UPSC NDA Math Model Paper 7

Given , the roots of the equation 3ax2+2bx+c=0 are in the ratio 2 : 3.
Let the roots be 2x and 3x.
Let α and β are the roots of any quadratic equation ax2+bx+c=0 then α+β=

−b

a

and α×β=

c

a

Sum of the roots = 2x + 3x = 5x
⇒5x=

−2b

3a

⇒x=

−2~b

15a

⇒x2=

4b2

225a2

.... (1)
Product of the roots =2x×3x=6x2
⇒

c

3a

=6x2 .... (2)
From equation |(1) and (2)

4b2

225a2

=

c

18a

⇒

4b2

25a

=

c

4

⇒8b2=25ac