Determinants Question 6
Question 6 - 2024 (30 Jan Shift 1)
Consider the system of linear equation $x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^{2} z=\mu^{2}+15$, where $\lambda, \mu \in R$. Which one of the following statements is NOT correct?
(1) The system has unique solution if $\lambda \neq \frac{1}{2}$ and $\mu \neq 1,15$
(2) The system is inconsistent if $\lambda=\frac{1}{2}$ and $\mu \neq 1$
(3) The system has infinite number of solutions if $\lambda=\frac{1}{2}$ and $\mu=15$
(4) The system is consistent if $\lambda \neq \frac{1}{2}$
Show Answer
Answer (2)
Solution
$x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda$
${ }^{2} z=\mu^{2}+15$
$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & 2 \lambda \\ 1 & 3 & 4 \lambda^{2}\end{array}\right|=(2 \lambda-1)^{2}$
For unique solution $\Delta \neq 0,2 \lambda-1 \neq 0,\left(\lambda \neq \frac{1}{2}\right)$
Let $\Delta=0, \lambda=\frac{1}{2}$
$\Delta _y=0, \Delta _x=\Delta _z=\left|\begin{array}{ccc}4 \mu & 1 & 1 \\ 10 \mu & 2 & 1 \\ \mu^{2}+15 & 3 & 1\end{array}\right|$
$=(\mu-15)(\mu-1)$
For infinite solution $\lambda=\frac{1}{2}, \mu=1$ or 15
