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

Let H$$_1$$, H$$_2$$, ..., H$$_n$$ be mutually exclusive and exhaustive events with P(H$$_i$$) > 0, i = 1, 2, ..., n. Let E be any other event with 0 < P(E) < 1.

Statement 1 : P(H$$_i$$ | E) > P(E | H$$_i$$). P(H$$_i$$) for $$i=1,2,...,n$$.

Statement 2 : $$\sum\limits_{i = 1}^n {P({H_i}) = 1} $$.

Tangents are drawn from the point (17, 7) to the circle $$x^2+y^2=169$$.

Statement 1 : The tangents are mutually perpendicular.

Statement 2 : The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $$x^2+y^2=338$$

Let the vector $$\overrightarrow {PQ} ,\overrightarrow {QR} ,\overrightarrow {RS} ,\overrightarrow {ST} ,\overrightarrow {TU} $$ and $$\overrightarrow {UP} $$, represent the sides of a regular hexagon.

Statement 1 : $$\overrightarrow {PQ} \times \left( {\overrightarrow {RS} + \overrightarrow {ST} } \right) \ne \overrightarrow 0 $$

Statement 2 : $$\overrightarrow {PQ} \times \overrightarrow {RS} = \overrightarrow 0 $$ and $$\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 $$