JEE MAIN - Mathematics (2020 - 4th September Evening Slot - No. 15)
If a and b are real numbers such that
$${\left( {2 + \alpha } \right)^4} = a + b\alpha $$
where $$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$ then a + b is equal to :
$${\left( {2 + \alpha } \right)^4} = a + b\alpha $$
where $$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$ then a + b is equal to :
33
9
24
57
Explanation
$$\alpha = \omega $$ as given $$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$
$$ \Rightarrow {(2 + \omega )^4} = a + b\omega \,({\omega ^3} = 1)$$
$$ \Rightarrow {2^4} + {4.2^3}\omega + {6.2^2}{\omega ^3} + 4.2.\,{\omega ^3} + {\omega ^4} = a + b\omega $$
$$ \Rightarrow 16 + 32\omega + 24{\omega ^2} + 8 + \omega = a + b\omega $$
$$ \Rightarrow 24 + 24{\omega ^2} + 33\omega = a + b\omega $$
$$ \Rightarrow - 24\omega + 33\omega = a + b\omega $$
$$ \Rightarrow a = 0,\,b = 9$$
$$ \Rightarrow {(2 + \omega )^4} = a + b\omega \,({\omega ^3} = 1)$$
$$ \Rightarrow {2^4} + {4.2^3}\omega + {6.2^2}{\omega ^3} + 4.2.\,{\omega ^3} + {\omega ^4} = a + b\omega $$
$$ \Rightarrow 16 + 32\omega + 24{\omega ^2} + 8 + \omega = a + b\omega $$
$$ \Rightarrow 24 + 24{\omega ^2} + 33\omega = a + b\omega $$
$$ \Rightarrow - 24\omega + 33\omega = a + b\omega $$
$$ \Rightarrow a = 0,\,b = 9$$
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