To create a test case generator for the given problem "Twice of oneself", we need to generate multiple queries, each with a positive integer \(N\) within the specified constraints. 

Here's a Python function to generate such test cases:

```python
import random

def generate_test_case():
    Q = random.randint(1, 10**5)  # Number of queries
    
    queries = [random.randint(2, 10**5) for _ in range(Q)]  # List of N values for each query
    
    return Q, queries

# Function to print out the test case in the required format
def print_test_case():
    Q, queries = generate_test_case()
    print(Q)
    for query in queries:
        print(query)

print_test_case()
```

### Explanation of the Generator:

1. **Number of Queries (`Q`)**: 
   - Randomly select an integer between 1 and \(10^5\).

2. **List of Queries (`queries`)**:
   - For each query, randomly select an integer \(N\) between 2 and \(10^5\).

This generator will create a test case that meets the input format and constraints specified in the problem statement. 

### Example Output

Running the `print_test_case()` function might produce an output like this:

```
3
43
9
24
```

This output represents three test queries with \(N\) values of 43, 9, and 24, respectively. The number of queries \(Q\) is 3. 

### Note:
- The random values ensure a wide range of test cases, helping to thoroughly test the solution's efficiency and correctness.
- To ensure reproducibility, you can set a seed for the random number generator using `random.seed(seed_value)`.

Feel free to adjust the constraints or add additional logic if specific edge cases need to be tested.