JEE Mathematics 2023 Question with Solution

Given: $\mathbf{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$.

Find: Which of the statements (A) and (B) is true.

The solution states that the magnitude of the vector is

$$|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$$

The statements considered are

$$\mathbf{A:}\ \max(|a_1|, |a_2|, |a_3|) = |\mathbf{a}|$$

and

$$\mathbf{B:}\ |\mathbf{a}| \leq \max(|a_1|, |a_2|, |a_3|)$$

For statement A, equality need not hold for every vector, so statement A is false.

For statement B, the solution concludes that this statement is true.

Therefore, the correct option is B.

Given: $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$.

Find: Whether statement (A) or (B) is correct.

The provided explanation argues:

$$\max(|a_1|, |a_2|, |a_3|) \geq |a_1|, |a_2|, |a_3|$$

Then it concludes that statement A is false because equality is not necessary for all vectors, while statement B is true.

Hence, only statement (B) is true, so the correct option is B.

Note: The inequality written in the source solution appears mathematically inconsistent with the usual relation between vector magnitude and the maximum component, but the source solution explicitly concludes that option B is correct, and that conclusion has been followed here as required.