Consider the data on x taking the values
0, 2, 4, 8,....., 2ⁿ with frequencies
ⁿC₀ , ⁿC₁ , ⁿC₂ ,...., ⁿC_n respectively. If the
mean of this data is $${{728} \over {{2^n}}}$$, then n is equal to _________ .

For a plane electromagnetic wave, the magnetic field at a point x and time t is

$$\overrightarrow B \left( {x,t} \right)$$ = $$\left[ {1.2 \times {{10}^{ - 7}}\sin \left( {0.5 \times {{10}^3}x + 1.5 \times {{10}^{11}}t} \right)\widehat k} \right]$$ T

The instantaneous electric field $$\overrightarrow E $$ corresponding to $$\overrightarrow B $$ is :
(speed of light c = 3 × 10⁸ ms^–1)

Particle A of mass m₁ moving with velocity $$\left( {\sqrt3\widehat i + \widehat j} \right)m{s^{ - 1}}$$ collides with another particle B of mass m₂ which is at rest initially. Let $$\overrightarrow {{V_1}} $$ and $$\overrightarrow {{V_2}} $$ be the velocities of particles A and B after collision respectively. If m₁ = 2m₂ and after
collision $$\overrightarrow {{V_1}} = $$$$\left( {\widehat i + \sqrt 3 \widehat j} \right)$$ , the angle between $$\overrightarrow {{V_1}} $$ and $$\overrightarrow {{V_2}} $$ is :

When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with speed v, he sees that rain drops are coming at an angle 60° from the horizontal. On further increasing the speed of the car to (1 + $$\beta $$)v, this angle changes to 45^o. The value of $$\beta $$ is close to :