Two waves are simultaneously passing through a string and their equations are :

y₁ = A₁ sin k(x $$-$$ vt), y₂ = A₂ sin k(x $$-$$ vt + x₀). Given amplitudes A₁ = 12 mm and A₂ = 5 mm, x₀ = 3.5 cm and wave number k = 6.28 cm^$$-$$1. The amplitude of resulting wave will be ................ mm.

A source of light is placed in front of a screen. Intensity of light on the screen is I. Two Polaroids P₁ and P₂ are so placed in between the source of light and screen that the intensity of light on screen is I/2. P₂ should be rotated by an angle of (degrees) so that the intensity of light on the screen becomes $${{3I} \over 8}$$.

If the maximum value of accelerating potential provided by a ratio frequency oscillator is 12 kV. The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is ...............

[m_p = 1.67 $$\times$$ 10^$$-$$27 kg, e = 1.6 $$\times$$ 10^$$-$$19C, Speed of light = 3 $$\times$$ 10⁸ m/s]

The acceleration due to gravity is found upto an accuracy of 4% on a planet. The energy supplied to a simple pendulum to known mass 'm' to undertake oscillations of time period T is being estimated. If time period is measured to an accuracy of 3%, the accuracy to which E is known as ..............%