To solve the question regarding Bohr's theory for hydrogen-like atoms, we will analyze each statement provided and determine its validity based on the principles of Bohr's model.
### Step-by-Step Solution:
1. **Understanding the Radius of the Orbit**:
- According to Bohr's theory, the radius of the nth orbit for hydrogen-like atoms is given by the formula:
\[
r_n = \frac{0.529 \, n^2}{Z} \, \text{Å}
\]
- Here, \( Z \) is the atomic number and \( n \) is the principal quantum number.
- As \( Z \) increases, the denominator increases, leading to a decrease in the radius \( r_n \).
**Conclusion**: The first statement, "if atomic number increases, radius of the orbit increases," is **false**.
**Hint**: Remember that the radius is inversely proportional to the atomic number \( Z \).
2. **Analyzing the Velocity of Electrons**:
- The velocity \( v_n \) of an electron in the nth orbit is given by:
\[
v_n = 2.18 \times 10^6 \frac{Z}{n} \, \text{m/s}
\]
- As the principal quantum number \( n \) increases, the velocity \( v_n \) decreases because \( n \) is in the denominator.
**Conclusion**: The second statement, "if principal quantum number of orbit increases, velocity of electrons in the orbit slightly increases," is **false**.
**Hint**: The velocity is inversely related to the principal quantum number \( n \).
3. **Effect of Mass on Electron Velocity**:
- The velocity of the electron in the nth orbit is independent of the mass of the electron. This can be derived from the formula for the radius and the quantization condition:
\[
MVR = \frac{n h}{2\pi}
\]
- Rearranging gives:
\[
V = \frac{n h}{2\pi M R}
\]
- Since the mass \( M \) appears in the denominator of the radius formula, the overall effect on velocity remains independent of mass.
**Conclusion**: The third statement, "if mass of revolving electron is increased, electron velocity in the orbit will remain exactly the same," is **true**.
**Hint**: The velocity is derived from the quantization condition and does not depend on the mass of the electron.
### Final Answer:
- The first statement is **false**.
- The second statement is **false**.
- The third statement is **true**.
