If the system of homogeneous equations 2x-y+z=0, x-2y+z=0, λ x-y+2z=0 has infinitely many solutions, then

Question

If the system of homogeneous equations 2x-y+z=0, x-2y+z=0, λ x-y+2z=0 has infinitely many solutions, then (A)  λ =5  (B)  λ =-5  (C)  λ ≠ ± 5  (D) None of these

Tardigrade Admin · Accepted Answer

If given system of equations have infinitely many solutions, then | beginmatrix 2 -1 1 1 -2 1 λ -1 2 endmatrix |=0 ⇒ 2(-4+1)+1(2-λ )+1(-1+2λ )=0 ⇒ -6+2-λ -1+2λ =0 ⇒ λ -5=0 ⇒ λ =5

Q. If the system of homogeneous equations $2 x - y + z = 0, x - 2 y + z = 0, λ x - y + 2 z = 0$ has infinitely many solutions, then

$λ = 5$

$λ = - 5$

$λ \neq = \pm 5$

None of these

If given system of equations have infinitely many solutions, then $∣ ∣ 21 λ - 1 - 2 - 1 112 ∣ ∣ = 0$
$\Rightarrow$ $2 (- 4 + 1) + 1 (2 - λ) + 1 (- 1 + 2 λ) = 0$
$\Rightarrow$ $- 6 + 2 - λ - 1 + 2 λ = 0$
$\Rightarrow$ $λ - 5 = 0$
$\Rightarrow$ $λ = 5$

Q. If the system of homogeneous equations 2x−y+z=0,x−2y+z=0,λx−y+2z=0 has infinitely many solutions, then

λ=5

λ=−5

λ=±5