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 root of the equation 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)=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 is a complex number such that alpha^(2)+alpha+1=0 then alpha^(31) is

If alpha is a complex number such that alpha^(2) - alpha + 1 = 0 , then alpha^(2011) =

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)=

If alpha is an imaginary root of the equation z^(n)-1=0 then 1+alpha+alpha^(2)+….. .+alpha^(n-1)=

Let alpha and beta be the roots of equation px^2 + qx + r = 0 , p != 0 .If p,q,r are in A.P. and 1/alpha+1/beta=4 , then the value of |alpha-beta| is :