The approximate value of $(3.978)^{3 / 2}$ is

The number of integral values of $k$, for which the equation $7 \cos x+5 \sin x=2 \mathrm{k}+1$ has a solution, is

The domain of definition of the function $f(x)$ given by the equation $2^x+2^y=2$ is

Let $\bar{a}=3 \hat{i}-\alpha \hat{j}+\hat{k}$ and $\bar{b}=\hat{i}+\alpha \hat{j}+3 \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$, is $8 \sqrt{3}$ sq. units, then $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}$ is equal to