If x,y and y are not all zero and if
ax+by+cz=0,bx+cy+az=0
and cx+ay+bz=0, then
prove that x:y:z=1:1:1 or `1:omega:omega^(2)"or" 1:omega^(2):omega`

If x, y, z are not all zero & if ax + by + cz=0, bx+ cy + az=0 & cx + ay + bz = 0 , then prove that x: y : z = 1 : 1 : 1 OR 1 :omega:omega^2 OR 1:omega^2:omega , where omega is one ofthe complex cube root of unity.

STATEMENT-1 : If a, b, c are distinct and x, y, z are not all zero and ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0 , then a + b + c = 0 and STATEMENT-2 : a^2 + b^2 + c^2 > ab + bc + ca , if a, b, c are distinct.

Show that x^(3) +y^(3) = (x + y) (omega x + omega^(2)y) (omega^(2)x + omega y)

The planes ax+4y+z=0,2y+3z-1=0 and 3x-bz+2=0 will

Prove that the lines ax+by + c = 0, bx+ cy + a = 0 and cx+ay+b=0 are concurrent if a+b+c = 0 or a+b omega + c omega^(2) + c omega = 0 where omega is a complex cube root of unity .

Without expanding at any stage, prove that |{:(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),1,omega):}| = 0

If omega!=1 is a cube root of unity and x+y+z!=0, then prove that |[x/(1+omega), y/(omega+omega^2),z/(omega^2+1)],[y/(omega+omega^2),z/(omega^2+1),x/(1+omega)],[(z)/(omega^2+1),x/(1+omega),y/(omega+omega^2)]|=0 if x=y=z

If |z| <= 1 and |omega| <= 1, show that |z-omega|^2 <= (|z|-|omega|)^2+(arg z-arg omega)^2

Given x=cy+bz,y=az+cx and z=bx+ay, then prove a^(2) +b^(2) +c^(2) +2abc =1.

If z=x+i y and w=(1-i z)//(z-i) and |w| = 1 , then show that z is purely real.