Let $$a,\,b \in R\,and\,{a^{2\,}} + {b^2} \ne 0$$. Suppose
$$S = \left\{ {Z \in C:Z = {1 \over {a + ibt}}, + \in R,t \ne 0} \right\}$$, where $$i = \sqrt { - 1} $$. Ifz = x + iy and z $$ \in $$ S, then (x, y) lies on

The value of

$$\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{\left( {k - 1} \right)\pi } \over 6}} \right)\sin \left( {{\pi \over 4} + {{k\pi } \over 6}} \right)}}} $$ is equal to

Let $$\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k$$ be a unit vector in $${{R^3}}$$ and
$$\widehat w = {1 \over {\sqrt 6 }}\left( {\widehat i + \widehat j + 2\widehat k} \right).$$ Given that there exists a vector $${\overrightarrow v }$$ in $${{R^3}}$$ such that $$\left| {\widehat u \times \overrightarrow v } \right| = 1$$ and $$\widehat w.\left( {\widehat u \times \overrightarrow v } \right) = 1.$$ Which of the following statement(s) is (are) correct?

Let $$P$$ be the image of the point $$(3,1,7)$$ with respect to the plane $$x-y+z=3.$$ Then the equation of the plane passing through $$P$$ and containing the straight line $${x \over 1} = {y \over 2} = {z \over 1}$$ is