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This question involves the computation of the Laplacian of \( \frac{1}{|\mathbf{r}|} \), where \( \mathbf{r} = x\hat{i} + y\hat{j} + 3\hat{k} \) and \( |\mathbf{r}| = \sqrt{x^2 + y^2 + 3^2} = \sqrt{x^2 + y^2 + 9} \).

### Step-by-step solution:

1. **Expression for the Laplacian**:  
   The Laplacian \( \nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) \) in three dimensions is given by:
   \[
   \nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) = \frac{\partial^2}{\partial x^2} \left(\frac{1}{|\mathbf{r}|}\right) + \frac{\partial^2}{\partial y^2} \left(\frac{1}{|\mathbf{r}|}\right) + \frac{\partial^2}{\partial z^2} \left(\frac{1}{|\mathbf{r}|}\right).
   \]

2. **General formula for \( \nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) \)**:
   The Laplacian of \( \frac{1}{|\mathbf{r}|} \) for \( \mathbf{r} \neq 0 \) is a well-known result:
   \[
   \nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) = 0, \quad \text{for } |\mathbf{r}| \neq 0.
   \]

   Since the question does not involve a delta function at the origin (distribution theory), we conclude:
   \[
   \nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) = 0.
   \]

### Final Answer:
\[
\boxed{(d) \, 0}
\]

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Would you like additional details or have further questions? Here are some related problems you might find interesting:

1. What is the divergence of the vector field \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^3} \)?  
2. Compute \( \nabla^2 |\mathbf{r}| \) for \( \mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k} \).  
3. Evaluate \( \nabla \cdot \mathbf{F} \) for \( \mathbf{F} = \nabla \left(\frac{1}{|\mathbf{r}|}\right) \).  
4. Determine \( \nabla^2 \frac{1}{|\mathbf{r}|^n} \) for \( n > 1 \).  
5. Verify if \( \mathbf{F} = -\nabla \left(\frac{1}{|\mathbf{r}|}\right) \) is irrotational.

**Tip**: The Laplacian often simplifies greatly for symmetric functions like \( \frac{1}{|\mathbf{r}|} \). Learning its standard results can save significant time!

If \( \mathbf{r} = x\hat{i} + y\hat{j} + 3\hat{k} \), then \( \nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) \) is?

Explore the solution to find \( \nabla^2 \left(\frac{1}{|\mathbf{r}|}\right) \), where \( \mathbf{r} = x\hat{i} + y\hat{j} + 3\hat{k} \). This problem highlights vector calculus concepts, including the Laplacian operator and symmetry properties of functions, suitable for undergraduate physics or mathematics students.

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