Math miscellaneousHard
Question
If ω is imaginary cube root of unity and (a + bω + cω2)2015 = (a + bω2 + cω) where a,b,c are unequal realnumbers, then a2 + b2 + c2 - ab - bc - ca is equal to -
Options
A.0
B.
C.1
D.-1
Solution
Let a + bω + cω2 = z
given equation is z2015 =
⇒ |z|2015 = || ⇒ |z |= 0 or 1
but |z| = 0 ⇒ a = b = c
|z| = 1 ⇒ z = 1
⇒ (a + bω + cω2)(a + bω2ω + cω) = 1
⇒ (a2 + b2 + c2 - ab - bc - ca) = 1
given equation is z2015 =
⇒ |z|2015 = |
| ⇒ |z |= 0 or 1but |z| = 0 ⇒ a = b = c
|z| = 1 ⇒ z
= 1⇒ (a + bω + cω2)(a + bω2ω + cω) = 1
⇒ (a2 + b2 + c2 - ab - bc - ca) = 1
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