Consider the following statements in respect of the determinant $$\left| {\matrix{ {{{\cos }^2}{\alpha \over 2}} & {{{\sin }^2}{\alpha \over 2}} \cr {{{\sin }^2}{\beta \over 2}} & {{{\cos }^2}{\beta \over 2}} \cr } } \right|$$ where $$\alpha$$, $$\beta$$ are complementary angles.

1. The value of the determinant is $${1 \over {\sqrt 2 }}\cos \left( {{{\alpha - \beta } \over 2}} \right)$$.

2. The maximum value of the determinant is $${1 \over {\sqrt 2 }}$$.

Which of the above statement(s) is/are correct?

If a, b, c are real numbers, then the value of the determinant $$\left| {\matrix{ {1 - a} & {a - b - c} & {b + c} \cr {1 - b} & {b - c - a} & {c + a} \cr {1 - c} & {c - a - b} & {a + b} \cr } } \right|$$ is

For the system of linear equations 2x + 3y + 5z = 9, 7x + 3y $$-$$ 2z = 8 and 2x + 3y + $$\lambda$$z = $$\mu$$

Under what condition does the above system of equations have infinitely many solutions?

For the system of linear equations 2x + 3y + 5z = 9, 7x + 3y $$-$$ 2z = 8 and 2x + 3y + $$\lambda$$z = $$\mu$$

Under what condition does the above system of equations have unique solutions?