The focal length of a thin biconvex lens is 20 cm. When an object is moved from a distance of 25 cm in front of it to 50 cm, the magnification of its image changes from m25 to m50. The ratio $${{{m_{25}}} \over {{m_{50}}}}$$ is __________.
An $$\alpha$$-particle and a proton are accelerated from the rest by a potential difference of 100 V. After this, their de Broglie wavelengths are $$\lambda$$$$\alpha$$ and $$\lambda$$p, respectively. The ratio $${{{\lambda _p}} \over {{\lambda _\alpha }}}$$, to the nearest integer, is _____________.
When two identical batteries of internal resistance 1 $$\Omega$$ each are connected in series across a resistor R, the rate of heat produced in R is J1. When the same batteries are connected in parallel across R, the rate is J2. If J1 = 2.25 J2, then the value of R in $$\Omega$$ is __________.
When two progressive waves $${y_1} = 4\sin (2x - 6t)$$ and $${y_2} = 3\sin \left( {2x - 6t - {\pi \over 2}} \right)$$ are superimposed, the amplitude of the resultant wave is __________.