Question

The value of `|alpha|` for which the system of equation
`alphax+y+z=alpha-1`
`x+alphay+z=alpha-1`
`x+y+alphaz=alpha-1`
has no solution , is `"____"`
A. either -2 or 1
B. -2
C. 1
D. not-2

Answer 1

Correct Answer - B
for no solution or infinitely many solutions
`|{:(alpha ,,-1,,-1),(1,,-alpha,,-1),(1,,-1,,-alpha):}|=0`
`" or " alpha(alpha^(2) -1)-1(alpha-1)+(1-alpha)=0`
`" or " alpha(alpha^(2)-1) -2alpha +2=0`
`" or " alpha(alpha -1) (alpha+1) -2(alpha -1)=0`
`" or " (alpha-1)(alpha^(2) +alpha -2) =0`
`" or " (alpha -1) (alpha+2)(alpha -1) =0`
`" or " (alpha-1)^(2)(alpha+2)=0`
`" or " alpha=1 ,1,-2`
But for `alpha =1` there are infinite solution. when `alpha=-2` we have
`-2x -y-z=-3`
`x+2y -z=-3`
`x-y +2z=-3`
Adding we get 0=-9 which is not true . Hence there is no solution.