$A$ transverse harmonic wave on a string is described by $y = 3 \sin(36t + 0.018x + \frac{\pi}{4})$,where $x$ and $y$ are in $cm$ and $t$ is in $s$. The least distance between two successive crests in the wave is .... $m$. (in $.5$)

If $y = \frac{e^x \log x}{x^2}$,then $\frac{dy}{dx} = $

The products formed in the reaction of phenol with $Br_2$ dissolved in $CS_2$ at $0^{\circ} C$ are

$A$ projectile is fired at a speed of $100 \, m/s$ at an angle of $37^\circ$ above the horizontal. At the highest point,the projectile breaks into two parts of mass ratio $1:3$,the smaller coming to rest. Then the distance of the heavier part from the launching point is ............... $m$.

$A$ block can slide on a smooth inclined plane of inclination $\theta$ kept on the floor of a lift. When the lift is descending with an acceleration $a$,the acceleration of the block relative to the incline is:

The equations of the circle which pass through the origin and make intercepts of lengths $4$ and $8$ on the $x$ and $y$-axes respectively are