In an experiment to find out the diameter of wire using screw gauge, the following observations were noted :

(A) Screw moves $$0.5 \mathrm{~mm}$$ on main scale in one complete rotation

(B) Total divisions on circular scale $$=50$$

(C) Main scale reading is $$2.5 \mathrm{~mm}$$

(D) $$45^{\text {th }}$$ division of circular scale is in the pitch line

(E) Instrument has 0.03 mm negative error

Then the diameter of wire is :

An object is projected in the air with initial velocity u at an angle $$\theta$$. The projectile motion is such that the horizontal range R, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be _________ degree.

If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth $$\alpha \mathrm{h}\left(\mathrm{h}<<\mathrm{R}_{\mathrm{e}}\right)$$ from the earth surface. The value of $$\alpha$$ will be _________.

(use $$\left.\mathrm{R}_{\mathrm{e}}=6400 \mathrm{~km}\right)$$

The pressure $$\mathrm{P}_{1}$$ and density $$\mathrm{d}_{1}$$ of diatomic gas $$\left(\gamma=\frac{7}{5}\right)$$ changes suddenly to $$\mathrm{P}_{2}\left(>\mathrm{P}_{1}\right)$$ and $$\mathrm{d}_{2}$$ respectively during an adiabatic process. The temperature of the gas increases and becomes ________ times of its initial temperature. (given $$\frac{\mathrm{d}_{2}}{\mathrm{~d}_{1}}=32$$)