A diwali cracker of mass 60 g, at rest, explodes into three pieces A, B and C of mass 10 g, 20 and 30 g respectively. After explosion velocities of A and B are 30 m/s along east and 20 m/s along north respectively. Find the instantaneous velocity of C.

Q: If a, b, c ∈ R then find a relation between a, b, c so that two quadratic equations
ax²+bx+c =0 and 1003x² 1505x + 2007 = 0 have a common root

Ans:
We observe that the discriminant of the quadratic equation 1003x²+ 1505x + 2007 =0 is (1505)² - 4 x 1003 x 2007 < 0.

⇒Roots of the equation are complex

We know that complex roots of an equation occur in conjugate pairs, and in this case whenever α be a common root between given equations, other root automatically becomes common

Hence the given equations have both roots common and consequently

which is the required relation