudes are equal, so the sum is not always greater than the difference.Thus, statement 3 is false.Answer:All three statements 1, 2, and 3 are not correct. Hence, the correct option is D (1, 2 and 3).