When a wave travels in a medium,the particle displacements are given by $y = a \sin 2\pi (bt - cx)$ where $a, b,$ and $c$ are constants. The maximum particle velocity will be twice the wave velocity if

$A$ ball is allowed to fall from a height of $10\, m$. If there is $40\%$ loss of energy due to impact,then after one impact,the ball will rise to a height of ......... $m$.

The minimum force required to start pushing a body up a rough (having coefficient of friction $\mu$) inclined plane is $F_1$, while the minimum force needed to prevent it from sliding down is $F_2$. If the inclined plane makes an angle $\theta$ with the horizontal such that $\tan \theta = 2\mu$, then the ratio $F_1 / F_2$ is

$A$ proton, a deuteron, and an $\alpha$-particle having the same momentum enter a region of uniform electric field between the parallel plates of a capacitor. The electric field is perpendicular to the initial path of the particles. Then the ratio of deflections suffered by them is

If the area of the triangle formed by the pair of lines $8x^2 - 6xy + y^2 = 0$ and the line $2x + 3y = a$ is $7$,then $a$ is equal to

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