State any two commonly used software tools for Time Series Analysis. (2Marks) Give an expression for \( X_{t}-5 X_{t-1}+7 X_{t-2}-3 X_{t-3} \quad \) in terms of second order di fferences.

Lösen Sie folgende Gleichungen für \( x \in \mathbb{R} \); (a) \( \sqrt{x+1}-x=-1 \) (b) \( 35+4^{x}-3 \cdot 2^{2+x}=0 \) (c) \( 2^{x-1}=\left(7^{x}\right)^{2} \) (d) \( \log _{2}(\sqrt{x})+\ln \left(\frac{1}{x^{3}}\right)=\frac{1}{\ln (2)}-6 \)

\( \lim _ { x \rightarrow 0 ^ { + } } \frac { \sqrt[ 3 ] { x \sin ^ { 2 } ( 2 x ) } ( \sqrt { x } - 1 ) } { e ^ { x } - \cos ( 3 x ) } \)

defermine the quadrant in which the angles lies

1. (10 points) Compute the product \( C \cdot\left(A^{T}+B\right) \) where \[ A=\left(\begin{array}{cc}-2 & 3 \\ 4 & -5 \\ 1 & 0\end{array}\right), B=\left(\begin{array}{ccc}4 & -7 & 1 \\ -5 & 6 & 2\end{array}\right), \quad C=\left(\begin{array}{cc}1 & 2 \\ 2 & 1 \\ 3 & 3\end{array}\right) \]

Find the real solutions of the equation. (Enter your answers as a comma-separated list. \( x=-8 \) (a) \( x^{3 / 5}=-8 \) (b) \( x^{2 / 3}=16 \) \( x=64 \) (c) \( x^{4 / 3}=-64 \) \( x=\square \)