If ω is a complex cube root of unity, then find the value of ((a+b ω+c ω2/c+a ω+b ω2)+(a+b ω+c ω2/b+c ω+a ω2))2.

Question

If ω is a complex cube root of unity, then find the value of ((a+b ω+c ω2/c+a ω+b ω2)+(a+b ω+c ω2/b+c ω+a ω2))2. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Required expression =[(ω2(a+b ω+c ω2)/a ω3+b ω4+c ω2)+(ω(a+b ω+c ω2)/a ω3+b ω+c ω2)]2 =[(ω2(a+b ω+c ω2)/a+b ω+c ω2)+(ω(a+b ω+c ω2)/a+b ω+c ω2)]2 =(ω2+ω)2=(-1)2 =1

Q. If ω is a complex cube root of unity, then find the value of (c+aω+bω2a+bω+cω2​+b+cω+aω2a+bω+cω2​)2.

Required expression =[aω3+bω4+cω2ω2(a+bω+cω2)​+aω3+bω+cω2ω(a+bω+cω2)​]2 =[a+bω+cω2ω2(a+bω+cω2)​+a+bω+cω2ω(a+bω+cω2)​]2 =(ω2+ω)2=(−1)2 =1

Q. If $ω$ is a complex cube root of unity, then find the value of $(\frac{a + bω + c ω ^{2}}{c + aω + b ω ^{2}} + \frac{a + bω + c ω ^{2}}{b + c ω + a ω ^{2}})^{2}$ .