UPSC NDA I 2025 Mathematics Paper

Given,
The equation is x2−x+1=0
We need to find the value of the following expression:
(x−‌

1

x

)2+(x−‌

1

x

)4+(x−‌

1

x

)8
The equation x2−x+1=0 is solved as follows:
x=‌

1±√−3

2

=eiπ∕3‌‌‌ or ‌‌‌x=e−iπ∕3
Now, substitute the value of X into the expression X−‌

1

X

:
x−‌

1

x

=i√‌3
Evaluate the powers of x−‌

1

x

Now, let's evaluate each term in the expression:
‌(x−‌

1

x

)2=(i√‌3)2=−3
‌(x−‌

1

x

)4=(−3)2=9
‌(x−‌

1

x

)8=92=81
Now, sum the values:
−3+9+81=87
∴ The value of the expression is 87 .