QJEEM21Madcaa: JEE Main 2021 Mathematics - Class 11 JEEM21Madcaa Solution: The sum of 162 th power of the roots of the equation x 3 − 2x 2 …

x³ $$-$$ 2x² + 2x $$-$$ 1 = 0

x = 1 satisfying the equation

$$\therefore$$ x $$-$$ 1 is factor of

x³ $$-$$ 2x² + 2x $$-$$ 1

= (x $$-$$ 1) (x² $$-$$ x + 1) = 0

x = 1, $$\frac{1+i\sqrt{3}}{2},\frac{1-i\sqrt{3}}{2}$$

x = 1, $$-$$ $$\omega$$², $$-$$$$\omega$$

Sum of 162^th power of roots

= (1)¹⁶² + ($$-$$$$\omega$$²)¹⁶² + ($$-$$$$\omega$$)¹⁶²

= 1 + ($$\omega$$)³²⁴ + ($$\omega$$)¹⁶²

= 1 + 1 + 1 = 3