Question

Match the LIST-I with LIST-II - Let $\vec{F}(x,y,z) = x\hat{i} + y\hat{j} + z\hat{k}$ and $\vec{G}(x,y,z) = xz\hat{i} + xy\hat{j} + yz\hat{k}$

Choose the correct answer from the options given below:

• A. A-IV, B-III, C-I, D-II
• B. A-IV, B-III, C-II, D-I
• C. A-III, B-IV, C-I, D-II
• D. A-II, B-IV, C-III, D-I

Hint:

For position vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$: $\text{div } \vec{r} = 3$ and $\text{curl } \vec{r} = \vec{0}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:

• Given \( \vec{F} = x\hat{i} + y\hat{j} + z\hat{k} \) and \( \vec{G} = xz\hat{i} + xy\hat{j} + yz\hat{k} \), match each quantity in List-I with its value in List-II.

Step 2: Detailed Explanation:

• A. \( \text{div}\,\vec{F} = 1 + 1 + 1 = 3 \), so A matches IV.

• B. \( \text{curl}\,\vec{F} = \vec{0} \) and \( \text{curl}\,\vec{G} = (z - 0)\hat{i} + (x - 0)\hat{j} + (y - 0)\hat{k} = z\hat{i} + x\hat{j} + y\hat{k} \), so \( \text{curl}(\vec{F} + \vec{G}) = z\hat{i} + x\hat{j} + y\hat{k} \), which matches III.

• C. \( \text{div}\,\vec{G} = z + x + y \), so C matches II.

• D. \( \text{div}(\vec{F}\times\vec{G}) = \vec{G}\cdot\text{curl}\,\vec{F} - \vec{F}\cdot\text{curl}\,\vec{G} = 0 - (xz + yx + zy) = -(xy + yz + zx) \), so D matches I.

• The matching is A-IV, B-III, C-II, D-I.

Step 3: Final Answer:
The correct option is (B) A-IV, B-III, C-II, D-I.