A velocity selector consists of electric field $$\vec{E}=E \,\hat{k}$$ and magnetic field $$\vec{B}=B \,\hat{j}$$ with $$B=12 \,m T$$. The value of $$E$$ required for an electron of energy $$728 \,\mathrm{e} V$$ moving along the positive $$x$$-axis to pass undeflected is :

(Given, mass of electron $$=9.1 \times 10^{-31} \mathrm{~kg}$$ )

Two masses $$M_{1}$$ and $$M_{2}$$ are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass $$M_{2}$$ is twice that of $$M_{1}$$, the acceleration of the system is $$a_{1}$$. When the mass $$M_{2}$$ is thrice that of $$M_{1}$$, the acceleration of the system is $$a_{2}$$. The ratio $$\frac{a_{1}}{a_{2}}$$ will be :

The area of cross section of the rope used to lift a load by a crane is $$2.5 \times 10^{-4} \mathrm{~m}^{2}$$. The maximum lifting capacity of the crane is 10 metric tons. To increase the lifting capacity of the crane to 25 metric tons, the required area of cross section of the rope should be :

(take $$g=10 \,m s^{-2}$$ )

If $$\vec{A}=(2 \hat{i}+3 \hat{j}-\hat{k})\, \mathrm{m}$$ and $$\vec{B}=(\hat{i}+2 \hat{j}+2 \hat{k}) \,\mathrm{m}$$. The magnitude of component of vector $$\vec{A}$$ along vector $$\vec{B}$$ will be ____________ $$\mathrm{m}$$.