Question

Match List-I with List-II. List-I:
A. \(\nabla\times\vec{r}\),
B. \(\operatorname{Div}(2x^2z\hat{i}-xy^2z\hat{j}+3yz^2\hat{k})\) at point \((1,1,1)\),
C. Curl of a vector is a,
D. Divergence of a vector is
A. List-II: I. Vector quantity, II. \(0\), III. Scalar quantity, IV. \(8\).

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

Hint:

Curl gives a vector and divergence gives a scalar.

Answer 1

The Correct Option is D

Concept:
Curl of a vector field gives a vector quantity, while divergence gives a scalar quantity.

Step 1: Evaluate \(\nabla\times\vec{r}\).
Position vector: \[ \vec{r}=x\hat{i}+y\hat{j}+z\hat{k} \] \[ \nabla\times\vec{r}=0 \] So: \[ A\rightarrow II \]

Step 2: Evaluate divergence. \[ \vec{F}=2x^2z\hat{i}-xy^2z\hat{j}+3yz^2\hat{k} \] \[ \nabla\cdot\vec{F} = \frac{\partial}{\partial x}(2x^2z) + \frac{\partial}{\partial y}(-xy^2z) + \frac{\partial}{\partial z}(3yz^2) \] \[ =4xz-2xyz+6yz \] At \((1,1,1)\): \[ =4-2+6=8 \] So: \[ B\rightarrow IV \]

Step 3: Curl and divergence nature. \[ \text{Curl of vector}=\text{Vector quantity} \] \[ C\rightarrow I \] \[ \text{Divergence of vector}=\text{Scalar quantity} \] \[ D\rightarrow III \] \[ \therefore \text{Correct Answer is (D)} \]