The coefficient of $x^n$ in $\frac{1-2x}{e^x}$ is:

An infinite number of identical capacitors each of capacitance $1 \mu F$ are connected as shown in the figure. Then,the equivalent capacitance between $A$ and $B$ is .......... $\mu F$.

When a resistor of $11 \Omega$ is connected in series with an electric cell,the current flowing in it is $0.5 \ A$. Instead,when a resistor of $5 \Omega$ is connected to the same electric cell in series,the current increases by $0.4 \ A$. The internal resistance of the cell is (in $Omega$)

At $25 \, ^oC$,the equilibrium $SO_2Cl_2(g) \rightleftharpoons SO_2(g) + Cl_2(g)$ is established in a closed vessel. When $Cl_2$ is added to this equilibrium mixture,which of the following statements is true for the system?
$(1)$ The concentrations of $SO_2Cl_2$ and $SO_2$ change.
$(2)$ More $Cl_2$ is produced.
$(3)$ The concentration of $SO_2$ decreases and $SO_2Cl_2$ increases.

Two stars of masses $M$ and $2M$ are at a distance $d$ apart and are revolving around their common center of mass. The angular velocity of the system of two stars is ($G$ is the universal gravitational constant):

Three charges $Q$,$+q$,and $+q$ are placed at the vertices of a right-angled isosceles triangle as shown. The net electrostatic potential energy of the configuration is zero if $Q$ is equal to