Let \( A=\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right] \) be a \( 2 \times 2 \) matrix. Let \( f_{A}: R^{2} \times R^{2} \rightarrow R \) defined by \( f_{A}(X, Y)=T A Y \) 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.

points. 4. If \( u=3 i+6 j-7 k, v=4 i-5 j+6 k \) and \( w=-2 i+3 j-9 k \) then show that \( (u \times v) \bullet w=(v \times w) \bullet u=(w \times u) \bullet v \)

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

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

A computer technician charges a fixed amount for a repair, plus an additiona What is the meaning of the initial amount for the equation? A. It means the technician charges \( \$ 45 \) per hour for the repair. B. It means the technician charges a fixed amount of \( \$ 45 \) for the repair. C. It means the technician charges a fixed amount of \( \$ 75 \) for the repair, D. It means the technician charges \( \$ 75 \) per hour for the repair.