UPSC NDA Math Model Paper 4

Using the expressions for the sum and the product of the roots, we have:
α+β=−p ... (1)
αβ=

p2

2

... (2)
α4+β4=r ... (3)
Squaring equation (1), we get:
α2+β2+2αβ=p2
Using equation (2), we get:
⇒α2+β2=0
Squaring again, we get:
⇒α4+β4+2α2β2=0
Using equations (2) and (3), we get:
⇒r=

p4

2

... (4)
The discriminant of the equation 2x2−4p2x+4p4−2r=0 is:
(−4p2)2−4(2)(4p4−2r)
=16p4−32p4+16r
Using equation (4), we get:
=16p4−32p4+8p4
=−8p4, which is always negative for non-zero real p.
Since the discriminant is <0, the roots are imaginary.