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 .

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.

If A =({:( 1,1,1),( 1,omega ^(2) , omega ),( 1 ,omega , omega ^(2)) :}) where omega is a complex cube root of unity then adj -A equals

Prove the following (a + b omega + c omega^(2))/(b + c omega + a omega^(2))= omega

The common roots of the equation x^(3) + 2x^(2) + 2x + 1 = 0 and 1+ x^(2008)+ x^(2003) = 0 are (where omega is a complex cube root of unity)

If the lines ax+by+c=0, bx+cy+a=0 and cx+ay+b=0 (a, b,c being distinct) are concurrent, then (A) a+b+c=0 (B) a+b+c=1 (C) ab+bc+ca=1 (D) ab+bc+ca=0

Prove that the value of determinant |{:(1,,omega,,omega^(2)),(omega ,,omega^(2),,1),( omega^(2),, 1,,omega):}|=0 where omega is complex cube root of unity .

If A is unimodular, then which of the following is unimodular? -A b. A^(-1) c. a d jA d. omegaA , where omega is cube root of unity

Let S = { z: x = x+ iy, y ge 0,|z-z_(0)| le 1} , where |z_(0)|= |z_(0) - omega|= |z_(0) - omega^(2)|, omega and omega^(2) are non-real cube roots of unity. Then

If a , b , c are real numbers, prove that |[a,b,c],[b,c,a],[c,a,b]|=-(a+b+c)(c+b w+c w^2)(a+b w^2+c w), where w is a complex cube root of unity.

If the lines (a-b-c) x + 2ay + 2a = 0 , 2bx + ( b- c - a) y + 2b = 0 and (2c+1) x + 2cy + c - a - b = 0 are concurrent , then the prove that either a+b+ c = 0 or (a+b+c)^(2) + 2a = 0