The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $$2\sqrt 2 $$ units in the south-westwardsdirection. Then its new position in the Argand plane is at the point represented by :

If P = $$\left[ {\matrix{ {{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr { - {1 \over 2}} & {{{\sqrt 3 } \over 2}} \cr } } \right],A = \left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\,\,\,$$

Q = PAP^T, then P^T Q²⁰¹⁵ P is :

If the equations x² + bx−1 = 0 and x² + x + b = 0 have a common root different from −1, then $$\left| b \right|$$ is equal to :

The number of distinct real roots of the equation,

$$\left| {\matrix{ {\cos x} & {\sin x} & {\sin x} \cr {\sin x} & {\cos x} & {\sin x} \cr {\sin x} & {\sin x} & {\cos x} \cr } } \right| = 0$$ in the interval $$\left[ { - {\pi \over 4},{\pi \over 4}} \right]$$ is :