If a takes two values for which \(\begin{vmatrix}2 & 1 & 3\\ 1 & a & -2\\ 0 & 3 & a\end{vmatrix} = 31\), then the sum of these two values is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{2}\)
Expanding along the first row:
\(\Delta = 2\big(a \cdot a - (-2)(3)\big) - 1\big(1 \cdot a - (-2)(0)\big) + 3\big(1 \cdot 3 - a \cdot 0\big)\)
\(\Delta = 2a^{2} + 12 - a + 9\)
\(\Delta = 2a^{2} - a + 21\)
Setting \(\Delta = 31\):
\(2a^{2} - a - 10 = 0\)
\((2a - 5)(a + 2) = 0\)
So \(a = \frac{5}{2}\) or \(a = -2\), and the sum of the values is
\( \displaystyle \frac{5}{2} + (-2) = \frac{1}{2} \)
Why the other options are incorrect
Option (1) \(-\displaystyle\frac{1}{2}\) — Incorrect. The sign is wrong; for \(2a^{2} - a - 10 = 0\) the sum of roots is \(-\frac{(-1)}{2} = \frac{1}{2}\).
Option (2) \(-5\) — Incorrect. This is the product of the two values, \(\frac{5}{2} \times (-2)\).
Option (3) 1 — Incorrect. The values are \(\frac{5}{2}\) and \(-2\), whose sum is \(\frac{1}{2}\), not 1.