CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Common roots of the equations z3+2z2+2z+1=0 and z1985+z100+1=0 are

A
ω,ω2
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
ω,ω3
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
ω2,ω3
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
None of these
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is A ω,ω2
The given equation z3+2z2+2z+1=0 can be rewritten as (z+1)(z2+z+1)=0. Its roots are 1,ω and ω2.
Let f(z)=z1985+z100+1
Putting z=1. ω and ω2 respectively, we get
f(1)=(1)1985+(1)100+10
Therefore, 1 is not a root of the equation f(z)=0.
Again, f(ω)=ω1985+ω100+1
=(ω3)661ω2+(ω3)33ω+1
=ω2+ω+1=0
Therefore, ω is a root of the equation f(z)=0.
Similarly, f(ω2)=0
Hence, ω and ω2 are the common roots.

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Properties of Conjugate of a Complex Number
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon