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.

Show that 1 + cot^2 alpha / (1+ cosec alpha) = cosec alpha

If the roots of the equation px ^(2) +qx + r=0, where 2p , q, 2r are in G.P, are of the form alpha ^(2), 4 alpha-4. Then the value of 2p + 4q+7r is :

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)

For a complex number Z, if one root of the equation Z^(2)-aZ+a=0 is (1+i) and its other root is alpha , then the value of (a)/(alpha^(4)) is equal to

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:

Find the range of real number alpha for which the equation z+alpha|z-1|+2i=0 has a solution.

Find the range of real number alpha for which the equation z+alpha|z-1|+2i=0 has a solution.

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=palpha^n+qbeta^n value alpha_9=

If alpha be a root of equation x^2+x+1=0 then find the vlaue of (alpha+ 1/alpha)+(alpha^2+1/alpha^2)^2+(alpha^3+1/alpha^3)^2+…+(alpha^6+1/alpha^6)^2

if (1+tan alpha )(1+tan4 alpha ) =2 where alpha in (0 , pi/16 ) then alpha equal to