QJEEM20Mad1f3: JEE Main 2020 Mathematics - Class 11 JEEM20Mad1f3 Solution: If a and b are real numbers such that where then a + b is equal …

Question

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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 :

Accepted Answer

\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

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