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:

Two strings $A$ and $B$ of lengths $L_A = 80 \ cm$ and $L_B = x \ cm$ respectively are used separately in a sonometer. The ratio of their densities $(d_A / d_B)$ is $0.81$. The diameter of $B$ is one-half that of $A$. If the strings have the same tension and fundamental frequency,the value of $x$ is:

The term independent of $x$ in the expansion of $\left(\sqrt{x}-\frac{2}{\sqrt{x}}\right)^{18}$ is

If $\tan \theta + \tan \left(\theta + \frac{\pi}{3}\right) + \tan \left(\theta + \frac{2\pi}{3}\right) = 3$,then which of the following is equal to $1$?

If $x^y=y^x$, then $x(x-y \log x) \frac{d y}{d x}$ is equal to :

$A$ lift is moving downwards with an acceleration equal to the acceleration due to gravity. $A$ body of mass $M$ kept on the floor of the lift is pulled horizontally. If the coefficient of friction is $\mu$,then the frictional resistance offered by the body is: