Let `alpha ` and `beta` be the roots of equation `x^2-(1-2a^2)x + (1-2a^2) =0` 1.) under what condition does the above equation have real roots. 1.) `a^2<1/2` 2.) `a^2>1/2` 3.) `a^2 <=1/2` 4.) `a^2>=1/2` 2.) Under what condition is `1/(alpha)^2 +1/(beta)^2< 1`?

Options

A. `a^(2) lt 1/2`
B. `a^(2) gt 1/2`
C. `a^(2) le 1/2`
D. `a^(2) ge 1/2`

Answer

D

Explanation

Using `ax^(2) + bx + c = 0 `
` a=1 , b = -(1 - 2a^(2)) & c = ( 1 - 2a^(2))`
For roots to be real,
` b^(2) - 4ac ge 0 `
` rArr [ - ( 1- 2a^(2))]^(2) - 4(1) ( 1 - 2a^(2)) ge0`
` rArr 4a^(4) + 4a^(2) - 3 ge 0 `
` rArr (2a^(2) - 1 ) ( 2a^(2) + 3 ) ge 0 `
` rArr a^(2) ge 1/2 or a^(2) le -3/2`