Find all values of \( K \) for which rank of matrix \( A=\left[\begin{array}{rrrr}K & -1 & 0 & 0 \\ 0 & K & -1 & 0 \\ 0 & 0 & K & -1 \\ -6 & 11 & -6 & 1\end{array}\right] \) is equal to \( 3 \quad[\mathrm{i} \)

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The input voltage to the \( R C \) circuit is given as \( \boldsymbol{x}(t)=t e^{-t / R C} u(t) \) and impulse response of the circuit is given by \( \mathbf{h}(\mathbf{t})=\frac{\mathbf{1}}{\mathbf{R C}} e^{-t / R C} \mathbf{u}(t) \) Find the output \( y(t) \).

\( \left( \begin{array} { l } { n } \\ { p } \end{array} \right) = \left( \begin{array} { c } { n - 2 } \\ { p } \end{array} \right) + 2 \left( \begin{array} { l } { n - 2 } \\ { p - 1 } \end{array} \right) + \left( \begin{array} { l } { n - 2 } \\ { p - 2 } \end{array} \right) \)

Sets \( A \) and \( B \) are subsets of the universal set \( U \). These sets are defined as follows. \[ \begin{array}{l}U=\{1,3,5,7,8\} \\ A=\{3,5,8\} \\ B=\{1,3\}\end{array} \] Find the following sets. Write your answer in roster form or as \( \varnothing \). (a) \( (A \cap B)^{\prime}=\square \) (b) \( A^{\prime} \cup B=\square \)

Sets \( A \) and \( B \) are subsets of the universal set \( U \). These sets are defined as follows. \[ \begin{array}{l}U=\{k, m, q, s, y, z\} \\ A=\{q, z\} \\ B=\{q, y\}\end{array} \] Find the following sets. Write your answer in roster form or as \( \varnothing \). (a) \( A \cup B^{\prime}=\square \) (b) \( A^{\prime} \cap B^{\prime}=\square \)