# FEB-2022 TAMIL NADU POLYTECHNIC BOARD EXAM ENGINEERING MATHEMATICS – 1(40012)QUESTION PAPER (Online) 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.
$\underline{PART\ -\ A}$
$1.\ \color{green}{If\ A =\begin{pmatrix} 5 & – 6 \\ 3 & 7\\ \end{pmatrix},\ find\ 3\ A?}\ \hspace{15cm}$

Solution: Refer Problem No. 6

$2. \ \color{green}{Find\ the\ value\ of\ (2i\ +\ i)(i\ +\ 3i)}\ \hspace{15cm}$

Solution: Refer Problem No. 2

$3.\ \color{green}{Show\ that \ tan (765^0)\ =\ 1}\ \hspace{15cm}$

Solution: Refer Problem Ex. 1

$4.\ \color{green}{Evaluate:\ \lim\ _{x\ \to\ 2}\ \frac{x^2\ -\ 2^2}{x\ -\ 2}}\ \hspace{15cm}$

Solution: Refer Problem No. 1

$5.\ \color{green}{Find\ the\ order\ and\ degree\ of\ the\ differential\ equation\ \frac{d^3y}{dx^3}\ -\ 5\ \frac{d^2y}{dx^2}\ +\ 6\ \frac{dy}{dx}\ +\ 7\ y\ =\ 0}\ \hspace{10cm}$

Solution: Refer Problem No. 1

$\underline{PART\ -\ B}$
$6.\ \color{green}{Prove\ that\ the\ matrix \begin{pmatrix} 8 & 16 \\ 6 & 12\\ \end{pmatrix},\ is\ singular.}\ \hspace{15cm}$

Solution: Refer Problem No. 8

$7.\ \color{green}{Find\ the\ Adjoint\ matrix\ of\ \begin{bmatrix} 5 & -6 \\ 3 & 2 \\ \end{bmatrix}}\ \hspace{15cm}$

Solution: Refer Problem No. 5

$8.\ \color{green}{Find\ the\ 4^{th}\ term\ in\ the\ expansion\ of\ (x^2\ +\ \frac{1}{x})^8}\ \hspace{15cm}$

Solution: Refer Problem No. 5

$9:\ \color{green}{Find\ the\ modulus\ and\ amplitude\ of\ \sqrt{3} – \ i}\ \hspace{18cm}$

Solution: Refer Problem No. 9

$10.\ \color{green}{If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of \ ω( ω\ +\ 1)}\ \hspace{15cm}$

Solution: Refer Problem No. 1

$11:\ \color{green}{Simplify\ \frac{cos⁡\ 5 θ + i sin⁡\ 5 θ} {cos⁡\ 3 θ – i sin⁡\ 3 θ}}\ \hspace{18cm}$

Solution: Refer Problem\ No.3

$12.\ \color{green}{Prove\ that\ Sin\ 40^{0}\ +\ Sin\ 20^{0}\ -\ Cos\ 10^{0}\ =\ 0}\ \hspace{15cm}$

Solution: Refer Problem No. 5

$13.\ \color{green}{Prove:\ Tan^{-1}\ (\frac{3x\ -\ x^3}{1\ -\ 3\ x^2})\ =\ 3\ Tan^{-1}}\ x\ \hspace{15cm}$

Solution: Refer Problem 2 under Part-C

$14:\ \color{green}{Prove\ that\ \frac{Sin\ 2A}{1\ +\ Cos\ 2A}\ =\ Tan\ A}\ \hspace{15cm}$

Solution: Refer Problem Ex. 1

$15.\ \color{green}{Evaluate:\ \lim\ _{x\ \to\ 0}\ \frac{Sin\ 2x}{Sin\ 4x}}\ \hspace{15cm}$

Solution: Refer Problem No. 4

$16.\ \color{green}{Find\ \frac{dy}{dx}\ if\ y\ =\ \frac{1}{x^2} +\ 3\ tan\ x\ -\ log\ x}\ \hspace{15cm}$

Solution: Refer Problem No. 1

$17.\ \color{green}{Find\ \frac{dy}{dx}\ if\ y\ =\ \frac{1\ +\ cos\ x}{1\ -\ cos\ x}}\ \hspace{15cm}$

Solution: Refer Problem No. 10

$18.\ \color{green}{Find\ \frac{d^2y}{dx^2}\ if\ y\ =\ x^4\ -\ 3\ x^3\ +\ 6\ x^2\ +\ 2\ x\ +\ 1}\ \hspace{15cm}$

Solution: Refer Problem No. 5

$19.\ \color{green}{Form\ the\ differential\ equation\ by\ eliminating\ the\ constant\ ‘a’\ in\ x^2\ +\ y^2\ =\ a^2}\ \hspace{15cm}$

Solution: Refer Problem No.1

$20.\ \color{green}{If\ u\ = 3\ x^3\ +\ 4\ \ y^3\ +\ 6\ x\ y\ ,\ find\ (i)\ \frac{∂u}{∂x}\ (ii)\ \frac{∂u}{∂y}}\ \hspace{15cm}$

Solution: Refer Problem No. 5

$\underline{PART\ -\ C}$
$21.\ A)\ i.\ \color{green}{Find\ the\ cofactor\ matrix\ of\ \begin{bmatrix} 2 & 3 & 4 \\ 1 & 2 & 3 \\ -1 & 1 & 2 \\ \end{bmatrix}}\ \hspace{15cm}$

Solution: Refer Example under cofactor element of matrix

$\hspace{1cm}\ ii.\ \color{green}{Solve\ by\ using\ Cramers\ Rule}\ \hspace{20cm}$
$\hspace{1cm}\ \color{green}{4\ x\ +\ y\ +\ z\ =\ 6,\ 2\ x\ -\ y\ -\ 2\ z\ =\ -\ 6\ and\ x\ +\ y\ +\ z\ =\ 3}\ \hspace{15cm}$

Solution: Refer Example 2

$(OR)$
$\hspace{0.5cm}\ B)\ i.\ \color{green}{If\ A =\begin{bmatrix} 2 & -3 & 8 \\ 21 & 6 & -6 \\ 4 & -33 & 19 \\ \end{bmatrix}\ ,\ B =\begin{bmatrix} 1 & -29 & -8 \\ 2 & 0 & 3 \\ 17 & 15 & 4 \\ \end{bmatrix}}\ \hspace{7cm}$$\color{green}{ Prove\ that\ (A\ +\ B)^T\ =\ A^T\ +\ B^T}\hspace{5cm}$

Solution: Refer Example 3 Under Addition and Subtraction of Matrices

$\hspace{1cm}\ ii.\ \color{green}{Find\ the\ term\ independent\ of\ x\ in\ the\ expansion\ of\ (x^2\ +\ \frac{1}{x})^{12}}\ \hspace{15cm}$

Solution: Refer Problem No. 9

$22.\ A)\ i.\ \color{green}{Find\ the\ modulus\ and\ amplitude\ of\ \frac{1}{2} + i\ \frac{\sqrt{3}}{2}}\ \hspace{18cm}$

Solution: Refer Problem No. 14

$\hspace{1cm}\ ii)\ \color{green}{Show\ that\ the\ complex\ numbers\ 1-2i,\ -1+4i,\ 5 +8i\ and\ 7+2i\ form\ a\ parallelogram}\ \hspace{10cm}$

Solution: Refer Problem No. 21

$(OR)$
$B)\ i)\ \ \color{green}{Simplify\ using\ DeMoivre’s\ theorem:\ \frac{(cos⁡\ 3θ + i sin⁡\ 3θ)^{-5}\ (cos⁡\ 2θ + i sin⁡\ 2θ)^4} {(cos⁡\ 4θ – i sin⁡\ 4θ)^{-2}\ (cos⁡\ 5θ – i sin⁡\ 5θ)^3}}\ \hspace{10cm}$

Solution: Refer Example 8

$\hspace{1cm}\ ii)\ \color{green}{Solve\ x^3\ -\ 1\ =\ 0}\ \hspace{18cm}$

Solution: Refer Example 4

$23.\ A)\ i.\ \color{green} {Prove\ that\ Sin^2\ A\ +\ Sin^2\ (60^0\ +\ A)\ +\ Sin^2\ (60^0\ -\ A)\ =\ \frac{3}{2}}\ \hspace{15cm}$

Solution: Refer Example 15

$\hspace{1cm}\ ii)\ \color{green}{If\ Cos\ A\ =\ \frac{1}{7} \ and\ Cos\ B\ =\ \frac{13}{14},\ Prove\ that\ A\ -\ B\ =\ \frac{π}{3}}\ \hspace{15cm}$

Solution: Refer Problem No. 2

$(OR)$
$B)\ i)\ \ \color{green} {Prove:\ Tan^{-1}\ (\frac{3x\ -\ x^3}{1\ -\ 3\ x^2})\ =\ 3\ Tan^{-1}\ x}\ \hspace{15cm}$

Solution: Refer Problem 2 under Part-C

$\hspace{1cm}\ ii)\ \color{green} {Prove\ that\ Sin\ 10^{0}\ Sin\ 30^{0}\ Sin\ 50^{0}\ Sin\ 70^{0}\ =\ \frac{1}{16}}\ \hspace{15cm}$

Solution: Refer Example 13

$24.\ A)\ i.\ \color{green} {\lim\ _{x\ \to\ 3}\ \frac{x^3\ -\ 3^3}{x^4\ -\ 3^4}}\ \hspace{15cm}$

Solution: Refer Example 4

$\hspace{1cm}\ ii)\ \color{green}{Differentiate\ the\ following\ with\ respect\ to\ x\ (i)\ if\ y\ =\ x^3\ (1\ +\ log\ x)\ (ii)\ y\ =\ \frac{x +\ Tan\ x}{Cos\ x}}\ \hspace{15cm}$

Solution: Refer Example 9

$(OR)$
$B)\ i)\ \ \color{green} {Differentiate\ the\ following\ with\ respect\ to\ x\ (i)\ if\ y\ =\ x^3\ Sin\ x\ Tan\ x\ (ii)\ y\ =\ \frac{x +\ 6}{x\ -\ 7}}\ \hspace{15cm}$

Solution: Refer Example 10

$\hspace{1cm}\ ii)\ \color{green}{Find\ \frac{dy}{dx}\ if\ x^3\ +\ y^3\ =\ 3\ a\ x\ y}\ \hspace{15cm}$

Solution: Refer Problem No. 14

$25.\ A)\ i.\ \color{green} {If\ y\ =\ x^2\ Sin\ x,\ prove\ that\ x^2\ y_2\ -\ 4\ x\ y_1\ +\ (x^2\ +\ 6)y\ =\ 0}\ \hspace{15cm}$

Solution: Refer Problem 7

$\hspace{1cm}\ ii)\ \color{green}{Eliminate\ the\ constant\ by\ differentiate\ twice\ y\ =\ a\ Cos\ x\ +\ b\ Sin\ x}\ \hspace{15cm}$

Solution: Refer Problem 5

$(OR)$
$B)\ i)\ \ \color{green} {If\ u\ =\ x^3\ +\ y^3\ +\ 3\ x\ y^2\ ,\ prove\ that\ x\ \frac{∂u}{∂x}\ +\ y\ \frac{∂u}{∂y}\ =\ 3\ u}\ \hspace{15cm}$

Solution: Refer Problem No. 5

$\hspace{1cm}\ ii)\ \color{green}{If\ u\ =\ x^3\ +\ y^3\ +\ 4\ x\ y,\ find\ the\ x^2\ \frac{∂^2u}{∂x^2}\ +\ y^2\ \frac{∂^2u}{∂y^2}}\ \hspace{15cm}$

Solution: Refer Problem No. 8