October-2022 TAMIL NADU POLYTECHNIC BOARD EXAM ENGINEERING MATHEMATICS – II(40022)QUESTION PAPER WITH SOLUTIONS

Answer all questions in PART- A. Each question carries one mark.
Answer any ten questions in PART- B. Each question carries two marks.
Answer all questions by selecting either A or B. Each question carries fifteen marks. (7 + 8)
Clarks Table and programmable calculators are not permitted.

\[\underline{PART\ -\ A}\]

\[1.\ \color{green}{Write\ down\ the\ formula\ for\ center\ and\ radius\ of\ the\ circle}\ \hspace{7cm}\]\[\color{green}{x^2\ +\ y^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ =\ 0}\ \hspace{8cm}\]

Soln: Refer Problem No. 3 under Analytical Geometry-II (Exercise Problems with Solutions)

\[2.\ \color{green}{Show\ that\ \overrightarrow{i}\ -\ 2\overrightarrow{j}\ -\ 4\overrightarrow{k}\ and\ 3\overrightarrow{i}\ -\ 6\overrightarrow{j}\ -\ 12\overrightarrow{k}\ are\ parallel}.\ \hspace{10cm}\]

Soln: Refer Example No. 15 under Product of Vectors

\[3.\ \color{green}{What\ is\ the\ value\ of\ \int\ \frac{f^! (x)}{f(x)} dx}\ \hspace{15cm}\]

Soln: Refer Example No. 4 METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

\[4.\ \color{green}{Evaluate: \int_1^2 \frac{dx}{x}}\ \hspace{20cm}\]

Soln: Refer Example No. 5 DEFINITE INTEGRALS

\[5.\ \color{green}{Find\ the\ Area\ bounded\ by\ the\ curve\ y\ =\ f(x)}\ \hspace{12cm}\]\[\color{green} {the\ X-axis\ and\ ordinates\ x\ =\ a\ and\ x\ =\ b}\ \hspace{5cm}\]

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

\[\underline{PART\ -\ B}\]

\[6.\ \color {green}{Find\ the\ equation\ of\ the\ circle\ with\ centre\ (0, -3)\ and\ radius\ 2\ units.}\ \hspace{5cm}\]

Soln: Refer Problem No. 2 under Analytical Geometry-II (Exercise Problems with Solutions)

\[7.\ \color {green}{Find\ the\ equation\ of\ the\ circle\ concentric\ with\ the\ circle\ x^2\ +\ y^2\ -\ 4\ x\ -\ 6\ y\ -\ 9\ =\ 0}\ \hspace{7cm}\]\[\color {green}{and\ passing\ through\ the\ point\ (-4 ,-5)}\ \hspace{5cm}\]

Soln: Refer Problem No. 10 under Analytical Geometry-II (Exercise Problems with Solutions)

\[8.\ \color{green}{What\ is\ the\ condition\ for\ the\ conic\ a\ x^2\ +\ 2\ h\ x\ y\ +\ b\ y^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ =\ 0}\ \hspace{7cm}\]\[\color {green} {represents\ a\ pair\ of\ straight\ lines}\ \hspace{5cm}\]

Soln: Refer Problem No. 6 under Conics (Exercise Problems with Solutions)

\[9.\ \color{green}{If\ the\ vectors\ 2\overrightarrow{i}\ -\ 3\overrightarrow{j}\ and\ 6\overrightarrow{i}\ -\ m\overrightarrow{j}\ are\ collinear,}\ \hspace{10cm}\]\[\color {green} {find\ the\ value\ of\ m}\ \hspace{5cm}\]

Soln: Refer Example No. 5 under Vector Introduction

\[10.\ \color{green}{If\ |\overrightarrow{a}| = 2,\ |\overrightarrow{b}|= 7\ and\ |\overrightarrow{a} × \overrightarrow{b}|=7,\ find\ the\ angle\ between\ \overrightarrow{a}\ and\ \overrightarrow{b}}.\ \hspace {3cm}\]

Soln: Refer Example No. 18 under Product of Vectors

\[11.\ \color{green}{Find\ the\ work\ done\ by\ the\ force\ \overrightarrow{i}\ -\ 7\overrightarrow{j}\ – 2\overrightarrow{k},}\ \hspace{8cm}\]\[\color{green}{when\ the\ displacement\ is\ 3\overrightarrow{i}\ -\ 5\overrightarrow{j}\ -\ 4\overrightarrow{k}}\ \hspace{7cm}\]

Soln: Refer Problem No. 12 under Product of two Vectors (Exercise Problems with solutions)

\[12.\ \color {green}{Evaluate: \int cos^2 x \ dx}\ \hspace{15cm}\]

Soln: Refer Example No. 12 INTEGRATION DECOMPOSITON METHOD (TEXT)

\[13.\ \color {green}{Evaluate: \int\ cot\ x\ dx}\ \hspace{15cm}\]

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

\[14.\ \color{green}{Evaluate:\ \int \frac{dx}{49\ +\ x^2}}\ \hspace{18cm}\]

Soln: Refer Example No. 2 (STANDARD INTEGRALS)

\[15.\ \color{green}{Evaluate:\ \int\ x\ log\ x\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 8 INTEGRATION BY PARTS

\[16.\ \color{green}{\int x^2\ e^x\ dx}\ \hspace{18cm}\]

Soln: Refer Problem No. 2 (BERNOULLI’S FORMULA – Excercise Problems with solutions)

\[17.\ \color {green}{Evaluate: \int_0^1 \frac {dx}{{\sqrt{1 – x^2}}}}\ \hspace{18cm}\]

Soln: Refer Example No. 3 DEFINITE INTEGRALS

\[18.\ \color {green}{Find\ the\ Area\ bounded\ by\ the\ curve\ y\ =\ x^2\ +\ x}\ \hspace{12cm}\]\[\color {green} {the\ X-axis\ between\ x\ =\ 0\ and\ x\ =\ 4}\ \hspace{5cm}\]

Soln: Refer Example No. 3 (AREA AND VOLUME)

\[19.\ \color{green}{Find\ the\ Volume\ of\ the\ solid\ formed\ when\ the\ area\ bounded\ by\ the\ curve\ y^2\ =\ 4\ x}\ \hspace{8cm}\]\[\color{green}{between\ x\ =\ 0\ and\ x\ =\ 1\ is\ rotated\ about\ the\ X\ -\ axis.}\ \hspace{5cm}\]

Soln: Refer Example No. 4 (AREA AND VOLUME)

\[20.\ \color{green}{Solve\ :\ \frac{y^2\ dy}{1\ +\ y^3}\ =\ \frac{dx}{x}}\ \hspace{18cm}\]

Soln: Refer Example No. 6 (FIRST ORDER DIFFERENTIAL EQUATION)

\[\underline{PART\ -\ C}\]

\[21.\ A)\ i.\ \color{green}{Find\ the\ equation\ of\ the\ circle}\ \hspace{10cm}\]\[\color{green}{having\ centre\ (2,-1)\ and\ passing\ through\ the\ point\ (8,7)}\ \hspace{5cm}\]

Soln: Refer Example No. 8 under Analytical Geometry-II

\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ equation\ of\ the\ circle\ passing\ through\ the\ origin\ and\ cuts\ orthogonally}\ \hspace{3cm}\]\[\color{green}{each\ of\ the\ circles\ x^2\ +\ y^2\ -\ 8\ y\ +\ 12\ =\ 0\ and\ x^2\ +\ y^2\ -\ 4\ x\ -\ 6\ y\ -\ 3\ =\ 0}\ \hspace{5cm}\]

Soln: Refer Problem No. 14 under Analytical Geometry-II (Exercise Problems with Solutions)

\[(OR)\]

\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Find\ the\ equation\ of\ the\ Ellipse\ with\ focus\ (-1,\ -3)\ and\ directrix\ x\ -\ 2y\ =\ 0\ and\ e\ =\ \frac{4}{5}}\ \hspace{5cm}\]

Soln: Refer Problem No. 7 under Conics (Exercise Problems with Solutions)

\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ value\ of\ α\ if\ \ α\ x^2\ -\ 10\ x\ y\ + \ 5\ x\ +\ 12\ y^2\ -\ 16y\ -\ 3\ =\ 0}\ \hspace{7cm}\]\[\color {green} {represents\ a\ pair\ of\ straight\ lines}\ \hspace{5cm}\]

Soln: Refer Example No. 9 under Conics

\[22.\ A)\ i.\ \color{green}{Prove\ that\ the\ points}\ \hspace{15cm}\]\[\color{green}{2\overrightarrow{i}\ + 3\overrightarrow{j}+ 4\overrightarrow{k}, 3\overrightarrow{i}\ + 4\overrightarrow{j}\ + 2\overrightarrow{k} and\ 4\overrightarrow{i}\ +\ 2 \overrightarrow{j}+ 3\overrightarrow{k}\ form\ an\ equilateral\ triangle}\]

Soln: Refer Problem No. 8 under Vector Introduction (Exercise Problems with Solutions)

\[\hspace{1cm}\ ii.\ \color{green}{Show\ that\ the\ points\ whose\ position\ vectors}\ \hspace{15cm}\]\[\color{green}{\overrightarrow{i}\ -\ 2\overrightarrow{j}\ -\ \overrightarrow{k},\ 2\overrightarrow{i}\ +\ 3\overrightarrow{j}\ +\ 3\overrightarrow{k}\ and\ -\ 7 \overrightarrow{j}\ -\ 5\overrightarrow{k}\ are\ collinear}\ \hspace{5cm}\]

Soln: Refer Problem No. 7 under Vector Introduction (Exercise Problems with Solutions)

\[(OR)\]

\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Prove\ that\ the\ vectors\ \overrightarrow{i}+2\overrightarrow{j}+ \overrightarrow{k},\ \overrightarrow{i} + \overrightarrow{j}- 3\overrightarrow{k}\ and\ 7\overrightarrow{i}-4\overrightarrow{j}+\overrightarrow{k}}\ \hspace{15cm}\]\[\color{green}{are\ perpendicular\ to\ each\ other.}\ \hspace{5cm}\]

Soln: Refer Problem No.17 Product of Two vectors Excercise Problems with solutions

\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ torque\ of\ the\ force\ 3\overrightarrow{i}+4\overrightarrow{j}+5\overrightarrow{k}\ acting\ through\ the\ point}\ \hspace{10cm}\]\[\color{green}{\overrightarrow{i}-2\overrightarrow{j}+3\overrightarrow{k} about\ the\ point\ 4\overrightarrow{i}- 3\overrightarrow{j}+\overrightarrow{k}.}\ \hspace{10cm}\]

Soln: Refer Problem No.25 Product of Two vectors Excercise Problems with solutions

\[23.\ A)\ i.\ \color{green}{Evaluate:\ \hspace{2cm}\ (i)\ \int ( tan\ x\ +\ cot\ x ) ^ 2\ dx\ \hspace{2cm}\ (ii)\ \int \sqrt{1\ +\ sin\ 2x}\ dx}\ \hspace{10cm}\]

Soln: For (i) Refer Example No. 17 INTEGRATION DECOMPOSITON METHOD (TEXT)

Soln: For (ii) Refer Example No. 10 INTEGRATION DECOMPOSITON METHOD (TEXT)

\[\hspace{1cm}\ ii.\ \color{green}{Evaluate:\ \hspace{2cm}\ (i)\ \int\ e^{sin^2\ x}\ sin\ 2x\ dx\ \hspace{2cm}\ (ii)\ \int\ (2x^2\ -\ 8 x\ +\ 5)^{11}\ (x\ -\ 2) dx}\ \hspace{10cm}\]

Soln: For (i) Refer Example No. 14 METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

Soln: For (ii) Refer Example No. 3 METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

\[(OR)\]

\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Evaluate:\ \hspace{2cm}\ (i)\ \int cos^3 x \ dx\ \hspace{2cm}\ (ii)\ \int\frac{(sin^{-1}\ x)^4}{\sqrt{1\ -\ x^2}}\ dx}\ \hspace{10cm}\]

Soln: For (i) Refer Example No. 18 INTEGRATION DECOMPOSITON METHOD (TEXT)

Soln: For (ii) Refer Example No. 17 METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

\[\hspace{1cm}\ ii.\ \color{green}{Evaluate:\ \hspace{2cm}\ (i)\ \int\frac{dx}{(2x + 3)^2\ +\ 49}\ \hspace{2cm}\ (ii)\ \int \frac{dx}{(x\ +\ 1)^2\ -\ 9}}\ \hspace{10cm}\]

Soln: For (i) Refer Problem No. 8 (STANDARD INTEGRALS- Excercise Problems with Solutions)

Son: For (ii) Refer Example No. 7 (STANDARD INTEGRALS)

\[24.\ A)\ i.\ \color{green}{Evaluate:\ \hspace{2cm}\ (a)\ \int x^n\ log\ x\ dx\ \hspace{2cm}\ (b)\ \int x^3\ sin\ x\ dx}\ \hspace{10cm}\]

Soln: For (i) Refer Example No. 10 INTEGRATION BY PARTS

Soln: For (ii) Refer Example No. 1 (BERNOULLI’S FORMULA)

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

Soln: For (i) Refer Example No. 4 (BERNOULLI’S FORMULA)

Soln: For (ii) Refer Example No. 9 (BERNOULLI’S FORMULA)

\[(OR)\]

\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Evaluate:\ \hspace{2cm}\ (i)\ \int\ x^2\ sin\ 4x\ dx\ \hspace{2cm}\ (ii)\ \int x^2\ e^{5x}\ dx}\ \hspace{10cm}\]

Soln: For (i) Refer Problem No. 4 (i) (BERNOULLI’S FORMULA – Excercise Problems with solutions)