JEE MAIN - Mathematics (2024 - 30th January Morning Shift - No. 9)
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 ?
The system has unique solution if $$\lambda \neq \frac{1}{2}$$ and $$\mu \neq 1,15$$
The system has infinite number of solutions if $$\lambda=\frac{1}{2}$$ and $$\mu=15$$
The system is consistent if $$\lambda \neq \frac{1}{2}$$
The system is inconsistent if $$\lambda=\frac{1}{2}$$ and $$\mu \neq 1$$
Explanation
$$x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda{ }^2 z=\mu^2+15$$,
$$\Delta=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 2 \lambda \\ 1 & 3 & 4 \lambda^2 \end{array}\right|=(2 \lambda-1)^2$$
For unique solution $$\Delta \neq 0,2 \lambda-1 \neq 0,\left(\lambda \neq \frac{1}{2}\right)$$
Let $$\Delta=0, \lambda=\frac{1}{2}$$
$$\begin{aligned} & \Delta_y=0, \Delta_x=\Delta_z=\left|\begin{array}{ccc} 4 \mu & 1 & 1 \\ 10 \mu & 2 & 1 \\ \mu^2+15 & 3 & 1 \end{array}\right| \\ & =(\mu-15)(\mu-1) \end{aligned}$$
For infinite solution $$\lambda=\frac{1}{2}, \mu=1$$ or 15
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