JEE MAIN - Mathematics (2011 - No. 20)
If $$\omega ( \ne 1)$$ is a cube root of unity, and $${(1 + \omega )^7} = A + B\omega \,$$. Then $$(A,B)$$ equals :
(1 ,1)
(1, 0)
(- 1 ,1)
(0 ,1)
Explication
$${\left( {1 + \omega } \right)^7} = A + B\omega ;\,\,\,\,{\left( { - {\omega ^2}} \right)^7} = A + B\omega $$
$$ - {\omega ^2} = A + B\omega ;\,\,\,\,\,\,\,\,\,\,1 + \omega = A + B\omega $$
$$ \Rightarrow A = 1,B = 1.$$
$$ - {\omega ^2} = A + B\omega ;\,\,\,\,\,\,\,\,\,\,1 + \omega = A + B\omega $$
$$ \Rightarrow A = 1,B = 1.$$