UPSC CDS I 2025 Mathematics Paper

First, rearrange the given equation into the standard quadratic form Ax2+Bx+C=0.
x+a+b=(‌

abx

ab+ax+bx

)
Multiply both sides by ( ab+ax+bx ):
‌⇒(x+a+b)(ab+ax+bx)=abx
‌⇒x(ab)+x(ax)+x(bx)+a(ab)+a(ax)+a(bx)+b(ab)+b(ax)+b(bx)=abx
‌⇒abx+ax2+bx2+a2b+a2x+abx+ab2+abx+b2x=abx
‌⇒ax2+bx2+abx+a2x+abx+abx+b2x−abx+a2b+ab2=0
‌⇒(a+b)x2+(a2x+b2x+2abx)+(a2b+ab2)=0
‌⇒(a+b)x2+(a2+b2+2ab)x+ab(a+b)=0
Recognize that (a2+b2+2ab)=(a+b)2 :
‌⇒(a+b)x2+(a+b)2x+ab(a+b)=0
‌⇒x2+(a+b)x+ab=0
Sum and product of the roots ( α and β ):
Sum of roots: α+β=(−‌

B

A

)=−‌

(a+b)

1

=−(a+b)
Product of roots: αβ=‌

C

A

=‌

ab

1

=ab
Finally, calculate (αβ+α+β) :
‌⇒(αβ+α+β)=ab+(−(a+b))
‌⇒(αβ+α+β)=ab−a−b