INTERNAL ASSESSMENT – I FOR ENGINEERING MATHEMATICS-II

\[\underline{PART\ -\ A}\]
\[Marks:\ 6\ ×\ 1\ =\ 6\]
\[1.\ Find\ the\ centre\ and\ radius\ of\ the\ circle\ x^2\ +\ y^2\ +\ 2\ x\ +\ 2\ y\ -\ 7\ =\ 0\ \hspace{15cm}\]
\[2.\ Show\ that\ the\ circles\ x^2\ +\ y^2\ -\ 4\ x\ +\ 2\ y\ +\ 5\ =\ 0\ \hspace{10cm}\]\[and\ x^2\ +\ y^2\ -\ 4\ x\ +\ 2\ y\ +\ 5\ =\ 0\ are\ concentric\ circles.\ \hspace{5cm}\]
\[3.\ Prove\ that\ the\ equation\ x^2\ +\ 6\ x\ y\ +\ 9y^2\ +\ 4\ x\ +\ 12\ y\ -\ 5\ =\ 0\ \hspace{10cm}\]\[is\ a\ parobala\ \hspace{14cm}\]
\[4.\ Find\ the\ Direction\ cosines\ and\ Direction\ ratios\ \hspace{10cm}\]\[of\ the\ vector\ 3\overrightarrow{i}\ + 4\overrightarrow{j}- 5\overrightarrow{k}\ \hspace{12cm}\]
\[5.\ Find\ the\ Unit\ vector\ along\ the\ vector\ 3\overrightarrow{i}\ + 4\overrightarrow{j}- 5\overrightarrow{k}\ \hspace{15cm}\]
\[\underline{PART\ -\ B}\]
\[Marks:\ 7\ ×\ 2\ =\ 14\]
\[7.\ Find\ the\ equation\ of\ the\ circle\ passing\ through\ the\ point\ (5, 4)\ \hspace{10cm}\]\[and\ concentric\ to\ the\ circle\ x^2\ +\ y^2\ -\ 4\ x\ +\ 2\ y\ +\ 5\ =\ 0\ \hspace{5cm}\]
\[8.\ Prove\ that\ the\ circles\ x^2\ +\ y^2\ -\ 4\ x\ +\ 6y\ +\ 4\ =\ 0\ and\ \hspace{10cm}\]\[and\ x^2\ +\ y^2\ +\ 2\ x\ +\ 4y\ +\ 4\ =\ 0\ cut\ orthogonally\ \hspace{5cm}\]
\[9.\ Find\ the\ equation\ of\ the\ parabola\ with\ focus\ at\ (1,\ -1)\ \hspace{10cm}\]\[and\ directrix\ x\ -\ y\ =\ 0.\ \hspace{5cm}\]
\[10.\ Show\ that\ the\ points\ whose\ position\ vectors\ \hspace{15cm}\]\[2\overrightarrow{i}\ – \overrightarrow{j} + 3\overrightarrow{k},\ 3\overrightarrow{i}\ – 5\overrightarrow{j} + \overrightarrow{k}\ and\ -\overrightarrow{i}\ +11 \overrightarrow{j}+ 9\overrightarrow{k}\ are\ collinear\ \hspace{5cm}\]
\[\underline{PART\ -\ C}\]
\[Marks:\ 2\ ×\ 15\ =\ 30\]
\[14.\ i)\ Find\ the\ equation\ of\ the\ circle\ passing\ through\ the\ point\ A(2.3)\ \hspace{10cm}\]\[and\ having\ its\ centre\ at\ (4,1)\ \hspace{5cm}\]
\[\hspace{1cm}\ ii)\ Prove\ that\ the\ circles\ x^2\ +\ y^2\ +\ 2x\ -\ 4y\ -\ 3\ =\ 0\ and\ \hspace{9cm}\]\[x^2\ +\ y^2\ -\ 8x\ +\ 6y\ +\ 7\ =\ 0\ touch\ each\ other.\ \hspace{5cm}\]
\[15.\ i)\ \ Prove\ that\ the\ points\ \hspace{15cm}\]
\[4\overrightarrow{i}\ +\ 2\overrightarrow{j}\ +\ 3\overrightarrow{k}, 3\overrightarrow{i}\ +\ 4\overrightarrow{j}\ +\ 2\overrightarrow{k}\ and\ 4\overrightarrow{i}\ +2 \overrightarrow{j}\ +\ 3\overrightarrow{k}\ form\ an\ equilateral\ triangle\]
\[\hspace{1cm}\ ii)\ Find\ the\ unit\ vector\ perpendicular\ to\ each\ of\ the\ vectors\ \hspace{9cm}\]\[2\overrightarrow{i} -\overrightarrow{j}+\overrightarrow{k} and\ 3\overrightarrow{i}+ 4\overrightarrow{j} -\overrightarrow{k}.\ \hspace{5cm}\]\[Also\ find\ the\ sine\ of\ the\ angle\ between\ the\ vectors .\ \hspace{5cm}\]