Q16
The equations \( 3 x ^ { 2 } - 5 x + p = 0 \) and \( 2 x ^ { 2 } - 2 x + q = 0 \) have one common root. The sum of the other roots of these two equations is
Answer: D
Key idea: Eliminate the \(x^2\) term to find the common root, then use the sums of roots.
Let \(\alpha\) be the common root, \(\beta\) the other root of the first equation and \(\gamma\) the other root of the second. By sums of roots, \(\alpha + \beta = \dfrac53\) and \(\alpha + \gamma = 1\), so
\[\beta + \gamma = \frac83 - 2\alpha.\]Since \(\alpha\) satisfies both equations, take (first) \(- \tfrac32\) (second):
\[(3\alpha^2 - 5\alpha + p) - (3\alpha^2 - 3\alpha + \tfrac32 q) = 0 \;\Rightarrow\; -2\alpha + p - \tfrac32 q = 0 \;\Rightarrow\; 2\alpha = p - \tfrac32 q.\]Therefore \(\beta + \gamma = \dfrac83 - p + \dfrac32 q\).
Hence, the answer is (D) \(\frac83 - p + \frac32 q\).