Answer
See work below.
Work Step by Step
a. Both of these equations are correct. For a capacitor, the charge and voltage are not independent because Q = CV, or V = Q/C.
Starting from the first equation, we can arrive at the second.
$$PE=\frac{CV^2}{2}=\frac{C}{2}(\frac{Q}{C})^2=\frac{Q^2}{2C}$$
b. The first equation is handy when the capacitance and voltage are known. Such a situation would arise when the capacitor is connected to a battery.
The second equation is to be used when the charge and capacitance are known. This might happen when an isolated capacitor is charged with a known Q.
c. We are told that the voltage is constant. Use the first equation provided for potential energy. The capacitance rises by a factor of K, the dielectric constant for paper.
$$PE_{new}=\frac{C_{new}V^2}{2}=\frac{KC_{old}V^2}{2}=K PE_{old}$$
The energy stored in the capacitor increased by a factor of 3.7, which is the dielectric constant for paper.
d. In this situation, the charge stays constant. Use the second equation provided for potential energy. The capacitance rises by a factor of K, the dielectric constant for quartz.
$$PE_{new}=\frac{Q^2}{2C_{new}}=\frac{Q^2}{2KC_{old}}=\frac{1}{K}\frac{Q^2}{2C_{old}}=\frac{PE_{old}}{K}$$
The energy stored in the capacitor decreased by a factor of 4.3, which is the dielectric constant for quartz. The new PE is 0.23 times the old PE.
