UPSC NDA I 2022 Mathematics Solved Paper

Concept:The dot product and cross product both satisfy the distributive law over vector addition, but the cross product does not satisfy the associative law.Explanation:Statement 1: Dot product is distributive over addition.For any vectors $\vec{a}$, $\vec{b}$, $\vec{c}$: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$.This can be verified by expanding components: Let $\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$, $\vec{b} = b_x\hat{i} + b_y\hat{j} + b_z\hat{k}$, $\vec{c} = c_x\hat{i} + c_y\hat{j} + c_z\hat{k}$.Then $\vec{a} \cdot (\vec{b} + \vec{c}) = a_x(b_x+c_x) + a_y(b_y+c_y) + a_z(b_z+c_z) = (a_x b_x + a_y b_y + a_z b_z) + (a_x c_x + a_y c_y + a_z c_z) = \vec{a}\cdot\vec{b} + \vec{a}\cdot\vec{c}$.Hence statement 1 is true.Statement 2: Cross product is distributive over addition.Similarly, $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$.Using component form, both sides yield the same result: $\hat{i}[a_y(b_z+c_z) - a_z(b_y+c_y)] - \hat{j}[a_x(b_z+c_z) - a_z(b_x+c_x)] + \hat{k}[a_x(b_y+c_y) - a_y(b_x+c_x)]$.Hence statement 2 is true.Statement 3: Cross product is not associative.For vectors $\vec{a}$ and $\vec{b}$ that are non‑zero and perpendicular: $(\vec{a} \times \vec{a}) \times \vec{b} = \vec{0} \times \vec{b} = \vec{0}$, but $\vec{a} \times (\vec{a} \times \vec{b})$ is a non‑zero vector perpendicular to $\vec{a}$. Thus $(\vec{a} \times \vec{a}) \times \vec{b} \neq \vec{a} \times (\vec{a} \times \vec{b})$.Therefore cross product is not associative; statement 3 is false.Answer:Options 1 and 2 only are correct. Correct answer is C (1 and 2 only).