Question

Consider the system of linear equations \( x + y + z = 4\mu \), \( x + 2y + 2\lambda z = 10\mu \), \( x + 3y + 4\lambda^2 z = \mu^2 + 15 \), where \( \lambda, \mu \in \mathbb{R} \). Which one of the following statements is NOT correct?

• A. The system has a unique solution if \( \lambda \neq \frac{1}{2} \) and \( \mu \neq 1, 15 \).
• B. The system has an infinite number of solutions if \( \lambda = \frac{1}{2} \) and \( \mu = 15 \).
• C. The system is inconsistent if \( \lambda = \frac{1}{2} \) and \( \mu \neq 1 \).
• D. The system is consistent if \( \lambda \neq \frac{1}{2} \).

Answer 1

The Correct Option is C

Write the system of equations in matrix form:

\[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 4\lambda \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4\mu \\ 10\mu \\ \mu^2 + 15 \end{bmatrix} \]

Let the coefficient matrix be \( A \):

\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 4\lambda \end{bmatrix} \]

Calculate the determinant of \( A \):

\[ \text{det}(A) = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 4\lambda \end{vmatrix} = (2\lambda - 1)^2 \]

For unique solutions, \( \text{det}(A) \neq 0 \) or \( \lambda \neq \frac{1}{2} \).

For infinite solutions, \( \lambda = \frac{1}{2} \), and consistency depends on the rank of the augmented matrix with specific values of \( \mu \).

The system is inconsistent if \( \lambda = \frac{1}{2} \) and \( \mu \neq 1 \)