The angle by which the coordinate axes are to be rotated about the origin so that the transformed equation of $\sqrt{3} x^2+(\sqrt{3}-1) x y-y^2=0$ is free from the $xy$ term is: (in $^{\circ}$)

If the axes are rotated by an angle of $30^{\circ}$ in the negative direction (clockwise) while keeping the origin fixed,what are the new coordinates of the point $(2, 1)$?

$A$ vector $\vec{a}$ has components $3p$ and $1$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If,with respect to the new system,$\vec{a}$ has components $p+1$ and $\sqrt{10}$,then a value of $p$ is equal to

If the axes are rotated through an angle $45^{\circ}$ in the positive direction without changing the origin,then the coordinates of the point $(\sqrt{2}, 4)$ in the old system are

If the axes are translated to the orthocentre of the triangle formed by the points $A(7,5), B(-5,-7), C(7,-7)$,then the coordinates of the incentre of the triangle in the new system are

When the axes are rotated through an angle $45^{\circ}$,the new coordinates of a point $P$ are $(1, -1)$. The coordinates of $P$ in the original system are