The mass of a spaceship is $1000\,kg$. It is to be launched from the earth's surface out into free space. The values of $g$ and $R$ (radius of earth) are $10\,m/s^2$ and $6400\,km$ respectively. The required energy for this work will be:

The spectrum of a black body at two temperatures $27^oC$ and $327^oC$ is shown in the figure. Let $A_1$ and $A_2$ be the areas under the two curves respectively. The value of $\frac{A_2}{A_1}$ is

If $S_n = 1^3 + 2^3 + \ldots + n^3$ and $T_n = 1 + 2 + \ldots + n$,then

If an electron in a hydrogen atom jumps from an orbit of level $n=3$ to an orbit of level $n=2$,then the emitted radiation frequency is (where $R=$ Rydberg's constant,$C=$ velocity of light).

$A$ galvanometer,having a resistance of $50 \Omega$,gives a full-scale deflection for a current of $0.05 \text{ A}$. The length in metres of a resistance wire of cross-sectional area $2.97 \times 10^{-2} \text{ cm}^2$ that can be used to convert the galvanometer into an ammeter which can read a maximum of $5 \text{ A}$ current is: (Specific resistance of the wire $= 5 \times 10^{-7} \Omega\text{-m}$)

$A$ proton, a deuteron, and an $\alpha -$ particle enter a region of perpendicular magnetic field (to their velocities) with the same kinetic energy. If $r_p, r_d,$ and $r_{\alpha}$ are the radii of the circular paths of these particles, then: