There are two values of 'a' which makes the value of determinant `|(1,-2,5),(2,a,-1),(0,4,2a)| = 86`, find the sum of these values of 'a'.

Options

A. 4
B. `-1`
C. 1
D. None of these

Answer

To solve the problem, we need to evaluate the determinant and set it equal to 86. We will follow these steps:

### Step 1: Write the determinant
We have the determinant given by:
\[ D = \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} \]

### Step 2: Expand the determinant
We can expand the determinant using the first row:
\[ D = 1 \cdot \begin{vmatrix} a & -1 \\ 4 & 2a \end{vmatrix} - (-2) \cdot \begin{vmatrix} 2 & -1 \\ 0 & 2a \end{vmatrix} + 5 \cdot \begin{vmatrix} 2 & a \\ 0 & 4 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
1. For the first determinant:
\[ \begin{vmatrix} a & -1 \\ 4 & 2a \end{vmatrix} = a \cdot 2a - (-1) \cdot 4 = 2a^2 + 4 \]

2. For the second determinant:
\[ \begin{vmatrix} 2 & -1 \\ 0 & 2a \end{vmatrix} = 2 \cdot 2a - (-1) \cdot 0 = 4a \]

3. For the third determinant:
\[ \begin{vmatrix} 2 & a \\ 0 & 4 \end{vmatrix} = 2 \cdot 4 - a \cdot 0 = 8 \]

### Step 4: Substitute back into the determinant
Now substituting these back into the determinant:
\[ D = 1 \cdot (2a^2 + 4) + 2 \cdot (4a) + 5 \cdot 8 \]
\[ D = 2a^2 + 4 + 8a + 40 \]
\[ D = 2a^2 + 8a + 44 \]

### Step 5: Set the determinant equal to 86
Now we set the determinant equal to 86:
\[ 2a^2 + 8a + 44 = 86 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 2a^2 + 8a + 44 - 86 = 0 \]
\[ 2a^2 + 8a - 42 = 0 \]

### Step 7: Simplify the equation
Dividing the entire equation by 2:
\[ a^2 + 4a - 21 = 0 \]

### Step 8: Factor the quadratic equation
Now we factor the quadratic:
\[ (a + 7)(a - 3) = 0 \]

### Step 9: Find the values of 'a'
Setting each factor to zero gives us:
\[ a + 7 = 0 \quad \Rightarrow \quad a = -7 \]
\[ a - 3 = 0 \quad \Rightarrow \quad a = 3 \]

### Step 10: Find the sum of the values of 'a'
Now we find the sum of the two values:
\[ -7 + 3 = -4 \]

### Final Answer
The sum of the two values of 'a' is:
\[ \boxed{-4} \]