In the experiment for measurement of viscosity \( \eta \) of a given liquid with a ball having radius \( R \), consider following statements:
A. Graph between terminal velocity \( V \) and \( R \) will be a parabola.
B. The terminal velocities of different diameter balls are constant for a given liquid.
C. Measurement of terminal velocity is dependent on the temperature.
D. This experiment can be utilized to assess the density of a given liquid.
E. If balls are dropped with some initial speed, the value of \( \eta \) will change.
Analyze each statement.
A: Incorrect, as the graph is not a parabola but rather a more complex function of radius and viscosity.
B: Incorrect, as terminal velocity varies with ball size and density.
C: Correct, as viscosity and terminal velocity are temperature-dependent.
D: Correct, as variations in terminal velocity can reflect differences in liquid density.
E: Correct, as the initial speed affects the drag force and settling time, influencing the measured viscosity.
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: