If `a ,b , c` are nonzero real numbers such that `|b cc a a b c a a bb c a bb cc a|=0,t h e n` `1/a+1/(bomega)+1/(comega^2)=0` b. `1/a+1/(bomega^2)+1/(comega^)=0` c. `1/(aomega)+1/(bomega^2)+1/c=0` d. none of these

If omega ne 1 is a cube root of unity, then show that (a+bomega+comega^(2))/(b+comega+aomega^(2))+(a+bomega+comega^(2))/(c+aomega+bomega^(2))=1

If a,b,c are real numbers such that 0 lt a lt 1, 0 lt b lt 1, 0 lt c lt 1, a + b + c = 2 , then prove that (a)/(1 - a) (b)/(1 - b) (c )/(1 - c) ge 8 .

Prove that =|1 1 1 a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

If |b+cc+a a+b a+bb+cc+a c+a a+bb+c|=k|a b cc a bb c a| , then value of k is 1 b. 2 c. 3 d. 4

Let a ,b ,a n dc be distinct nonzero real numbers such that (1-a^3)/a=(1-b^3)/b=(1-c^3)/c dot The value of (a^3+b^3+c^3) is _____________.

If p=a+bomega+comega^2,q=b+comega+aomega^2,a n dr=c+aomega+bomega^2,w h e r ea ,b ,c!=0a n domega is the complex cube root of unity, then

Let omega= e^((ipi)/3) and a, b, c, x, y, z be non-zero complex numbers such that a+b+c = x, a + bomega + comega^2 = y, a + bomega^2 + comega = z .Then, the value of (|x|^2+|y|^2|+|y|^2)/(|a|^2+|b|^2+|c|^2)

Two systems of rectangular axes have the same origin. If a plane cuts them at distance a ,b ,c and a^prime ,b^(prime),c ' from the origin, then a. 1/(a^2)+1/(b^2)+1/(c^2)+1/(a^('2))+1/(b^('2))+1/(c^('2))=0 b. 1/(a^2)-1/(b^2)-1/(c^2)+1/(a^('2))-1/(b^('2))-1/(c^('2))=0 c. 1/(a^2)+1/(b^2)+1/(c^2)-1/(a^('2))-1/(b^('2))-1/(c^('2))=0 d. 1/(a^2)+1/(b^2)+1/(c^2)+1/(a^('2))+1/(b^('2))+1/(c^('2))=0

Show that: |a b-cc+b a+c b c-a a-bb+a c|=(a+b+c)(a^2+b^2+c^2)dot

If a, b, c are the sides of a scalene triangle such that |[a , a^2 , a^(3)-1 ],[b , b^2 , b^(3)-1],[ c, c^2 , c^(3)-1]|=0 , then the geometric mean of a, b c is