The displacement-time graph for two particles $A$ and $B$ are straight lines inclined at angles of $30^o$ and $60^o$ with the time axis. The ratio of velocities of $V_A : V_B$ is

Two ideal slits $S_1$ and $S_2$ are at a distance $d$ apart and are illuminated by light of wavelength $\lambda$ passing through an ideal source slit $S$ placed on the line through $S_2$ as shown. The distance between the plane of the slits and the source slit is $D$. $A$ screen is held at a distance $D$ from the plane of the slits. The minimum value of $d$ for which there is darkness at $O$ is

$A$ student measures the distance traversed in free fall of a body,initially at rest in a given time. He uses this data to estimate $g$,the acceleration due to gravity. If the maximum percentage errors in measurement of the distance and the time are $e_1$ and $e_2$ respectively,the percentage error in the estimation of $g$ is

$A$ rectangular loop of length $l$ and breadth $b$ is placed at a distance of $x$ from an infinitely long wire carrying current $i$ such that the direction of the current is parallel to the breadth. If the loop moves away from the current-carrying wire in a direction perpendicular to it with a velocity $v$,the magnitude of the induced emf in the loop is: ($\mu_0=$ permeability of free space)

$A$ gramophone record is revolving with an angular velocity $\omega$. $A$ coin is placed at a distance $R$ from the centre of the record. The static coefficient of friction is $\mu$. The coin will revolve with the record if

If $(a + \sqrt{2} b \cos x)(a - \sqrt{2} b \cos y) = a^2 - b^2$,where $a > b > 0$,then $\frac{dx}{dy}$ at $(\frac{\pi}{4}, \frac{\pi}{4})$ is