Read the following information carefully and answer the questions given below.

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $${x^2} - (1 - 2{a^2})x + (1 - 2{a^2}) = 0$$.

Under what condition does the above equation have real roots?

Read the following information carefully and answer the questions given below.

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $${x^2} - (1 - 2{a^2})x + (1 - 2{a^2}) = 0$$.

Under what condition is $${1 \over {{\alpha ^2}}} + {1 \over {{\beta ^2}}} < 1$$?

Read the following information carefully and answer the questions given below.

2x² + 3x $$-$$ $$\alpha$$ = 0 has roots $$-$$2 and $$\beta$$ while the equation x² $$-$$ 3mx + 2m² = 0 has both roots positive, where $$\alpha$$ > 0 and $$\beta$$ > 0.

What is the value of $$\alpha$$?

Read the following information carefully and answer the questions given below.

2x² + 3x $$-$$ $$\alpha$$ = 0 has roots $$-$$2 and $$\beta$$ while the equation x² $$-$$ 3mx + 2m² = 0 has both roots positive, where $$\alpha$$ > 0 and $$\beta$$ > 0.

If $$\beta$$, 2, 2 m are in GP, then what is the value of $$\beta$$$$\sqrt m $$?