A redioctive nucleus decays as follows :
`Xoverset(alpha)(to)X_(1)overset(beta)(to)X_(2)overset(alpha)(to)X_(3)overset(alpha)(to)X_(4)`
If the atomic number and the mass number of X are 72 and 180 then the mass number and atomic number of `X_4` are-

Options

A. `172, 70`
B. `171, 69`
C. `172, 69`
D. `172, 68`

Answer

To solve the problem, we need to track the changes in the atomic number and mass number of the radioactive nucleus X as it undergoes a series of decays: two alpha decays and one beta decay.

### Step-by-Step Solution:

1. **Identify Initial Values:**
- Atomic number (Z) of X = 72
- Mass number (A) of X = 180

2. **First Decay (Alpha Decay):**
- In alpha decay, the mass number decreases by 4 and the atomic number decreases by 2.
- New mass number (A1) = 180 - 4 = 176
- New atomic number (Z1) = 72 - 2 = 70

3. **Second Decay (Beta Decay):**
- In beta decay, the mass number remains the same, but the atomic number increases by 1.
- New mass number (A2) = 176 (remains the same)
- New atomic number (Z2) = 70 + 1 = 71

4. **Third Decay (Alpha Decay):**
- Again, in alpha decay, the mass number decreases by 4 and the atomic number decreases by 2.
- New mass number (A3) = 176 - 4 = 172
- New atomic number (Z3) = 71 - 2 = 69

5. **Fourth Decay (Alpha Decay):**
- In this final alpha decay, the mass number decreases by 4 and the atomic number decreases by 2.
- New mass number (A4) = 172 - 4 = 168
- New atomic number (Z4) = 69 - 2 = 67

6. **Final Result:**
- The mass number of X4 = 168
- The atomic number of X4 = 67

### Conclusion:
The mass number and atomic number of X4 are 168 and 67, respectively.