JEE Main 2005
Exam JEE Main
Year 2005
Subject Mathematics
Chapter Determinants JEE Main 2005 · Mathematics · Determinants
The system of equations aligned & α x+y+z=α-1 & x+α y+z=α-1 & x+y+α...
The system of equations
α x + y + z = α − 1 x + α y + z = α − 1 x + y + α z = α − 1 \begin{aligned}
& \alpha x+y+z=\alpha-1 \\
& x+\alpha y+z=\alpha-1 \\
& x+y+\alpha z=\alpha-1
\end{aligned} αx + y + z = α − 1 x + α y + z = α − 1 x + y + α z = α − 1
has no solution, if
α \alpha α is
Options A − 2 -2 − 2 B either − 2 -2 − 2 C not -2 D 1
Step-by-step solution
\alphax + y + z = α − 1 x + \alphay + z = α − 1 x + y + z α = α − 1 Δ = ∣ α 1 1 1 α 1 1 1 α ∣ = α ( α 2 − 1 ) − 1 ( α − 1 ) + 1 ( 1 − α ) = α ( α − 1 ) ( α + 1 ) − 1 ( α − 1 ) − 1 ( α − 1 ) ⇒ ( α − 1 ) [ α 2 + α − 1 − 1 ] = 0 ⇒ ( α − 1 ) [ α 2 + α − 2 ] = 0 [ α 2 + 2 α − α − 2 ] = 0 ( α − 1 ) [ α ( α + 2 ) − 1 ( α + 2 ) ] = 0 ( α − 1 ) = 0 , α + 2 = 0 ⇒ α = − 2 , 1 ; but α ≠ 1 \begin{aligned}& \alphax+y+z=\alpha-1 \\
& x+\alphay+z=\alpha-1 \\
& x+y+z \alpha=\alpha-1 \\
& \Delta=\left|\begin{array}{ccc}
\alpha& 1 & 1 \\
1 & \alpha& 1 \\
1 & 1 & \alpha
\end{array}\right|\\
& =\alpha\left(\alpha^2-1\right)-1(\alpha-1)+1(1-\alpha)\\
& =\alpha(\alpha-1)(\alpha+1)-1(\alpha-1)-1(\alpha-1) \\
& \Rightarrow(\alpha-1)\left[\alpha^2+\alpha-1-1\right]=0 \\
& \Rightarrow(\alpha-1)\left[\alpha^2+\alpha-2\right]=0 \\
& {\left[\alpha^2+2 \alpha-\alpha-2\right]=0} \\
& (\alpha-1)[\alpha(\alpha+2)-1(\alpha+2)]=0 \\
& (\alpha-1)=0, \alpha+2=0 \Rightarrow \alpha=-2,1 ; \text{ but } \alpha \neq1
\end{aligned} \alphax + y + z = α − 1 x + \alphay + z = α − 1 x + y + z α = α − 1 Δ = α 1 1 1 α 1 1 1 α = α ( α 2 − 1 ) − 1 ( α − 1 ) + 1 ( 1 − α ) = α ( α − 1 ) ( α + 1 ) − 1 ( α − 1 ) − 1 ( α − 1 ) ⇒ ( α − 1 ) [ α 2 + α − 1 − 1 ] = 0 ⇒ ( α − 1 ) [ α 2 + α − 2 ] = 0 [ α 2 + 2 α − α − 2 ] = 0 ( α − 1 ) [ α ( α + 2 ) − 1 ( α + 2 )] = 0 ( α − 1 ) = 0 , α + 2 = 0 ⇒ α = − 2 , 1 ; but α = 1
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