Engineering Mathematics II Problems: Practice Questions

Here, you can find a list of engineering mathematics questions that can be treated as practice problems in end-sem exam.

For the study material of Engineering Mathematics II, please visit this LINK.

Engineering Mathematics II Assignments

The students of Sections D, G and J need to submit the below assignments on sequence, series and complex analysis.

Sequence and Series

Q1. Discuss the convergence of the sequences:

(i) $\left\{\dfrac{n^n}{(n-1)^n}\right\}$ [Solution] (ii) $\{(-1)^n\}$ [Solution] (iii) $\left\{\dfrac{(-1)^n}{n} + 2\right\}$ [Solution]

Q2: Define monotonically increasing and decreasing sequence. Show $\left \{ \dfrac{n+1}{2n+1} \right \}$ is a monotonically decreasing sequence.

Q3: Define a bounded sequence. Show the sequence $\left \{ (-1)^{n+1} \dfrac{n+1}{2n+1} \right \}$ is bounded.

Q3. Find the value of following series

(i) $1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\cdots$

(ii) $\dfrac{1}{1\cdot 2}+\dfrac{1}{2\cdot 3}+\dfrac{1}{3\cdot 4}+\cdots$

(iii) $1+\dfrac{1}{1\cdot 2}+\dfrac{1}{2}+\dfrac{1}{2\cdot 3}$ $+\dfrac{1}{2^2}+\dfrac{1}{3\cdot 4}+\dfrac{1}{2^3}+\cdots$

Q4. Prove the series $\displaystyle \sum_{n=1}^\infty \dfrac{n}{n+1}$ cannot converge.

Q5. State limit comparison test, ratio test and root test for an infinite series. Test the convergence of the following series:

1. $\sum \sin \left( \dfrac{1}{n} \right)$ [Full Solution]
2. $\sum \dfrac{1}{n} \sin \left( \dfrac{1}{n} \right)$ (1 OR 2) [Full Solution]
3. $\sum_{n=1}^\infty \dfrac{\sin nx}{n^2} \, (x>0)$
4. $\sum_{n=1}^\infty \dfrac{n}{n^2+\sqrt{n}}$.
5. $\sum_{n=2}^\infty \dfrac{n^{n^2}}{(n-1)^{n^2}}$. [Full Solution]

Q6: State Leibnitz’s test. Discuss the convergence of the series

(i) $1-\dfrac{1}{3}+\dfrac{1}{5}-$ $\dfrac{1}{7}+\cdots$ (OR) (ii) $1-\dfrac{1}{2^2}+\dfrac{1}{3^2}-$ $\dfrac{1}{4^2}+\cdots$

Q7: Define absolute and conditional convergence of a series. Give an example.

Q8. (i) Find the Fourier series representation of the function

$f(x) = \begin{cases} \pi+x, ~~ & \text{ if} ~~-\pi<x<0 \\ 0 ~~ & \text{ if} ~~0<x<\pi. \end{cases}$

(OR)

(ii) Find Fourier series expansion of

$f(x) = \begin{cases} -k ~~ & \text{ if} ~~-\pi<x<0 \\ k ~~ & \text{ if} ~~0<x<\pi, \quad \quad \end{cases}$ where $k$ is a constant.

(iii) Find Fourier sine and cosine series representation of the function $f(x)=5$ if $0 \leq x \leq 5$. [Solution]

Q9. Find the radius of convergence for the power series $\displaystyle \sum_{n=1}^\infty \dfrac{n!}{n^n}x^n$. Also, find the interval of convergence. [Full Solution]

Complex Analysis

Q1. Find the real and imaginary parts of $\dfrac{\overline{z}}{z}$. Prove that $f(z)=\overline{z}$ is not differentiable at $z=0$.

Q2. Show the function f(z) defined below is not continuous at z=0. [Full Solution]

$f(z)=\begin{cases} \dfrac{Im(z)}{|z|} & \text{if } z \neq 0 \\ 0 & \text{if } z=0. \end{cases} \quad$

Q3. State Cauchy-Riemann (C-R) equations.

(i) Show the function

f(z) = f(x, y) $=\begin{cases} \dfrac{x^2y^5(x+iy)}{x^4+y^{10}} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{if } (x,y)=(0,0). \end{cases}$

satisfies C-R equations at z=0; but not differentiable (analytic) at z=0.

(ii) Show that function $f(z) = \sqrt{|\text{Re}(z) \text{Im}(z)|}$ [that is, $f(x,y)=\sqrt{|xy|}$] satisfies C-R equation at $z=0$ [i.e, $(x,y)=(0,0)$], but not differentiable (analytic) at $z=0$. [Full Solution]

Q4:

1. Evaluate $\displaystyle \int_C \dfrac{1}{z} ~dz$ where C is the unit circle |z|=1.
2. Compute $\displaystyle \int_C \text{Re}(z) ~dz$ where C is the curve parameterized by $x=t, y=t^3$ joining from (-1, -1) to (1, 1).

Q5. State Cauchy integral theorem and formulas. Evaluate the integral

(i) $\displaystyle\int_{|z|=1} \dfrac{1-z}{z(z-1)(z-2)} dz$

(ii) $\displaystyle\int_{|z|=4} \dfrac{\cos z}{z^2+\pi^2} dz$

(ii) $\displaystyle\int_{|z|=3} \dfrac{e^{2z}}{(z+1)^4} dz$

Q6. Define a harmonic function. Show the function $u(x,y)=x+y^3-3x^2y$ is a harmonic function. Find its conjugate harmonic.

Q7. Find the singular points of the functions

(i) $f_1(z)=\dfrac{\sin(\pi z)}{(z-1)^2}$ (OR)

(ii) $f_2(z)=\dfrac{1}{z^2(z^2-a^2)}$ where $a\neq 0$ [Full Solution]

and discuss the nature of the singularities. Also, find the residues at those points.

The students of Section H need to submit the below assignments on Differential Equation.

Differential Equation

Q1. Find the order and degree of

$\left(\dfrac{d^4y}{dx^4} \right)^5+5y^4\left(\dfrac{d^3y}{dx^3} \right)^2=x^6$.

Q2: Find the value of $k$ for which the differential equation $\left(xy^2+kx^2y\right)dx + \left(x+y\right)x^2dy=0$ is exact and solve it for this value of $k$.

Q3: Eliminate A and B to form an ODE, where A and B are constants.

1. y = Ae^x + Be^-x
2. y = e^-x(A + Bx)
3. y = A cosx + B sinx

Q4. Find integrating factor of $\cos^2 x \dfrac{dy}{dx}+y=\tan x$. Hence, solve it.

Q5: Solve $(x^2+y^2)~dx=xy~dy$

Q6. Solve $\dfrac{dy}{dx}+x \sin 2y=x^3 \cos^2 y$

Q7. Solve $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2 y^6$. [Full Solution]

(OR) $\dfrac{dy}{dx}+y \tan x= y^3 \sec x$ [Full Solution]

Q8. Find the particular integral $\dfrac{1}{D^2-1} e^x$.

Q9. Find the complementary function and the particular integral of

(i) $\left(D^2+4\right)y= \cos^2{x}$. (ii) $\dfrac{d^2y}{dx^2}+2 \dfrac{dy}{dx}+y=e^{x}+\sin x$ (iii) $\dfrac{d^2y}{dx^2}-4 \dfrac{dy}{dx}+4y=x^2+e^{2x}$ (OR) (iv) $\dfrac{d^2y}{dx^2}-4 \dfrac{dy}{dx}+4y=x^2e^{2x}$.

Q10. Solve the Cauchy-Euler differential equation

$x^2 \dfrac{d^2y}{dx^2}-3x \dfrac{dy}{dx}+4y=\sin(\ln x)$

Q11. Using the method of variation of parameters, solve

(i) $\dfrac{d^2y}{dx^2}+y=\sec x$ (Or)

(ii) $\dfrac{d^2y}{dx^2}+4y=\tan 2x$.

Q12. If a force $F(x, y)=x^2y \vec{i}+2xy \vec{j}$ displaces a particle in the $xy$-plane from $(0, 0)$ to $(1,4)$ along a curve $y=2x^2$, then find the work done.

Q13. Evaluate $\displaystyle \iint_S \vec{A} \cdot \hat{n}~ds$ where $\vec{A}=18z \vec{i}-12\vec{j}+3y \vec{k}$ and $S$ is the part of the plane $2x+3y+6z=12$ including in the first octant.

Q14. State Green’s theorem. Using this theorem, evaluate $\displaystyle \int_C (xy^2dx+xydy)$ where $C$ is the closed curve described counter clockwise of the triangle with the vertices $(0,0), (1,0), (1,1)$.

Q15. State Gauss’s Divergence theorem. Using this theorem, evaluate

$\displaystyle \iint_S \vec{F} \cdot \vec{n}~ds$

where $\vec{F}=(2x-z) \vec{i}+x^2y\vec{j}-xz^2 \vec{k}$ and $S$ is the surface of the cube bounded by $x=0, x=1$, $y=0, y=1$, $z=0, z=1$.