The value of $\lambda $ for which the given system of equations will have infinitely many solutions $x+y -6z =0; \lambda x -y +2z=0;-3x+\lambda y +2z=0,$ is /are :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $\lambda = -\frac{5}{3}$ or 1
for infinite solutions
$\begin{vmatrix}1&1&-6\\λ&-1&2\\-3&λ&2\end{vmatrix}=0$
$C_2→C_2-C_1$
$C_3→C_3+6C_1$
$\begin{vmatrix}1&0&0\\λ&-1-λ&2+6λ\\-3&λ+3&-16\end{vmatrix}=0$
$16(1+λ)-2(λ+3)^2=0$
or $(λ+3)^2-8(λ+1)=0⇒λ^2+6λ+9-8λ-8=0$
$⇒λ^2-2λ+1=0$
$16(1+λ)-2(λ+3)(1+3λ)=0$
$8+8λ=3λ^2+3+10λ$
$3λ^2-5+2λ=0$
$⇒3λ^2+5λ-3λ-5=0$
$(3λ+5)(λ-1)=0$
$λ=1,-\frac{5}{3}$