UPSC NDA 2 2021 Mathematics Solved Paper

Concept:The angle $\theta$ between two vectors $\vec{a}$ and $\vec{b}$ is given by $\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}$.Explanation:We are given:$\vec{a} + 3\vec{b} = 3\hat{i} - \hat{j}$ ...(1)$2\vec{a} + \vec{b} = \hat{i} - 2\hat{j}$ ...(2)Multiply equation (1) by 2: $2\vec{a} + 6\vec{b} = 6\hat{i} - 2\hat{j}$.Subtract equation (2) from this result:$(2\vec{a} + 6\vec{b}) - (2\vec{a} + \vec{b}) = (6\hat{i} - 2\hat{j}) - (\hat{i} - 2\hat{j})$This simplifies to $5\vec{b} = 5\hat{i}$, so $\vec{b} = \hat{i}$.Substitute $\vec{b}$ into equation (1):$\vec{a} + 3(\hat{i}) = 3\hat{i} - \hat{j}$ → $\vec{a} = (3\hat{i} - \hat{j}) - 3\hat{i} = -\hat{j}$.Thus $\vec{a} = -\hat{j}$ and $\vec{b} = \hat{i}$.Compute the dot product: $\vec{a}\cdot\vec{b} = (-\hat{j})\cdot(\hat{i}) = 0$.The magnitudes are $|\vec{a}| = 1$ and $|\vec{b}| = 1$.Using the formula: $\cos\theta = \frac{0}{1\cdot 1} = 0$, so $\theta = \cos^{-1}0 = \frac{\pi}{2}$.Answer:The angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{2}$, which corresponds to option D.