The decreasing order of nucleophilicity among the following nucleophiles is:
$(I)$ $CH_3-COO^-$
$(II)$ $CH_3-O^-$
$(III)$ $CH_3-CH_2^-$
$(IV)$ $CH_3-SO_3^-$

Let $S = \{0, 1, 2, 3, \ldots, 100\}$. The number of ways of selecting $x, y \in S$ such that $x \neq y$ and $x + y = 100$ is

Two blocks of masses $1 ~kg$ and $2 ~kg$ are connected by a metal wire going over a smooth pulley as shown in the figure. The breaking stress of the metal is $2 \times 10^9 ~N/m^2$. What should be the minimum radius of the wire used if it is not to break? Take $g = 10 ~m/s^2$.

Bond energy of covalent $O-H$ bond in water is

An electron moving with a speed $u$ along the positive $x-$ axis at $y = 0$ enters a region of uniform magnetic field $\vec{B} = -B_0\hat{k}$ which exists to the right of the $y-$ axis. The electron exits from the region after some time with the speed $v$ at coordinate $y$. Then: