UPSC 2014 CDS I Math Paper

m+n=−

b

a

and mn=

c

a

Sum of the roots =

m2+1

m

+

n2+1

n

=−

b

a

+

(− b a )

( c a )

=−

(ab+bc)

a

c
Product of the roots
=

m2+1

m

×

n2+1

n

=

m2n2+m2+n2+1

mn

=

m2n2+(m+n)2−2mn+1

mn

So, the required equation is x2−(Sum of the roots)x + Product of the roots = 0
=0
Alternative Method:Consider a=1,b=0,c=–1. Then we have
x2−1=0⇒x=±1
∴m=1,n=−1
∴

m2+1

m

=2,

n2+1

n

=−2
∴ Equation is x2−4=0
Also,
⇒−x2+4=0⇒x2−4=0