SOLUTIONS TO INTERNAL ASSESSMENT – I FOR ENGINEERING MATHEMATICS-II

\[\underline{PART\ -\ A}\]
\[Marks:\ 6\ ×\ 1\ =\ 6\]
\[1.\ \color{Red}{Find\ the\ centre\ and\ radius\ of\ the\ circle\ x^2\ +\ y^2\ +\ 2\ x\ +\ 2\ y\ -\ 7\ =\ 0}\ \hspace{15cm}\]

Soln: Refer Example 1 under General Equation of Circle

\[2.\ \color{red}{Show\ that\ the\ circles\ x^2\ +\ y^2\ -\ 4\ x\ +\ 2\ y\ +\ 5\ =\ 0}\ \hspace{10cm}\]\[\color{red}{and\ x^2\ +\ y^2\ -\ 4\ x\ +\ 2\ y\ +\ 5\ =\ 0\ are\ concentric\ circles.}\ \hspace{5cm}\]

Soln: Refer Example 1 under Family of Circles

\[3.\ \color{red}{Prove\ that\ the\ equation\ x^2\ +\ 6\ x\ y\ +\ 9y^2\ +\ 4\ x\ +\ 12\ y\ -\ 5\ =\ 0}\ \hspace{10cm}\]\[\color{red}{is\ a\ parobala}\ \hspace{14cm}\]

Soln: Refer Example 1 under General equation of Conic

\[4.\ \color{red}{Find\ the\ Direction\ cosines\ and\ Direction\ ratios}\ \hspace{10cm}\]\[\color{red}{of\ the\ vector\ 3\overrightarrow{i}\ + 4\overrightarrow{j}- 5\overrightarrow{k}}\ \hspace{12cm}\]

Soln: Refer Example 1 under Direction cosines and Direction ratios

\[5.\ \color{red}{Find\ the\ Unit\ vector\ along\ the\ vector\ 3\overrightarrow{i}\ + 4\overrightarrow{j}- 5\overrightarrow{k}}\ \hspace{15cm}\]

Soln: Refer Example 1 under Unit vector

\[6.\ \color{red}{Find\ the\ value\ of\ m\ if\ the\ vectors\ 3\overrightarrow{i} -\overrightarrow{j} + 5\overrightarrow{k}}\ \hspace{10cm}\]\[\color{red}{and\ -6\overrightarrow{i}+ m\overrightarrow{j}+4\overrightarrow{k} are\ perpendicular}\ \hspace{12cm}\]

Soln: Refer Example 2 under Scalar product

\[\underline{PART\ -\ B}\]
\[Marks:\ 7\ ×\ 2\ =\ 14\]
\[7.\ \color{red}{Find\ the\ equation\ of\ the\ circle\ passing\ through\ the\ point\ (5, 4)\ \hspace{10cm}\]\[and\ concentric\ to\ the\ circle\ x^2\ +\ y^2\ -\ 4\ x\ +\ 2\ y\ +\ 5\ =\ 0}\ \hspace{5cm}\]

Soln: Refer Example 2 under Family of Circles

\[8.\ \color{red}{Prove\ that\ the\ circles\ x^2\ +\ y^2\ -\ 4\ x\ +\ 6y\ +\ 4\ =\ 0\ and}\ \hspace{10cm}\]\[\color{red}{and\ x^2\ +\ y^2\ +\ 2\ x\ +\ 4y\ +\ 4\ =\ 0\ cut\ orthogonally}\ \hspace{5cm}\]

Soln: Refer Example 1 under Orthogoanal Circles

\[9.\ \color{red}{Find\ the\ equation\ of\ the\ parabola\ with\ focus\ at\ (1,\ -1)}\ \hspace{10cm}\]\[\color{red}{and\ directrix\ x\ -\ y\ =\ 0.}\ \hspace{5cm}\]

Soln: Refer Example 1 under Equation of Parobola

\[10.\ \color{red}{Show\ that\ the\ points\ whose\ position\ vectors}\ \hspace{15cm}\]\[\color{red}{2\overrightarrow{i}\ – \overrightarrow{j} + 3\overrightarrow{k},\ 3\overrightarrow{i}\ – 5\overrightarrow{j} + \overrightarrow{k}\ and\ -\overrightarrow{i}\ +11 \overrightarrow{j}+ 9\overrightarrow{k}\ are\ collinear}\ \hspace{5cm}\]

Soln: Refer Example 1 under Condition of 3 position vectors

\[11.\ \color{red}{Find\ the\ area\ of\ the\ parellelogram\ whose\ adjacent\ sides\ are\ 3\overrightarrow{i}- \overrightarrow{k}\ \hspace{10cm}\]\[and\ \overrightarrow{i}+ \overrightarrow{j}+\overrightarrow{k}.}\ \hspace{12cm}\]

Soln: Refer Example 2 under Scalar product

\[12.\ \color{red}{Find\ the\ work\ done\ by\ the\ force\ 3\overrightarrow{i}+ 5\overrightarrow{j}+ 7\overrightarrow{k},}\ \hspace{10cm}\]\[\color{red}{when\ the\ displacement\ is\ 2\overrightarrow{i}- \overrightarrow{j} +\overrightarrow{k}}\ \hspace{12cm}\]

Soln: Refer Example 2 under Vector product

\[\underline{PART\ -\ C}\]
\[Marks:\ 2\ ×\ 15\ =\ 30\]
\[14.\ \color{red}{i)\ Find\ the\ equation\ of\ the\ circle\ passing\ through\ the\ point\ A(2.3)}\ \hspace{10cm}\]\[\color{red}{and\ having\ its\ centre\ at\ (4,1)}\ \hspace{5cm}\]

Soln: Refer Example 3 under General Equation of Circle

\[\hspace{1cm}\ \color{red}{ii)\ Prove\ that\ the\ circles\ x^2\ +\ y^2\ +\ 2x\ -\ 4y\ -\ 3\ =\ 0\ and}\ \hspace{9cm}\]\[\color{red}{x^2\ +\ y^2\ -\ 8x\ +\ 6y\ +\ 7\ =\ 0\ touch\ each\ other.}\ \hspace{5cm}\]

Soln: Refer Example 1 under Contact of Circle

\[15.\ \color{red}{i)\ \ Prove\ that\ the\ points}\ \hspace{15cm}\]
\[\color{red}{4\overrightarrow{i}\ +\ 2\overrightarrow{j}\ +\ 3\overrightarrow{k}, 3\overrightarrow{i}\ +\ 4\overrightarrow{j}\ +\ 2\overrightarrow{k}\ and\ 4\overrightarrow{i}\ +2 \overrightarrow{j}\ +\ 3\overrightarrow{k}\ form\ an\ equilateral\ triangle}\]

Soln: Refer Example 2 under Condition of 3 position vectors

\[\hspace{1cm}\ \color{red}{ii)\ Find\ the\ unit\ vector\ perpendicular\ to\ each\ of\ the\ vectors}\ \hspace{9cm}\]\[\color{red}{2\overrightarrow{i} -\overrightarrow{j}+\overrightarrow{k} and\ 3\overrightarrow{i}+ 4\overrightarrow{j} -\overrightarrow{k}.}\ \hspace{5cm}\]\[\color{red}{Also\ find\ the\ sine\ of\ the\ angle\ between\ the\ vectors .}\ \hspace{5cm}\]
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