## Detroit Mercy Engineering Ph.D. Mathematics Qualifying Examination

Spring 2021 Exam Date and Time: Monday, March 1, 11am to 5pm US Eastern Time.This exam is conducted by the Department of Mathematics, University of Detroit Mercy. Currently the math faculty responsible for the exam are Prof. Abhijit Dasgupta and Prof. Nart Shawash.

Students pursuing Engineering PhD at Detroit Mercy are required to pass this exam, and they should attempt the exam early in their program.

Detailed information about the examination is given below, including a sample exam. However,

all information here is subject to change without notice and this page may not reflect the latest changes. For the latest information, contact the math faculty in charge of the exam.

## General information about the exam

- This six-hour exam is offered two times a year, one near the end of Fall term (early December), and the other near the end of the Winter/Spring term (April). It is given usually on the last day of the dead week (before final exams begin), during 9am to 3pm, or 10am to 4pm, or 11am to 5pm.
- Students planning to take the exam are urged to start preparing well in advance (preferably at least one full term preceding the exam), and be in touch both with their doctoral advisors and with the math department faculty in charge of the exam to make sure they are aware of all information regarding the exam.
- Students planning to take the Fall exam should register by September 30, and those planning to take the Winter/Spring exam should register by January 31. Registration consists of (i) notifying both their advisors and the math faculty in charge of the exam, and (ii) getting confirmation from them that they will be offered the exam.

## Format of the exam

The current exam format is as follows.

- The exam consists of 16 problems, with 4 problems from each of the following four subject areas:

Vector CalculusDifferential EquationsLinear Algebra(Linear Algebra Practice ProblemsNew!)Probability and Statistics- Students are required to answer a total of 8 problems, choosing at least one problem from each of the four areas above. If a student answers more than 8 problems, only the first 8 will be graded. Each problem is worth 10 points.
- The exam is usually open book, but only books approved by the math faculty are allowed. Caculators are allowed, but no other electronics is allowed. In particular, computers of any kind (e.g. laptops or tablets) and cellular phones are prohibited.

## Standard Textbooks

The suggestedtextbooksare the ones used in Detroit Mercy courses for the above four areas, namely:

Vector Calculus:Calculus: Multivariable, by Anton, Bivens, Davis. 9th Ed. (Wiley) 2009. ISBN: 978-0-470-18346-5.Differential Equations:Differential Equations: An Introduction to Modern Methods and Applications, 2nd ed., by James R. Brannan and Willam E. Boyce. Publisher: Wiley. ISBN: 978-0-470-45824-2.Linear Algebra:Elementary Linear Algebra, 6th Ed, by Ron Larson and David Falvo, ISBN-13 = 978-0-618-78376-2, ISBN-10 = 0-618-78376-8.The exam covers the entire book, with heavy emphasis on Chapters 4, 5, 6, and 7Statistics:Probability and Statistics for Engineering and the Sciences, by Jay L. Devore, 7th Edition, 2008. ISBN-10: 0495382175, ISBN-13: 978-0-495-38217-1.The exam covers Chapters 2, 3 (skip 3.5), 4 (skip 4.5 and 4.6), 5, 7, 8, and 12, with heavy emphasis on 4.3, 5.4, 5.5, 7.2, and 7.3.Kreyszig's comprehensive textbookAdvanced Engineering Mathematicscovers essentially all of the topics of the exam.

## Topics Covered

The exam coversstandard undergraduate topics in the four areasfound in the above textbooks, namely:

Vector Calculus:Standard topics of vector calculus up to and including Green's Theorem, the Gauss Divergence Theorem, and Stokes Theorem, with applications.Differential Equations:Standard topics in Ordinary Differential Equations including linear systems and the phase space, stability of linear autonomous systems, and Laplace transforms.Linear Algebra:Gaussian and Gauss-Jordan elimination, elementary matrices and inverses, determinants, vector spaces, subspaces, linear independence and span, bases and dimension, matrix rank theory, change of basis, inner products, orthogonal complements, angles, Gram-Schmidt process, linear transformations, the matrix, kernel and range of a transformation, change of basis and similarity, eigenvalues and eigenvectors, diagonalization, symmetric matrices and orthogonal diagonalization. (Linear Algebra Practice ProblemsNew!)Probability and Statistics:Discrete and continuous random variables, the binomial and normal distributions, properties and applications of the normal distribution, joint distributions, covariance, correlation, distribution of the sample mean and the central limit theorem, distribution of linear combinations, confidence intervals, normal populations, population mean and proportion, hypothesis testing (single sample), p-values, simple linear regression and correlation.

## Sample Exam

Instructions:Answer 8 of the 16 questions,choosing at least one question from each of the four parts.

Part I: Vector Calculus

- Verify Green's Theorem \[ \int\limits_R\hspace{-1.4ex}\int\left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right)\,dx \,dy =\oint\limits_C (F_1\, dx + F_2\, dy)\] by evaluating both sides for the vector field \(\mathbf{F} =[x^2+y^2, x^2-y^2]\) and the region \(R: 1\leq y \leq 2-x^2\), where the path along the boundary of \(R\) is oriented counterclockwise.

- Evaluate the surface integral \(\displaystyle \int\limits_S \hspace{-1.6ex} \int \hspace{-3.6ex} \subset\hspace{-0.5ex}\supset \mathbf{F\cdot n} \, dA \) using the Divergence Theorem, showing details of your work.
\(\mathbf{F} = [\sin y \,,\, \cos x \,,\, \cos z]\); \(\;S\) is the surface of the cylinder and two disks given by \(x^2+y^2\leq 4\), \(|z|\leq 2\).

- Evaluate the flux integral \(\displaystyle\iint\limits_S \mathbf{ F\cdot n} \, dA\) for the following data. Sketch the surface \(S\), and show details of your work.
\(\mathbf{F} = [0 \,,\, \sin y \,,\, \cos z]\); \(\;S\) is the portion of the cylinder \(x=y^2\) restricted by \(0\leq y \leq \frac{\pi}{4}\) and \(0\leq z \leq y\).

- Calculate the line integral \(\displaystyle\oint\limits_C \mathbf{F\cdot r'} \, ds\) using Stokes's theorem for the given \(\mathbf{F}\) and \(C\). Assume the Cartesian coordinates to be right-handed and the \(z\)-component of the surface normal to be nonnegative. Show details of your work.
\(\mathbf{F} = [y^2, \, x^2, \, z+x]\); \(\;C\) is around the triangle with vertices \((0,0,0)\), \((1,0,0)\), \((1,1,0)\).

Part II: Differential Equations

- Use the method of
undetermined coefficientsto find the solution of the initial value problem \[ y'' + 9y = 2\sin 3t, \qquad y(0) = 2, \; y'(0) = -1. \]

- Consider the following system \[ \mathbf{x}' = \left(\! \begin{array}{cc} 1 & -5 \\ 1 & -3 \end{array}\! \right) \mathbf{x} \]

- Find the general solution of the given system of ODEs.
- Sketch the phase portrait for the system above. [Indicate whether the tip of the state vector \(\mathbf{x}(t)\) moves in counterclockwise or clockwise direction. A random guess has 50% chance of being correct, so there will be no credit for guessing. You have to explain
why.Hint: Use the system matrix to find the velocity vector \(\mathbf{x}'(t_0)\).]- Find the IVP solution to the system in the first part if \(\mathbf{x}(0)= \left(\! \begin{array}{c} 1 \\ 1 \end{array} \! \right)\).

Hint: \(A\cos \omega t + B\sin \omega t = \sqrt{A^2+B^2}\cos\left( \omega t -\delta\right)\), where \(\tan \delta =\frac{B}{A}\).- Make a rough sketch of component plots of the specific solution in the last part.

- Consider the linear system \[ dx/dt = a_{11}x+a_{12}y\,, \qquad dy/dt = a_{21}x+a_{22}y, \] where \(a_{11}, a_{12}, a_{21}, a_{22}\) are real constants. Let \(p=a_{11}+a_{22}\), \(q=a_{11}a_{22}-a_{12}a_{21}\), and \(\Delta =p^2-4q\). Observe that \(p\) and \(q\) are the trace and determinant, respectively, of the coefficient matrix of the given system. Show that the critical point \((0,0)\) is a

- Node if \(q \gt 0\) and \(\Delta \geq 0\);
- Saddle point if \(q \lt 0\);
- Spiral point if \(p \neq 0\) and \(\Delta \lt 0\);
- Center if \(p=0\) and \(q \gt 0\).
Hint: These conclusions can be obtained by studying the eigenvalues \(\lambda_1\) and \(\lambda_2\). It may also be helpful to establish, and then to use, the relations \(\lambda_1 \lambda_2 = q\) and \(\lambda_1 +\lambda_2 = p\).

- (
Impulse Functions.) Find the solution of the IVP and. \[ y''+4y=\delta(t-4\pi), \quad y(0)=\frac{1}{2}, \; y'(0)=0. \]draw its graph

Part III: Linear Algebra

- Let \(T \colon \mathbf{R}^3 \to \mathbf{R}^3\) be the linear transformation whose matrix relative to the standard basis of \(\mathbf{R}^3\) is \[ \left[ \begin{array}{rrr} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{array} \right]. \] Let \(T^2\) be the transformation defined by \(T^2(\mathbf{x}) := T(T(\mathbf{x}))\), \(T^3\) be the transformation defined by \(T^3(\mathbf{x}) := T(T^2(\mathbf{x}))\), and so on. Find the matrix of \(T^{1000}\) relative to the standard basis of \(\mathbf{R}^3\), explaining your work.

- In \(\mathbf{R}^3\), let \(S\) be the subspace spanned by the vectors \(\mathbf{u} := (1, 0, 1)\) and \(\mathbf{v} := (−1, 1, 0)\).

- Find the angle that the vector \(\mathbf{i} := (1,0,0)\) makes with the subspace \(S\).
- Find the orthogonal complement \(S^{\perp}\) of the subspace \(S\).
- Find an orthonormal basis \(B\) for \(\mathbf{R}^3\) such that two vectors in \(B\) are in \(S\).

- Let \(T \colon \mathbf{R}^5 \to \mathbf{R}^3\) be a linear transformation whose standard matrix is \(A\), where \[ A = \left[ \begin{array}{rrrrr} 1 & -2 & -1 & \,\;1 & 9\:\\ 2 & -4 & 0 & 0 & 8\:\\ -1 & 2 & 1 & 3 & -5\: \end{array}\right]. \]

- Find a basis for the range of \(T\) (\(\text{range}(T)\)) and determine the rank of \(T\).
- Find a basis for kernel of \(T\) (i.e., the null-space of \(T\)) and determine the nullity of \(T\).
- Is the last column of \(A\) in the span of the first three columns? Why?
- Are the last three columns of \(A\) linearly independent? Why?
- What is the dimension of the subspace spanned by the first three columns?

- (Diagonalization.)

- Prove that if \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are eigenvectors of a \(3 \times 3\) matrix with distinct eigenvalues \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\), then the vectors \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) form a basis of \(\mathbf{R}^3\).
- Consider the matrix \[ A = \left[ \begin{array}{rrr} 6 & 0 & 0\\ 5 & 4 & 0\\ 3 & 2 & 1 \end{array}\right]. \] Use the first part to explain why \(A\) is diagonalizable.
- Find a basis of \(\mathbf{R}^3\) with respect to which \(A\) is diagonalizable.
- Is \(A\) orthogonally diagonalizable? why?

Part IV: Probability and Statistics

- Exactly 80% of the cars in a certain state are properly insured. If a random sample of 2500 cars from that state is drawn, what is the
approximate probabilitythat

- At least 78% of the cars in the sample are properly insured?
- At least 81% of the cars in the sample are properly insured?
- Between 79% and 81% of the cars in the sample are properly insured?
Show and explain all your work, including the principles of approximation involved.

- The fuel efficiencies for two types of cars are denoted by \(X\) and \(Y\) (in mpg). Suppose these variables are independent and normally distributed with \(\mu_X = 25\), \(\mu_Y = 23\), and \(\sigma_X = \sigma_Y = 4\). One each from the two types of cars are chosen at random.

- Find the probability that the average fuel efficiency \((X + Y)/2\) for the two cars exceeds 25 mpg.
- Find the probability that the difference in fuel efficiencies between the two cars is at most 3 mpg.

- In a survey of 10,000 randomly selected voters in a certain state, 6,437 voters said that they support ballot Proposal A. Calculate and interpret approximate 95% and 99% confidence intervals for the actual proportion of voters in that state who support Proposal A.

- A zoologist studied nine different wolf packs and reported that the average pack size was 7.3 with a standard deviation of 2.1. Assume a normal distribution.

- Find a 95% prediction interval for the average pack size.
- Use a 5% confidence level to test the hypothesis that the average pack size is more than 6.5. In particular,

- Clearly state the null and the alternative hypotheses;
- Find the appropriate test statistic and its value;
- Find the rejection region; and
- Clearly state the final conclusion.
- Find the \(p\)-value.
## Past Examination Dates and Time

Fall 2020: No Exam!

Spring 2020: No Exam!

Fall 2019 Exam Date and Time: Monday, Dec 9, 10am to 4pm.

Spring 2019: No Exam!

Fall 2018: No Exam!

Spring 2018 Exam Date and Time: Monday, May 7, 10am to 4pm.

Fall 2017 Exam Date, Time: Monday, December 11, 10am to 4pm.

Spring 2016 Exam Date and Time: Monday, April 25, 10am to 4pm.

Fall 2015 Exam Date and Time: Monday, December 14, 10am to 4pm.

Spring 2015 Exam Date and Time: Monday, April 20, 10am to 4pm.

Fall 2014 Exam Date and Time: Friday, December 5, 9am to 3pm.

Last updated: 2021-02-05 16:11:30 EST