Engineering Math Problems, Practice Final Exam

Engineering Math Problems: In this page, you can find a list of Engineering Math Problems which can be useful for practice for final exam.

Eng. Math II Content: Link

Sequence, Series & Complex Analysis

Q1: What is a bounded sequence. Show the sequence {x_n} where $x_n=\dfrac{2n-1}{4n}$ is bounded.

Q2: State ratio test and root test for a series. Also state Leibnitz test for alternating series.

Q3: Test the convergence of the series:

1. $\sum_{n=1}^\infty \dfrac{n^{p-1}}{n!}, \,\, (p>1)$
2. $\sum_{n=1}^\infty \dfrac{n!5^n}{n^n}$
3. $1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\cdots$
4. $1-\dfrac{1}{2^2}+\dfrac{1}{3^2}-\dfrac{1}{4^2}+\cdots$

Q4: Find the limit of the sequence {x_n} where $x_n=\dfrac{3n(2n+2)}{(2n+1)(n-3)}$. i.e, find $\lim\limits_{n \to \infty} x_n$. Is this sequence convergent?

Q5: What is a conditionally convergent sequence? Show the series is conditionally convergent: $\dfrac{1}{2}-\dfrac{2}{5}+\dfrac{3}{10}-\dfrac{4}{17}+\dfrac{5}{26}-\cdots$.

Q6: Find the Fourier series of the following functions:

1. $f(x)=e^{ax}$ in the interval $-\pi\leq x \leq \pi$.
2. $f(x)=x\sin x$ in the interval $-\pi\leq x \leq \pi$ and hence prove that $\dfrac{1}{3}-\dfrac{1}{3 \times 5}+\dfrac{1}{5 \times 7}-\cdots=\dfrac{\pi-2}{4}$.
3. $f(x)=x-x^2$ in the interval $-1<x<1$.

Q7: State Cauchy-Riemann equations. Show the converse of this theorem is not true by explaining the example: $f(z)=\sqrt{|xy|}$ where z=x+iy, satisfies the Cauchy-Riemann equations at (0,0) but $f'(0)$ does not exist. [Solution]

Q8: Evaluate the following integrals:

1. $\displaystyle \int_{|z|=5} \dfrac{z^2-2z+1}{(z-1)z^2(z-2)}dz$
2. $\displaystyle \int_{|z|=4} \dfrac{e^z}{z^2+\pi^2}dz$
3. $\displaystyle \int_i^{3-i} (2xy-iy^2)dz$ along the straight line joining the points $z=i$ and $z=3-i$.

Q9: Find the analytic function whose real part is $u=x^3-3xy^2+3x^2-3y^2+2x+1$.

Q10: Find the residues of the following functions:

1. $f(z)=\dfrac{2z}{\cos z}$
2. $f(z)=\cot \pi z$
3. $f(z)=\dfrac{\cot \pi z}{(z-a)^2}$.

Eng. Math II Content: Link

Differential Equation and Vector Calculus

Q1: Show the set {e^x, e^2x, e^3x} is linearly independent. What about the set {ln x, ln x², ln x³}?

Q2: Show that the following differential equation is inexact and hence find the solution: $(2xy+x^2)dy=(3y^2+2xy)dx$.

Q3: Find the general solutions of the Bernoulli’s differential equations:

1. $x\dfrac{dy}{dx}-y=y^2\ln x$.
2. $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2y^6$

Q4: Find the general solutions of the Cauchy-Euler equations:

1. $x^2\dfrac{d^2y}{dx^2}-2y=2\ln(x)$
2. $x^2\dfrac{d^2y}{dx^2}+2x\dfrac{dy}{dx}-6y=\cos 2(\ln x)$

Q5: Solve the following differential equations:

1. $\dfrac{dy}{dx}=\dfrac{y}{x}+\tan\left(\dfrac{y}{x}\right)$
2. $(x^2-y^2)dx-xydy=0$.
3. $\dfrac{dy}{dx}=\sin(x+y)$.
4. $2 \dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-10y=0$, $y(0)=2, y'(0)=-1$.
5. $(D^2-4D+4)y=x^3$ where $D\equiv \dfrac{d}{dx}$.

Q6: Using the method of variation of parameter, solve the differential equations

1. (D²+4D+4)y = e^-2x
2. $\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}=e^x\sin x$.

Q7: State Green’s theorem. Using Green’s theorem, evaluate $\displaystyle \int_C (x^2y dx+x^2dy)$ where $C$ is the boundary described counterclockwise of the triangle with vertices (0,0), (1,0), (1,1).

Q8: Problem related to Work Done by a Force (see class note)

Q9: State and Applications of Gauss Divergence Theorem (see class note)

Engineering Mathematics II Mid Term Practice Problems

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

Q1: Define a convergent, divergent and oscillatory sequence with an example.

Q2: Write down the definition of a bounded sequence with examples.

Q3: What is a monotonically increasing/decreasing sequence? Show that the sequence $\left\{ \dfrac{n+1}{2n+3} \right\}$ is monotonically increasing.

Q4: State ratio and root test for a series. Test the convergence of the following series:

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

Q5: Test the convergence of the series $\dfrac{2}{1 \cdot 3 \cdot 5}+\dfrac{4}{3 \cdot 5 \cdot 7}+\dfrac{6}{5 \cdot 7 \cdot 9}+\cdots$

Q6: What is an alternating series? State Leibnitz’s test for an alternating series. Using this test or otherwise, find the values of $x$ for which the series

$x-\dfrac{x^2}{2^2}+\dfrac{x^3}{3^2}-\dfrac{x^4}{4^2}+\cdots$

converges.

Q7: Find the radius of convergence of the series: $1+2x+2^2 x^2+2^3x^3+\cdots$.

Q8: Write down the orders and degrees of the differential equations listed below:

1. $\frac{dy}{dx}=\sin^2(x+y)$
2. $\frac{d^2y}{dx^2}+\frac{dy}{dx}=\sqrt{x+y}$.

Q9: Eliminate A and B to form an ODE: [Full Solution]

1. y=Ax+A³
2. y=e^-x(Acosx + Bsinx).

Q10: Find the value of $m$ for which the differential equation

$\left(xy^2+mx^2y\right)dx + \left(x+y\right)x^2dy=0$

is exact and solve it for this value of $m$.

Q11: What is an integrating factor of a ODE? For the inexact differential equation xdy-ydx=0, write down all possible integrating factors. [Note: Both 1/x² and 1/y² are integrating factors. Verify it.]

Q12: Solve:

1. $\dfrac{dy}{dx}=\dfrac{y}{x}+\tan\left(\dfrac{y}{x}\right)$
2. $(x^2-y^2)dx-xydy=0$.
3. $\dfrac{dy}{dx}=\sin(x+y)$.
4. $x\dfrac{dy}{dx}=y-\sqrt{x^2+y^2}$ (homogeneous differential equation)
5. $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2y^6$

Q13: Solve the differential equation $\frac{dy}{dx}+y \cot x=\text{cosec}~ x$. [View Solution]

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