Introducing the state-of-the-art Free Matrix Calculator, your go-to web resource for performing a variety of matrix operations with ease. This tool acts as your virtual helper while handling complex matrix computations, helping you with addition, subtraction, multiplication, and determining determinants and transposes.
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Examining the Matrix Universe:
Key components of mathematical landscapes, matrices display a systematic collection of numbers or symbols arranged in rows and columns. With its intuitive layout, our Online Matrix Calculator facilitates quick and precise computations for professionals, scholars, and students studying physics, statistics, computer graphics, and related scientific subjects.
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In the realm of mathematics, a matrix stands as a structured arrangement of numbers, symbols, or expressions, meticulously organized in rows and columns. Matrices serve as fundamental tools widely applied across scientific domains such as physics, computer graphics, probability theory, statistics, calculus, and numerical analysis.
Typically denoted as m × n, the dimensions of a matrix, let's say A, signify that it contains m rows and n columns. An element within a matrix is often referred to using a variable with two subscripts, representing its location in the matrix. For instance, ai,j, where i equals 1 and j equals 3, denotes a1,3 as the specific value in the first row and third column of the matrix.
Matrix operations such as addition, multiplication, and subtraction are analogous to elementary arithmetic and algebraic operations. However, they entail distinct characteristics and adhere to particular limitations. Descriptions of the matrix operations are delineated below.
Matrix Addition
Matrix addition can only be executed on matrices of identical size, i.e., both matrices being of m × n dimensions. For example, one can add multiple matrices of sizes 3 × 3, 1 × 2, or 5 × 4, but cannot add matrices of differing dimensions, such as a 2 × 3 and a 3 × 2 matrix.
When matrices are of matching size, the addition is accomplished by summing the corresponding elements in the matrices. For instance, considering two matrices, A and B, with elements ai,j and bi,j, the resulting matrix, C, is formulated by adding each element and placing the outcome in the corresponding position in the new matrix:
exapmle for addition of matrix
A =
1
2
3
4
; B =
5
6
7
8
In the above matrices, a1,1 = 1; a1,2 = 2; b1,1 = 5; b1,2 = 6; etc. add the corresponding , elements to obtain ci,j. Adding the values in corresponding rows and columns:
a1,1 + b1,1 = 1 + 5 = 6 = c1,1
a1,2 + b1,2 = 2 + 6 = 8 = c1,2
a2,1 + b2,1 = 3 + 7 = 10 = c2,1
a2,2 + b2,2 = 4 + 8 = 12 = c2,2
Thus, matrix C is:
C =
6
8
10
12
Matrix Subtraction
Matrix subtraction closely resembles matrix addition, with the key distinction being the subtraction of values instead of addition. Similar to matrix addition, the matrices involved in subtraction must be of equal size.
Subtraction is conducted by deducting the elements in corresponding rows and columns of the matrices:
[Example of matrix subtraction]
Matrix Multiplication
Scalar Multiplication
Matrices can be multiplied by a scalar by multiplying every element in the matrix by the scalar value. For instance, given a matrix A and a scalar c, the product of c and A is derived by multiplying each element in A by the scalar.
[Example of scalar multiplication]
Matrix-Matrix Multiplication
The process of multiplying two (or more) matrices is more intricate than scalar multiplication. To multiply two matrices, the number of columns in the first matrix must align with the number of rows in the second matrix. For instance, a 2 × 3 matrix can be multiplied by a 3 × 4 matrix, but not a 2 × 3 matrix by a 4 × 3 matrix
Use following link for previous year question papers of BCA
Use following link for previous year question papers of B.Sc (Computer Science)
Paper-II, Digital Communication and Networking, ELC- 232
Notes of Digital Communication and Networking.
Chap. 1 Introduction to Electronics Communication.
Chap. 2 Modulation and Demodulation.
Chap. 3 Multiplexing Spectrum spreading and media access control.
Below here 3 chapter notes are provided.
Chap. 1 Introduction to Electronics Communication.
UNIT 1: Introduction to Electronic Communication (9)
Introduction to Communication: Elements of Communication system, types of noise sources,
Electromagnetic spectrum, signal and channel bandwidth,
Types of communication: simplex, half duplex, full duplex, baseband and broadband,
Serial communication: asynchronous and synchronous,
Information Theory: Information entropy, rate of information (data rate, baud rate), channel
capacity, Nyquist theorem, Signal to noise ratio, Noise Figure, Shannon
theorem,
Error handling codes: Necessity, Hamming code, CRC
Chap. 2 Modulation and Demodulation.
UNIT 2: Modulation and Demodulation (5)
Introduction to modulation and demodulation: Concept and need of modulation and demodulation,
Digital Modulation techniques: Pulse Code Modulation (PCM), FSK, QPSK, QAM.
Chap. 3 Multiplexing Spectrum spreading and media access control.
UNIT 3: Multiplexing, Spectrum Spreading and Media Access Control (12)
Multiplexing techniques: Frequency division multiplexing, wavelength division multiplexing, Time
division multiplexing
Spread Spectrum techniques: Frequency hopping Spread Spectrum, Direct Sequence Spread
Spectrum
Media Access Control (MAC):
Random Access Protocol: ALOHA, CSMA, CSMA/CD, CSMA/CA,
Controlled Access Protocols: Reservation, Polling, Token passing,
Channelization Protocols: FDMA, TDMA, CDMA.
UNIT 4: Computer Networking (10)
Introduction to computer networks
Types of networks : LAN, MAN, WAN, Wireless networks, Switching, Internet,
Network topology : point to point, Star, Ring, Bus, Mesh, Tree, Daisy Chain, Hybrid
Network devices : Repeater, Switch, Networking cables, Router, Bridge, Hub, Brouter, Gateway.
Wired LANs:-
Ethernet: Ethernet protocol, standard Ethernet, 100 MBPS Ethernet, Gigabit Ethernet, 10 Gigabit
Ethernet,
Computer network model: OSI and TCP/IP.
Recommended books:
1.Communication Electronics: Principles and Applications, Frenzel, Tata Mc Graw Hill
publication, 5th edition.
2. Data Communication and Networking, Forouzan, Mc Graw Hill publication, 5th edition
3. Computer Networks, Tanenbaum, pHI publication, 5th edition.
Chapter Name and Download link
Chap. 1 Introduction to Electronics Communication.
Chap. 2 Modulation and Demodulation.
Chap. 3 Multiplexing Spectrum spreading and media access control.
Use following link for previous year question papers of BCA
Use following link for previous year question papers of B.Sc (Computer Science)
UNIT- 1: Basics of Microcontroller and Intel 8051 architecture [08]
Introduction to microcontrollers, difference in controller and processor.
Architecture of 8051, Internal block diagram, Internal RAM organization, SFRS, pin functions
of 8051, I/O port structure and Operation, External Memory Interface.
UNIT-2: Programming model of 8051 [10]
Instruction classification, Instruction set, Addressing Modes: Immediate, register, direct,
indirect and relative, assembler directives (ORG, END), features with examples, I/O Bit and
Byte programming using assembly language for LED and seven segment display (SSD)
interfacing.
Introduction to 8051 programming in C.
UNIT- 3: Timer /Counter, Interrupts [10]
Timer / counter: TMOD, TCON, SCON, SBUF, PCON Registers, Timer modes, programming
for time delay using mode 1 and mode 2.
Interrupts: Introduction to interrupt, Interrupt types and their vector addresses, Interrupt enable
register and interrupt priority register (IE, IP)
UNIT- 4: Interfacing, Serial Communication [08]
Programming of serial port without interrupt, Serial Communication: Synchronous and
asynchronous serial communication, Use of timer to select baud rate for serial communication.
Interfacing : ADC, DAC, LCD, stepper motor
Recommended books:
1. 8051 microcontroller and Embedded system using assembly and C : Mazidi and McKinley,
Pearson publications
2. The 8051 microcontroller – Architecture, programming and applications: K.Uma Rao
and Andhe Pallavi, Pearson publications.
Use following link for previous year question papers of BCA
Use following link for previous year question papers of B.Sc (Computer Science)
1.1 Division Algorithm (without Proof),
1.2 G.C.D. using division algorithm and expressing it as linear combination
1.3 Euclid’s lemma
1.4 Equivalence relation (revision), Congruence relation on set of integers, Equivalence
class partition
Unit 2. Groups [03 Lectures]
2.1 Binary Operation
2.2 Group: Definition and Examples
2.3 Elementary Properties of Groups
Unit 3. Finite Groups and Subgroups [10 Lectures]
3.1 Order of a group, order of an element
3.2 Examples (Zn, +) and (U(n), *)
3.3 Subgroup definition, Finite subgroup test, subgroups of Zn
3.4 Generator, cyclic group, finding generators of Zn( Corollary 3,4 without proof)
3.5 Permutation group, definition, composition of two permutations, representation
as product of disjoint cycles, inverse and order of a permutation, even/ odd
permutation
3.6 Cosets: Definition, Examples and Properties, Lagrange Theorem(without Proof)
Unit 4. Groups and Coding Theory [18 Lectures]
4.1 Coding of Binary Information and Error detection
4.2 Decoding and Error Correction
4.3 Public Key Cryptography
Text Books:-
1. Contemporary Abstract Algebra By J. A, Gallian (Seventh Edition)
Unit 1:Chapter 0, Unit 2: Chapter 2, Unit 3: Chapter 3 ,4, 5 and 7
2. Discrete Mathematical Stuctures By Bernard Kolman, Robert C. Busby and Sharon
Ross (6th Edition) Pearson Education Publication
Unit 4: Chapter 11
Use following link for previous year question papers of BCA
Use following link for previous year question papers of B.Sc (Computer Science)
1. A textbook of Computer Based Numerical and Statistical Techniques, by A. K.
Jaiswal and Anju Khandelwal. New Age International Publishers.
Unit 1: Chapter 2: Sec. 2.1, 2.5, 2.7
Unit 2: Chapter 3: Sec. 3.1, 3.2, 3.4, 3.5, Chapter 4: Sec. 4.1, 4.2, 4.3,
Chapter 5: Sec. 5.1, 5.2, 5.4, 5.5
Unit 3: Chapter 6: Sec. 6.1, 6.3, 6.4, 6.5, 6.6, 6.7
Unit 4: Chapter 7: Sec. 7.1, 7.4, 7.5, 7.6
Reference Books:-
1. S.S. Sastry; Introductory Methods of Numerical Analysis, 3rd edition,
Prentice Hall of India, 1999.
2. H.C. Saxena; Finite differences and Numerical Analysis, S. Chand and
Company.
3. K.E. Atkinson; An Introduction to Numerical Analysis, Wiley Publications.
4. Balguruswamy; Numerical Analysis.
Use following link for previous year question papers of BCA
Use following link for previous year question papers of B.Sc (Computer Science)
Mathematics python practical SYBSC (computer science) semester-III
Based basic arithmetic operations and Numerical techniques.
Here Practical questions with there solutions (code) are provided.
which is useful for basic python learner students and Students from SPPU at Second year of computer science course.
Practicals:
Practical 1: Introduction to Python, Python Data Types-I (Unit 1)
Q.1. Write a python code to evaluate sin, cos at 3.14.
2. Write a python function that calculate square of a given number.
2A. Write a python function that calculate Multiplication of given number.
3. Write a python function that calculates sum and products of three numbers.
4. Write a python function that calculates area and circumference of a circle if
radius is given.
5. Write a python function that calculates roots of the quadratic equation ax^2
+ bx + c = 0.
6. Find the value of the following expression if x and y are true and z is false, by
using python.
Practical 2: Python Data Types- II (Unit 2)
1. Given a strings s1 = “Hello” and s2 = “Hi”. Show the results of the following
string expressions.
2. The two lists are given: l1= [1, 2, 3, 4, 5] and l2=[a, b, c , d , e] Show the results
of the following string expressions.
3. Two tuples are given: t1= ('p','u','n','e')and t2= ('m','u','m','b','a','i'). Show the r
esults of the following tuple expressions.
4. Write a python program to print vowels of user entered sting.
5. Write a python program to count number of characters of string.
6. Write a python program to reverse the tuple.
7. Write a python program using tuple to swap the values of two variables.
8. Write a python program to add ‘ing’ at the end of a given sting.
9. Write a python program to change a given string to a new string where first
and last characters have been exchanged.
10. Write a python program to get the largest number from a list.
Practical 3: Control statements in Python-I (Unit 3- 3.1, 3.2)
1.Write a function that prints weather number is divisible by another
number.
2. Write a function that test weather number is divisible by 3, 5 and 7.
3. Write a function that gives given number is positive, negative or zero.
4. Print Fibonacci numbers less than 1000.
5. Find sum of first 98 natural numbers.
6. Find number of integer between 0 and 1000, which are multiple of 11.
7. Define Euler’s phi function in python and hence find phi (120).
Practical 4: Control statements in Python-II (Unit 3- 3.3)
1. Using type() find type of
i) 123
ii) 15.6
iii) Pune
iv) [1,2,3]
v) (2,4,6)
2. Using id() Find id of
i) 123
ii) 15.6
iii) Pune
iv) [1,2,3] .
3. Convert:
i) 30 to float
ii) 123.32 to int
iii) 3.14159 to string
4. Using math.pow find 2^10.
5. Using user defined function (name it as ‘twice’) print your name twice.
6. Using user defined function in above question no.5 print word Python 8
times.
1. Using sympy module declare the vectors u =(1 2 3) v =( -7 0 3) Find
i. u+v
ii. u-v
iii. 6u
iv. 5v
v. 6u + 5v
2. Using sympy module declare any 3x3 matrix B and find
i. type(B)
ii. eigenvalues of B
3. Declare following matrices using sympy module
i. 3x5 zero matrix
ii. diagonal matrix (1,2,4)
iii. 5x4 one’s matrix.
4. Using sympy module declare any two 3x3 matrices A and B. Find
i. D1+D2
ii. D1-D2
iii. D1*D1
iv. D1*D2
v. D1**4
5. Using sympy module declare any two 3x3 matrices A and B. Find
i. A^(-1)
ii. B^(-1)
6. Using sympy module declare any 3x3 matrix A. print
i. first row
ii. third column
iii. first and second column
7. Using sympy module declare any 4x3 matrix A.
i. delete second column
ii. delete third column
iii. insert fourth row as (1, 1)
Practical 6: Application : Determinants, system of Linear Equations (Unit 4- 4.4, 4.5)
1. Using sympy module declare any 3x4 matrix and find transpose.
2. Using sympy module declare any 3x3 matrix A and find determinant of A.
3. Using sympy module declare any 3x3 matrix A and find reduced row echelon
form of A.
4. Using sympy module declare any two 3x3 matrices A and find nullspace of
A.
5. Using sympy module declare any two 3x3 matrices A and find columnspace
of A.
6. Using sympy module declare any two 4x4 matrices A and find rank of A.
7. Using sympy module declare any two 4x4 matrices A and find nullspace,
columnspace and rank of A.
Practical 7: Application : System of equations (Unit 4- 4.5)
1. Using sympy module and linslove() command, solve the following system of
equations.
2. Using sympy module and gauss-Jordan method, solve the following system
x + 2y + 3z = 3, 4x + 5y + 6z = 6, 7x + 8y + 10z = 9
3. Using sympy module and LU decomposition, solve the following system
6x + 18y + 3z = 3, 42 + 12y + z = 19, 4x + 15y + 3z = 0
4. Solve the following system by gauss-elimination method.
5. Solve the following system by LU decomposition method.
6. Solve the following system by gauss-elimination method ( i,e using
linsolve() command) .
1. Using sympy module find eigenvalues of the following matrix.
2. Using sympy module declare any 4x4 matrix and find eigenvalues of that
matrix.
3. Using sympy module find eigenvalues of the following matrix.
4. Using sympy module find eigenvectors of the following matrix.
5. Using sympy module find eigenvectors of the following matrix.
6. Using sympy module find eigenvectors of the following matrix.
1. Using sympy module decide if following the matrix is diagonalizable or not.
2. Using sympy module decide if following the matrix is diagonalizable or not.
If yes find matrix P and D (where P^(-1)AP = D).
3. Using sympy module find matrix P and D.
4. Using sympy module find find matrix P and D.
5. Using python decide whether matrix is diagonalizable, if yes find matrix P
and D.
6. Using python find matrix P and D.
1. Using newton-Raphson method find root of equation f(x) = x^(3) − 5x + 1 with
x0 = 0. 5 correct to three decimal places(error e=0.00001).
2. Using newton-Raphson method find root of equation xlog10(x) = 12. 34 with
x0 = 10.
3. Using newton-Raphson method find approximate value of √5 correct to ten
decimal places.
4. Using false-position method find approximate root of f(x) = xex − cos (x) in
interval (0, 1).
5. Using false-position method find approximate root of f(x) = tan(x) − 2x in
interval (1.1, 1.2).
6. Using false-position method find approximate value of √5 correct to ten
decimal places.
MTC- 243 Python Programming Language -II (mathematics) practical
Practical No. and name
Questions
Solutions
Practical 1: Graph Plotting (Unit 1 – 1.1 – 1.3)
Practical 2: Graph Plotting (Unit 1 – 1.4 – 1.7)
Practical 3: Application to Computational Geometry (Unit 2 – 2.1)
Practical 4: Application to Computational Geometry (Unit 2 – 2.2)
Practical 5: Application to Computational Geometry (Unit 2 – 2.3)
Practical 6: Study of Graphical aspects of Two dimensional transformation matrix using matplotlib
Practical 7: Study of Graphical aspects of Three dimensional transformation matrix using matplotlib
Practical 8: Study of Graphical aspects of Three dimensional transformation matrix using matplotlib
Practical 9: Study of effect of concatenation of Two dimensional and Three dimensional transformations
Practical 10: Generation of Bezier curve using given control points
Practical 11: Study of Operational Research in Python (Unit 3.1)
Practical 12: Study of Operational Research in Python (Unit 3.2)
S.Y.BSc(Comp.Sci.) Python practical 01
Title: 2D Graphs
1) Write a Python program to plot 2D graph of the functions f(x) = x^2 and g(x)= x^3 in [−1, 1].
2) Write a Python program to plot 2D graph of the functions f(x) = log10(x) in
the interval [0, 5].
3) Using Python plot the graph of function f(x) = sin(x) on the interval [0, 2Ï€].
4) Using Python plot the graph of function f(x) = sin^(-1)(x) on the interval [−1,1].
5) Using Python, plot the graph of function f(x) = sin(x) − e^x + 3x − log10(x) on the Interval [0, Ï€].
6) Plot the graph of f(x) = x^5 in [0, 5] with red dashed line with circle markers.
7) Plot the graphs of sin x, cos x, e^x and x^2 in [0, 5] in one figure with (2 × 2) subplots.
8) Write a python program to Plot 2D X-axis and Y-axis black color and in the
same diagram plot green triangle with vertices [5, 4], [7, 4], [6, 6].
9) Plot the graph of y = e ^((−x)^2) in [−5, 5] with red dashed-points line with Upward
Pointing triangle.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 01. solution.
SYBSc (comp. sci.) Python Practical -1
PYTHON Practical no-2 SYBSC (Comp. Sci)
For Solution Click Here
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 02. solution.
To download answers click here
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 03. solution.
Python Practical No. 4 SYBSC ( Comp. Sci.)
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 11. solution.
Practical no4 Solution
4fvv
Python Practical no. 5 Sybsc (comp.sci)
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 05. solution.
SYBSC python practical no.05
Python practical No. 06
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 06. solution.
python practical no. 6
Python practical no. 07
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics)
practical No. 07. solution.
Python practical no. 07
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics)
practical No. 07. solution.
SYBSC Python practical no. 08
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 08. solution.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 09. solution.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 10. solution.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 10. solution.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 11. solution.
These practical based on linear programming problems. Here we solve linear programming problems using python. It is very easy and interesting find the solution of LPP by python programming. In this practical we solve the 6 questions of LPP by using python programming. Which is very useful for mathematics students to solve LPP.
Python programming save the time and get good results.
Some questions for practice.
.
1) Minimize: Z = 6x + 7y
4x + y ≥ 40,
2x + 3y ≥ 90,
x, y ≥ 0
2) Maximize Z = 2x + 3y
x + y ≤ 30,
x ≤ 20, y ≤ 12
x, y ≥ 0
Practical No. 12
Practical No. 12 Solution
S.Y.BSc(Comp.Sci.) Python practical 01
Title: 2D Graphs
1) Write a Python program to plot 2D graph of the functions f(x) = x^2 and g(x)= x^3 in [−1, 1].
2) Write a Python program to plot 2D graph of the functions f(x) = log10(x) in
the interval [0, 5].
3) Using Python plot the graph of function f(x) = sin(x) on the interval [0, 2Ï€].
4) Using Python plot the graph of function f(x) = sin^(-1)(x) on the interval [−1,1].
5) Using Python, plot the graph of function f(x) = sin(x) − e^x + 3x − log10(x) on the Interval [0, Ï€].
6) Plot the graph of f(x) = x^5 in [0, 5] with red dashed line with circle markers.
7) Plot the graphs of sin x, cos x, e^x and x^2 in [0, 5] in one figure with (2 × 2) subplots.
8) Write a python program to Plot 2D X-axis and Y-axis black color and in the
same diagram plot green triangle with vertices [5, 4], [7, 4], [6, 6].
9) Plot the graph of y = e ^((−x)^2) in [−5, 5] with red dashed-points line with Upward
Pointing triangle.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 01. solution.
SYBSc (comp. sci.) Python Practical -1
PYTHON Practical no-2 SYBSC (Comp. Sci)
For Solution Click Here
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 02. solution.
To download answers click here
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 03. solution.
Python Practical No. 4 SYBSC ( Comp. Sci.)
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 11. solution.
Practical no4 Solution
4fvv
Python Practical no. 5 Sybsc (comp.sci)
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 05. solution.
SYBSC python practical no.05
Python practical No. 06
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 06. solution.
python practical no. 6
Python practical no. 07
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics)
practical No. 07. solution.
Python practical no. 07
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics)
practical No. 07. solution.
SYBSC Python practical no. 08
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 08. solution.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 09. solution.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 10. solution.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 10. solution.
SYBSc(C S) Sem - IV MTC- 243 Python Programming Language -II (mathematics) practical No. 11. solution.
These practical based on linear programming problems. Here we solve linear programming problems using python. It is very easy and interesting find the solution of LPP by python programming. In this practical we solve the 6 questions of LPP by using python programming. Which is very useful for mathematics students to solve LPP.
Python programming save the time and get good results.
Some questions for practice.
.
1) Minimize: Z = 6x + 7y
4x + y ≥ 40,
2x + 3y ≥ 90,
x, y ≥ 0
2) Maximize Z = 2x + 3y
x + y ≤ 30,
x ≤ 20, y ≤ 12
x, y ≥ 0
Practical No. 12
Practical No. 12 Solution
Use following link for previous year question papers of BCA
Use following link for previous year question papers of B.Sc (Computer Science)
