JEE MAIN - Mathematics (2020 - 6th September Evening Slot - No. 9)
The sum of distinct values of $$\lambda $$ for which the
system of equations
$$\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lambda z = 0$$
$$\left( {\lambda - 1} \right)x + \left( {4\lambda - 2} \right)y + \left( {\lambda + 3} \right)z = 0$$
$$2x + \left( {3\lambda + 1} \right)y + 3\left( {\lambda - 1} \right)z = 0$$
has non-zero solutions, is ________ .
$$\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lambda z = 0$$
$$\left( {\lambda - 1} \right)x + \left( {4\lambda - 2} \right)y + \left( {\lambda + 3} \right)z = 0$$
$$2x + \left( {3\lambda + 1} \right)y + 3\left( {\lambda - 1} \right)z = 0$$
has non-zero solutions, is ________ .
Answer
3
Explanation
$$\left| {\matrix{
{\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr
{\lambda - 1} & {4\lambda - 2} & {\lambda + 3} \cr
2 & {3\lambda + 1} & {3\left( {\lambda - 1} \right)} \cr
} } \right|$$ = 0
R2 $$ \to $$ R2 – R1
R3 $$ \to $$ R3 – R1
$$\left| {\matrix{ {\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr 0 & {\lambda - 3} & { - \lambda + 3} \cr {3 - \lambda } & 0 & {\lambda - 3} \cr } } \right| = 0$$
C1 $$ \to $$ C1 + C3
$$\left| {\matrix{ {3\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr { - \lambda + 3} & {\lambda - 3} & { - \lambda + 3} \cr 0 & 0 & {\lambda - 3} \cr } } \right| = 0$$
$$ \Rightarrow $$ ($$\lambda $$ - 3) [(3$$\lambda $$ - 1) ($$\lambda $$ - 3) – (3 – $$\lambda $$) (3$$\lambda $$ + 1)] = 0
$$ \Rightarrow $$ ($$\lambda $$ – 3) [3$$\lambda $$2 – 10$$\lambda $$ + 3 –(8$$\lambda $$ –3$$\lambda $$2 + 3)] = 0
$$ \Rightarrow $$ ($$\lambda $$ – 3) (6$$\lambda $$2 – 18$$\lambda $$) = 0
$$ \Rightarrow $$ (6$$\lambda $$) ($$\lambda $$ – 3)2 = 0
$$ \Rightarrow $$ $$\lambda $$ = 0, 3
$$ \therefore $$ sum of values of $$\lambda $$ = 0 + 3 = 3
R2 $$ \to $$ R2 – R1
R3 $$ \to $$ R3 – R1
$$\left| {\matrix{ {\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr 0 & {\lambda - 3} & { - \lambda + 3} \cr {3 - \lambda } & 0 & {\lambda - 3} \cr } } \right| = 0$$
C1 $$ \to $$ C1 + C3
$$\left| {\matrix{ {3\lambda - 1} & {3\lambda + 1} & {2\lambda } \cr { - \lambda + 3} & {\lambda - 3} & { - \lambda + 3} \cr 0 & 0 & {\lambda - 3} \cr } } \right| = 0$$
$$ \Rightarrow $$ ($$\lambda $$ - 3) [(3$$\lambda $$ - 1) ($$\lambda $$ - 3) – (3 – $$\lambda $$) (3$$\lambda $$ + 1)] = 0
$$ \Rightarrow $$ ($$\lambda $$ – 3) [3$$\lambda $$2 – 10$$\lambda $$ + 3 –(8$$\lambda $$ –3$$\lambda $$2 + 3)] = 0
$$ \Rightarrow $$ ($$\lambda $$ – 3) (6$$\lambda $$2 – 18$$\lambda $$) = 0
$$ \Rightarrow $$ (6$$\lambda $$) ($$\lambda $$ – 3)2 = 0
$$ \Rightarrow $$ $$\lambda $$ = 0, 3
$$ \therefore $$ sum of values of $$\lambda $$ = 0 + 3 = 3
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