# Available Subjects

Number theory: Number theory is known as the queen of the mathematics. There are many categories in number theory, such as elementary number theory, cryptography, geometry of numbers, Algebraic number theory, etc. Number theory has a lot of applications in day to day life such as cryptography is used for encoding and decoding of secret messages or ciphers. Number theory is used in the fields of banking, solving highly computational commands etc.

Sample Questions MATLAB is a short form of matrix laboratory. It’s a software to analyse data, to develop algorithms and to create models and applications. It performs matrix manipulations, function plotting and so much higher mathematical operations. Millions of engineers and scientist use MATLAB. MATLAB has a big range of applications in each field such as in mathematics, engineering, machine learning, in day to day life, such as signal processing, communications, image and video processing, etc. In mathematics MATLAB is used to create and solve matrices and linear algebra problems, in polynomials and interpolation, to analyse the data, differential equations, etc Graph Theory: This subject is about the study of graphs. It is concerned with graphs, coloring of graphs, networking, connectivity, graph enumeration, computational problems, geometry, knot theory, etc. Graph theory is useful in day to day life, such as in DNA sequencing, in designing the traffic network, in web designing, traffic lights, social networks, in matching problems, google maps, etc.

Sample Questions: The sub-subject “differential equations” is a branch of Mathematics that involves the function and its derivatives. The function represents the physical quantity and the derivative represents the rate of change. The differential equation established a relationship between them.
A differential equation is known by its order and the order is determined by their derivative. A first derivative equation is a first order differential equation, the second derivative equation is a second order differential equation. Differential equation has several types: Ordinary differential equation, partial differential equation, linear or non-linear differential equation.

Sample Questions: The sub-subject “Partial differential equations” is a branch of Mathematics that involves an equation which contains the function of several variable and its partial derivatives. A partial derivative of a function is a derivative of the function of two or more variable with respect to the more than one variable.
A partial differential equation is known by its order and the order is determined by their derivative. A first derivative equation is first order partial differential equation, the second derivative equation is second order partial differential equation. A partial differential equation has several types: Linear PDE, Quasi-linear PDE, Nonlinear PDE.

Sample Questions: Abstract algebra is also known as modern algebra. The most common divisions of abstract algebra are, commutative algebra, representation theory, and homological algebra. Abstract algebra includes study of group theory, ring theory, vector spaces, lattices and algebras. Real life applications of abstract algebra are medical imaging (algebraic topology is used to convert 3D information from the data out of a scanner), coordination between robots etc.

Sample Questions: Topology is the mathematical study of the properties of the object that are preserved under some transformations (deformation, twisting, stretching etc.) called as homeomorphism. For instance, a circle can be transformed into an ellipse by stretching. Thus, making them topologically equivalent. The most common branches of topology are, algebraic topology, differential topology and low-dimensional topology. Applications of topology are in many fields. It is used in computer science to encode the data set in the form of a specific version of Betti number called as a barcode. Under cosmology, topology can be used to determine the shape of the universe (known as space-time topology).

Sample Questions:  Numerical analysis is a vast subject that includes analysis of techniques in order to get approximate solutions to difficult problems and the study of algorithms & design. Numerical analysis includes, interpolation, extrapolation, regression, optimization, evaluating integrals, computing values of function etc. Numerical analysis is used in solving difficult engineering problems (solution of higher order differential equations, evaluations of integrals).

Sample Questions: Functional analysis is the branch of mathematics that is derived from the study of vector spaces equipped with some limitations (example, inner product, norm, topology, etc.) Functional analysis includes, Normed vector spaces which further includes Hilbert spaces and Banach spaces.  Hahn-Banach theorem one of the important theorems in functional analysis is used in financial mathematics in the Fundamental theorem of asset pricing. Another important application is signal analysis and data compression.
Sample Questions:
Example 1. What is bank reconciliation statement?   Example 2. Explain the importance of Bank Reconciliation Statement and give real examples explaining all possible items that make differences between cash balance in general ledger and bank statement balance.   Example 3. When Cash is presented on a Balance Sheet, what are the details of the Cash A/C? How does a Bank Reconciliation Statement factor into the Cash Balance? Please be specific in your details.   Example 4. A firm’s bank reconciliation statement shows a book balance of \$32,740, an NSF check of \$1,350, and a service charge of \$95. Its adjusted book balance is
• \$31,485.
• \$34,185.
• \$33,995.
• \$31,295.
Complex analysis is a category of analysis that deals with the complex numbers. In simpler terms it is the study of complex functions of a complex variable and their algebra, geometry dealing with derivatives, integration, and other practical applications. Various important results include Cauchy Integral theorem, Morera’s Theorem, Liouville’s Conformality Theorem etc.   Complex analysis has its real-life applications in mainly engineering, specifically nuclear, electrical, mechanical and aerospace engineering.

Sample Questions: The sub-subject “Differential geometry” is a branch of mathematics that uses the performances of differential & integral calculus and multilinear algebra.  The concepts that are mainly covered in differential geometry are the geometry of plane curves, curvature of curves and surfaces etc. An important application of differential geometry is in the field of economics. It is used to investigate the geometrical structures. Also, it is applied in engineering to solve the problems related to digital signaling processes.

Sample Questions: Real analysis is a subdivision of pure mathematics that sets the ground work for numerous other subfields, such as probability, calculus and differential equations.Consecutively, real analysis is grounded on fundamental concepts from number theory and topology. A strong experience in calculus, logical reasoning and proofs is needed to study real analysis.

Sample Questions: Engineering mathematics is one major sub-subject in Mathematics which is the building block of all the streams and lays foundation of the concepts for all the students of graduates studying in the sciences and engineering. Engineering mathematics consists of all the topics in differential calculus, integral calculus, linear algebra and differential equations with applications to various engineering problems.

Sample Questions: Algebra – is a generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic.
Sample Questions:
1. Apply the product rule for exponents, if possible. 25) (-2×3)(9x10y6) 26) 5 5 -3 -3 0 4

2. A recipe calls for 1/2 cups of sugar. You find that you only have 1/18 cups of sugar left. What fraction of the recipe can you make? Your answer must be a fraction written in simplest terms. If your answer is a whole number, then give your answer as a fraction with 1 in the denominator. Cups of sugar
Algebra – is a generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic.
Sample Questions:
1. Apply the product rule for exponents, if possible. 25) (-2×3)(9x10y6) 26) 5 5 -3 -3 0 4

2. A recipe calls for 1/2 cups of sugar. You find that you only have 1/18 cups of sugar left. What fraction of the recipe can you make? Your answer must be a fraction written in simplest terms. If your answer is a whole number, then give your answer as a fraction with 1 in the denominator. Cups of sugar
Precalculus – A course of study taken as a prerequisite for the study of calculus, usually involving advanced algebra and trigonometric functions.
Sample Questions:
1. Write the following in terms of sin e and cos θ; then simplify if possible. (Leave your answer in terms of sin θ and/or cos θ.) csc θ tan θ Need Help?

2. Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if po log
Geometry – The branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues.
Sample Questions:
1. Congruent Inscribed Angles Proof.How would you prove that two angles in a circle that inscribe the same arc are congruent?Is it possible to prove it without using the “Congruent Inscribed Angles Theorem”

2.a. fx,y)-(e,cos(xy Let c(t) be a path with c(0) (0,0) and c'(0) 1,2). What is the tangent vector to f o clt) at t 0 b. fr,y)-esinx-y Let c(t) be a path with c(0) (0,0) and c'(0) (2,-3). What is the tangent vector to fo clt) at t0
Trigonometry – The branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.
Sample Questions:
1. Why is the magnitude of the sum of two vectors less than or equal to the sum of the magnitudes of each vector? Please be as detailed as possible.   2. 1)Explain why you will get an error for cos-1(-1.008). Specific.2) Solve algebraically, 2sin(3x)-1=0, give the first exact positive solutions.
Single variable calculus is a branch of calculus that deals with functions of a single variable. The topics which are involved within single variable calculus are limits and continuity, applications of graphing, rates and approximations, differentiation and integration of functions in a single variable.

Sample Questions:- Multi variable itself defines its meaning that more than one variable. Multi variable is the extended form of calculus of single variable to more than one variable. It studies the function of more than one variable. It deals with so many topics such as Gradient, flux, vector field, surface integral Jacobian, Differential operator, Contour integral etc. There are lots of applications of multi variable calculus in day to day life, such as, in the stock market, in engineering, social science, to analyse deterministic systems etc.

Sample Questions:- The sub-subject “Vector calculus” is a branch of Mathematics that disturbed with differentiation and integration of vector fields, primarily in 3 – dimensional. Sometimes {\displaystyle \mathbb {R} ^{3}.}Sometimes vector calculus is used as an alternative of multivariable calculus, which includes partial differentiation and multiple integration along with calculus. Vector calculus plays a crucial part in differential geometry as well as study of partial differential equations.

Sample Questions: Advanced calculus is the further advanced extension of calculus that deals with the functions of one variable together with the linear algebra. Main topics included in advanced calculus are vector spaces, Differential calculus, Reiman integration etc. Advanced calculus has its main applications in field like statistics majorly.

Sample Questions: Discrete Math Discrete mathematics is a category of mathematics that deals with the objects discretely or rather distinctly. Unlike the term ‘continuous’ used in real analysis, the values in discrete are treated separately. The study of Discrete mathematics involves the concepts like sets, relations and functions, graph theory, Boolean algebra, probability, counting theory etc. The main applications of Discrete mathematics are in the computer science field for computing, programming languages, algorithms, cryptography, etc.

Sample Questions: Finite mathematics is the part of calculus. It is concerned with limits, finite number of variables, linear programming, elementary matrix algebra, introduction to probability, mathematics of finance etc. Finite mathematics gives a survey of mathematical techniques and has many applications in day to day life, such as linear programming is used for transportation optimization, Probability is used in weather prediction, sports strategies etc.

Sample Questions: Applied Mathematics is concerned with mainly focused on the applications of Mathematics. It deals with the applications of calculus, differential equations. etc. Important applications of applied part are in the field of business, life sciences, social science, computer science, etc.

Sample Questions: Biology is a category of mathematics that deals with the study and theoretical analysis of living organisms. The mathematical problems that require the knowledge of biology comes under this division of mathematics. Main topics that come under this are Mainly related to real life problems that relates to cell biology, ecology, evolution, molecular biology, etc. One of the most common and well-known problem it deals with is Cancer-Inspired Free Boundary Problem.
Sample Questions:
1. A drug is given at an infusion rate of 50 mg/h. The drug concentration value determined at 3.2 h after the start of the infusion is 8 mg/L. Assuming the patient has 5.4 L of blood, estimate the half-life of this drug. Round your answer to two decimal places.

2. In a sorority with 40 members, 18 take Mathematics, 5 take both Mathematics and biology, and 8 take neither Mathematics nor Biology. How many take Biology but not Mathematics.

3. What is the difference between discrete and distributed time delay in mathematical biology? advantage and disadvantage?

4. (a) Why Is Mathematical Biology So Hard?

(b) What is one example of a biological question to solve using mathematics?
Business mathematics is the arithmetic used by profitable firms to evident and accomplish corporate maneuvers.  Commercial organizations make use of mathematics in financial analysis, forecasting, inventory management, accounting and marketing. Business management can be done more efficiently by making use of more advanced mathematics like linear programming, calculus and matrix algebra.
Sample Questions:
1. What did you learn in business mathematics class especially related to payroll?

2. John invested \$15,000 in a mutual fund for 4 years ago. If the fund grew at the rate of 9.8%/year compounded monthly, what would John’s account be worth today?

3. Determine the utilization and the efficiency for each of these situations: A. A loan processing operation that processes an average of 7 loans per day. The operation has a design capacity of 10 loans per day and an effective capacity of 8 loans per day. B. A furnace repair team that services an average of four furnaces a day if the design capacity is six furnaces a day and the effective capacity is five furnaces a day. C. Would you say that systems that have higher efficiency ratios than other systems will always have higher utilization ratios than those other systems? Explain.

4. WDH Incorporation has borrowed \$1,000,000 to purchase new factory equipment and the company must repay the loan in 5 years, quarterly at 7% interest rate and interest rate per period is 1.75%. Find the quarterly payment to be made by the company to discharge the loan in time.
OR is signifies research on operations. OR is a scientific method of providing explicit quantitative understanding and assessment of complex situations for better decision making. In operation research, problems are broken down into basic components solved by constructing mathematical models and deriving the solution of the model. Operation research used for solving different types of problems like problem dealing with the waiting line, problem dealing with the allocation of materials etc. Operation research has a several types of methods like linear programming, network flow programming, integer programming, non-linear programming, and dynamic programming.

Sample Questions: Sample Questions:
1. Consider a statistical decision (e, M,, L) with sample space X where Θ-(01, θ2), H and ye [0,0.25)). Find the minimax decision. in R2 plane. Further show that the class of all non-randomized Bayes decisions.

2. In a point estimation problem e-(0,) A (0, and X follows Poisson distribution with parameter 8 and a sample of size one is made available. Show that the estimator T0X) – X is not Bayes but a generalized Bayes under quadratic loss.
Sample Questions:
Joe Zilch is practicing basketball by repeatedly making attempts (shots) to put the ball in the basket. We label his first shot as random variable (RV) XI , second shot as X2,…, nth shot as X,, etc. When he takes the nth shot, he either makes a basket (X, -1) or misses (X,-0 ). He finds that the result of any shot x, depends on the outcome of his last two shots X -2 and X- as follows:

P(X-1 l he missed both of his last two shots) 1/2

P(X-1 l he made one of his last two shots) 2/3

P(X1 l he made both of his last two shots) – 3/4

a). Show how Joe’s basketball play may be modeled using a Markov chain. How many states are needed? (Hint: Define a state as the outcome of his last two shots). Draw a labeled state transition diagram or trellis describing the process

b). Find the transition matrix P for the process

c). Given that he missed his first two shots, find the joint probability that Joe makes shots number 3 and 4. d). Joe made his first two shots with probability 0.5 and missed his first two shots with probability 0.5. Given these facts, find the joint probability that Joe makes shots number 3 and 4

e). Find the probability that Joe makes any single shot in the long run.
Sample Questions:
1.  A recent article in the Arizona Republic indicated that the mean selling price of the homes in the area is more than \$220,000. Can we conclude that the mean selling price in the Goodyear, AZ, area is more than \$220,000? Use the .01 significance level. What is the p-value?
Sample Questions:
1. Let r є N. Let X1,X2, be identically distributed random variables having finite mean m, which are r-dependent, i.e. such that XkXk,.,Xk, are independent whenever kiti > ki +r for each i. (Thus, independent random variables are 0-dependent.) Prove that with probability one, X Xi – m as n -oo. Hint: Break up the sum Ση! Xi into r different sums.
Sample Questions:
1. Bob, Kevin, and Jack are playing the following game: a single player will toss 5 fair coins. Assume the tosses are independent , and let H = 1 and T 0. Before the coin tosses, without consulting each other, each player chooses some number of the tosses (at least one) and calculates the sum (mod 2) of the tosses. That is the player’s score for example, supposed Kevin chose the second, third and fourth tosses, and Jack chose the first, second, and last toss. If the following coin tossing sequence occurs: HTHHT then Kevin’s score is 0 and Jack’s score is 1. Assume that each player does not select exactly the same subset of tosses.
(a) Show that the scores of the three players are pairwise independent
(b) Show that the scores of Bob, Kevin, and Jack are not always mutually independent. (A counter example when they are not independent is sufficient, but obviously explain your answer.)