Discrete Mathematics & Graph Theory
- Find the inverse of the matrix A = [[1,2,3],[0,4,5],[1,0,6]] using the adjoint method. (10 Marks)
Ans 1.
Introduction
Another matrix that can be multiplied with a square matrix to produce the identity matrix is called its inverse matrix. Invertibility of a matrix is related to the determinant of the matrix as not all matrices have inverses. One of the standard methods for calculating the inverse of a matrix is the adjoint method, which is applicable to small square matrices, e.g., a three by three matrix. This approach is based on cofactors and the determinant of matrix.
Meaning of the Adjoint Method
The adjoint method is to use the relationship that the inverse of a matrix is equal to the adjoint of the matrix divided by the
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- Find the solution of the following system using gauss elimination: 3x + 4y + z = 10, 2x – 3y + 5z = -9, x + 2y – z = 6. (10 Marks)
Ans 2.
Introduction
There are many different ways to solve a system of linear equations with multiple unknowns; one of the most reliable methods is the “Gauss Elimination”. This is a way of transforming the given system to an upper triangular form by using a series of row operations. After reducing the system to this simplified form, the values of the unknowns can be determined step by step by back substitution.
Idea used behind the Gauss Elimination
The concept behind Gauss elimination is that it eliminates one term from each of the equations below the pivot equation. This is accomplished by adding and/or subtracting appropriate multiples of one equation to another until a specific variable is eliminated. This is continued until the system has the form of an upper triangular matrix, with the last equation having only one
- Let f:A -> B, g:B -> C, and h:C -> D. If f(x) = x + 4, g(y) = 3y and h(z) = z – 1, find the composition h o g o f(x), also check the bijective ness of mappings. (10 Marks)
Ans 3.
Introduction
Compositions of functions are functions that are applied in succession. The concept is very prevalent in mathematics, where the result of a function is used as the input to another function. In addition to identifying the composition, it is important to see if each mapping is bijective, that is, one to one and onto simultaneously.
Meaning
- Check this (p -> q) v (not q -> not p) = True or not. (10 Marks)
Ans 4.
Introduction
In logic, a statement that is always true no matter what truth values its individual parts take is called a tautology. A truth table is created for all possible combinations of truth values of the statements in the logical expression to determine if it is a tautology. If the expression is true in every row of the truth table, it is true and confirmed as a tautology.
The conditional and contrapositive
- The table shows the height distribution (in cm) of students in a school: 140-150: 5, 150-160: 10, 160-170: 12, 170-180: 8, 180-190: 5. Find the median height of the students. (10 Marks)
Ans 5.
Introduction
In grouped data (where the data is given in intervals and the number of items in each interval), the most common approach for locating the median value can no longer be applied. However, the median must be calculated using a special formula for continuous grouped frequency distributions. This formula is to find the class where the median actually belongs and is based on the cumulative frequency.
For a grouped
- Explain the degree of a vertex in an undirected graph and binary trees. (10 Marks)
Ans 6.
Introduction
In discrete mathematics, an important class of structures are graphs and trees, which are used to model relationships among various objects. The degree of a vertex is one of the fundamental characteristics that are evaluated in a graph and it is a measure of the number of connections that a given vertex has with other vertices. There is another type of structure based on
Python Programming
- a) Explain the importance of exception handling in developing reliable Python applications. Demonstrate how multiple exceptions can be handled within a program using an appropriate example. b) Discuss the different file modes used in Python for reading, writing, appending, and updating files, with suitable examples. (5+5 Marks)
Ans 1.
Introduction
Sometimes, a Python program runs into these kind of unexpected scenarios that include invalid operations, missing files or wrong input. Two basic programming skills that are used to create applications that act reliably when things go wrong are exception handling and file handling.
Exception Handling is used to handle the exceptions.
Exception handling is a way to prevent a program from crashing abruptly when it encounters an error at runtime and instead handle the error. This prevents the application from becoming unstable and provides a normal message to the user rather than an ugly error trace. It also facilitates segregating the normal
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- a) Differentiate between implicit and explicit type conversion in Python. Explain how each affects data processing and program execution with suitable examples. b) Explain membership and identity operators in Python. Differentiate between them with suitable examples. (5+5 Marks)
Ans 2.
Python is flexible with data types, and also offers special operators to test for membership and identity of objects. Learning type conversion and these operators would help a programmer to write correct and predictable code.
Type Conversion – Implicit and Explicit
Implicit type conversion is the automatic conversion of one type into another type when an operation is performed between them, without the programmer’s code. This typically occurs when a small
- a) Explain the role of lists as mutable data structures in Python. Compare the working of insert(), append(), and extend() methods with suitable examples, and identify situations where each method is most appropriate. b) Explain the role of view functions in a Flask web application. Differentiate between static and dynamic routes and illustrate their implementation with suitable examples. (5+5 Marks)
Ans 3.
Introduction
Lists are used for a lot of tasks in python because data can be modified in them once they’ve been created. As a lightweight web framework, Flask determines the behavior of a web application based on the request based on the function to be called and the route set.
Lists are Mutable Data Structures
The reason a list in Python is termed as mutable is that the elements of a list can be altered, added or deleted without creating a new object in memory, after the list is created. This is unlike immutable
- a) How are arguments different from parameters? Explain keyword arguments and variable-length arguments in detail. b) Explain importance of regular expression in Text processing. Discuss different quantifiers with examples. (5+5 Marks)
Ans 4.
Python functions are parameterized and arguments determine the flow of data into a function. Regular expressions are also a powerful tool for searching and processing text according to regular patterns.
Arguments versus Parameters
A parameter is a variable that appears in the parentheses of the function definition, and can be given the value to be substituted later. When a function is called, an argument is the actual value passed to the function. To put it simply, Parameters are part of the function’s definition, and Arguments are part of the function’s call. Keyword arguments are used to pass arguments to a function by the parameter name; the order in which they are written is not important. Star and double star symbols can be used to pass a
- a) Discuss the role of game loops and event handling in game development. b) Explain the steps for establishing a database connection in Python. Discuss the role of SAVEPOINT in handling partial transaction failures with a suitable example. (5+5 Marks)
Ans 5.
There are two vastly different fields, game development and database handling, and both fields require continuous well structured processes in python. Game loops provide a smooth running game and database connections provide safe data storage and manipulation for a program.
Handling game loops and events
The game loop is the most essential component of a game program that is constantly repeated while the game is
- a) Describe the role of the Pandas library in data preprocessing. Discuss the data preprocessing steps in detail. b) Explain the concepts of Continuous Integration and Continuous Deployment/Delivery (CI/CD) in the context of Python application development. Discuss how a CI/CD pipeline can support automated testing, code-quality checks, and application deployment. (5+5 Marks)
Ans 6.
In many cases, modern Python programming includes tasks such as getting raw data ready for analysis and deploying and testing applications automatically. The first task can be achieved using the Pandas library, and the second task can be achieved using CI/CD pipelines.
Concept and Application
Pandas is used for data preprocessing in various ways.
Pandas is a
Programming & Problem-Solving using C
- (a) Explain the structure of a C program. (b) Explain input/output functions (printf, scanf). (5+5 Marks)
Ans 1.
Introduction
All the C program have a fixed structure which the C compiler expects and it communicates with the user principally through input and output functions. The first step to learning C programming correctly is to understand these two basics.
Structure of a C Program
A typical C program starts with preprocessor directives, typically consisting of header files required by the program to execute, e.g., stdio dot h for input-output. This is followed by the main function, the beginning point of execution of any C program and the point from which program execution actually
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- What are the major components of a C program? (10 Marks)
Ans 2.
A C program is divided into different parts, each of which is essential to understanding and implementing the program correctly by the compiler. A learner who has a knowledge of each of these components reads and writes C programs with more clarity.
Concept and Application
Preprocessor Directives
Preprocessor directives are instructions that are processed before the actual compilation begins, and they usually start with the hash symbol. The most frequently used directive is include
- Define operators in C and explain their types with suitable C Programs. (10 Marks)
Ans 3.
Introduction
Operators in C are special symbols that tell the compiler to perform a specific mathematical, logical or other kind of operation on the given values. They are the fundamental units that are used when writing expressions and conditions in a program.
Meaning of Operators
An operator is a symbol that indicates a particular operation on one or more values (operands) and returns a value, the result of that operation. C offers a variety of operators, which can be classified into various categories according to the type of operations they perform.
Arithmetic Operators
Arithmetic operators are symbols that are used to perform basic mathematical calculations, including the addition, subtraction, multiplication, division and remainder operations. The operators are regularly used in
- (a) Explain call by value and call by address. (b) Explain the difference between library and user-defined functions. (5+5 Marks)
Ans 4.
Data can be supplied to functions in various ways and there are some that are already provided by the compiler or can be created by the programmer. Both these ideas are important while designing a well structured C program.
Call by Value and Call by Address.
For call by value, a copy of the actual argument is passed to the function so that if the value of the argument is changed within the function, it will not affect the value of the argument in the calling program. Call by address: Memory address of actual argument passed to the function using pointer, both the pointer passed to the function and the actual argument are pointing to the same memory
- What is the difference between declaration and definition of function? (10 Marks)
Ans 5.
Introduction
In C, there are two concepts that are related but distinct – declaration and definition, related to functions. Many beginners get the two terms mixed up, but they are used for different purposes in a program.
Meaning of Function Declaration
A function declaration, also known as a function prototype, serves to inform the compiler of the name of the function, the value type returned by the function, and the type of parameter that it accepts, but not the actual function body. It is typically declared in the “top” of the program, before the main function, so that the compiler will know what type the function is, and what to expect in terms of calling it, even if the entire function will only be used later in the program. A declaration is
- (a) Explain advantages and disadvantages of recursion. (b) Explain function prototype and its importance. (5+5 Marks)
Ans 6.
Introduction
Two important concepts about functions in C are recursion and function prototypes, both of which impact the way in which a program is written and its performance when it runs.
Advantages and Disadvantages of Recursion
Recursion is a method that involves a function calling itself to get the answer by repeatedly solving smaller instances of the same problem. The beauty of the recursive approach is that recursive code is shorter and easier to read when a problem is naturally recursive, such as computing a factorial
Relational Database Management System
- Explain the different types of Database Management Systems based on data model, number of users, number of sites, cost, and purpose. Also compare Centralised and Distributed Database Systems with suitable examples. (5+5 Marks)
Ans 1.
DBMS can be categorised in various ways, based on the type of data they store and the ways in which they are utilised. In addition to the above classification, it is equally important to compare centralised and distributed databases as they are the two options used by various organizations.
Based on Data Model and Number of Users:
Depending on the data model, the DBMS can be classified into hierarchical, network, relational and object oriented, with the relational model being the most popular since it represents data as simple
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- Explain the concept of Query Optimisation. Discuss the role of heuristics and query transformation rules in improving query execution efficiency. (5+5 Marks)
Ans 2.
Introduction
In the world of databases, query optimisation plays a crucial role in database management, as the same query might be answered in various ways, with some being much more efficient than others. The database system attempts to determine the most efficient manner in which to execute a query prior to its execution.
Query
- Explain normalization in DBMS and also explain each normal form (1NF to 5NF) with examples. (5+5 Marks)
Ans 3.
Introduction
In Database Design, Normalization is a process to structure the data to prevent the data being stored in multiple places or problems with inserting, updating or deleting data records. It is performed in a steps by steps method called the normal forms, where each normal form is more
- Discuss data fragmentation, data replication, distributed transactions, and the Two-Phase Commit (2PC) protocol with suitable examples. (10 Marks)
Ans 4.
Introduction
Distributed database systems distribute data and processing across multiple locations, leading to several important concepts: fragmentation, replication, and coordinated transaction handling across locations.
Data Fragmentation
Data fragmentation is the practice of breaking up a database into smaller components or fragments and placing these
- Explain Parallel Database Architecture and discuss I/O Parallelism, Inter-Query Parallelism, and Intra-Query Parallelism. (10 Marks)
Ans 5.
Introduction
Parallel database systems consist of several processors and storage devices that cooperate together in order to process large quantities of data and queries more rapidly than using a system with only one processor. The parallelism may occur in several different ways, depending on the component of the system to be parallelized.
Parallel Database Architecture
- Explain the different lock modes. Discuss how hierarchical database elements are managed using locks. (10 Marks)
Ans 6.
Introduction
Locking is a popular method in databases for managing access to resources when concurrent users wish to read or write to the same data. There are various locking modes that exist to find a compromise between
Data Visualization
- Define Data Visualization and explain its importance in decision-making. Discuss any four types of data used in visualization with suitable examples. (5+5 Marks)
Ans 1.
Introduction
Data visualization refers to the process of representing data using visual tools like charts or graphs, which allows for the easy visualization of patterns and trends. It contributes greatly to the data-driven decision making of organisations.
Meaning and Importance of Data Visualization
Data visualization is the use of visual content like charts, graphs and maps to help people comprehend information more quickly and effortlessly than by reading numbers in a table. It plays a crucial role in decision-making as it enables managers and analysts to easily identify trends, compare values and uncover any unexpected patterns that may not be apparent in extensive datasets. A well-designed visualization can convey a finding to audiences who may lack a deep technical understanding, and
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- A company has the following monthly sales data for four products: Product A, Product B, Product C and Product D. Explain how you would use Excel to create an appropriate visualization for comparing their sales. Also discuss the situations in which a Bar Chart and Pie Chart would be suitable. (5+5 Marks)
Ans 2.
A very popular tool used to rapidly visualize business information is the spreadsheet program called Excel. Comparing the monthly sales of a few different products can be more readily apparent if a company selects the proper chart type in Excel.
To create a Visualization in Excel:
To compare the sales of the four
- Explain the role of Python in Data Visualization. Differentiate between Matplotlib and Seaborn. Further, explain how a Scatter Plot, Histogram, and Heat Map can be used to extract meaningful insights from a dataset. (5+5 Marks)
Ans 3.
Introduction
Python is a popular language for data visualization due to its ability to use various visualization libraries to generate complex and customizable visualizations directly from code. Of these, Matplotlib and Seaborn are the most popular libraries, and each is appropriate for somewhat different purposes.
The role of Python in Data visualisation.
With the Python, a user can interact with the data set, prepare it as needed, and create a visualisation as part of the same process without having to change tools. The Python code can be reused, and so the same visualization can be reproduced automatically whenever the data that it is based on changes, particularly useful for periodic reports. Python also has a variety of chart
- What is meant by effective data visualization? Explain the major principles that should be followed while designing a visualization. Discuss how a poorly designed chart can lead to incorrect interpretation of data. (5+5 Marks)
Ans 4.
Introduction
While it is correct to say that good data visualization is about making a chart, it is also correct to say that it’s about making a chart that communicates the right message without confusion. By adhering to some design principles, charts that mislead the viewer can be avoided.
Meaning of Effective Data Visualization
Good data visualization is a chart or graph that renders the data in a comprehensible,
- A project manager wants to visualize project activities and their duration, while an analyst wants to understand the distribution of employee salaries. Which Excel charts would you recommend in each case? Explain the working and applications of Gantt Chart and Box-and-Whisker Plot with suitable examples. (5+5 Marks)
Ans 5.
Introduction
Depending on the business situation, there are various types of charts for meaningful representation of data. A project manager who wants to monitor a set of activities throughout time and an analyst who wants to analyze salary distribution would require two very different types of visualization in Excel.
Visualization for Project Activities
If the project manager desires to see the activities and time frame of the project, then a Gantt Chart is the best recommendation in excel. Each activity is represented by a horizontal bar in a Gantt chart, with the length of the bar indicating the duration of the activity and the position of the bar on the chart indicating
- A retail organization has three years of daily sales data. Explain how you would use Python to clean, prepare, visualize, and analyse the data over time. Discuss the role of time-series plots, trend analysis, and interactive dashboards in presenting the findings to management. (5+5 Marks)
Ans 6.
Python offers a viable tool for cleaning, preparing, visualizing and analyzing large volumes of daily sales data accumulated over many years to make better management decisions, which retail organizations frequently collect.
Cleaning and Preparing the Sales Data
In Python, the initial stage would be to load the three years of daily sales data into a dataframe, possibly using the Pandas library. Duplicate entries would be removed, sales values would be dealt with, (such as nulls) and the date column would be converted to a date data type so that time based analysis