$A$ tuning fork of frequency $392 \ Hz$ resonates with $50 \ cm$ length of a string under tension $T$. If the length of the string is decreased by $2\%$,keeping the tension constant,the number of beats heard when the string and the tuning fork are made to vibrate simultaneously is:

$A$ load of $1 ~kg$ weight is attached to one end of a steel wire of area of cross-section $3 ~mm^2$ and Young's modulus $10^{11} ~N/m^2$. The other end is suspended vertically from a hook on a wall,then the load is pulled horizontally and released. When the load passes through its lowest position,the fractional change in length is $(g = 10 ~m/s^2)$.

How many litres of oxygen (at $STP$) are required for complete combustion of $39 \text{ g}$ of liquid benzene? (Atomic weights: $C=12, O=16, H=1$)

If $y=\frac{\log _e x}{x}$ and $z=\log _e x$,then $\frac{d^2 y}{d z^2}+\frac{d y}{d z}$ is equal to

The distribution of relative intensity $I(\lambda )$ of blackbody radiation from a solid object versus the wavelength $\lambda $ is shown in the figure. If the Wien displacement law constant is $2.9 \times 10^{-3}\ mK$,what is the approximate temperature of the object in $K$?

The mass of a planet is half that of the earth and the radius of the planet is one-fourth that of earth. If we plan to send an artificial satellite from the planet,the escape velocity will be,(escape velocity on earth $v_e = 11 \ km \ s^{-1}$) (in $km \ s^{-1}$)