JEE MAIN - Mathematics (2021 - 27th July Morning Shift - No. 14)
Let $$\alpha$$, $$\beta$$ be two roots of the
equation x2 + (20)1/4x + (5)1/2 = 0. Then $$\alpha$$8 + $$\beta$$8 is equal to
equation x2 + (20)1/4x + (5)1/2 = 0. Then $$\alpha$$8 + $$\beta$$8 is equal to
10
100
50
160
Explanation
x2 + (20)1/4x + (5)1/2 = 0
$$ \Rightarrow $$ x2 + $$\sqrt 5$$ = - (20)1/4x
Squaring both sides, we get
$${\left( {{x^2} + \sqrt 5 } \right)^2} = \sqrt {20} {x^2}$$
$$ \Rightarrow $$ x4 = $$-$$5 $$\Rightarrow$$ x8 = 25
$$ \Rightarrow $$ $$\alpha$$8 + $$\beta$$8 = 50
$$ \Rightarrow $$ x2 + $$\sqrt 5$$ = - (20)1/4x
Squaring both sides, we get
$${\left( {{x^2} + \sqrt 5 } \right)^2} = \sqrt {20} {x^2}$$
$$ \Rightarrow $$ x4 = $$-$$5 $$\Rightarrow$$ x8 = 25
$$ \Rightarrow $$ $$\alpha$$8 + $$\beta$$8 = 50
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