Question

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax² + bx + c = 0 and −ax² + bx + c = 0 has real roots.

Answer 1

SOLUTION :  

Given : ax² + bx + c = 0 …………(1)

and - ax² + bx + c = 0…………..(2)

On comparing the given equation with Ax² + Bx + C = 0  

Let D1 & D2 be the discriminants of the two given equations .

For eq 1 :  

Here, A = a  , B =  b , C = c

D(discriminant) = B² – 4AC

D1 = B² – 4AC = 0

D1 = (b)² - 4 × a × C

D1 = b² - 4ac ………….…(3)

For eq 2 :  

- ax² + bx + c = 0

Here, A = -a  , B =  b , C = c

D(discriminant) = B² – 4AC

D2 = (b)² - 4 × -a × c

D2 = b²  +  4ac…. …………(4)

Given : Roots are real for both the Given equations i.e D ≥ 0.

D1 ≥ 0  

b² - 4ac ≥ 0  

[From eq 3]

b²  ≥ 4ac  …………..(5)

D2 ≥ 0

b²  +  4ac ≥ 0 …………….(6)

From eq 5 & 6 ,  We proved that at least one of the given equation has real roots.

[Given : a,b,c are real number and ac ≠0]

HOPE THIS ANSWER WILL HELP YOU..

Answer 2

Answer:

A thin paper of thickness 0⋅02 mm having a refractive index 1⋅45 is pasted across one of the slits in a Young’s double slit experiment. The paper transmits 4/9 of the light energy falling on it. (a) Find the ratio of the maximum intensity to the minimum intensity in the fringe pattern. (b) How many fringes will cross through the centre if an identical paper piece is pasted on the other slit also? The wavelength of the light used is 600 nm.