If $\overline{\mathrm{a}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=2 \hat{i}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ are two vectors, then the angle between the vectors $3 \overline{\mathrm{a}}+5 \overline{\mathrm{~b}}$ and $5 \overline{\mathrm{a}}+3 \overline{\mathrm{~b}}$ is

Let $A=\left[\begin{array}{cc}1 & 2 \\ -1 & 4\end{array}\right]$ and $A^{-1}=\alpha \mathrm{I}+\beta \mathrm{A}, \alpha, \beta \in \mathbb{R}$, I is the identity matrix of order 2 , then $4(\alpha-\beta)$ is

$$\lim _\limits{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}=$$

If $\overline{\mathrm{a}}$ is perpendicular to $\bar{b}$ and $\bar{c},|\bar{a}|=2$, $|\overline{\mathrm{b}}|=3,|\overline{\mathrm{c}}|=4$ and the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $\frac{\pi}{3}$, then $\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]=$