SOLUTIONS TO MODEL EXAM FOR ENGINEERING MATHEMATICS-II

DR.B.R.AMBEDKAR POLTECHNIC COLLEGE, YANAM

MODEL EXAM

ENGINEERING MATHEMATICS – II

Time – 3 hours (Maximum Marks – 100)
Note:
  1. Answer all questions in PART- A. Each question carries one mark. 
  2. Answer any ten questions in PART- B. Each question carries two marks.
  3. Answer all questions by selecting either A or B. Each question carries fifteen marks.
  4. Clarks Table and programmable calculators are not permitted.
\[\underline{PART\ -\ A}\ (5× 1\ =\ 5)\]
\[1.\ \color{Red}{Find\ the\ centre\ and\ radius\ of\ the\ circle\ x^2\ +\ y^2\ +\ 4\ x\ +\ 4\ y\ -\ 1\ =\ 0}\ \hspace{15cm}\]

Soln: Refer Problem No.3 under Analytical Geometry Excercise problems with solutions

\[2.\ \color{Red}{Find\ the\ Unit\ vector\ along\ the\ vector\ 3\overrightarrow{i}\ + 4\overrightarrow{j}- 5\overrightarrow{k}}\ \hspace{15cm}\]

Soln: Refer Example No. 1 under Unit Vector (Vector Introduction)

\[3.\ \color{Red}{Evaluate: \int(x^2 -x + 5)\ dx}\ \hspace{15cm}\]

Soln: Refer Example No. 1 INTEGRATION DECOMPOSITON METHOD (TEXT)

\[4.\ \color{Red}{Evaluate\ :\ \int\ \frac{2x}{1 + x^2}\ dx}\ \hspace{15cm}\]

Soln: Refer Example No. 3 METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

\[5.\ \color{Red}{Evaluate\ :\ \int_1^2 x^2\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 1 DEFINITE INTEGRALS

\[\underline{PART\ -\ B}\ (10× 2\ =\ 20)\]
\[6.\ \color{Red}{Prove\ that\ the\ circles\ x^2\ +\ y^2\ -\ 4\ x\ +\ 6y\ +\ 4\ =\ 0\ and}\ \hspace{7cm}\]\[\color{Red}{and\ x^2\ +\ y^2\ +\ 2\ x\ +\ 4y\ +\ 4\ =\ 0\ cut\ orthogonally}\ \hspace{5cm}\]

Soln: Refer Example No. 1 under Orthogonal Circles (Analytical Geometry-II

\[7.\ \color{Red}{Find\ the\ equation\ of\ the\ circle\ passing\ though\ the\ point\ (5,4)}\ \hspace{7cm}\]\[ \color{Red} {and\ concentric\ to\ the\ circle\ x^2\ +\ y^2\ -\ 8\ x\ +\ 12y\ +\ 15\ =\ 0}\ \hspace{5cm}\]

Soln: Refer Example No. 2 under Family of Circles (Analytical Geometry-II

\[8.\ \color{Red}{Prove\ that\ the\ equation\ x^2\ +\ 6\ x\ y\ +\ 9y^2\ +\ 4\ x\ +\ 12\ y\ -\ 5\ =\ 0}\ \hspace{7cm}\]\[ \color{Red} {is\ a\ parobala}\ \hspace{5cm}\]

Soln: Refer Example No. 1 under General Equation of Conic (Conics)

\[9.\ \color{Red}{Find\ the\ equation\ of\ the\ parabola\ with\ focus\ at\ (1,\ -1)}\ \hspace{8cm}\]\[ \color{Red} {and\ directrix\ x\ -\ y\ =\ 0.}\ \hspace{5cm}\]

Soln: Refer Example No. 1 under Equation of Parobola (Conics)

\[10.\ \color{Red}{Show\ that\ the\ points\ whose\ position\ vectors}\ \hspace{12cm}\]\[ \color{Red} {2\overrightarrow{i}\ +\ 3\overrightarrow{j}\ -\ 5\overrightarrow{k},\ 3\overrightarrow{i}\ +\ \overrightarrow{j}\ -\ 2\overrightarrow{k}\ and\ 6\overrightarrow{i}\ -\ 5 \overrightarrow{j}\ +\ 7\overrightarrow{k}\ are\ collinear}\ \hspace{3cm}\]

Soln: Refer Example No. 1 under Conditions for Vectors (Vector Introduction)

\[11.\ \color{Red}{Find\ the\ value\ of\ m\ if\ the\ vectors\ 3\overrightarrow{i} -\overrightarrow{j} + 5\overrightarrow{k}}\ \hspace{10cm}\]\[ \color{Red} {and\ -6\overrightarrow{i}+ m\overrightarrow{j}+4\overrightarrow{k} are\ perpendicular}\ \hspace{5cm}\]

Soln: Refer Example No. 3 under Scalar Product of Vectors (Product of Vectors)

\[12.\ \color{Red}{Find\ the\ area\ of\ parallelogram\ whose\ adjacent\ sides\ are}\ \hspace{10cm}\]\[ \color{Red} {3\overrightarrow{i}- \overrightarrow{k} and\ \overrightarrow{i}+ \overrightarrow{j}+\overrightarrow{k}}\ \hspace{5cm}\]

Soln: Refer Example No. 2 under Vector Product of Vectors (Product of Vectors)

\[13.\ \color{Red}{Find\ the\ moment\ of\ the\ force\ 3\overrightarrow{i}+\overrightarrow{k}\ acting\ through\ the\ point}\ \hspace{10cm}\]\[ \color{Red} {\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k} about\ the\ point\ 2\overrightarrow{i}+ \overrightarrow{j}-2\overrightarrow{k}.}\ \hspace{5cm}\]

Soln: Refer Example No. 1 under Application of Vector Product (Product of Vectors)

\[14.\ \color{Red}{Evaluate: \int cos^3 x \ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 7 under Trigonometry Related Formulae (INTEGRATION DECOMPOSITON METHOD (TEXT))

\[15.\ \color{Red}{Evaluate:\ \int \frac{dx}{4x^2 + 9}}\ \hspace{18cm}\]

Soln: Refer Example No. 2 (STANDARD INTEGRALS)

\[16.\ \color{Red}{Evaluate:\ \int x\ sin\ x\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 1 INTEGRATION BY PARTS

\[17.\ \color{Red}{\int x^3\ cos\ x\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 2 (BERNOULLI’S FORMULA)

\[18.\ \color{red}{Find\ the\ Area\ bounded\ by\ the\ curve\ y\ =\ x^2}\ \hspace{12cm}\]\[\color{red}{the\ X-axis\ and\ ordinates\ x\ =\ 0\ and\ x\ =\ 2}\ \hspace{7cm}\]

Soln: Refer Example No. 1 (AREA AND VOLUME)

\[19.\ \color{red}{Solve:\ \frac{dy}{dx}\ =\ \frac{1\ +\ y^2}{1\ +\ x^2}}\ \hspace{18cm}\]
\[20.\ \color{red}{Find\ the\ integrating\ factor\ of\ \frac{dy}{dx}\ -\ \frac{3}{x}\ y\ =\ x^2}\ \hspace{15cm}\]

Soln: Refer Example No. 2 (LINEAR TYPE DIFFERENTIAL EQUATION)

\[\underline{PART\ -\ C}\ (5× 15\ =\ 75)\]
\[21.\ A)\ i.\ \color{Red}{Find\ the\ equation\ of\ the\ circle\ passing\ through\ the\ point\ A(2.-3)}\ \hspace{7cm}\]\[ \color{Red} {and\ having\ its\ centre\ at\ C(- 5,1)}\ \hspace{5cm}\]

Soln: Refer Example No. 3 under General Equation of Circle (Analytical Geometry-II

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

Soln: Refer Example No. 1 Under Contact of Circles (Analytical Geometry-II

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{Red}{Prove\ that\ equation\ 6\ x^2\ +\ 13\ x\ y\ +\ 6\ y^2\ +\ 8\ x\ +\ 7\ y\ +\ 2\ =\ 0}\ \hspace{7cm}\]\[ \color{Red} {represents\ a\ pair\ of\ straight\ lines}\ \hspace{5cm}\]

Soln: Refer Example No. 1 under Condition for General Equation of Conic to represent Pair of Straight lines (Conics)

\[\hspace{1cm}\ ii.\ \color{Red}{Find\ c\ if\ \ 2\ x^2\ +\ 3\ x\ y\ -\ 2\ y^2\ -\ 5\ x\ +\ 5\ y\ +\ c\ =\ 0}\ \hspace{7cm}\]\[ \color{Red} {represents\ a\ pair\ of\ straight\ lines}\ \hspace{5cm}\]

Soln: Refer Example No. 2 under Condition for General Equation of Conic to represent Pair of Straight lines (Conics)

\[22.\ A)\ i.\ \color{Red}{Prove\ that\ the\ points\ whose\ position\ vectors\ are}\ \hspace{7cm}\]\[ \color{Red} {3\overrightarrow{i}\ – \overrightarrow{j}+ 6\overrightarrow{k}, 5\overrightarrow{i}\ – 2\overrightarrow{j}+ 7\overrightarrow{k} and\ 6\overrightarrow{i}\ -5 \overrightarrow{j}+ 2\overrightarrow{k}\ form\ a\ right\ angled\ triangle}\]

Soln: Refer Example No. 3 under Conditions for Vectors (Vector Introduction)

\[\hspace{1cm}\ ii.\ \color{Red}{Find\ the\ projection\ of\ the\ vector\ 3\overrightarrow{i}+ \overrightarrow{j}- 2\overrightarrow{k} on\ 7\overrightarrow{i}+ \overrightarrow{j}+2\overrightarrow{k}.}\ \hspace{7cm}\]\[ \color{Red} {Also\ find\ the\ angle\ between\ them}\ \hspace{5cm}\]

Soln: Refer Example No. 2 under Angle between vectors using scalar product (Product of Vectors)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{Red}{Find\ the\ unit\ vector\ perpendicular\ to\ each\ of\ the\ vectors\ 2\overrightarrow{i} -\overrightarrow{j}+\overrightarrow{k} and\ 3\overrightarrow{i}+ 4\overrightarrow{j} -\overrightarrow{k}}\ \hspace{7cm}\]\[ \color{Red} {Also\ find\ the\ sine\ of\ the\ angle\ between\ the\ vectors .}\ \hspace{5cm}\]

Soln: Refer Example No. 5 under Vector Product of Vectors (Product of Vectors)

\[\hspace{1cm}\ ii.\ \color{Red}{A\ particle\ acted\ on\ by\ the\ forces\ 3\overrightarrow{i}+ 2\overrightarrow{j}- 3\overrightarrow{k} and\ \overrightarrow{i}+ 7\overrightarrow{j}+7\overrightarrow{k} acting\ on\ the\ particle}\]\[ \color{Red} {displaces\ the\ particle\ from\ the\ point\ \overrightarrow{i}+ 2\overrightarrow{j}+ 3\overrightarrow{k} to\ the\ point\ 3\overrightarrow{i}- 5\overrightarrow{j}+4\overrightarrow{k}.}\ \hspace{5cm}\]\[\color{Red}{Find\ the\ total\ work\ done\ by\ the\ forces.}\ \hspace{5cm}\]

Soln: Refer Example No. 2 under Application of Scalar Product (Product of Vectors)

\[23.\ A)\ i.\ \color{Red}{Evaluate:\ \hspace{2cm}\ (i)\ \int \frac{sin^2 x}{ 1- cos\ x} \ dx\ \hspace{2cm}\ (ii)\ \int sin 5x\ cos 2x\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 4 and Example No. 8 under Trigonometry Related Formulae (INTEGRATION DECOMPOSITON METHOD (TEXT))

\[\hspace{1cm}\ ii.\ \color{red}{Evaluate:\ \hspace{2cm}\ (i)\ \int\ \frac{(2x + 1)}{ x^2 + x + 1}\ dx\ \hspace{2cm}\ (ii)\ \int\ \frac{sec^2(log x)}{ x}\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 5 and Example No.7 METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{red}{Evaluate:\ \hspace{2cm}\ (i)\ \int\ \frac{2x + 9}{ x^2 + 9x + 30}\ dx\ \hspace{2cm}\ (ii)\ \int\ sin^3x\ cos x\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 6 and Example No.8 METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

\[\hspace{1cm}\ ii.\ \color{red}{Evaluate:\ \hspace{2cm}\ (i)\ \int \frac{dx}{x^2 – 36}\ dx\ \hspace{2cm}\ (ii)\ \int \frac{dx}{(3x + 2)^2 + 16}}\ \hspace{10cm}\]

Soln: Refer Example No. 5 and Example No. 4 (STANDARD INTEGRALS)

\[24.\ A)\ i.\ \color{Red}{Evaluate:\ \hspace{2cm}\ (i)\ \int x^2\ log\ x\ dx\ \hspace{2cm}\ (ii)\ \int x\ cos\ 5x\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 7 and No. 6 INTEGRATION BY PARTS

\[\hspace{1cm}\ ii.\ \color{red}{Evaluate:\ \hspace{2cm}\ (i)\ \int x^2\ cos\ 2x\ dx\ \hspace{2cm}\ (ii)\ \int x^3\ e^{2x}\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 4 and Example No. 6 (BERNOULLI’S FORMULA)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{red}{Evaluate:\ \hspace{2cm}\ (i)\ \int x^2\ sin\ 2x\ dx\ \hspace{2cm}\ (ii)\ \int x^2\ e^{5x}\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 3 and Example No. 5 (BERNOULLI’S FORMULA)

\[\hspace{1cm}\ ii.\ \color{red}{Evaluate: \int_0^\frac{\pi}{2} (2 + sin x)^3 cos x \ dx}\ \hspace{15cm}\]

Soln: Refer Example No. 9 DEFINITE INTEGRALS

\[24.\ A)\ i.\ \color{red}{Find\ the\ volume\ of\ right\ circular\ cone\ \ of\ height\ ‘h’\ and\ base\ radius\ ‘r’\ by\ integration}\ \hspace{15cm}\]

Soln: Refer Example No. 6 (AREA AND VOLUME)

\[\hspace{1cm}\ ii.\ \color{Red}{Find\ the\ Area\ bounded\ by\ the\ curve\ y\ =\ x^2\ -\ 6x\ +\ 8}\ \hspace{10cm}\]\[\color{red}{and\ the\ X-axis}\ \hspace{5cm}\]

Soln: Refer Example No. 6 (AREA AND VOLUME)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{Red}{Solve:\ (1\ +\ e^x)\ sec^2\ y\ dy\ -\ e^x\ tan\ y\ dx\ =\ 0}\ \hspace{15cm}\]

Soln: Refer Example No. 7 (FIRST ORDER DIFFERENTIAL EQUATION)

\[\hspace{1cm}\ ii.\ \color{red}{Solve:\ \frac{dy}{dx}\ -\ \frac{2x}{1\ +\ x^2}\ y\ =\ (1\ +\ x^2)}\ \hspace{15cm}\]

Soln: Refer Example No. 5 (LINEAR TYPE DIFFERENTIAL EQUATION)

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