Following are some questions posed by the teacher in the mathematics classroom:

A. What is the area of the rectangle whose one side is 5 cm and perimeter is 30 cm?

B. Find a set of numbers whose median is 4.

C. List all prime numbers between 0-8.

D. Tell me anything mathematical information you know about rectangles.

The correct option is A & C are closed ended and B & D are open ended questions.

To understand why this is the correct classification, let us analyze the nature of closed-ended and open-ended questions in a mathematics classroom:

1. Closed-ended questions: These are questions that have a fixed, unique answer or a limited, specific set of correct answers. They usually do not leave scope for creative or multiple valid responses.

2. Open-ended questions: These are questions that can have multiple correct answers, different approaches to solve, or open-ended interpretations. They encourage creative thinking and conceptual exploration.

Now, let us analyze each of the given questions step-by-step:

Question A: "What is the area of the rectangle whose one side is 5 cm and perimeter is 30 cm?"
Let the length of one side be $l=5\text{ cm}$ and the perimeter be $P=30\text{ cm}$.
Using the perimeter formula $P=2(l+b)$:

$30=2(5+b)$

$15=5+b\Rightarrow b=10\text{ cm}$

Therefore, Area = $l\times b=5\times 10=50{\text{ cm}}^2$.
Since there is only one definite numerical answer ($50{\text{ cm}}^2$), Question A is a closed-ended question.

Question B: "Find a set of numbers whose median is 4."
There are infinitely many sets of numbers that can have a median of 4. For example:

Set 1: {3, 4, 5}

Set 2: {1, 2, 4, 6, 8}

Set 3: {4, 4, 4}

Since there are multiple correct responses and possibilities, Question B is an open-ended question.

Question C: "List all prime numbers between 0-8."
The prime numbers between 0 and 8 are strictly 2, 3, 5, and 7.
Since this question requires a specific, fixed set of numbers with no other valid possibilities, Question C is a closed-ended question.

Question D: "Tell me anything mathematical information you know about rectangles."
Students can share a wide variety of mathematical facts, such as:

- Opposite sides are equal and parallel.

- All interior angles are $90°$.

- Diagonals are equal in length and bisect each other.

- Area = $\text{length}\times \text{width}$.

Because students can provide diverse, open-ended mathematical facts, Question D is an open-ended question.

Conclusion:
- A and C are closed-ended questions.
- B and D are open-ended questions.