Let a complex number `alpha,alpha!=1,` be a rootof hte euation `z^(p+q)-z^p-z^q+1=0,w h e r ep ,q` are distinct primes. Show that either `1+alpha+alpha^2++alpha^(p-1)=0or1+alpha+alpha^2++alpha^(q-1)=0` , but not both together.

Let a complex number alpha,alpha!=1, be a rootof hte euation z^(p+q)-z^(p)-z^(q)+1=0, where p,q are distinct primes.Show that either 1+alpha+alpha^(2)+...+alpha^(p-1)=0 or 1+alpha+alpha^(2)+...+alpha^(q-1)=0 but not both together.

If tan alpha+sin alpha=p and tan alpha-sin alpha=q tehn p^2-q^2=

If alpha,beta are roots of the equation x^(2)-px+q=0, then find the value of (i)alpha^(2)(alpha^(2)beta^(-1)-beta)+beta^(2)(beta^(2)alpha^(-1)-alpha)

If alpha,beta be the roots of the equation x^(2)-px+q=0 then find the equation whose roots are (q)/(p-alpha) and (q)/(p-beta)

The system of equations alpha x-y-z=alpha-1 , x-alpha y-z=alpha-1,x-y-alpha z=alpha-1 has no solution if alpha

If alpha,beta are the roots of the equation x^(2)+px+q=0, then -(1)/(alpha),-(1)/(beta) are the roots of the equation.

Let p,q be integers and let alpha,beta be the roots of the equation x^(2)-2x+3=0 where alpha!=beta For n=0,1,2,......, Let alpha_(n)=p alpha^(n)+q beta^(n) value alpha_(9)=

Let alpha be a root of the equation x ^(2) - x+1=0, and the matrix A=[{:(1,1,1),(1, alpha , alpha ^(2)), (1, alpha ^(2), alpha ^(4)):}] and matrix B= [{:(1,-1, -1),(1, alpha, - alpha ^(2)),(-1, -alpha ^(2), - alpha ^(4)):}] then the vlaue of |AB| is: