UPSC NDA Math Model Paper 10

Here, α and β are the roots of the equation x2+x+1=0
α=

−1+√12−4

2

=

−1+√3i

2

and β=

−1−√3i

2

...... (Using‌x=

−b±√b2−4ac

2a

)
α=ω
and β=ω2
In equation x2−x+1=0
Roots =

1±√12−4

2

=

1±√3i

2

So, roots are −β and −α, i.e., −ω2 and −ω
Sum of roots in quadratic equation x2−x+1=0 , is 1
Assume roots are α7 and β13
α7+β13=((−ω)7+(−ω2)13)
=−((ω3)2+(ω3)8ω2)
=−(ω+ω2)
=−(−1)
=1
Hence, option (1) is correct.