The Cartesian coordinates of the point whose polar coordinates are $\left(\frac{1}{2}, 120^{\circ}\right)$ are:

If a square $ABCD$,where $A(0,0), B(2,0), C(2,2)$ and $D(0,2)$ undergoes the following transformations successively,then the final figure would be a:
$(i)$ $f_1(x, y) \longrightarrow (y, x)$
(ii) $f_2(x, y) \longrightarrow (x+3y, y)$
(iii) $f_3(x, y) \longrightarrow \left(\frac{x-y}{2}, \frac{x+y}{2}\right)$

If $(-\sqrt{2}, \sqrt{2})$ are the Cartesian coordinates of a point,then its polar coordinates are $.....$

The point to which the origin is to be shifted to eliminate $x$ and $y$ terms of the equation $4x^2+9y^2-8x+36y+4=0$ is

To which point should the origin be shifted in order to eliminate the first-degree terms ($x$ and $y$ terms) from the equation $4x^2 + 9y^2 - 8x + 36y + 4 = 0$?

$A$ square,of each side $2$,lies above the $x-$axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle $30^o$ with the positive direction of the $x-$axis,then the sum of the $x-$coordinates of the vertices of the square is