The Correct Option is C
Solution and Explanation
Approach: The answer must hold for any valid $p,q$, so manufacture one concrete pair of equations with a known common root, compute the actual sum of the other roots, and see which option reproduces that number. This dodges all the abstract algebra.
Step 1: Force a common root, say $\alpha = 1$. Plug into the first equation: $3(1) - 5(1) + p = 0 \Rightarrow p = 2$. Plug into the second: $2(1) - 2(1) + q = 0 \Rightarrow q = 0$.
Step 2: First equation $3x^2 - 5x + 2 = 0$ has roots summing to $5/3$, so the other root $\beta = 5/3 - 1 = 2/3$. Second equation $2x^2 - 2x = 0$ has roots $0$ and $1$, so the other root $\gamma = 0$. Actual sum of other roots $= \dfrac{2}{3} + 0 = \dfrac{2}{3}$.
Step 3: Now evaluate each option at $p = 2,\ q = 0$ and match $2/3$: option 1 gives $5/3 - 2 = -1/3$; option 2 gives $8/3 + 2 = 14/3$; option 3 gives $8/3 - 2 + 0 = 2/3$ — match; option 4 gives $2 + 0 - 1 = 1$. Only option 3 hits $2/3$.
Final answer: $\dfrac{8}{3} - p + \dfrac{3}{2}q$ — option 3.