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=

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=

For alpha=1 , the pair of linear equations alpha x+3y=-7 and 2x+6y=14 has infinitely many solutions.

If f(alpha)=[[1,alpha,alpha^2],[alpha,alpha^2,1],[alpha^2,1,alpha]] then find the value of f(3^(1/3))

Prove that |{: (1,alpha,alpha^2), (alpha,alpha^2, 1),(alpha^2, 1,alpha) :}|=-(1-alpha^3)^2

If f(alpha) = |{:(1,alpha,alpha^2),(alpha,alpha^2,1),(alpha^2,1,alpha):}| , then find the value of f(3^(1/3)) .

If f(alpha)=|{:(1,alpha,alpha^(2)),(alpha,alpha^(2),1),(alpha^(2),1,alpha):}| , then f(root(3)(3)) is equal to

If alpha is a non real root of x^(6)=1 then alpha^(5)+alpha^(3)+alpha+1hline alpha^(2)+1

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

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