Two point white dots are $1\, mm$ apart on a black paper. They are viewed by an eye with a pupil diameter of $3\, mm$. Approximately,what is the maximum distance in $m$ at which these dots can be resolved by the eye? [Take wavelength of light $\lambda = 500\, nm$]

The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following transformations successively.
$I$. Moves to the new point under the given translation of origin.
$II$. Translated through $2$ units along the negative direction of the new $X$-axis.
$III$. Rotated through an angle $\frac{\pi}{4}$ about the origin of the new system in the clockwise direction. The final position of the point $(7,5)$ is

An ideal gas is taken through the cycle $A \to B \to C \to A$,as shown in the figure. If the net heat supplied to the gas in the cycle is $5 \ J$,the work done by the gas in the process $C \to A$ is ....... $J$.

$A$ wire of length $2$ units is cut into two parts which are bent respectively to form a square of side $= x$ units and a circle of radius $= r$ units. If the sum of the areas of the square and the circle so formed is minimum,then:

The radius of gyration of a disc of mass $50\,g$ and radius $2.5\,cm$,about an axis passing through its centre of gravity and perpendicular to the plane,is ....... $cm$.

Which of the following complex ions exhibits optical isomerism? ($en = 1,2$-diaminoethane)