Discussion
If a, b, and c are the roots of \(2x^{3} - 3x^{2} + px - 1 = 0\) and sum of the two roots is 1, the value of \(p\) is:
Answer: C
The value is \(p = 3\).
For \(2x^{3} - 3x^{2} + px - 1 = 0\), the sum of the roots is \(-(-3)/2 = 3/2\). If two of the roots add up to 1, the third root is \(3/2 - 1 = 1/2\). Substituting \(x = 1/2\):
\[ \begin{aligned} 2\left(\tfrac{1}{8}\right) - 3\left(\tfrac{1}{4}\right) + \tfrac{p}{2} - 1 &= 0 \\ \tfrac{1}{4} - \tfrac{3}{4} - 1 + \tfrac{p}{2} &= 0 \\ \tfrac{p}{2} = \tfrac{3}{2}, \quad p &= 3 \end{aligned} \]
With \(p = 2\), \(p = 1\) or \(p = 1/3\), \(x = 1/2\) would not satisfy the equation, so no pair of roots would sum to 1.
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