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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 445
Question 445 of 472
Medium
MCQ
Consider the system of linear equations \[x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15\] where \[\lambda, \mu \in \mathbf{R}\]. Which one of the following statements is NOT correct ?
A
The system has unique solution if \[\lambda \neq \frac{1}{2}\] and \[\mu \neq 1,15\]
B
The system has infinite number of solutions if \[\lambda=\frac{1}{2}\] and \[\mu=15\]
C
The system is consistent if \[\lambda \neq \frac{1}{2}\]
D
The system is inconsistent if \[\lambda=\frac{1}{2}\] and \[\mu \neq 1\]
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Official solution
\[x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda{ }^2 z=\mu^2+15\], \[\Delta=\left|\[
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\begin{array}{|ccccccc|}1 & 1 & 1 \\ 1 & 2 & 2 \lambda \\ 1 & 3 & 4 \lambda^2\end{array}
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\]\right|=(2 \lambda-1)\^2\] For unique solution \[\Delta \neq 0,2 \lambda-1 \neq 0,\left(\lambda \neq \frac{1}{2}\right)\] Let \[\Delta=0, \lambda=\frac{1}{2}\] \[
\begin{aligned} & \Delta_y=0, \Delta_x=\Delta_z=\left|\[\begin{array}{|ccccccc|}4 \mu & 1 & 1 \\ 10 \mu & 2 & 1 \\ \mu^2+15 & 3 & 1\end{array}\end{aligned}
\]\right| \\ \& =(\mu-15)(\mu-1) \] For infinite solution \[\lambda=\frac{1}{2}, \mu=1\] or 15
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