Given data:
* The frequency of the string vibration is f = 200 Hz.
Solution:
(A). The frequency of vibration in terms of the length of the string is,
![f=\frac{\sqrt[]{T}}{L}](https://img.qammunity.org/2023/formulas/physics/college/wsoclt2upk9kyfn1fbj0nf94fva6dnib2j.png)
If the length of the string is decreased to 1/4 of its actual value, then the frequency of the string is,
![\begin{gathered} f^(\prime)=\frac{\sqrt[]{T}}{(L)/(4)} \\ f^(\prime)=\frac{4\sqrt[]{T}}{L} \\ f^(\prime)=4f \end{gathered}](https://img.qammunity.org/2023/formulas/physics/college/cfzzttnlrbhcl1l6jpgpikytgwj5pv7mdh.png)
Substituting the known values,

Thus, the frequency of the vibration in the given case is 800 Hz.
(B). If the tension of the string is quadrupled, that is its value becomes 4 times the actual value, then the frequency of the vibration is,
![\begin{gathered} f^(\prime)=\frac{\sqrt[]{4T}}{L} \\ f^(\prime)=\frac{2\sqrt[]{T}}{L} \\ f^(\prime)=2f \end{gathered}](https://img.qammunity.org/2023/formulas/physics/college/vasnw44s26t8wbrf7wsqvm0eswm6miejja.png)
Substituting the known values,

Thus, the frequency of the vibration becomes 400 Hz.