For a real number `alpha` ,if the system `[[1,alpha,alpha^(2)],[alpha,1,alpha],[alpha^(2),alpha,1]][[x],[y],[z]]=[[1],[-1],[1]]]` of linear equations, has infinitely many solutions, then `1+alpha+alpha^(2)=`

For a real number alpha, if the system [1alphaalpha^2alpha1alphaalpha^2alpha1][x y z]=[1-1 1] of linear equations, has infinitely many solutions, then 1+alpha+alpha^2=

Prove that |1alphaalpha^2alphaalpha^2 1alpha^2 1alpha|=-(1-alpha^3)^2dot

If alpha is a non-real cube root of -2 , then the value of |(1,2 alpha,1),(alpha^(2),1,3 alpha^(2)),(2,2 alpha,1)| , is

Let A(alpha)=[(cos alpha, 0,sin alpha),(0,1,0),(sin alpha, 0, cos alpha)] and [(x,y,z)]=[(0,1,0)] . If the system of equations has infinite solutions and sum of all the possible value of alpha in [0, 2pi] is kpi , then the value of k is equal to

If alpha is nonreal and alpha= (1)^(1/5) then the find the value of 2^(|1+alpha+alpha^2+alpha^-2-alpha^-1|)

If alpha is an imaginary fifth root of unity, then log_(2)|1+alpha+alpha^(2)+alpha^(3)-1/alpha|=

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 solutions, is ________.

If alpha is a non-real fifth root of unity, then the value of 3^(|1+alpha+alpha^(2),alpha^(-2)-alpha^(-1)| , is

If a variable takes the discrete values alpha+4 , alpha -(7)/(2), alpha-(5)/(2), alpha-2,alpha-3,alpha+(1)/(2), alpha-(1)/(2), alpha+5(alpha gt 0) , then the median is

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 "____"