Answer
(a) The tension should increase by a factor of 4
(b) The length of the string should decrease by a factor of 2
Work Step by Step
(a) We know that $f = \frac{v}{\lambda}$. If the length of the string doesn't change, then $\lambda$ doesn't change. To play twice the frequency, the speed of the wave along the string must double.
$v \propto \sqrt{T}$
To double the speed, we need to increase the tension $T$ by a factor of 4.
(b) We know that $f = \frac{v}{\lambda}$. If the tension of the string doesn't change, then $v$ doesn't change. To play twice the frequency, the wavelength must be half the original wavelength.
The wavelength of the fundamental frequency is $2L$. To have a wavelength of $L$, the length of the string must be $\frac{L}{2}$ which is half the original length. We need to decrease the length by a factor of 2.
