Let O be the origin and A be the point $${z_1} = 1 + 2i$$. If B is the point $${z_2}$$, $${\mathop{\rm Re}\nolimits} ({z_2}) < 0$$, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?

If the system of linear equations.

$$8x + y + 4z = - 2$$

$$x + y + z = 0$$

$$\lambda x - 3y = \mu $$

has infinitely many solutions, then the distance of the point $$\left( {\lambda ,\mu , - {1 \over 2}} \right)$$ from the plane $$8x + y + 4z + 2 = 0$$ is :

Let A be a 2 $$\times$$ 2 matrix with det (A) = $$-$$ 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be :

The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = y^a is $${{364} \over 3}$$, is equal to :