If the system of equation
`2x + y- =5`
`2x-5y +lambda =mu`
`x+ 2y -5z =7`
has infinitely many solutions, then `(lambda+mu)^2 +(lambda -mu)^2` is equal to

Options

A. `904`
B. `912`
C. `916`
D. `920`

Answer

To solve the problem, we need to analyze the system of equations given:

1. \( 2x + y - z = 5 \) (Equation 1)
2. \( 2x - 5y + \lambda z = \mu \) (Equation 2)
3. \( x + 2y - 5z = 7 \) (Equation 3)

We need to find the condition under which this system has infinitely many solutions. For a system of linear equations to have infinitely many solutions, the determinant of the coefficients must be zero, and the ratios of the constants must be equal.

### Step 1: Write the coefficient matrix and calculate its determinant.

The coefficient matrix \( A \) is:

\[ A = \begin{bmatrix} 2 & 1 & -1 \\ 2 & -5 & \lambda \\ 1 & 2 & -5 \end{bmatrix} \]

The determinant \( D \) of this matrix can be calculated using the formula for the determinant of a 3x3 matrix:

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

Substituting the values:

\[ D = 2((-5)(-5) - (2)(\lambda)) - 1((2)(-5) - (1)(\lambda)) - 1((2)(2) - (1)(-5)) \]

Calculating each term:

1. \( (-5)(-5) = 25 \)
2. \( (2)(\lambda) = 2\lambda \)
3. \( (2)(-5) = -10 \)
4. \( (1)(\lambda) = \lambda \)
5. \( (2)(2) = 4 \)
6. \( (1)(-5) = -5 \)

Now substituting back into the determinant formula:

\[ D = 2(25 - 2\lambda) - 1(-10 - \lambda) - 1(4 + 5) \]

This simplifies to:

\[ D = 2(25 - 2\lambda) + 10 + \lambda - 9 \]
\[ D = 50 - 4\lambda + 10 + \lambda - 9 \]
\[ D = 51 - 3\lambda \]

Setting the determinant \( D = 0 \) for infinitely many solutions:

\[ 51 - 3\lambda = 0 \]
\[ 3\lambda = 51 \]
\[ \lambda = 17 \]

### Step 2: Calculate \( D_1 \) and set it to zero.

Next, we calculate \( D_1 \) using the modified matrix:

\[ D_1 = \begin{vmatrix} 5 & 1 & -1 \\ 2 & -5 & 17 \\ 7 & 2 & -5 \end{vmatrix} \]

Calculating \( D_1 \):

\[ D_1 = 5((-5)(-5) - (2)(17)) - 1((2)(-5) - (7)(17)) - 1((2)(2) - (7)(-5)) \]

Calculating each term:

1. \( (-5)(-5) = 25 \)
2. \( (2)(17) = 34 \)
3. \( (2)(-5) = -10 \)
4. \( (7)(17) = 119 \)
5. \( (2)(2) = 4 \)
6. \( (7)(-5) = -35 \)

Now substituting back into the determinant formula:

\[ D_1 = 5(25 - 34) - 1(-10 - 119) - 1(4 + 35) \]
\[ D_1 = 5(-9) + 129 - 39 \]
\[ D_1 = -45 + 129 - 39 \]
\[ D_1 = 45 \]

Setting \( D_1 = 0 \):

\[ -45 + 129 - 39 = 0 \]
\[ 3\mu = 39 \]
\[ \mu = -13 \]

### Step 3: Calculate \( (\lambda + \mu)^2 + (\lambda - \mu)^2 \)

Now we have \( \lambda = 17 \) and \( \mu = -13 \).

Calculating \( \lambda + \mu \):

\[ \lambda + \mu = 17 - 13 = 4 \]

Calculating \( \lambda - \mu \):

\[ \lambda - \mu = 17 + 13 = 30 \]

Now we calculate:

\[ (\lambda + \mu)^2 + (\lambda - \mu)^2 = 4^2 + 30^2 \]
\[ = 16 + 900 = 916 \]

Thus, the final answer is:

\[ \boxed{916} \]