In a Young's double slit experiment, the intensity at a point is $$\left(\frac{1}{4}\right)^{\text {th }}$$ of the maximum intensity, the minimum distance of the point from the central maximum is _________ $$\mu \mathrm{m}$$. (Given : $$\lambda=600 \mathrm{~nm}, \mathrm{~d}=1.0 \mathrm{~mm}, \mathrm{D}=1.0 \mathrm{~m}$$)

If $$\vec{a}$$ and $$\vec{b}$$ makes an angle $$\cos ^{-1}\left(\frac{5}{9}\right)$$ with each other, then $$|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|$$ for $$|\vec{a}|=n|\vec{b}|$$ The integer value of $$\mathrm{n}$$ is _________.

Two persons pull a wire towards themselves. Each person exerts a force of $$200 \mathrm{~N}$$ on the wire. Young's modulus of the material of wire is $$1 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$$. Original length of the wire is $$2 \mathrm{~m}$$ and the area of cross section is $$2 \mathrm{~cm}^2$$. The wire will extend in length by _________ $$\mu \mathrm{m}$$.

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $$4 \mathrm{~m}, 2 \mathrm{~ms}^{-1}$$ and $$16 \mathrm{~ms}^{-2}$$ at a certain instant. The amplitude of the motion is $$\sqrt{x}, \mathrm{~m}$$ where $$x$$ is _________.