One mole of an ideal gas is allowed to expand reversibly and adiabatically from a temperature of $27\,^{\circ}C$. If the work done during the process is $3\,kJ$,then the final temperature of the gas is ......$K$ $(C_V = 20\,J/K)$.

$A$ sonometer wire resonates with a given tuning fork,forming standing waves with five antinodes between the two bridges when a mass of $9 \, kg$ is suspended from the wire. When this mass is replaced by a mass $M$,the wire resonates with the same tuning fork,forming three antinodes for the same positions of the bridges. The value of $M$ is .... $kg$.

At a certain place,the horizontal component of the Earth's magnetic field is $B_0$ and the angle of dip is $45^o$. The total intensity of the field at that place will be:

$A$ mercury drop of radius $1 ~cm$ is sprayed into $10^6$ drops of equal size. The energy expended in joules is (Surface tension of mercury is $460 \times 10^{-3} ~N/m$)

The equation of the image of the circle $x^2 + y^2 + 16x - 24y + 183 = 0$ by the line mirror $4x + 7y + 13 = 0$ is

$y + x^2 = \frac{dy}{dx}$ has the solution