A hyperbola, having the transverse axis of the length $$2\sin \theta $$, is confocal with the ellipse $$3{x^2} + 4{y^2} = 12$$. Then its equation is

The number of distinct real values of $$\lambda$$, for which the vectors $$ - {\lambda ^2}\widehat i + \widehat j + \widehat k,\widehat i - {\lambda ^2}\widehat j + \widehat k$$ and $$\widehat i + \widehat j - {\lambda ^2}\widehat k$$ are coplanar, is :

A man walks a distance of 3 units from the origin towards the north-east (N 45$$^\circ$$E) direction. From there, he walks a distance of 4 units towards the north-west (N 45$$^\circ$$W) direction to reach a point P. Then the position of P in the Argand plane is

The number of solutions of the pair of equations

$$2{\sin ^2}\theta - \cos 2\theta = 0$$

$$2{\cos ^2}\theta - 3\sin \theta = 0$$

in the interval $$[0,2\pi]$$ is