JEE MAIN - Mathematics (2019 - 12th January Evening Slot - No. 10)
The set of all values of $$\lambda $$ for which the system of linear equations
x – 2y – 2z = $$\lambda $$x
x + 2y + z = $$\lambda $$y
– x – y = $$\lambda $$z
has a non-trivial solutions :
x – 2y – 2z = $$\lambda $$x
x + 2y + z = $$\lambda $$y
– x – y = $$\lambda $$z
has a non-trivial solutions :
is an empty set
contains more than two elements
is a singleton
contains exactly two elements
Explanation
$$\left| {\matrix{
{\lambda - 1} & 2 & 2 \cr
1 & {2 - \lambda } & 1 \cr
1 & 1 & 1 \cr
} } \right| = 0$$
$$ \Rightarrow {\left( {\lambda - 1} \right)^3} = 0 \Rightarrow \lambda = 1$$
$$ \Rightarrow {\left( {\lambda - 1} \right)^3} = 0 \Rightarrow \lambda = 1$$
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