If $f(x) = \left|\begin{array}{ccc} 1 & x & x+1 \\ 2x & x(x-1) & x(x+1) \\ 3x(x-1) & x(x-1)(x-2) & (x-1)x(x+1) \end{array}\right|$,then $f(2012)$ is equal to:

The circle $4x^2 + 4y^2 - 12x - 12y + 9 = 0$

If $u=f(r)$,where $r^2=x^2+y^2$,then $\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)$ is equal to

If the $x$-coordinate of a point $P$ on the line joining the points $Q(2, 2, 1)$ and $R(5, 2, -2)$ is $4$,then the $z$-coordinate of $P$ is:

When two sound sources of the same amplitude but of slightly different frequencies $v_1$ and $v_2$ are sounded simultaneously,the sound one hears has a frequency equal to

All wires have the same resistance $r$. The equivalent resistance between $A$ and $B$ is $R$ when the keys $K_1$ and $K_2$ are open. Now,if the keys are closed,what will be the new equivalent resistance?