MODEL EXAM QUESTION PAPER WITH SOLUTIONS FOR ENGINEERING MATHEMATICS – 1

\[\underline{PART\ -\ A}\]
\[1.\ Find\ the\ cofactor\ of\ 4\ in\ the\ matrix\ \begin{bmatrix} 1 & 0 & -1 \\ 2 & 3 & 4 \\ 7 & 8 & -2 \\ \end{bmatrix}\ \hspace{10cm}\]
\[\color {black}{Solution:}\ Refer\ Example\ 1\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/11/applications-of-matrices-and-determinantstext/

\[2.\ Find\ the\ rank\ of\ the\ matrix\ \begin{bmatrix} 5 & 2 \\ 6 & 3 \\ \end{bmatrix}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Example\ 1\ under\ Rank\ of\ a\ Matrix\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/09/05/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text/

\[3.\ If\ Z_1 = 1 + 4i,\ Z_2 = -3 + 6i,\ find\ 3\ Z_1\ +\ 2\ Z_2.\ \hspace{18cm}\]
\[\color {black}{Solution:}\ Example\ 3\ under\ Algebra\ of\ Complex\ Numbers\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/09/26/unit-ii-complex-numbers/

\[4.\ If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of \ ω ^4\ +\ ω ^5\ +\ ω ^6\ \hspace{16cm}\]
\[\color {black}{Solution:}\ Example\ 3\ under\ Roots\ of\ Complex\ Numbers\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/10/16/unit-ii-complex-numbers-3/

\[5.\ Find\ \frac{dy}{dx}\ if\ y\ =\ 8\ e^x\ -\ 4\ cosec\ x\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 1\ \hspace{15cm}\]

https://yanamtakshashila.com/2022/01/03/differentiation-excercise/

\[\underline{PART\ -\ B}\]
\[\color {black} {6.)\,Prove\ that}\ \begin{bmatrix} 1 & -1 & 2 \\ 2 & -2 & 4 \\ 3 & -3 & 6 \\ \end{bmatrix}\ is\ a\ singular\ matrix\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Example\ c\ under\ Singular\ and Non-Singular\ of\ a\ Matrix\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/08/03/unit-i-algebra/

\[7.\ Find\ the\ inverse\ of\ \begin{bmatrix} 1 & -1 \\ -2 & 0 \\ \end{bmatrix}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Example\ 1\ under\ Inverse\ of\ a\ Matrix\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/09/05/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text/

\[8.\ Find\ the\ 6^{th}\ term\ in\ the\ expansion\ of\ (x\ +\ \frac{1}{x})^{10}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Example\ 3\ under\ Binomial\ Theorem\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/09/16/unit-i-algebra-2/

\[9.\ Find\ the\ Real\ and\ Imaginary\ parts\ of\ \frac{2+ 3i}{4+ 5i}\ \hspace{18cm}\]

https://yanamtakshashila.com/2021/09/26/unit-ii-complex-numbers/

\[10.\ If\ a = cos⁡\ x + i sin⁡\ x,\ b = cos⁡\ y + i sin⁡\ y,\ find\ ab\ and\ \frac{1}{ab}\ \hspace{18cm}\]
\[\color {black}{Solution:}\ Example\ 3\ under\ De-Moivre’s\ Theorem\ \hspace{15cm}\]
https://yanamtakshashila.com/2021/10/13/unit-ii-complex-numbers-2
\[11.\ Find\ the\ value\ of\ \frac{Tan\ {20}^0\ +\ Tan\ {25}^0}{1\ -\ Tan\ {20}^0\ Tan\ {25}^0}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Example\ 3\ under\ Compound\ Angles\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/10/21/unit-ii-complex-numbers-4/

\[12:\ Prove\ that\ Cos^4\ A\ -\ Sin^4\ A\ =\ Cos\ 2A\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Example\ 5\ under\ Multiple\ and\ Sub-multiple\ angles\ \hspace{15cm}\]
https://yanamtakshashila.com/2021/10/27/3-2-multiple-and-sub-multiple-angles/
\[13.\ Express\ Sin\ 5A\ -\ Sin\ 3A\ as\ product\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Example\ 2\ under\ Sum\ and\ Product\ Formulae\hspace{15cm}\]

https://yanamtakshashila.com/2021/10/31/3-3-sum-and-product-formulae/

\[14.\ Evaluate:\ \lim\ _{x\ \to\ 2}\ \frac{x^4\ -\ 2^4}{x\ -\ 2}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Example\ 2\ under\ Limits\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/11/08/limits/

\[15.\ Find\ \frac{dy}{dx}\ if\ y\ =\ e^x\ log\ x\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Example\ 6\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/11/13/differentiation/

\[16.\ Find\ \frac{dy}{dx}\ if\ y\ =\ (x^2\ +\ 5)\ Cos\ x\ e^{-\ 2x}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 2\ \hspace{15cm}\]

https://yanamtakshashila.com/2022/01/03/differentiation-excercise/

\[17.\ If\ y\ =\ log(Sin\ x),\ find\ \frac{dy}{dx}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Example\ 3\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/11/18/differentiation-methods/

\[18.\ Form\ the\ differential\ equation\ of\ y^2\ =\ 4\ a\ x\ by\ eliminating\ the\ constant\ ‘a’\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 1\ \hspace{15cm}\]

https://yanamtakshashila.com/2022/01/03/successive-differentiation-excercise/

\[19.\ If\ u\ = 2\ x^3\ +\ 4\ \ y^3\ +\ 2\ x\ y\ ,\ find\ \frac{∂u}{∂x}\ and\ \frac{∂u}{∂y}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Example\ 3\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/03/partial-differentiation/

\[20.\ If\ u\ =\ x^3\ +\ y^3\ ,\ find\ \frac{∂^2u}{∂x^2}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 1\ \hspace{15cm}\]

https://yanamtakshashila.com/2022/01/03/partial-differentiation-excercise/

\[\underline{PART\ -\ C}\]
\[\color {black} {21.\ A)\ i.}\ Solve\ by\ using\ Cramers\ Rule\ \hspace{20cm}\]
\[3x – y + 2z=8,\ x – y + z = 2\ and\ 2x + y – z = 1\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 2\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/11/applications-of-matrices-and-determinantstext/

\[ii.\ Find\ the\ inverse\ of\ \begin{bmatrix} 3 & 1 & -1\\ 2 & -2 & 0 \\ 1 & 2 & -1 \\ \end{bmatrix}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 6\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/11/applications-of-matrices-and-determinantstext/

\[\color {black} {B)\ i.}\ Find\ the\ rank\ of\ the\ matrix\ \begin{bmatrix} 1 & 2 & 3 & 2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 & 5 \\ \end{bmatrix}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Example\ 4\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/09/05/largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text/

\[ii.\ Find\ the\ middle\ terms\ in\ the\ expansion\ of\ (x^3\ +\ \frac{2}{x^3})^{11}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 1\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/15/binomial-theorem-revision/

\[\color {black} {22.\ A)\ i.}\ Find\ the\ modulus\ and\ amplitude\ of\ \frac{-\ 3\ +\ i}{-\ 1\ +\ i}\ \hspace{18cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 1\ \hspace{15cm}\]

https://yanamtakshashila.com/2022/01/02/algebra-of-complex-numbers-excersice/

\[ii.\ Show\ that\ the\ complex\ numbers\ 2+i,\ 4+3i,\ 2+5i\ and\ 3i\ form\ a\ square\ \hspace{10cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 11\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/09/26/unit-ii-complex-number/

\[\color {black} {B)\ i.}\ Simplify\ using\ DeMoivre’s\ theorem:\ \frac{(cos⁡\ 2θ – i sin⁡\ 2θ)^7\ (cos⁡\ 3θ + i sin⁡\ 3θ)^{-5}} {(cos⁡\ 4θ + i sin⁡\ 4θ)^2\ (cos⁡\ 5θ – i sin⁡\ 5θ)^{-6}}\ \hspace{10cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 2\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/16/de-movires-theorem-revision/

\[ii.\ Solve\ x^5 +\ x^3\ +\ x^2\ +\ 1\ =\ 0\ \hspace{18cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 6\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/10/16/unit-ii-complex-numbers-3/

\[\color {black} {23.\ A)\ i.}\ If\ A\ and\ B\ are\ acute\ angles\ and\ Sin\ A\ =\ \frac{1}{\sqrt{10}},\ Sin\ B\ =\ \frac{1}{\sqrt{5}},\ \hspace{15cm}\]\[ Prove\ that\ (A\ +\ B)\ =\ \frac{π}{4}\ \hspace{10cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 6\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/10/21/unit-ii-complex-numbers-4/

\[ii.\ If\ A\ +\ B\ =\ {45}^0 \ Prove\ that\ (1\ +\ Tan\ A)\ (1\ +\ Tan\ B)\ =\ 2.\ \hspace{15cm}\]\[Hence\ deduce\ the\ value\ of\ Tan\ \ 22\ {\frac{1}{2}}^0\ \hspace{13cm}\]

https://yanamtakshashila.com/2021/10/21/unit-ii-complex-numbers-4/

\[\color {black} {B)\ i.}\ Prove\ that\ \frac{Sin\ 3\ \theta}{Sin\ \theta}\ -\ \frac{Cos\ 3\ \theta}{Cos\ \theta}\ =\ 2\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 1\ \hspace{15cm}\]

https://yanamtakshashila.com/2022/01/02/multiple-and-sub-multiple-angles-excercise/

\[\color {black} {24.\ A)\ i.}\ Evaluate:\ (i)\ \lim\ _{x\ \to\ 3}\ \frac{x^6\ -\ 3^6}{x\ -\ 3}\ \hspace{1cm}\ (ii)\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 7\ θ}{Sin\ 2\ θ}\ \hspace{7cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 1\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/23/limits-revision/

\[ii.\ Find\ \frac{dy}{dx}\ (i)\ if\ y\ =\ e^x\ log\ x\ Cos\ x\ (ii)\ y\ =\ \frac{x^2\ +\ Tan\ x}{x\ -\ Sin\ x}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 9\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/11/13/differentiation/

\[\color {black} {B)\ i.}\ \frac{dy}{dx}\ (i)\ if\ y\ =\ (2\ x\ +\ 1)(3\ x\ -\ 7)(4\ -\ 9\ x)\ (ii)\ y\ =\ \frac{e^x\ +\ Sin\ x}{1\ -\ Cos\ x}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 2\ \hspace{15cm}\]

https://yanamtakshashila.com/2022/01/03/differentiation-excercise/

\[ii..\ Find\ \frac{dy}{dx}\ if (i)\ \ x^2\ +\ y^2\ +\ 2\ x\ +\ 3\ y\ =\ 0\ \hspace{2cm}\ and\ (ii)\ y\ =\ a\ +\ x\ e^y\ \hspace{2cm}\]
\[\color {black}{Solution:}\ Refer\ Q.\ No. 14th\ and\ 15th\ \hspace{10cm}\]

https://yanamtakshashila.com/2021/11/18/differentiation-methods/

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

https://yanamtakshashila.com/2021/11/23/successive-differentiation/

\[ii.\ If\ u\ = \frac{x^3\ y^3}{x^3\ +\ y^3} ,\ Show\ that\ x\ \frac{∂u}{∂x}\ +\ y\ \frac{∂u}{∂y}\ =\ 3\ u\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 5\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/03/partial-differentiation/

\[\color {black} {B)\ i.}\ If\ y\ =\ a\ Cos(log\ x)\ +\ b\ Sin\ (log\ x),\ prove\ that\ x^2\ y_2\ +\ x\ y_1\ +\ y\ =\ 0\ \hspace{10cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 7\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/11/23/successive-differentiation/

\[ii.\ If\ u\ =\ x^3\ -\ 2\ x^2\ y\ +\ 3\ x\ y^2\ +\ y^3\ ,\ find\ \frac{∂^2u}{∂x^2}\ and\ \frac{∂^2u}{∂y^2}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Refer\ Problem\ 6\ \hspace{15cm}\]

https://yanamtakshashila.com/2021/12/03/partial-differentiation/