Question

If there are two values of ‘a’ which makes determinant
\(\Delta=\begin{bmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \\ \end{bmatrix}=86\)
Then the sum of these numbers

• A. -4
• B. 4
• C. 9
• D. 5

Answer 1

The Correct Option is A

We are given the determinant: \[ \Delta = \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86 \] We expand this determinant along the third row since it has a zero: \[ \Delta = 0 \cdot \begin{vmatrix} -2 & 5 \\ a & -1 \end{vmatrix} - 4 \cdot \begin{vmatrix} 1 & 5 \\ 2 & -1 \end{vmatrix} + 2a \cdot \begin{vmatrix} 1 & -2 \\ 2 & a \end{vmatrix} \] Calculate the 2×2 determinants: \[ \begin{vmatrix} 1 & 5 \\ 2 & -1 \end{vmatrix} = (1)(-1) - (5)(2) = -1 - 10 = -11 \] \[ \begin{vmatrix} 1 & -2 \\ 2 & a \end{vmatrix} = (1)(a) - (-2)(2) = a + 4 \] So: \[ \Delta = -4(-11) + 2a(a + 4) = 44 + 2a^2 + 8a \] Set equal to 86: \[ 2a^2 + 8a + 44 = 86 \Rightarrow 2a^2 + 8a - 42 = 0 \Rightarrow a^2 + 4a - 21 = 0 \] Solving the quadratic: \[ a = \frac{-4 \pm \sqrt{16 + 84}}{2} = \frac{-4 \pm \sqrt{100}}{2} = \frac{-4 \pm 10}{2} \Rightarrow a = 3, -7 \] Sum of the values of a: \( 3 + (-7) = -4 \) 

Answer 2

We are given the determinant:

\(\Delta=\begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86\)

Let's calculate the determinant:

\(\Delta = 1 \cdot \begin{vmatrix} a & -1 \\ 4 & 2a \end{vmatrix} - (-2) \cdot \begin{vmatrix} 2 & -1 \\ 0 & 2a \end{vmatrix} + 5 \cdot \begin{vmatrix} 2 & a \\ 0 & 4 \end{vmatrix}\)

\(\Delta = 1(2a^2 - (-4)) + 2(4a - 0) + 5(8 - 0)\)

\(\Delta = 2a^2 + 4 + 8a + 40\)

\(\Delta = 2a^2 + 8a + 44\)

We are given that \(\Delta = 86\), so

2a² + 8a + 44 = 86

2a² + 8a - 42 = 0

Divide by 2:

a² + 4a - 21 = 0

This is a quadratic equation in 'a'. We can find the sum of the roots using the formula for the sum of roots of a quadratic equation (ax² + bx + c = 0), which is -b/a.

In this case, a = 1, b = 4, and c = -21. Therefore, the sum of the roots (the two values of 'a') is -4/1 = -4.

The two values of a can be solved by factoring

a² + 4a - 21 = (a+7)(a-3)

a = -7 or a = 3

-7 + 3 = -4

Answer: -4