Solid $Ba(NO_3)_2$ is gradually dissolved in a $1.0 \times 10^{-3} \ M \ Na_2CO_3$ solution. At what concentration of $Ba^{2+}$ will a precipitate begin to form? ($K_{sp}$ for $BaCO_3 = 5.1 \times 10^{-9}$)

Living organisms can be distinguished from non-living things on the basis of which ability,excluding exceptions?

Let $\overline{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\overline{b}=\hat{i}+\hat{j}$. Let $\overline{c}$ be a vector such that $|\overline{c}-\overline{a}|=3$,$|(\overline{a} \times \overline{b}) \times \overline{c}|=3$,and the angle between $\overline{c}$ and $\overline{a} \times \overline{b}$ is $30^{\circ}$. Then $\overline{a} \cdot \overline{c}$ is equal to

$A$ simple pendulum has a periodic time $T_1$ when it is on the surface of the Earth of radius $R$. Its periodic time is $T_2$ when it is taken to a height $R$ above the Earth's surface. The value of $\frac{T_2}{T_1}$ is

The length of the common chord of the two circles $x^2+y^2-4y=0$ and $x^2+y^2-8x-4y+11=0$ is

One litre of oxygen at a pressure of $1 ~atm$ and two litres of nitrogen at a pressure of $0.5 ~atm$ are introduced into a vessel of volume $1 ~L$. If there is no change in temperature,the final pressure of the mixture of gas (in $atm$) is