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Sample Topics

### Number Theory

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

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

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: ### Differential equations

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: ### Partial differential equations

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

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

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

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

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

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: ### Differential Geometry

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

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

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.
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