A sonometer wire of resonating length $$90 \mathrm{~cm}$$ has a fundamental frequency of $$400 \mathrm{~Hz}$$ when kept under some tension. The resonating length of the wire with fundamental frequency of $$600 \mathrm{~Hz}$$ under same tension _______ $$\mathrm{cm}$$.

In a single slit experiment, a parallel beam of green light of wavelength $$550 \mathrm{~nm}$$ passes through a slit of width $$0.20 \mathrm{~mm}$$. The transmitted light is collected on a screen $$100 \mathrm{~cm}$$ away. The distance of first order minima from the central maximum will be $$x \times 10^{-5} \mathrm{~m}$$. The value of $$x$$ is :

A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is $$\frac{x}{5}$$. The value of $$x$$ is _________.

A wire of resistance $$20 \Omega$$ is divided into 10 equal parts, resulting pairs. A combination of two parts are connected in parallel and so on. Now resulting pairs of parallel combination are connected in series. The equivalent resistance of final combination is _________ $$\Omega$$.