Electrons with de-Broglie wavelength $\lambda$ fall on the target of an $X$-ray tube. The cut-off wavelength of the emitted $X$-ray is:

Starting from rest,a body slides down a $45^\circ$ inclined plane in twice the time it takes to slide down the same distance in the absence of friction. The coefficient of friction between the body and the inclined plane is

$A$ magnetized wire of magnetic moment $M$ and length $L$ is bent into an arc of a circle subtending an angle of $60^\circ$ at the center. The new magnetic moment is:

$A$ string of mass $m$ and length $l$ hangs from the ceiling as shown in the figure. $A$ wave in the string moves upward. $v_A$ and $v_B$ are the speeds of the wave at points $A$ and $B$ respectively. Then $v_B$ is

If the best and the worst paper never appear together,then in how many ways can six examination papers be arranged?

Emulsification of $10 \, mL$ of oil in water produces $24 \times 10^{18}$ droplets. If the surface tension at the oil-water interface is $0.03 \, J \, m^{-2}$ and the area of each droplet is $12.5 \times 10^{-16} \, m^2$,the energy spent in the formation of oil droplets is $.... \, J$.