At $25^{\circ}C$,the solubility product of $Mg(OH)_2$ is $1.0 \times 10^{-11}$. At what $pH$ value will the precipitation of $Mg^{2+}$ ions begin in a solution containing $0.001 \ M \ Mg^{2+}$ ions?

If the roots of the equation $4x^3 - 12x^2 + 11x + k = 0$ are in arithmetic progression,then $k$ is equal to

What type of life cycle in plants is indicated by the given figure?

$A$ galvanometer $G$ deflects full scale when a potential difference of $0.50 \ V$ is applied. The internal resistance of the galvanometer $r_g$ is $25 \ \Omega$. An ammeter is constructed by incorporating the galvanometer and an additional shunt resistance $R_S$. The ammeter deflects full scale when a measurement of $2.0 \ A$ is made. The resistance $R_S$ is closest to ............. $\Omega$.

Match the following (Take the relative strength of the strongest fundamental forces in nature as $1$):
| List-$I$ (Fundamental forces in nature) | List-$II$ (Relative strength) |
| :--- | :--- |
| $(A)$ Strong nuclear force | $(e)$ $10^{-2}$ |
| $(B)$ Weak nuclear force | $(f)$ $1$ |
| $(C)$ Electromagnetic force | $(g)$ $10^{10}$ |
| $(D)$ Gravitational force | $(h)$ $10^{-13}$ |
| | $(i)$ $10^{-39}$ |
The correct match is:

If $\int \frac{dx}{(1+\sqrt{x}) \sqrt{x-x^2}} = \frac{A \sqrt{x}}{\sqrt{1-x}} + \frac{B}{\sqrt{1-x}} + C$,where $C$ is a real constant,then $A+B$ equals to