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

  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. (7 + 8)
    Clarks Table and programmable calculators are not permitted.
\[\underline{PART\ -\ A}\]
\[1.\ \color{green}{What\ is\ the\ equation\ of\ the\ circle\ with\ centre\ at\ origin\ and\ radius\ 1\ unit?}\ \hspace{15cm}\]

Soln: Refer Example No. 1 under Equation of circle with centre and radius (Analytical Geometry-II

\[2.\ \color{green}{Write\ the\ condition\ for\ two\ vectors\ to\ be\ perpendicular}\ \hspace{18cm}\]

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

\[3.\ \color{green}{Evaluate:\ \int \frac{dx}{x^2 + 4}}\ \hspace{20cm}\]

Soln: Refer Example No. 1 (STANDARD INTEGRALS)

\[4.\ \color{green}{Evaluate\ :\ \int_0^1( x^2\ -\ 2)\ dx}\ \hspace{20cm}\]

Soln: Refer Example No. 2 DEFINITE INTEGRALS

\[5.\ \color{green}{Solve:\ \frac{dy}{dx}\ =\ e^x}\ \hspace{18cm}\]

Soln: Refer Example No. 3 (FIRST ORDER DIFFERENTIAL EQUATION)

\[\underline{PART\ -\ B}\]
\[6.\ \color {green}{Show\ that\ the\ straight\ line\ 4\ x-\ y\ =\ 17\ passes\ through}\ \hspace{15cm}\]\[\color{green}{the\ centre\ of\ the\ circle\ x^2\ +\ y^2\ -\ 8\ x\ +\ 2\ y\ +\ 3\ =\ 0}\ \hspace{5cm}\]

Soln: Refer Example No. 6 under Equation of Cirrcle with centre and radus (Analytical Geometry-II

\[7.\ \color {green}{Find\ the\ centre\ and\ radius\ of\ the\ circle\ x^2\ +\ y^2\ -\ 4\ x\ +\ 8\ y\ -\ 5\ =\ 0}\ \hspace{5cm}\]

Soln: Refer Example No. 4 under General Equation of Cirrcle (Analytical Geometry-II

\[8.\ \color{green}{Check\ whether\ the\ conic\ 2\ x^2\ -\ 16\ x\ y\ +\ 8\ y^2\ -\ y\ +\ 3\ = 0\ represent\ a\ hyperbola}\ \hspace{5cm}\]

Soln: Refer Example No. 3 under General Equation of Conic (Conics)

\[9.\ \color{green}{Find\ the\ area\ of\ parallelogram\ whose\ adjacent\ sides\ are}\ \hspace{10cm}\]\[ \color{green} {-\ \overrightarrow{i}\ + 2\overrightarrow{j}\ +\ 4\overrightarrow{k}\ and\ \overrightarrow{i}\ – \overrightarrow{j}\ -\ \overrightarrow{k}}\ \hspace{5cm}\]

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

\[10.\ \color{green}{Can\ a\ vector\ have\ direction\ angles\ 30^0,\ 45^0,\ 60^0}\ \hspace{10cm}\]

Soln: Refer Example No. 2 under Direction cosines and ratios (Vector Introduction)

\[11.\ \color{green}{Find\ the\ Dot\ product\ of\ \overrightarrow{i}+\overrightarrow{j}\ +\ \overrightarrow{k}\ ,\ \overrightarrow{i}\ + 3\overrightarrow{k}}\ \hspace{10cm}\]

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

\[12.\ \color {green}{Evaluate: \int(3\ x^2\ -\ 5\ sec^2\ x\ +\ \frac{7}{x})\ dx}\ \hspace{15cm}\]

Soln: Refer Example No. 9 INTEGRATION DECOMPOSITON METHOD (TEXT)

\[13.\ \color {green}{Evaluate: \int\ \frac{3x^2}{x^3\ +\ 1}\ dx}\ \hspace{15cm}\]

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

\[14.\ \color{green}{Evaluate:\ \int \frac{dx}{9\ -\ x^2}}\ \hspace{18cm}\]

Soln: Refer Example No. 7 (STANDARD INTEGRALS)

\[15.\ \color{green}{Evaluate:\ \int x\ cos\ 5x\ dx}\ \hspace{18cm}\]

Soln: Refer Example No. 7 INTEGRATION BY PARTS

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

Soln: Refer Problem No. 5 (ii) (BERNOULLI’S FORMULA – Excercise Problems with solutions)

\[17.\ \color {green}{Evaluate: \int_0^\frac{\pi}{4} sec^2 x\ dx}\ \hspace{18cm}\]

Soln: Refer Problem No. 4 (DEFINITE INTEGRALS – Excercise Problems with solutions)

\[18.\ \color {green}{Find\ the\ Order\ and\ degree\ of\ the\ differential\ equation}\ \hspace{12cm}\]\[\color {green}{x^2\ y^{II}\ +\ x(y^I)^3\ +\ y\ =\ 0}\ \hspace{7cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[Order\ =\ 2\ \hspace{5cm}\ degree\ =\ 1\]
\[19.\ \color{green}{Find\ the\ Volume\ of\ the\ solid\ formed\ when\ the\ area\ bounded\ by\ the\ curve\ y^2\ =\ 4\ x}\ \hspace{8cm}\]\[\color{green}{between\ x\ =\ 0\ and\ x\ =\ 1\ is\ rotated\ about\ the\ X\ -\ axis.}\ \hspace{5cm}\]

Soln: Refer Example No. 2 (AREA AND VOLUME)

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

Soln: Refer Example No. 5 (FIRST ORDER DIFFERENTIAL EQUATION)

\[\underline{PART\ -\ C}\]
\[21.\ A)\ i.\ \color{green}{Find\ the\ equation\ of\ the\ circle\ on\ the\ line\ joining\ the\ points\ (2,3),\ (-\ 4,\ 5)\ as\ diameter.}\]\[\color{green}{Aslo\ find\ the\ centre\ and\ radius\ of\ the\ circle}\ \hspace{5cm}\]

Soln: Refer Example No. 1 under Equation of circle joining two end points as diameter (Analytical Geometry-II

\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ equation\ of\ the\ circle\ passing\ through\ the\ origin\ and\ cuts\ orthogonally}\ \hspace{3cm}\]\[\color{green}{each\ of\ the\ circles\ x^2\ +\ y^2\ -\ 6\ x\ +\ 8\ =\ 0\ and\ x^2\ +\ y^2\ -\ 2\ x\ -\ 2\ y\ -\ 7\ =\ 0}\ \hspace{5cm}\]

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

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Find\ the\ equation\ of\ the\ Ellipse\ with\ focus\ (2,\ 3)\ and\ directrix\ x\ =\ 7\ and\ e\ =\ \frac{1}{2}}\ \hspace{5cm}\]

Soln: Refer Example No. 2 under Equation of Conic (Conics)

\[\hspace{1cm}\ ii.\ \color{green}{Show\ that\ the\ second\ degree\ equation}\ \hspace{15cm}\]\[\color{green}{12\ x^2\ +\ 7\ x\ y\ -\ 10\ y^2\ +\ 13\ x\ +\ 45\ y\ -\ 35\ =\ 0}\ \hspace{7cm}\]\[\color{green}{in\ x\ and\ y\ represents\ a\ pair\ of\ straight\ lines}\ \hspace{5cm}\]

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

\[22.\ A)\ i.\ \color{green}{{Prove\ that\ the\ points}\ \hspace{15cm}}\ \hspace{7cm}\]\[\color{green} {\overrightarrow{j}\ +\ 10 \overrightarrow{k}\ ,\ 7\overrightarrow{i}\ +\ 6\overrightarrow{j}\ +\ 6\overrightarrow{k} and\ -\ 4\overrightarrow{i}\ +\ 9 \overrightarrow{j}\ +\ 6 \overrightarrow{k}\ form\ an\ isosceles\ triangle}\]

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

\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ value\ of\ ‘P\ such\ the\ that\ the\ vectors\ 2\overrightarrow{i}\ – 3\overrightarrow{j}\ +\ 5\overrightarrow{k},\ p\overrightarrow{i}\ +\ 2\overrightarrow{j}\ -\ \overrightarrow{k}\ and}\ \hspace{7cm}\]\[ \color{green} {3\overrightarrow{i}\ -\ \overrightarrow{j}\ +\ 4\overrightarrow{k}\ lie\ on\ the\ same\ plane}\ \hspace{5cm}\]

Soln: Refer Example No. 1 under Scalar triple Product of Vectors (Product of Vectors)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Find\ the\ area\ of\ the\ triangle\ formed\ by\ the\ points\ whose\ position\ vectors}\ \hspace{15cm}\]\[\color{green}{3\overrightarrow{i}\ -\ 2\overrightarrow{j}\ +\ \overrightarrow{k}\ ,\ \overrightarrow{i}\ -\ 3\overrightarrow{j}\ +\ 5\overrightarrow{k}\ and\ 2\overrightarrow{i}\ +\ \overrightarrow{j}\ -\ 4\overrightarrow{k}}\ \hspace{5cm}\]

Soln: Refer Problem No.19 Product of Two vectors Excercise Problems with solutions

\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ Magnitude\ of\ the\ moment\ of\ the\ force\ 6\overrightarrow{i}\ +\ \overrightarrow{j}\ -\ \overrightarrow{k}}\ \hspace{7cm}\]\[ \color{green} {acting\ along\ the\ point\ (0,\ 1,\ -\ 1)\ about\ the\ point\ (4,\ 3,\ -\ 1)}\ \hspace{5cm}\]

Soln: Refer Example No. 2 under Moment of Force about a point (Product of Vectors)

\[23.\ A)\ i.\ \color{green}{Evaluate:\ \hspace{2cm}\ (a)\ \int (3x + 2 ) ( x + 1) \ dx\ \hspace{2cm}\ (b)\ \int sin 3x\ sin x\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 6 under basic formulae and Example No. 9 under Trigonometry Related Formulae (INTEGRATION DECOMPOSION METHOD (TEXT))

\[\hspace{1cm}\ ii.\ \color{green}{Evaluate:\ \hspace{2cm}\ (a)\ \int\ sec^7\ x\ tan\ x\ dx\ \hspace{2cm}\ (b)\ \int\ \frac{e^x}{e^x\ +\ 5}\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 10 and Example No.6 METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Evaluate:\ \hspace{2cm}\ (a)\ \int\ \frac{2ax + b}{{\sqrt{(ax^2 + bx + c)}}}\ dx\ \hspace{2cm}\ (b)\ \int\ \frac{dx}{(7x + 1)^2 + 25}}\ \hspace{10cm}\]

Soln: Refer Example No. 11 in METHODS OF INTEGRATION – INTEGRATION BY SUBSTITUTION

Soln: Refer Example No. 6 (i) (STANDARD INTEGRALS- Excercise Problems with Solutions)

\[\hspace{1cm}\ ii.\ \color{green}{Evaluate:\ \hspace{2cm}\ (a)\ \int \frac{dx}{(4x\ +\ 1)^2 – 36}\ dx\ \hspace{2cm}\ (b)\ \int \frac{dx}{{\sqrt{4\ -\ 81x^2}}}}\ \hspace{10cm}\]

Soln: Refer Example No. 6 (STANDARD INTEGRALS)

Soln: Refer Example No. 6 (ii) (STANDARD INTEGRALS- Excercise Problems with Solutions)

\[24.\ A)\ i.\ \color{green}{Evaluate:\ \hspace{2cm}\ (a)\ \int x^n\ log\ x\ dx\ \hspace{2cm}\ (b)\ \int x\ e^{-\ 7x}\ dx}\ \hspace{10cm}\]

Soln: Refer Example No. 9 and No. 3 INTEGRATION BY PARTS

\[\hspace{1cm}\ ii.\ \color{green}{Evaluate:\ \hspace{2cm}\ (a)\ \int x^2\ cos\ 6x\ dx\ \hspace{2cm}\ (b)\ \int x^2\ e^{-9x}\ dx}\ \hspace{10cm}\]

Soln: Refer Problem No. 6 (BERNOULLI’S FORMULA – Excercise Problems with solutions)

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

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Evaluate: \int_0^\frac{\pi}{2} cos\ 3 x\ cos\ x\ dx}\ \hspace{15cm}\]

Soln: Refer Example No. 8 DEFINITE INTEGRALS

\[\hspace{1cm}\ ii.\ \color{green}{Evaluate:\ \hspace{2cm}\ (a)\ \int (x\ +\ 3)\ sin\ 7x\ dx\ \hspace{2cm}\ (b)\ \int x^3\ e^{-3x}\ dx}\ \hspace{10cm}\]

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

Soln: For (b) Refer Example No. 8 (BERNOULLI’S FORMULA)

\[25.\ A)\ i.\ \color{green}{Find\ the\ Volume\ of\ the\ solid\ generated\ by\ the\ area\ enclosed\ by\ the\ curve\ y^2\ =\ x(x\ -\ 1)^2}\ \hspace{10cm}\]\[\color{green}{and\ the\ X-axis\ when\ rotated\ about\ X- axis}\ \hspace{15cm}\]

Soln: Refer Example No. 8 (AREA AND VOLUME)

\[\hspace{1cm}\ ii.\ \color{green}{Solve:\ \frac{dy}{dx}\ +\ y\ cot\ x\ =\ sin^3\ x}\ \hspace{15cm}\]

Soln: Refer Example No. 6 (LINEAR TYPE DIFFERENTIAL EQUATION)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Solve:\ (1\ +\ e^y)\ sec^2\ x\ dx\ +\ 5\ e^y\ tan\ x\ dy\ =\ 0}\ \hspace{15cm}\]

Soln: Refer Example No. 10 (FIRST ORDER DIFFERENTIAL EQUATION)

\[\hspace{1cm}\ ii.\ \color{green}{Solve:\ \frac{dy}{dx}\ +\ 2y\ tan\ x\ =\ e^{tan\ x}}\ \hspace{15cm}\]

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

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