В координатном пространстве заданы векторы \( \vec{a}(-1 ; 2 ; 0) \) и \( \vec{b}(0,1 ;-2) \). айдите: а) координаты суммы \( (\vec{p}=\vec{a}+\vec{b}) \) і \( (\vec{q}=\vec{a}-\vec{b}) \) разности этих векторов; б) одули векторов \( \vec{a}, \vec{b}, \vec{p} \) и \( \vec{q} ; \) в) скалярное пронзведение \( \alpha \) векторов \( \vec{a} \) п \( \vec{b} ; \) г) угол \( \varphi \) ежду векторами \( \vec{a} \) і \( \vec{b} \).

Find the general solution of the recurrence relation \( x_{n+1}=10 x_{n}+18 n \). The general solution of \( x_{n+1}=10 x_{n}+18 n \) is \( x_{n}=\square \). (Type an expression using \( A \) as the constant and \( n \) as the variable index.)

Find DTFT of \[ x(n)=\left(\frac{1}{2}\right)^{n-1} u(n-1) \]

1. Diberikan \( H \) subgroup dari grup \( G \) dan \( x, y \in G \). Buktikan bahwa \[ H x=H y \text { jika dan hanya jika } x y^{-1} \in H \] 2. Diberikan \( H \) subgroup dari grup G dan \( \mathrm{x}, y \in G \) maka \( x H=y H \) atau \( x H \cap y H=\emptyset \). Buktikan! 3. Diberikan \( H \) subgroup dari grup \( G \) maka himpunan semua koset kiri yaitu \( \{x H \mid x \in G\} \) membentuk partisi pada G . Buktikan! 4. Diberikan \( H \) subgroup dari grup G dan \( x \in G \), maka order dari \( H \) sama dengan order dari \( x H \) dan sama dengan order dari \( H x \) yaitu: \[ |H|=|x H|=|H x| \]

Write short notes on the following; a) Isoquants b) Isoquant lines c) Marginal rates of technical substitution d) Production function QUESTION Two Discuss the reasons for the growth in size of firms and outline the ways can grow QuESTION THREE Consider a production function of the firm Q = AL, where O < \( \mathrm{a}<1 \) Required; a) Find the marginal product of L (MPL) b) Express the MPL in terms of \( \mathrm{L}, \mathrm{L} \), and Q c) Determine the slope of MPL d) What is the sign of the slope of MPL? e) Determine whether the MPL increases, diminishes or stays const increases. QUESTION Four Given the production function; \( \mathrm{Q}=200-\mathrm{L}^{2}-\mathrm{K}^{2} \) And corresponding art function \( \mathrm{C}=2 \mathrm{~L}+3 \mathrm{~K} \) The firm wishes to maximize its output given a cost outlay of \( \mathrm{C}=30 \) Required a) Write down the maximization problem for the firm b) Find the levels of K and L for which Q is maximized

6. 6. Calcula el número cromático del grafo bipartito K_\{3,4\}.