The system of equations
`-kx+3y-14z=25`
`-15x+4y-kz=3`
`-4x+y+3z=4`
is consistent for all k is set

Options

A. `R-{13}`
B. `R-{-11,13}`
C. `R-{-11,11}`
D. R

Answer

To determine the values of \( k \) for which the given system of equations is consistent, we need to analyze the determinant of the coefficient matrix. The system of equations is:

1. \(-kx + 3y - 14z = 25\)
2. \(-15x + 4y - kz = 3\)
3. \(-4x + y + 3z = 4\)

We can express this system in the matrix form \( AX = B \), where \( A \) is the coefficient matrix and \( B \) is the constant matrix.

### Step 1: Write the Coefficient Matrix
The coefficient matrix \( A \) is given by:

\[ A = \begin{pmatrix} -k & 3 & -14 \\ -15 & 4 & -k \\ -4 & 1 & 3 \end{pmatrix} \]

### Step 2: Calculate the Determinant of \( A \)
To find when the system is consistent, we need to ensure that the determinant of \( A \) is not equal to zero. We calculate the determinant \( \Delta \) of matrix \( A \):

\[ \Delta = \begin{vmatrix} -k & 3 & -14 \\ -15 & 4 & -k \\ -4 & 1 & 3 \end{vmatrix} \]

Using the determinant formula for a \( 3 \times 3 \) matrix:

\[ \Delta = -k \begin{vmatrix} 4 & -k \\ 1 & 3 \end{vmatrix} - 3 \begin{vmatrix} -15 & -k \\ -4 & 3 \end{vmatrix} - 14 \begin{vmatrix} -15 & 4 \\ -4 & 1 \end{vmatrix} \]

Calculating each of the \( 2 \times 2 \) determinants:

1. \(\begin{vmatrix} 4 & -k \\ 1 & 3 \end{vmatrix} = (4)(3) - (-k)(1) = 12 + k\)
2. \(\begin{vmatrix} -15 & -k \\ -4 & 3 \end{vmatrix} = (-15)(3) - (-k)(-4) = -45 - 4k\)
3. \(\begin{vmatrix} -15 & 4 \\ -4 & 1 \end{vmatrix} = (-15)(1) - (4)(-4) = -15 + 16 = 1\)

Substituting back into the determinant:

\[ \Delta = -k(12 + k) - 3(-45 - 4k) - 14(1) \]

Expanding this:

\[ \Delta = -12k - k^2 + 135 + 12k - 14 \]

Simplifying:

\[ \Delta = -k^2 + 121 \]

### Step 3: Set the Determinant Not Equal to Zero
For the system to be consistent, we require:

\[ -k^2 + 121 \neq 0 \]

This simplifies to:

\[ k^2 \neq 121 \]

### Step 4: Solve for \( k \)
The solutions to \( k^2 = 121 \) are:

\[ k = 11 \quad \text{or} \quad k = -11 \]

Thus, the values of \( k \) that make the system inconsistent are \( k = 11 \) and \( k = -11 \). Therefore, the system is consistent for all \( k \) except these two values.

### Final Answer
The system of equations is consistent for all \( k \) except \( k = 11 \) and \( k = -11 \).

---