Question

The system of linear equations
$x+y+z=154x+6y+7z=24k^{2}x+6y+7z=k+22$

Then the incorrect statement about the system of equations is

• A. The system of equations has no solution for exactly one value of $k$.
• B. For $k=0$ the system has unique solution.
• C. For $k=\pm$ 2, the system has no solution
• D. There exists $k\in R$ such that the system has no solution.

Answer 1

The correct option is C For $k=\pm$ 2, the system has no solution

$\left|\begin{array}{ccc}1& 1& 1\\ 4& 6& 7\\ k^2& 6& 7\end{array}\right|=0\Rightarrow 1(42-42)-1(28-7k^2)+1(24-6k^2)=0\Rightarrow k=\pm 2$

For $k=2$ , plane $2$ and plane $3$ are same, so infinite solution.

For $k=-2$ , plane $2$ and plane $3$ are parallel, hence, no solution.

For $k=0$
$\left|\begin{array}{ccc}1& 1& 1\\ 4& 6& 7\\ 0& 6& 7\end{array}\right|=-4\ne 0$
$\Rightarrow$ The system of equations will have a unique solution.