ASSIGNMENT – I FOR ENGINEERING MATHEMATICS – I

\[\LARGE {\color {red}{Questions\ from\ Unit-I\ and\ Unit-II}}\]
\[Date:\ 24-10-2021\ \hspace{10cm}\ Max\ Marks:\ 10\ ×\ 2\ =\ 20m\]
\[\color {black} {1.}\ Solve\ the\ following\ equations\ using\ Cramers\ Rule\ \hspace{20cm}\]\[3x + y – z=2,\ 2x – y + 2z = 6\ and\ 2x + y – 2z = -2\ \hspace{10cm}\]
\[2.\ Find\ the\ inverse\ of\ \begin{bmatrix} 1 & 1 & -1\\ 2 & 1 & 0 \\ -1 & 2 & 3 \\ \end{bmatrix}\ \hspace{15cm}\]
\[3.\ Find\ the\ rank\ of\ the\ matrix\ \begin{bmatrix} 5 & 3 & 14 & 4 \\ 0 & 1 & 2 & 1 \\ 1 & -1 & 2 & 0 \\ \end{bmatrix}\ \hspace{15cm}\]
\[\color {black} {4\ .}\ Find\ the\ middle\ term\ in\ the\ expansion\ of\ (2x\ -\ \frac{x^2}{4})^{16}\ \hspace{15cm}\]
\[\color {black} {5.}\ Find\ the\ coefficient\ of\ x^5\ in\ the\ expansion\ of\ (x\ -\ \frac{1}{x})^{11}\ \hspace{15cm}\]
\[\color {black} {6 .}\ Find\ the\ term\ independent\ of\ x\ in\ the\ expansion\ of\ (4x^3 + \frac{3}{x^2})^{20}\ \hspace{15cm}\]
\[7.\ Express\ \frac{(1+ i)(1 + 2i)}{1+ 3i}\ in\ the\ form\ of\ a+ib\ \hspace{18cm}\]
\[8\ Find\ the\ modulus\ and\ amplitude\ of\ \frac{-3 + i}{-1 + i}\ \hspace{18cm}\]
\[9.\ In\ an\ Argand\ diagram\ Show\ that\ the\ complex\ numbers\ 3+2i,\ 5+4i,\ 3+6i\ and\ 1 + 4i\ form\ a\ square\ \hspace{10cm}\]
\[10.\ Simplify\ using\ DeMoivre’s\ theorem:\ \frac{(cos⁡\ 3θ + i sin⁡\ 3θ)^4\ (cos⁡\ 4θ + i sin⁡\ 4θ)^2} {(cos⁡\ 2θ + i sin⁡\ 2θ)^5\ (cos⁡\ 5θ + i sin⁡\ 5θ)^3}\ \hspace{10cm}\]