5. Let \( A=\left[\begin{array}{ll}1 & 3 \\ 2 & 3\end{array}\right] \) be a \( 2 \times 2 \) matrix. Let \( f_{A}: R^{2} \times R^{2} \rightarrow R \) defined by \( f_{A}(X, Y)= \) 'XAY is a symmetric bilinear form where \( X, Y \) are column vectors in \( R^{2} \). Then find the matrix of \( f_{A} \) with respect to the basis (i) \( \{(1,0),(0,1)\} \) and (ii) \( \{(1,1),(1,-1)\} \).

Prove that for three sets \( \mathrm{A}, \mathrm{B} \) and C . \( (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime} \) \( (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} \) Define Cartesian nroduct of two sets.

(xii) Costs in objective function in phase 2 of two phase method are (a) Actual coefficients (b) Zero (c) -1 (d) None of these (xiii) The right hand side constant of a constreintin a primal problem appears in the corresponding dual as : (a) A coefficient in the objective function (b) A right-hand side constant of a constraint in a (c) An input-output coefficient (d) None of these

2. Вычислить и построить на комплексной плоскости число \( \sqrt[3]{\left(-\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^{2}} \). \( \left(\begin{array}{cc}1 & i \sqrt{3})^{3}\end{array}\right. \)

1. Find the Eigen values of the matrix \( A=\left[\begin{array}{lll}2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2\end{array}\right] \)

You run an animal shelter facility and are asked to purchase food supply for the next month. You normally purchase two types of food, Purina and Natural Choice. Purina costs \( \$ 180 \) per bag. Each bag weighs 80 pounds and can feed as many as six animals per month. The Natural Choice bag costs \( \$ 110 \), weighs 60 pounds, and can feed four animals per month. There is a new type of food, Pedigree, which you must consider purchasing. Pedigree food costs \( \$ 110 \) per bag. Each bag of Pedigree weighs 180 pounds and can feed as many as 16 animals per month. You have been given a maximum budget of \( \$ 2,800 \) for this purchase, although you don't have to spend that much. The facility has room for no more than 2,000 pounds. You can only buy whole bags of each food. Create an adequate mode and solve it in Excel to maximize the number of animals to be fed during the month.