jee-main 2025 Q12

jee-main · India · session1_23jan_shift1 Matrices Linear System and Inverse Existence
If the system of equations $$( \lambda - 1 ) x + ( \lambda - 4 ) y + \lambda z = 5$$ $$\lambda x + ( \lambda - 1 ) y + ( \lambda - 4 ) z = 7$$ $$( \lambda + 1 ) x + ( \lambda + 2 ) y - ( \lambda + 2 ) z = 9$$ has infinitely many solutions, then $\lambda ^ { 2 } + \lambda$ is equal to
(1) 6
(2) 10
(3) 20
(4) 12
If the system of equations
$$( \lambda - 1 ) x + ( \lambda - 4 ) y + \lambda z = 5$$
$$\lambda x + ( \lambda - 1 ) y + ( \lambda - 4 ) z = 7$$
$$( \lambda + 1 ) x + ( \lambda + 2 ) y - ( \lambda + 2 ) z = 9$$
has infinitely many solutions, then $\lambda ^ { 2 } + \lambda$ is equal to\\
(1) 6\\
(2) 10\\
(3) 20\\
(4) 12