Engineering Mathematics I

The following are the assignments for the upcoming exam on Engineering Mathematics I. Please solve the questions and submit.

Study Material:

Rolle’s Theorem	Characteristic Equation of a Matrix
nth Derivative	Cayley-Hamilton Theorem

** End Sem Syllabus: Unit I , II, III, and IV.

Unit I and IV: Differential Calculus & Vector Algebra

Note: This section is for Sec G and Sec L.

Q1: Show that the following limits do not exist.

1. lim_x→1 [x] where [x] denotes the greatest integer not exceeding x.
2. lim_x→0 $\dfrac{|x|}{x}$.

Q2: If $y=(ax+b)^m$, then find $y_n$. For this, you study the article nth Derivative.

Q3: If $y=e^{m \sin^{-1}x}$, then show that

$(1-x^2)y_{n+2} -(2n+1)xy_{n+1}$ $- (n^2+m^2)y_n =0.$

Q3: If $y=\tan^{-1}x$, then show that

$(1+x^2)y_{n+2} +2x(n+1)y_{n+1}$ $+ n(n+1)y_n =0.$

Q4: Find the maximum and minimum values of the functions:

1. $y=x^3-3x^2-9x+12$.
2. $y=2x^3-21x^2+36x-20$.

Q5: Using Taylor’s theorem, find the series expansion of $f(x)=e^x$ in ascending powers of $x$.

Q6: Using Taylor’s theorem, find the series expansion of $f(x)=\sin x$ in powers of $(x -\dfrac{\pi}{2})$.

Q7: Verify the Rolle’s theorem for the function $f(x)=2x^3+x^2-4x-2$ on the interval $[-\sqrt{2}, \sqrt{2}]$.

Q8: Verify Lagrange’s mean value theorem for the function $f(x)=x(x-1)(x-2)$ on the interval $[0, \frac{1}{2}]$.

Q8: Let $u =\dfrac{x^2y^2}{x+y}$. Apply Euler’s theorem to find $x \dfrac{\partial{u}}{\partial{x}} + y \dfrac{\partial{u}}{\partial{y}}$. Hence, show $x^2 \dfrac{\partial^2{u}}{\partial{x^2}} + 2xy\dfrac{\partial^2{u}}{\partial{x}\partial{y}}$ $+y^2\dfrac{\partial^2{u}}{\partial{y^2}}=6u$.

Q9: Find the divergence and curl of $\vec{v}=x^2yz\hat{i}+3xy^2\hat{j}$ $+(xz-yz)\hat{k}$ at (1, -1, 1).

Q10: Find $p$ so that the vectors $\vec{a}=\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=\hat{i}+2\hat{j}-\widehat{k,}$ and $\vec{c}=3\hat{i}+p\hat{j}+5\hat{k}$ are coplanar.

Q11: Find the unit normal to the surface $x^3-y^3+3xyz=1$ at (1,2,-1).

Unit II and III: Integral Calculus & Linear Algebra

Note: This section is for Sec J and Sec F.

Q1: Find the reduction formulae for $\int \sec^n x \ dx$ and $\int \tan^n x \ dx$. using this, evaluate $\int_0^{\pi/4}\tan^4 x \ dx$

Q2: Compute the integral using the beta gamma function:

$\int_0^{\frac{\pi}{2}} \sin^4 x \cos^3 x \ dx$.

Q3: Find the area of the surface generated by revolving the parabola 𝑦²=2𝑎𝑥 about X-axis bounded by 𝑥=𝑎.

Q4: What is a Hermitian matrix and a skew-Hermitian matrix? Express the following matrix as sum of a Hermitian and a skew-Hermitian matrix:

\begin{equation} A = \begin{pmatrix} 1+i & -i \\ -3+i & 5 \end{pmatrix}. \end{equation}

Q5: Find the values of 𝜆 and 𝜇 for which the following system of linear equations has i) no solution, ii) unique solution, and iii) infinitely many solutions:

𝑥+𝑦+𝑧 = 1

𝑥+2𝑦-𝑧 = 𝜇

5𝑥+7𝑦+𝜆𝑧 = 𝜇².

Q6: Find the values of 𝜆 and 𝜇 for which the following system of linear equations has i) no solution, ii) unique solution, and iii) infinitely many solutions:

𝑥+2𝑦+𝑧 = 1

2𝑥+𝑦+3𝑧 = 𝜇

𝑥−4𝑦+𝜆𝑧 = 𝜇+1.

Q7: Check the diagonalizability of the following matrix:

$A = \begin{pmatrix} 1 & 1 & -2 \\ -1 & 2 & 1 \\ 0 & 1 & -1 \end{pmatrix}$ $\quad$ or $\quad$ $A = \begin{pmatrix}-2&0&0\\5&2&6\\3&4&7 \end{pmatrix}.$

Q8: Find the eigenvalue and eigenvectors of the matrix

\begin{equation} A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 0 & 1 \end{pmatrix}. \end{equation}

Q9: State Cayley-Hamilton theorem. Verify this theorem for the matrix:

$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{pmatrix}$ $\quad$ or $\quad$ $A= \begin{pmatrix}1&-1&2 \\ 1&5&-3 \\ 3&5&7 \end{pmatrix}.$

Q10: Find the inverse of the following matrix using Cayley-Hamilton theorem.

\begin{equation} A = \begin{pmatrix} 1 & 0 & 2 \\ 0 & -1 & 1 \\ 0 & 1 & 0 \end{pmatrix}. \end{equation}

Q11: Find the inverse of the following matrix using elementary row operations (Gauss-Jordon method):

$A = \begin{pmatrix} 1 & -1 & 5 \\ 0 & 2 & 3 \\ 0 & 0 & 3 \end{pmatrix}$ $\quad$ or $\quad$ $A=\begin{pmatrix} 1&-1&-2 \\ -1&2&4 \\ 0&0&3 \end{pmatrix}$.