June-2022 TAMIL NADU POLYTECHNIC BOARD EXAM ENGINEERING MATHEMATICS – I(40012)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.
\[\underline{PART\ -\ A}\]
\[1.\ \color{green}{Evaluate\ \begin{vmatrix} sin\ \theta & -\ cos\ \theta \\ cos\ \theta & sin\ \theta \\ \end{vmatrix}}\ \hspace{15cm}\]

Solution: Refer Example 3 Under Determinant of 2nd Order of Matrices and Determinants

\[2.\ \color{green}{Find\ the\ value\ of\ i^3\ +\ i^5}\ \hspace{18cm}\]

Solution: Refer Problem No. 2 under Algebra of Complex Numbers (Excercise problems with soltuions)

\[3.\ \color{green}{Prove\ that\ \frac{Sin\ 2A}{1\ +\ Cos\ 2A}\ =\ Tan\ A}\ \hspace{20cm}\]

Solution: Refer Example. 10

\[4.\ \color{green}{Evaluate:\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 3θ}{θ}}\ \hspace{20cm}\]

Solution: Refer Example. 6

\[5.\ \color{green}{ if\ y\ =\ x^2\ +\ 6\ x\ -\ 15\ Find\ \frac{d^2y}{dx^2}}\ \hspace{20cm}\]

Solution: Problem No.2

\[\underline{PART\ -\ B}\]
\[6.\ \color {green}{Prove\ that\ the\ matrix\ \begin{pmatrix} 2 & 3 & – 1 \\ 4 & 6 & 5 \\ 6 & 2 & 1 \\ \end{pmatrix}\ is\ a\ non\ -\ singular}\ \hspace{8cm}\]

Solution: Refer Example 4 Under Singular and Non-Singular Matrix of Matrices and Determinants

\[7.\ \color {green}{If\ A =\begin{pmatrix} 6 & 5 \\ -3 & 8 \\ \end{pmatrix}\ and\ B =\begin{pmatrix} 2 & 3 \\ 4 & -1 \\ \end{pmatrix}\ ,\ Find\ BA}\ \hspace{8cm}\]

Solution: Refer Example 2 Under Multiplication of Matrices of Matrices and Determinants

\[8.\ \color{green}{Find\ the\ value\ of\ m\ if\ \begin{vmatrix} 3 & 4 & -2 \\ -3 & 6 & 2 \\ 4 & 1 & m \\ \end{vmatrix}\ =\ 0}\ \hspace{11cm}\]

Solution: Refer Example 2 Under Determinant of 3rd Order of Matrices and Determinants

\[9.\ \color{green}{Simplify\ (cos\ 15^0 + i sin⁡\ 15^0 )^6}\ \hspace{18cm}\]

Solution: Refer Example 1 under De-Moivre’s Theorem (Text)

\[10.\ \color{green}{Find\ the\ cube\ roots\ on\ unity}\ \hspace{18cm}\]

Solution: Refer Example 5 under Roots of Complex Numbers (Text)

\[11.\ \color{green}{Solve\ x^2\ -\ 1\ =\ 0}\ \hspace{18cm}\]

Solution: Refer Example 4 under Roots of Complex Numbers (Text)

\[12.\ \color {green}{Prove\ that\ Tan^{-1}\ (\frac{1}{x})\ =\ cot^{-1}\ x}\ \hspace{18cm}\]

Solution: Refer Example No. 15 under Inverse Trigonometric Functions (Text)

\[13.\ \color {green}{ If\ tan\ θ\ =\ 3,\ find\ tan\ 3θ}\ \hspace{18cm}\]

Solution: Refer Example. 20

\[14.\ \color {green}{ Prove\ that}\ Cos\ 10^{0}\ +\ Cos\ 70^{0}\ =\ \sqrt{3}\ Cos\ 40^{0}\ \hspace{18cm}\]

Solution: Refer Example. 3

\[15.\ \color {green}{Evaluate:\ \lim\ _{x\ \to\ a}\ \frac{x^3\ -\ a^3}{x\ -\ a}}\ \hspace{18cm}\]

Solution: Refer Example. 1

\[16.\ \color {green}{Find\ \frac{dy}{dx}\ if\ y\ =\ x^3\ log\ x}\ \hspace{18cm}\]

Solution: Refer Problem. 6

\[17.\ \color {green}{If\ xy\ =\ c^2\, Find\ \frac{dy}{dx}}\ \hspace{18cm}\]

Solution: Refer Example 16.

\[18.\ \color {green}{Find\ \frac{d^2y}{dx^2}\ if\ y\ =\ Sec\ x}\ \hspace{18cm}\]

Solution: Refer Example 5 .

\[19.\ \color {green}{Form\ the\ differential\ equation\ by\ eliminating\ the\ constant\ ‘a’\ in\ x^2\ +\ y^2\ =\ a^2}\ \hspace{18cm}\]

Solution: Refer Example 2.

\[20.\ \color {green}{If\ u\ =\ x^3\ +\ y^3\ +\ 3\ xy}\ ,\ \color {red} {find\ \frac{∂^2u}{∂x^2}}\hspace{18cm}\]

Solution: Refer Problem No. 5.

\[\underline{PART\ -\ C}\]
\[21.\ A)\ i.\ \color{green}{Find\ the\ inverse\ of\ the\ matrix}\ \begin{bmatrix} 3 & 4 & 1 \\ 0 & -1 & 2 \\ 5 & -2 & 6 \\ \end{bmatrix}\ \hspace{15cm}\]

Solution: Refer Example 3 under Inverse of matrix of Application of Matrices and Determinants (Text)

\[\hspace{1cm}\ ii.\ \color{green}{Solve\ the\ following\ equations\ using\ Cramers\ Rule}\ \hspace{12cm}\]\[\color{green}{3x\ -\ y\ +\ 2z\ =\ 8,\ x\ +\ y\ +\ z\ =\ 2\ and\ 2x\ +\ y\ -\ z\ =\ -\ 1}\ \hspace{7cm}\]

Solution: Refer Problem No. 11 of Application of Matrices and Determinants (Excercise Problems)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Find\ the\ rank\ of\ the\ matrix\ \begin{pmatrix} 1 & 2 & -1 & 3 \\ 2 & 4 & -4 & 7 \\ -1 & -2 & -2 & -2 \\ \end{pmatrix}}\ \hspace{15cm}\]

Solution: Refer Example 4 under Rank of matrix of Application of Matrices and Determinants (Text)

\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ coefficient\ of\ x^5\ in\ the\ expansion\ of\ (ax\ +\ \frac{b}{x})^{11}}\ \hspace{10cm}\]
\[22.\ A)\ i.\ \color{green}{Find\ the\ Real\ and\ Imaginary\ parts\ of\ \frac{(1+ i)(2 – i)}{1+ 3i}}\ \hspace{15cm}\]

Solution: Refer Problem No. 11 under Algebra of Complex Numbers (Excercise problems with soltuions)

\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ value\ of\ \frac{(cos⁡\ x\ -\ i sin⁡\ x)^3\ (cos⁡\ 3x\ +\ i sin⁡\ 3x)^5} {(cos⁡\ 2x\ – i sin⁡\ 2x)^5\ (cos⁡\ 5x\ + i sin⁡\ 5x)^7}\ when\ x\ =\ \frac{2\pi}{13}}\ \hspace{10cm}\]

Solution: Refer Example 9 under De-Moivre’s Theorem (Text)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Show\ that\ the\ points\ 3 + 2i,\ 5 + 4i,\ 3 + 6i\ and\ 1 + 4i\ in\ an\ Arand\ diagram\ form\ a\ square}\ \hspace{5cm}\]

Solution: Refer Problem No. 12 under Algebra of Complex Numbers (Excercise problems with soltuions)

\[\hspace{1cm}\ ii.\ \color{green}{Solve\ x^8\ -\ x^5\ +\ x^3\ -\ 1\ =\ 0}\ \hspace{15cm}\]

Solution: Refer Example 8 under Roots of Complex Numbers (Text)

\[23.\ A)\ i.\ \color{green}{If\ Sin\ A\ =\ \frac{8}{17} \ and\ sin\ B\ =\ \frac{5}{13},\ Show\ that\ Sin(A\ +\ B)\ =\ \frac{171}{221}}\ \hspace{15cm}\]

Solution: Refer Problem No. 12

\[\hspace{1cm}\ ii.\ \color{green}{Prove\ that\ Cos\ 10^{0}\ Cos\ 30^{0}\ Cos\ 50^{0}\ Cos\ 70^{0}\ =\ \frac{3}{16}}\ \hspace{15cm}\]

Solution: Refer Example No. 12 under Inverse Trigonometric Functions (Text)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Prove\ that\ \frac{SinA\ +\ Sin2A\ +\ Sin3A}{CosA\ +\ Cos2A\ +\ Cos3A}\ =\ Tan2A}\ \hspace{15cm}\]

Solution: Refer Example No. 12 under Inverse Trigonometric Functions (Text)

\[\hspace{1cm}\ ii.\ \color{green}{Prove\ that\ Cos^{-1}\ (4x^3\ -\ 3x)\ =\ 3\ Cos^{-1}\ x}\ \hspace{15cm}\]

Solution: Refer Example No. 18 under Inverse Trigonometric Functions (Text)

\[24.\ A)\ i.\ \color{green}{Evaluate:\ \lim\ _{θ\ \to\ 0}\ \frac{5Sin\ 6θ}{3Sin\ 2θ}}\ \hspace{15cm}\]

Solution: Refer Example. 8

\[\hspace{1cm}\ ii.\ \color{green}{Find \frac{dy}{dx}\ (i)\ if\ y\ =\ (x\ +\ 1)(x\ +\ 2)(x\ +\ 3)\ \hspace{2cm}\ (ii)\ y\ =\ \frac{x^2\ +\ 1}{e^x}}\ \hspace{15cm}\]

Solution: Refer Example. 11

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Find\ \frac{dy}{dx}\ (i)\ if\ y\ =\ log\ x\ (2x\ +\ 1)\ \sqrt{x} \ \hspace{2cm}\ (ii)\ y\ =\ \frac{ax\ +\ b}{cx\ +\ d}}\ \hspace{5cm}\]

Solution: Refer Problem. 16

\[\hspace{1cm}\ ii.\ \color{green}{Find \frac{dy}{dx}\ (i)\ if\ y\ =\ Sin(3x\ +\ 4)\ \hspace{2cm}\ (ii)\ if\ x^3\ +\ y^3\ =\ 3}\ \hspace{15cm}\]

Solution: Refer Example No. 2 and No. 14 under Differentiation Methods (Text)

\[25.\ A)\ i.\ \color{green}{Form\ a\ differential\ equation\ by\ eliminating\ the\ constant\ A\ and\ B\ from\ y\ =\ A\ Cos\ 5x\ +\ B\ Sin\ 5x}\ \hspace{15cm}\]

Solution: Refer Example. 4 under Successive Differentiation (Text)

\[\hspace{1cm}\ ii.\ \color{green}{If\ y\ =\ \frac{cos\ x}{x}\ prove\ that\ x\ y_2\ +\ 2\ y_1\ +\ xy\ =\ 0}\ \hspace{15cm}\]

Solution: Refer Example. 12 under Successive Differentiation (Text)

\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{ If\ u\ = 2\ x^3\ +\ 3\ x^2y\ +\ 4\ x y^2\ +\ 4\ y^3,\ find\ \frac{∂u}{∂x}\ and\ \frac{∂u}{∂y}}\ \hspace{15cm}\]

Solution: Refer Problem No. 5 under Partial Differentiation (Excercise problems with soltuions)

\[\hspace{1cm}\ ii.\ \color{green}{If\ u\ =\ 2\ x^3\ -\ 3\ x^2y\ +\ 3\ x y^2\ +\ 5\ y^3,\ find\ the\ value\ of\ \ x\ \frac{∂u}{∂x}\ +\ y\ \frac{∂u}{∂y}}\ \hspace{15cm}\]

Solution: Refer Example 5 under Partial Differentiation (Text)

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