Seminar – 1 for Enginnering Mathematics – II

$Group\ 1:\ Find\ the\ equation\ of\ the\ circle\ passing\ through\ the\ point\ A(2.3)\ \hspace{7cm}$$and\ having\ its\ centre\ at\ (4,1)\ \hspace{5cm}$
$Group\ 2:\ Prove\ that\ the\ circles\ x^2\ +\ y^2\ -\ 8\ x\ +\ 6y\ -\ 23\ =\ 0\ and\ \hspace{7cm}$$and\ x^2\ +\ y^2\ -\ 2\ x\ -\ 5y\ +\ 16\ =\ 0\ cut\ orthogonally\ \hspace{5cm}$
$Group\ 3:\ Prove\ that\ equation\ 2\ x^2\ -\ 7\ x\ y\ +\ 3\ y^2\ +\ 5\ x\ -\ 5\ y\ +\ 2\ =\ 0\ \hspace{7cm}$$represents\ a\ pair\ of\ straight\ lines\ \hspace{5cm}$
$Group\ 4:\ Show\ that\ the\ points\ whose\ position\ vectors\ \hspace{15cm}$$2\overrightarrow{i}\ – \overrightarrow{j} + 3\overrightarrow{k},\ 3\overrightarrow{i}\ – 5\overrightarrow{j} + \overrightarrow{k}\ and\ -\overrightarrow{i}\ +11 \overrightarrow{j}+ 9\overrightarrow{k}\ are\ collinear\ \hspace{5cm}$
$Group\ 5:\ Prove\ that\ the\ points\ \hspace{15cm}$
$2\overrightarrow{i}\ + 3\overrightarrow{j}+ 4\overrightarrow{k}, 3\overrightarrow{i}\ + 4\overrightarrow{j}+ 2\overrightarrow{k} and\ 4\overrightarrow{i}\ +2 \overrightarrow{j}+ 3\overrightarrow{k}\ form\ an\ equilateral\ triangle$
$Group\ 6:\ If\ \overrightarrow{a}\ = \ 8\overrightarrow{i}\ +\ 4\overrightarrow{j}\ – 3\overrightarrow{k}\ and\ \overrightarrow{b}\ =\ 2\overrightarrow{i}\ -\ 3\overrightarrow{j}\ +\ 2\overrightarrow{k},\ \hspace{15cm}$$find\ the\ projection\ of\ \overrightarrow{a}\ on\ \overrightarrow{b}\ .\ Also\ find\ the\ angle\ between\ them\ \hspace{5cm}$
$Group\ 7:\ Find\ the\ work\ done\ by\ the\ force\ \overrightarrow{i}\ +\ 3\overrightarrow{j}\ – \ \overrightarrow{k}\ when\ it\ displaces\ a\ particle\ \hspace{15cm}$$from\ the\ point\ 2\overrightarrow{i}\ -\ 6\overrightarrow{j}\ +\ 7\overrightarrow{k}\ to\ the\ point\ 3\overrightarrow{i}\ -\ \overrightarrow{j}\ – \ 5\overrightarrow{k}\ \hspace{8cm}$
$Group\ 8:\ Find\ the\ moment\ of\ the\ force\ 6\overrightarrow{i}\ +\ \overrightarrow{j}\ +\ \overrightarrow{k}\ acting\ through\ the\ point\ \hspace{10cm}$$\overrightarrow{i}\ +\ 2\overrightarrow{j}\ +\ 3\overrightarrow{k} about\ the\ point\ -\overrightarrow{i}\ – \overrightarrow{j}\ +\ \overrightarrow{k}.\ \hspace{10cm}$