If $$\overrightarrow a $$ and $$\overrightarrow b $$ are vectors in space given by $$\overrightarrow a = {{\widehat i - 2\widehat j} \over {\sqrt 5 }}$$ and $$\overrightarrow b = {{2\widehat i + \widehat j + 3\widehat k} \over {\sqrt {14} }},$$ then find the value of $$\,\left( {2\overrightarrow a + \overrightarrow b } \right).\left[ {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow a - 2\overrightarrow b } \right)} \right].$$

Let $$f$$ be a real-valued differentiable function on $$R$$ (the set of all real numbers) such that $$f(1)=1$$. If the $$y$$-intercept of the tangent at any point $$P(x,y)$$ on the curve $$y=f(x)$$ is equal to the cube of the abscissa of $$P$$, then find the value of $$f(-3)$$

Let $$P,Q,R$$ and $$S$$ be the points on the plane with position vectors $${ - 2\widehat i - \widehat j,4\widehat i,3\widehat i + 3\widehat j}$$ and $${ - 3\widehat i + 2\widehat j}$$ respectively. The quadrilateral $$PQRS$$ must be a

Equation of the plane containing the straight line $${x \over 2} = {y \over 3} = {z \over 4}$$ and perpendicular to the plane containing the straight lines $${x \over 3} = {y \over 4} = {z \over 2}$$ and $${x \over 4} = {y \over 2} = {z \over 3}$$ is