The imaginary number $i$ is defined such that $i^{2} = - 1$ . What is the value of $(1 - i \sqrt{5}) (1 + i \sqrt{5})$ ?

\sqrt{5}

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5

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6

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\sqrt{6}

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Solution

The correct option is C $6$
Imaginary number is $i = \sqrt{- 1}$

Value of $(1 - i \sqrt{5}) (1 + i \sqrt{5})$

It is in the form of $(a + b) (a - b) = a^{2} - b^{2}$

So, $(1 - i \sqrt{5}) (1 + i \sqrt{5}) = {(1)}^{2} - {(i \sqrt{5})}^{2}$

$\Rightarrow 1 - (i^{2} \times 5)$

$\Rightarrow 1 - (- 1 \times 5)$

$\Rightarrow 6$

$(1 - i \sqrt{5}) (1 + i \sqrt{5}) = 6$

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{[\frac{- 1 + i \sqrt{3}}{2}]}^{6} + {[\frac{- 1 - i \sqrt{3}}{2}]}^{6} + {[\frac{- 1 + i \sqrt{3}}{2}]}^{5} + {[\frac{- 1 - i \sqrt{3}}{2}]}^{5} =

{[\frac{- 1 + i \sqrt{3}}{2}]}^{6} + {[\frac{- 1 - i \sqrt{3}}{2}]}^{6} + {[\frac{- 1 + i \sqrt{3}}{2}]}^{5} + {[\frac{- 1 - i \sqrt{3}}{2}]}^{5} =

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