A ball at $O$ is in equilibrium as it is attached with two strings $AO$ and $DO,$ which are tied at $A$ and $D.$ $AO=DO=a\sqrt{5}$. The charges at $A,B,C$ and $D$ are $+q, +Q, +2Q$ and $-q,$ respectively. Find the correct options. The ball at $O$ is positively charged.

A charged cork of mass $m$ suspended by a light string is placed in a uniform electric field of strength $\overrightarrow{E}=\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)\times {10}^5 \mathrm{N} {\mathrm{C}}^{-1}$ as shown in the figure. If in equilibrium position tension in the string is $\frac{2mg}{\left(1+\sqrt{3}\right)}$, then angle $\alpha$ with the vertical can be

The following figure shows a block of mass $m$ suspended from a fixed point by means of a vertical spring. The block is oscillating simple harmonically and carries a charge $q.$ There also exists a uniform electric field in the region. Consider four different cases. The electric field is zero in case $1$, $\frac{mg}{q}$ downward in case $2$, $\frac{mg}{q}$ upward in case $3$, and $\frac{2mg}{q}$ downward in case $4$. The speed at the mean position is the same in all cases. Select the correct alternative(s).