Question

Let $S$ be the set of real values of $k$ for which the system of equations \[ kx + y + z = k \] \[ x + ky + z = k \] \[ x + y + kz = k \] has no solution. Then $|S|$ is

• A. 1
• B. 2
• C. 0
• D. $\infty$

Hint:

For symmetric systems, $k = 1$ often represents a case of identical planes (infinitely many solutions), while the other root (here, $k = -2$) represents parallel but distinct conditions (no solution).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the number of real values of $k$ for which the given system of linear equations is inconsistent (has no solution).

Step 2: Key Formula or Approach:
For a system of linear equations to have no solution, the determinant of the coefficients ($D$) must be 0.
If $D = 0$, the system can either have infinitely many solutions or no solution. We must analyze both cases.

Step 3: Detailed Explanation:

• Let us write the determinant of the coefficient matrix:
\[ D = \begin{vmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{vmatrix} \]

• Evaluating the determinant:
\[ D = k(k^2 - 1) - 1(k - 1) + 1(1 - k) \]
\[ D = k(k - 1)(k + 1) - 2(k - 1) \]
\[ D = (k - 1)[k(k + 1) - 2] = (k - 1)(k^2 + k - 2) \]
\[ D = (k - 1)^2(k + 2) \]

• Setting $D = 0$ gives the critical values:
\[ k = 1 \quad \text{or} \quad k = -2 \]

• Let us analyze Case 1: $k = 1$:
The system of equations becomes:
\[ x + y + z = 1 \]
\[ x + y + z = 1 \]
\[ x + y + z = 1 \]
This represents a single plane, which has infinitely many solutions. Thus, $k = 1 \notin S$.

• Let us analyze Case 2: $k = -2$:
The system of equations becomes:
\[ -2x + y + z = -2 \quad \text{--- (Eq 1)} \]
\[ x - 2y + z = -2 \quad \text{--- (Eq 2)} \]
\[ x + y - 2z = -2 \quad \text{--- (Eq 3)} \]
Adding all three equations:
\[ (-2 + 1 + 1)x + (1 - 2 + 1)y + (1 + 1 - 2)z = -2 - 2 - 2 \]
\[ 0x + 0y + 0z = -6 \implies 0 = -6 \]
This is a contradiction, which means the system has no solution. Thus, $k = -2 \in S$.

• Therefore, the set $S$ contains only one element, $S = \{-2\}$, so $|S| = 1$.



Step 4: Final Answer:
The number of elements in $S$ is 1.