If the initial velocity in horizontal direction of a projectile is unit vector $$\hat{i}$$ and the equation of trajectory is $$y=5 x(1-x)$$. The $$y$$ component vector of the initial velocity is ______________ $$\hat{j}$$. ($$\mathrm{Take}$$ $$\left.\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$$

A disc of mass $$1 \mathrm{~kg}$$ and radius $$\mathrm{R}$$ is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be $$4 \sqrt{\frac{x}{3 R}} \,\operatorname{rad}{s}^{-1}$$ where $$x=$$ ____________. $$\left(g=10 \mathrm{~ms}^{-2}\right)$$

In an experiment to determine the Young's modulus of wire of a length exactly $$1 \mathrm{~m}$$, the extension in the length of the wire is measured as $$0.4 \mathrm{~mm}$$ with an uncertainty of $$\pm\, 0.02 \mathrm{~mm}$$ when a load of $$1 \mathrm{~kg}$$ is applied. The diameter of the wire is measured as $$0.4 \mathrm{~mm}$$ with an uncertainty of $$\pm \,0.01 \mathrm{~mm}$$. The error in the measurement of Young's modulus $$(\Delta \mathrm{Y})$$ is found to be $$x \times 10^{10}\, \mathrm{Nm}^{-2}$$. The value of $$x$$ is _________________. $$\left(\right.$$take $$\mathrm{g}=10 \mathrm{~ms}^{-2}$$ )

When a car is approaching the observer, the frequency of horn is $$100 \mathrm{~Hz}$$. After passing the observer, it is $$50 \mathrm{~Hz}$$. If the observer moves with the car, the frequency will be $$\frac{x}{3} \mathrm{~Hz}$$ where $$x=$$ ________________.