A radioactive element `A` decays by the sequence and with half-lives given below:
`A underset(alpha)overset(30 "min"rarr B underset(2 beta)overset(2 "days")rarr C`
Which of the following statement about this system are correct?

Options

A. The mass number of `B` is greater then `A`.
B. After two hours, less than 10% of the initial `A` is left.
C. Maximum amount of `B` present at any time is less than 50% of the initial amount of `A`.
D. The atomic numbers of `A` and `C` are the same.

Explanation

To solve the problem regarding the radioactive decay of element A, we will analyze the decay sequence and the statements provided.

### Step-by-Step Solution:

1. **Understanding the Decay Sequence**:
- Element A decays to element B via alpha decay.
- Element B further decays to element C via two beta decays.
- The half-life for the decay of A to B is 30 minutes, and the half-life for the decay of B to C is 2 days.

2. **Analyzing the First Statement**:
- The first statement claims that the mass number of B is greater than that of A.
- In alpha decay, an alpha particle (which consists of 2 protons and 2 neutrons, hence a mass of 4) is emitted. Therefore, the mass number of B (let's denote it as A_B) will be:
\[ A_B = A_A - 4 \]
- This means the mass number of B is less than that of A. Thus, this statement is **false**.

3. **Analyzing the Second Statement**:
- The second statement claims that after 2 hours, less than 10% of the initial amount of A is left.
- Convert 2 hours to minutes: \(2 \text{ hours} = 120 \text{ minutes}\).
- Calculate the number of half-lives that have passed:
\[ \text{Number of half-lives} = \frac{120 \text{ minutes}}{30 \text{ minutes}} = 4 \]
- The remaining amount of A after 4 half-lives can be calculated using the formula:
\[ N_t = N_0 \left(\frac{1}{2}\right)^n = N_0 \left(\frac{1}{2}\right)^4 = \frac{N_0}{16} \]
- 10% of the initial amount \(N_0\) is \( \frac{N_0}{10} \).
- Since \( \frac{N_0}{16} < \frac{N_0}{10} \), this statement is **true**.

4. **Analyzing the Third Statement**:
- The third statement claims that the maximum amount of B at any time is less than 50% of the initial amount of A.
- Initially, when A decays to B, the maximum amount of B will occur just after A has decayed significantly.
- However, since B is also decaying (with a half-life of 2 days), we need to analyze the situation.
- The maximum amount of B will occur before it starts decaying significantly. However, since B is produced from A and A decays relatively quickly, it is likely that B will not exceed 50% of A at any point.
- Therefore, this statement is **true**.

5. **Analyzing the Fourth Statement**:
- The fourth statement claims that the atomic numbers of A and C are the same.
- Let’s denote the atomic number of A as Z_A.
- After alpha decay, the atomic number of B will be:
\[ Z_B = Z_A - 2 \]
- After two beta decays, the atomic number of C will be:
\[ Z_C = Z_B + 2 = (Z_A - 2) + 2 = Z_A \]
- Thus, the atomic numbers of A and C are indeed the same, making this statement **true**.

### Conclusion:
The correct statements are:
- Statement B: After 2 hours, less than 10% of the initial amount of A is left. (True)
- Statement C: The maximum amount of B at any time is less than 50% of the initial amount of A. (True)
- Statement D: The atomic numbers of A and C are the same. (True)

### Final Answer:
The correct statements are B, C, and D.