Home
Class 12
MATHS
If alpha, beta in C are distinct roots o...

If `alpha, beta in C` are distinct roots of the equation `x^2+1=0` then `alpha^(101)+beta^(107)` is equal to

A

2

B

`-1`

C

0

D

1

Text Solution

Verified by Experts

The correct Answer is:
D
`x^(2) - x + 1 =0`
`rArr x = (1 pm sqrt(-3))/(2)`
`= (1pm isqrt(3))/(2)`
`=-[(-1 pm isqrt(3))/(2)]`
`therefore x = - omega, -omega^(2)` where `omega` is imaginary cube root of unity.
Let `alpha = - omega and beta= - omega^(2)`
` therefore alpha^(101) + beta^(107)= (-omega )^(101) + (-omega^(2))^(107)`
`= -[omega^(101) + omega^(214)]`
` =-[omega^(99) omega^(2) + omega^(213)omega ]`
`=-[omega^(2) + omega]`
` =-[omega^(2) + omega]`
`=- (-1) = 1" "("as" 1+ omega+omega^(2) =0)`
If we consider `alpha = - omega^(2) and beta = -omega`, we get the same results.
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE|Exercise JEE Advanced Previous Year|14 Videos

  • COMPLEX NUMBERS

    CENGAGE|Exercise Question Bank|30 Videos

  • COMPLEX NUMBERS

    CENGAGE|Exercise Exercise (Numerical)|30 Videos

  • CIRCLES

    CENGAGE|Exercise Question Bank|32 Videos

  • CONIC SECTIONS

    CENGAGE|Exercise Solved Examples And Exercises|91 Videos

Similar Questions

Explore conceptually related problems

If alpha, beta in C are the distinct roots of the equation x^(2)-x+1=0 , then alpha^(101)+beta^(107) is equal to

if alpha and beta are the roots of the equation 2x^(2)-5x+3=0 then alpha^(2)beta+beta^(2)alpha is equal to

If alpha,beta are roots of the equation ax^(2)-bx-c=0, then alpha^(2)-alpha beta+beta^(2) is equal to-

If alpha,beta are the roots of the equation x^(2)-p(x+1)-c=0, then (alpha+1)(beta+1)=

If alpha and beta are the roots of the equation x^(2)-2x+4=0, then alpha^(9)+beta^(9) is equal to

If alpha and beta are the roots of the equation x^(2)-x+1=0, alpha^(2009)+beta^(2009 is equal to

If alpha and beta are two distinct roots of the equation a cos x+b sin x=10, then tan((alpha+beta)/(2)) is equal to