Let S be the set of all λ∈R for which the system of linear equations 2x – y + 2z = 2 x – 2y + λz = –4 x + λy + z = 4

Question

Let S be the set of all $\lambda\in R$ for which the system of linear equations

2x – y + 2z = 2

x – 2y + λz = –4

x + λy + z = 4

has no solution. then the set S

(1) is a singleton

(2) contains exactly two elements

(3) contains more than two elements

(4) is an empty set

Answer 1

Ans. (2) contains exactly two elements

Sol. For no. solution Δ = 0 and at least one of Δ₁, Δ₂, Δ₃ is non-zero.

$\Delta=\begin{vmatrix}2&-1&2\\1&-2&\lambda\\1&\lambda&1\end{vmatrix}=-(\lambda-1)(2\lambda+1)$

$\Delta_1=\begin{vmatrix}2&-1&2\\-4&-2&\lambda\\4&\lambda&1\end{vmatrix}=-2(\lambda^2+6\lambda-4)$

$\Delta=0\implies\lambda=1,-{1\over2}$

Hence, $S=\{1,-{1\over2}\}$