An infinitely long rod lies along the axis of a concave mirror of focal length $f$. The near end of the rod is at a distance $u > f$ from the mirror. Its image will have a length:

Consider the following gaseous equilibrium in a closed container of volume $V$ at $T \text{ K}$:
$P_{2(g)} + Q_{2(g)} \rightleftharpoons 2PQ_{(g)}$.
$2 \text{ moles}$ each of $P_{2(g)}$, $Q_{2(g)}$, and $PQ_{(g)}$ are present at equilibrium. Now, $1 \text{ mole}$ each of $P_{2}$ and $Q_{2}$ are added to the equilibrium while keeping the temperature at $T \text{ K}$. The number of moles of $P_{2}$, $Q_{2}$, and $PQ$ at the new equilibrium, respectively, are:

The catalyst used for the polymerization of olefins is....

Along the $x$-axis,three charges $\frac{q}{2}, -q$ and $\frac{q}{2}$ are placed at $x=0, x=a$ and $x=2a$ respectively. The resultant electric potential at a point $P$ located at a distance $r$ from the charge $-q$ (where $r > a$) is ($\varepsilon_0$ is the permittivity of free space):

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 35x + 2 = 0$,then the value of $(2\alpha - 35)^3 \cdot (2\beta - 35)^3$ is equal to ......

$A$ piece of semiconductor is connected in series in an electric circuit. On increasing the temperature,the current in the circuit will