The maximum area (in square units) of a rectangle whose vertices lie on the ellipse $${x^2} + 4{y^2} = 1\,\,$$ is

For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the angle between the hypotenuse and the side is

The maximum value of $$f\left( x \right) = 2{x^3} - 9{x^2} + 12x - 3$$
in the interval $$\,0 \le x \le 3$$ is __________.

The Taylor series expansion of $$3$$ $$sin$$ $$x$$ $$+2cos$$ $$x$$ is