If there are two values of a which makes determinant,
`Delta=|(1,-2,5),(2,a,-1),(0,4,2a)|=86` then the sum of these number is

Options

A. `4`
B. `5`
C. `-4`
D. `9`

Explanation

To solve the problem, we need to calculate the determinant of the given matrix and set it equal to 86. The matrix is:

\[ \Delta = \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} \]

### Step 1: Calculate the Determinant

We can calculate the determinant using the formula for a 3x3 matrix:

\[ \Delta = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where the matrix is represented as:

\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \]

In our case:
- \( a = 1, b = -2, c = 5 \)
- \( d = 2, e = a, f = -1 \)
- \( g = 0, h = 4, i = 2a \)

Substituting these values into the determinant formula:

\[ \Delta = 1(a \cdot 2a - (-1) \cdot 4) - (-2)(2 \cdot 2a - (-1) \cdot 0) + 5(2 \cdot 4 - a \cdot 0) \]

### Step 2: Simplify Each Term

Calculating each term step-by-step:

1. **First Term**:
\[ 1(a \cdot 2a - (-1) \cdot 4) = 1(2a^2 + 4) = 2a^2 + 4 \]

2. **Second Term**:
\[ -(-2)(2 \cdot 2a - 0) = 2(4a) = 8a \]

3. **Third Term**:
\[ 5(2 \cdot 4 - 0) = 5 \cdot 8 = 40 \]

### Step 3: Combine All Terms

Now, combine all the terms:

\[ \Delta = 2a^2 + 4 + 8a + 40 \]
\[ \Delta = 2a^2 + 8a + 44 \]

### Step 4: Set the Determinant Equal to 86

Now we set the determinant equal to 86:

\[ 2a^2 + 8a + 44 = 86 \]

### Step 5: Rearrange the Equation

Rearranging gives:

\[ 2a^2 + 8a + 44 - 86 = 0 \]
\[ 2a^2 + 8a - 42 = 0 \]

### Step 6: Simplify the Quadratic Equation

Dividing the entire equation by 2:

\[ a^2 + 4a - 21 = 0 \]

### Step 7: Factor the Quadratic Equation

Now we can factor the quadratic:

\[ (a + 7)(a - 3) = 0 \]

### Step 8: Solve for \( a \)

Setting each factor to zero gives us:

1. \( a + 7 = 0 \) → \( a = -7 \)
2. \( a - 3 = 0 \) → \( a = 3 \)

### Step 9: Find the Sum of the Values of \( a \)

Now we find the sum of the two values of \( a \):

\[ -7 + 3 = -4 \]

### Final Answer

Thus, the sum of the two values of \( a \) is:

\[ \boxed{-4} \]