Let S be the set of all real values of k for which the system of linear equations x+y+z=2;2x+y−z=3;3x+2y+kz=4 has a unique solution. Then S is

Let S be the set of all real values of k for which the system of linear equations $x + y + z = 2$ ; $2 x + y - z = 3; 3 x + 2 y + k z = 4$ has a unique solution. Then S is

A
an empty set
B
equal to R
C
equal to {0}
D
equal to R - {0}

Solution:

For unique solution

$|\begin{array}{l} 1 1 1 \\ 2 1 - 1 \\ 3 2 k \end{array}| \neq 0$

$\begin{array}{l} = 1 (k + 2) - 1 (2 k + 3) + 1 (4 - 3) \neq 0 \\ = - k + 2 - 3 + 1 \neq 0 \Rightarrow k \neq 0 \\ ∴ S = R - \{0\} \end{array}$

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