FEB-2022 TAMIL NADU POLYTECHNIC BOARD EXAM ENGINEERING MATHEMATICS – 1(40012)QUESTION PAPER (Online) WITH SOLUTIONS

\[\underline{PART\ -\ A}\]
\[1.\ If\ A =\begin{pmatrix} 5 & – 6 \\ 3 & 7\\ \end{pmatrix},\ find\ 3\ A?\ \hspace{15cm}\]

Solution: Refer Problem No. 6

\[2. \ Find\ the\ value\ of\ (2i\ +\ i)(i\ +\ 3i)\ \hspace{15cm}\]

Solution: Refer Problem No. 2

\[3.\ Show\ that \ tan (765^0)\ =\ 1\ \hspace{15cm}\]

Solution: Refer Problem Ex. 1

\[4.\ Evaluate:\ \lim\ _{x\ \to\ 2}\ \frac{x^2\ -\ 2^2}{x\ -\ 2}\ \hspace{15cm}\]

Solution: Refer Problem No. 1

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

Solution: Refer Problem No. 1

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

Solution: Refer Problem No. 8

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

Solution: Refer Problem No. 5

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

Solution: Refer Problem No. 5

\[9:\ Find\ the\ modulus\ and\ amplitude\ of\ \sqrt{3} – \ i\ \hspace{18cm}\]

Solution: Refer Problem No. 9

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

Solution: Refer Problem No. 1

\[11:\ Simplify\ \frac{cos⁡\ 5 θ + i sin⁡\ 5 θ} {cos⁡\ 3 θ – i sin⁡\ 3 θ}\ \hspace{18cm}\]

Solution: Refer Problem\ No.3

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

Solution: Refer Problem No. 5

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

Solution: Refer Problem 2 under Part-C

\[Ex\ 14:\ Prove\ that\ \frac{Sin\ 2A}{1\ +\ Cos\ 2A}\ =\ Tan\ A\ \hspace{15cm}\]

Solution: Refer Problem Ex. 1

\[15.\ Evaluate:\ \lim\ _{x\ \to\ 0}\ \frac{Sin\ 2x}{Sin\ 4x}\ \hspace{15cm}\]

Solution: Refer Problem No. 4

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

Solution: Refer Problem No. 1

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

Solution: Refer Problem No. 10

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

Solution: Refer Problem No. 5

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

Solution: Refer Problem No.1

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

Solution: Refer Problem No. 5

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

Solution: Refer Example under cofactor element of matrix

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

Solution: Refer Example 2

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

Solution: Refer Example 3 Under Addition and Subtraction of Matrices

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

Solution: Refer Problem No. 9

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

Solution: Refer Problem No. 14

\[ii.\ Show\ that\ the\ complex\ numbers\ 1-2i,\ -1+4i,\ 5 +8i\ and\ 7+2i\ form\ a\ parallelogram\ \hspace{10cm}\]

Solution: Refer Problem No. 21

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

Solution: Refer Example 8

\[ii)\ Solve\ x^3\ -\ 1\ =\ 0\ \hspace{18cm}\]

Solution: Refer Example 4

\[\color {black} {23.\ A)\ i.}\ Prove\ that\ Sin^2\ A\ +\ Sin^2\ (60^0\ +\ A)\ +\ Sin^2\ (60^0\ -\ A)\ \hspace{15cm}\]

Solution: Refer Example 15

\[ii.\ If\ Cos\ A\ =\ \frac{1}{7} \ and\ Cos\ B\ =\ \frac{13}{14},\ Prove\ that\ A\ -\ B\ =\ \frac{π}{3}\ \hspace{15cm}\]

Solution: Refer Problem No. 2

\[B)\ i.\ \ Prove:\ Tan^{-1}\ (\frac{3x\ -\ x^3}{1\ -\ 3\ x^2})\ =\ 3\ Tan^{-1}\ x\ \hspace{15cm}\]

Solution: Refer Problem 2 under Part-C

\[ii).\ Prove\ that\ Sin\ 10^{0}\ Sin\ 30^{0}\ Sin\ 50^{0}\ Sin\ 70^{0}\ =\ \frac{1}{16}\ \hspace{15cm}\]

Solution: Refer Example 8

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

Solution: Refer Example 4

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

Solution: Refer Example 9

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

Solution: Refer Example 10

\[ii.\ Find\ \frac{dy}{dx}\ if\ x^3\ +\ y^3\ =\ 3\ a\ x\ y\ \hspace{15cm}\]

Solution: Refer Problem No. 14

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

Solution: Refer Problem 7

\[ii.\ Eliminate\ the\ constant\ by\ differentiate\ twice\ y\ =\ a\ Cos\ x\ +\ b\ Sin\ x\ \hspace{15cm}\]

Solution: Refer Problem 5

\[\color {black} {B)\ i.}\ If\ u\ =\ x^3\ +\ y^3\ +\ 3\ x\ y^2\ ,\ prove\ that\ x\ \frac{∂u}{∂x}\ +\ y\ \frac{∂u}{∂y}\ =\ 3\ u\ \hspace{15cm}\]

Solution: Refer Problem No. 5

\[ii.\ If\ u\ =\ x^3\ +\ y^3\ +\ 4\ x\ y,\ find\ the\ x^2\ \frac{∂^2u}{∂x^2}\ +\ y^2\ \frac{∂^2u}{∂y^2}\ \hspace{15cm}\]

Solution: Refer Problem No. 8

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