Question

Consider the system of equations \[ x+y+z=4\mu,\qquad x+2y+2z=10\mu,\qquad x+3y+4\lambda z=\mu^2+15, \] where \(\lambda\) and \(\mu\) are real numbers. Which of the following is correct?

• A. The system is consistent for all values of \(\lambda\) and \(\mu\).
• B. The system has infinite number of solutions if \(\lambda\neq\frac34\) and \(\mu=15\).
• C. The system is inconsistent if \(\lambda=\frac34,\ \mu\neq1\) and \(\mu\neq15\).
• D. The system has unique solutions if \(\lambda=\frac34\) and \(\mu\neq15\).

Hint:

For a system of linear equations: \[ \boxed{|A|\neq0 \Longrightarrow \text{Unique solution}} \] If \[ \boxed{|A|=0,} \] then compare the ranks of the coefficient matrix and augmented matrix to determine whether the system has infinitely many solutions or is inconsistent.

Answer 1

The Correct Option is C