The sum 1 + 2 . 3 + 3 . 3² + ......... + 10 . 3⁹ is equal to :

Let p be the plane passing through the intersection of the planes $$\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 5$$ and $$\overrightarrow r \,.\,\left( {2\widehat i - \widehat j + \widehat k} \right) = 3$$, and the point (2, 1, $$-$$2). Let the position vectors of the points X and Y be $$\widehat i - 2\widehat j + 4\widehat k$$ and $$5\widehat i - \widehat j + 2\widehat k$$ respectively. Then the points :

A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is :

Water is being filled at the rate of 1 cm³ / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm² / sec) at which the wet conical surface area of the vessel increases is