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Question

Mathematics Question on linear inequalities

Let the system of linear equations x + 2y + z = 2, αx + 3y – z = α, –αx + y + 2z = –α be inconsistent. Then α is equal to :

A

52\frac{5}{2}

B

−52-\frac{5}{2}

C

72\frac{7}{2}

D

−72-\frac{7}{2}

Answer

−72-\frac{7}{2}

Explanation

Solution

x\+2y+z=2x \+ 2y + z = 2

αx\+3y–z=ααx \+ 3y – z = α

–αx+y\+2z=–α–αx + y \+ 2z = –α

Δ=∣121\α3−1\−α12∣=1(6+1)−2(2α−α)+1(α+3α)Δ=\begin{vmatrix}1&2&1\\\α&3&−1\\\−α&1&2\end{vmatrix}=1(6+1)−2(2α−α)+1(α+3α)

= 7+2α7 + 2α

Δ=0⇒α=−72Δ=0⇒α=−\frac{7}{2}

Δ1=∣221\α3−1\−α12∣=14+2α≠0for  α=−72Δ_1=\begin{vmatrix}2&2&1\\\α&3&−1\\\−α&1&2\end{vmatrix}=14+2α≠0 for \;α=−\frac{7}{2}

α=−72α = - \frac{7}{2}

Hence, the correct option is (D): −72- \frac{7}{2}