Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat{i} + 2\hat{j} + \hat{k}$ and $-3\hat{i} - 2\hat{j} + \hat{k}$,respectively. The centre of mass of this system has a position vector:

If $A(1,2,3), B(3,7,-2), C(6,7,7)$ and $D(-1,0,-1)$ are points in a plane, then the vector equation of the line passing through the centroids of $\triangle ABD$ and $\triangle ACD$ is

The temperature of the system decreases in the process of

If the circle $x^2+y^2+6x-2y+k=0$ bisects the circumference of the circle $x^2+y^2+2x-6y-15=0$,then $k$ is equal to

The $P-V$ diagram of a system undergoing thermodynamic transformation is shown in the figure. The work done on the system in going from $A$ $\rightarrow B$ $\rightarrow C$ is $50 \ J$ and $20 \ cal$ of heat is given to the system. The change in internal energy between $A$ and $C$ is ...... $J$.

Electrons ejected from the surface of a metal,when light of certain frequency is incident on it,are stopped fully by a retarding potential of $3 \ V$. Photoelectric effect in this metallic surface begins at a frequency $6 \times 10^{14} \ s^{-1}$. The frequency of the incident light in $s^{-1}$ is : [Planck's constant $= 6.4 \times 10^{-34} \ J \ s$,charge on the electron $= 1.6 \times 10^{-19} \ C$]