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 , the infinitely many solutions , then `1+alpha + alpha^2`=

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 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

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 f(alpha,beta)=|(cos alpha,-sin alpha,1),(sin alpha,cos alpha,1),(cos(alpha+beta),-sin(alpha+beta),1)|, then

If x sin alpha = y cos alpha, prove that : x/(sec 2alpha) + y/(cosec 2 alpha) = x

cos alpha cos 2 alpha cos 4 alpha cos 8 alpha = (1)/(16), if alpha = 24 ^(@)