Question

If \( \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86 \), then the sum of all possible values of \( a \) is

• A. \( 4 \)
• B. \( 5 \)
• C. \( -4 \)
• D. \( 9 \)

Hint:

When asked for the sum of roots, save time by using Vieta's formulas instead of fully solving for each root.
Always look for the row or column with the most zeros to expand the determinant more efficiently.

Answer 1

The Correct Option is C

Concept:
• Expansion of a \( 3 \times 3 \) determinant using minors and cofactors.
• Vieta's formulas for a quadratic equation \( ax^2 + bx + c = 0 \): Sum of roots is \( -b/a \).

Step 1: Expand the determinant along Row 1
\[ \Delta = -1 \begin{vmatrix} a & -1 4 & 2a \end{vmatrix} - (-2) \begin{vmatrix} -2 & -1 \\ 0 & 2a \end{vmatrix} + 5 \begin{vmatrix} -2 & a \\ 0 & 4 \end{vmatrix} \]
Calculate the \( 2 \times 2 \) determinants:
\[ \Delta = -1(a(2a) - (-1)(4)) + 2((-2)(2a) - (-1)(0)) + 5((-2)(4) - a(0)) \]
\[ \Delta = -1(2a^2 + 4) + 2(-4a) + 5(-8) \]
\[ \Delta = -2a^2 - 4 - 8a - 40 \]
\[ \Delta = -2a^2 - 8a - 44 \]

Step 2: Set the expression equal to -86 and solve the quadratic
\[ -2a^2 - 8a - 44 = -86 \]
Add 86 to both sides:
\[ -2a^2 - 8a + 42 = 0 \]
Divide the entire equation by -2 to simplify:
\[ a^2 + 4a - 21 = 0 \]

Step 3: Find the sum of values of \( a \)
For a quadratic equation \( x^2 + px + q = 0 \), the sum of roots is \( -p \).
Here, the equation is in the variable \( a \) with coefficient \( p = 4 \).
Sum of possible values of \( a = -4 \).

Step 4: Verification (Optional)
Factoring the quadratic: \( a^2 + 7a - 3a - 21 = 0 \implies (a+7)(a-3) = 0 \).
The roots are \( a = -7 \) and \( a = 3 \).
Sum \( = -7 + 3 = -4 \). The result is consistent.