This document contains formulas and definitions related to mathematics for Class 12. It covers topics such as relations and functions, inverse trigonometric functions, matrices, determinants, and continuity and differentiability. Some key points include definitions of relations like reflexive, symmetric, and transitive relations. It also provides formulas for inverse trigonometric functions and their properties. Matrices are defined including operations like transpose, addition, and multiplication. Determinants are defined for matrices of various orders.
Physics Practical File - with Readings | Class 12 CBSESaksham Mittal
The document appears to be a scanned collection of pages from a book or manual. It contains images of many pages with text and diagrams but no clear overall narrative or topic. As a scanned document, it provides visual copies of written content but no coherent summary can be extracted from the random assortment of pages.
The document outlines the terms and conditions for a home loan agreement between a lender and borrower. It specifies details such as the loan amount, interest rate, repayment schedule, borrower obligations, default conditions, and foreclosure procedures. The lender agrees to provide a loan to the borrower to purchase a home, and the borrower agrees to repay the loan amount plus interest according to the payment schedule described.
This document describes a library management system project created by Darshit Rajeshbhai Vaghasiya. The project uses Python and MySQL to create tables for books, book issues, and returns. It allows adding, issuing, returning, and deleting books. Functions and source code are provided to perform these tasks. Sample outputs and tables demonstrate the functionality of the project.
This document provides an overview of an alternating current (AC) generator. It includes sections on the principle, construction, theory of operation, circuit diagram, expression for induced electromotive force (emf), and applications. The key components of an AC generator are an armature coil that rotates in a magnetic field, slip rings to draw current from the rotating coil, and brushes that supply the output. As the coil rotates, the changing magnetic flux induces an alternating current in the coil. The maximum induced emf is expressed as ε = ε° sin(ωt). Applications include power generation and distribution, vehicles, appliances, and portable generators.
The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
(1) Chiranjeet Samantaray completed an investigatory project on studying the variation of current using a light dependent resistor (LDR) under the guidance of his physics teacher. (2) The project investigated how the current in a circuit containing an LDR varies with changes in the power and distance of an incandescent light source illuminating the LDR. (3) The results showed that the LDR resistance decreases and current increases with higher light intensity from decreasing the distance or increasing the power of the light source.
TOPIC-To investigate the relation between the ratio of :-1. Input and outpu...CHMURLIDHAR
TOPIC-To investigate the relation between the ratio of :-1. Input and output voltage.2. Number of turnings in the secondary coil and primary coil of a self made transformer.
TO STUDY THE QUANTITY OF CASEIN PRESENT IN DIFFERENT SAMPLES OF MILKAnkitSharma1903
This document is a certificate and report for a school science project on studying the quantity of casein in different milk samples. It was completed by Ankit Sharma, a class 12 student, under the guidance of his teacher Mr. S.C. Jatt. The report includes an introduction on milk and casein, the aim, requirements, procedure, observations, and conclusions of the experiment. It found that different milk samples contain varying percentages of casein, with buffalo milk containing the highest at 4.20% and cow milk the lowest at 3.00%.
Physics Practical File - with Readings | Class 12 CBSESaksham Mittal
The document appears to be a scanned collection of pages from a book or manual. It contains images of many pages with text and diagrams but no clear overall narrative or topic. As a scanned document, it provides visual copies of written content but no coherent summary can be extracted from the random assortment of pages.
The document outlines the terms and conditions for a home loan agreement between a lender and borrower. It specifies details such as the loan amount, interest rate, repayment schedule, borrower obligations, default conditions, and foreclosure procedures. The lender agrees to provide a loan to the borrower to purchase a home, and the borrower agrees to repay the loan amount plus interest according to the payment schedule described.
This document describes a library management system project created by Darshit Rajeshbhai Vaghasiya. The project uses Python and MySQL to create tables for books, book issues, and returns. It allows adding, issuing, returning, and deleting books. Functions and source code are provided to perform these tasks. Sample outputs and tables demonstrate the functionality of the project.
This document provides an overview of an alternating current (AC) generator. It includes sections on the principle, construction, theory of operation, circuit diagram, expression for induced electromotive force (emf), and applications. The key components of an AC generator are an armature coil that rotates in a magnetic field, slip rings to draw current from the rotating coil, and brushes that supply the output. As the coil rotates, the changing magnetic flux induces an alternating current in the coil. The maximum induced emf is expressed as ε = ε° sin(ωt). Applications include power generation and distribution, vehicles, appliances, and portable generators.
The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
(1) Chiranjeet Samantaray completed an investigatory project on studying the variation of current using a light dependent resistor (LDR) under the guidance of his physics teacher. (2) The project investigated how the current in a circuit containing an LDR varies with changes in the power and distance of an incandescent light source illuminating the LDR. (3) The results showed that the LDR resistance decreases and current increases with higher light intensity from decreasing the distance or increasing the power of the light source.
TOPIC-To investigate the relation between the ratio of :-1. Input and outpu...CHMURLIDHAR
TOPIC-To investigate the relation between the ratio of :-1. Input and output voltage.2. Number of turnings in the secondary coil and primary coil of a self made transformer.
TO STUDY THE QUANTITY OF CASEIN PRESENT IN DIFFERENT SAMPLES OF MILKAnkitSharma1903
This document is a certificate and report for a school science project on studying the quantity of casein in different milk samples. It was completed by Ankit Sharma, a class 12 student, under the guidance of his teacher Mr. S.C. Jatt. The report includes an introduction on milk and casein, the aim, requirements, procedure, observations, and conclusions of the experiment. It found that different milk samples contain varying percentages of casein, with buffalo milk containing the highest at 4.20% and cow milk the lowest at 3.00%.
chemistry investigatory project on food adulterationappietech
This chemistry project certificate summarizes Sharath Nair's research project on detecting common food adulterants under the guidance of his teacher Rakhi Phathak. The project includes an introduction on the history and issues of food adulteration, objectives to study common adulterants in different foods, acknowledgments, contents listing the sections, experiments conducted to detect adulterants in fats/oils, sugar, and spices, and precautions consumers can take to avoid adulterated foods.
This document describes the construction and application of a Wheatstone bridge circuit. It begins by introducing Wheatstone bridges and their inventor. It then discusses the key components of a Wheatstone bridge, including four resistors where one has an unknown value. The working principle is explained, where balancing the resistor ratios results in no current through the galvanometer. Example circuits are provided. Applications include measuring light, pressure, strain and more. Limitations include inaccuracies under unbalanced conditions and limited resistance ranges.
This document appears to be a chemistry project file submitted by Nikhil Dwivedi, a class 12 science student, on the topic of studying food adulterants. The file includes a certificate, acknowledgements, index, objective, introduction on food adulteration and laws, and details of experiments conducted to detect adulterants in foods like oils, fats, sugar and spices. The introduction provides background on food adulteration issues and legislation in India and other countries to protect consumers from health risks of adulterated foods.
Chemistry Practical Record Full CBSE Class 12 Muhammad Jassim
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Full wave rectifier Physics Investigatory ProjectSaksham Mittal
The document outlines the terms and conditions for a rental agreement between John Doe and Jane Doe for the lease of an apartment located at 123 Main St from January 1, 2023 through December 31, 2023. Key details include the monthly rent amount, late fees, repairs and maintenance responsibilities, entry rules, lease extension and termination terms, and liability waivers.
This document describes an investigatory project on investigating the relationship between the input and output voltage of a transformer. It includes an introduction describing transformers, the objectives of investigating the ratio of input/output voltages and primary/secondary coil turns. The document outlines the theory of transformer operation, required apparatus, procedures followed, applications of transformers, sources of error, conclusions and references. The student aims to build self-made transformers and measure voltages and currents to determine the relationships.
Here's my Mathematics Board Practical File. I hope you find it as useful as it was to me. I constantly got complimented for my file from internal as well as external teachers so I thought of sharing my work with all of you. This file is however of CBSE class 12th 2020-2021 syllabus.
The document discusses the benefits of meditation for reducing stress and anxiety. Regular meditation practice can help calm the mind and body by lowering heart rate and blood pressure. Studies have shown that meditating for just 10-20 minutes per day can have significant positive impacts on both mental and physical health.
“To estimate the charge induced on each one of the two identical Styrofoam (o...VanshPatil7
This document is a certificate certifying that Vansh Patil of class 12th at SNBP International School completed a physics project on estimating the charge induced on two identical Styrofoam balls suspended vertically using Coulomb's Law, in partial fulfillment of a CBSE exam. The student thanks their physics teachers Miss Luna and Miss Ruchita for guidance. The project report includes an introduction to transformers, the theory behind them, sample circuit diagrams, observation tables showing measurements taken, results discussing relationships between voltage ratios and coils, and applications of transformers such as in voltage regulators.
Computer science class 12 project on Super Market BillingHarsh Kumar
This document is Harsh Kumar's final project report for the subject of Computer Science in Class XII. It details the development of a "Super Market Billing" software project under the guidance of his teacher, Mr. Manoj Kumar Singh. The report contains documentation of the project's features, code, and output. Harsh Kumar declares that all coding was the result of his personal efforts for his school's CBSE examination.
Rectifier class 12th physics investigatory projectndaashishk7781
This document is a physics project submitted by Ashish Kumar to his teacher, Mr. C.S. Jha, on the topic of rectifiers. It describes constructing a full wave bridge rectifier to convert alternating current (AC) to direct current (DC). The project aims to understand rectification and explain center tapped and bridge full wave rectification. It details the circuit components used, including a transformer, diodes, capacitors and resistor. The document explains how the full wave bridge rectifier works during each half cycle to allow current flow in one direction only, producing a pulsating DC output that is filtered by the capacitors. Testing showed the rectifier produced a 12V DC current.
Maths Class 12 Probability Project PresentationAaditya Pandey
The document discusses the concept of probability. It defines probability as the likelihood of an event occurring based on the number of possible outcomes. It provides an example of calculating the probability of picking a red ball from a basket containing balls of different colors. The document then discusses key terms related to probability like sample space, sample point, events, mutually exclusive events, and exhaustive events. It also explains the concepts of conditional probability and Bayes' theorem along with examples. It discusses the multiplication theorem of probability and the concept of independent events.
Computer Science Investigatory Project Class 12Self-employed
The document describes a project report submitted by Rahul Kushwaha on a railway ticket reservation system. It includes certificates from the guide and examiner approving the report. The report contains sections describing the header files used, files generated, the working of the program, the coding, output screens, and conclusion. It was submitted for a computer science class and thanks the guide, principal, parents and classmates for their support.
CLASS 12 PHYSICS PROJECT - Measuring current using halfwave rectifierMathesh T
The document describes an experiment to measure current using a half-wave rectifier. A student named Mathesh from VELS Vidhyashram School designed the experiment for their class 12 physics project. The experiment involves connecting a diode, capacitor, voltmeter and resistance box in a circuit to form a half-wave rectifier. By measuring the voltage output across different resistances and using Ohm's law, the average current of 0.264 Amperes was calculated. The summary concludes that a half-wave rectifier is rarely used in practice due to its high ripple factor but is cheap and simple to construct.
This document describes a chemistry student's school project on detecting food adulterants. It includes a certificate signed by the teacher, acknowledgements, introduction on food adulteration and laws, objectives, theory on common adulterants and tests to detect them in samples. The experiments describe tests to detect adulterants in oils/fats, sugar, chili powder, turmeric and pepper. Observations of sample tests show no adulterants detected. The conclusion stresses the importance of selecting non-adulterated food for health.
This document discusses functions and their properties. It defines a function as a special relation where each first element is paired with exactly one second element. Functions are represented as sets of ordered pairs. The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values. Functions can be represented graphically and through equations, and can be transformed through shifts, reflections, and stretching/shrinking. Common function families include linear, quadratic, exponential, and trigonometric functions.
chemistry investigatory project on food adulterationappietech
This chemistry project certificate summarizes Sharath Nair's research project on detecting common food adulterants under the guidance of his teacher Rakhi Phathak. The project includes an introduction on the history and issues of food adulteration, objectives to study common adulterants in different foods, acknowledgments, contents listing the sections, experiments conducted to detect adulterants in fats/oils, sugar, and spices, and precautions consumers can take to avoid adulterated foods.
This document describes the construction and application of a Wheatstone bridge circuit. It begins by introducing Wheatstone bridges and their inventor. It then discusses the key components of a Wheatstone bridge, including four resistors where one has an unknown value. The working principle is explained, where balancing the resistor ratios results in no current through the galvanometer. Example circuits are provided. Applications include measuring light, pressure, strain and more. Limitations include inaccuracies under unbalanced conditions and limited resistance ranges.
This document appears to be a chemistry project file submitted by Nikhil Dwivedi, a class 12 science student, on the topic of studying food adulterants. The file includes a certificate, acknowledgements, index, objective, introduction on food adulteration and laws, and details of experiments conducted to detect adulterants in foods like oils, fats, sugar and spices. The introduction provides background on food adulteration issues and legislation in India and other countries to protect consumers from health risks of adulterated foods.
Chemistry Practical Record Full CBSE Class 12 Muhammad Jassim
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Full wave rectifier Physics Investigatory ProjectSaksham Mittal
The document outlines the terms and conditions for a rental agreement between John Doe and Jane Doe for the lease of an apartment located at 123 Main St from January 1, 2023 through December 31, 2023. Key details include the monthly rent amount, late fees, repairs and maintenance responsibilities, entry rules, lease extension and termination terms, and liability waivers.
This document describes an investigatory project on investigating the relationship between the input and output voltage of a transformer. It includes an introduction describing transformers, the objectives of investigating the ratio of input/output voltages and primary/secondary coil turns. The document outlines the theory of transformer operation, required apparatus, procedures followed, applications of transformers, sources of error, conclusions and references. The student aims to build self-made transformers and measure voltages and currents to determine the relationships.
Here's my Mathematics Board Practical File. I hope you find it as useful as it was to me. I constantly got complimented for my file from internal as well as external teachers so I thought of sharing my work with all of you. This file is however of CBSE class 12th 2020-2021 syllabus.
The document discusses the benefits of meditation for reducing stress and anxiety. Regular meditation practice can help calm the mind and body by lowering heart rate and blood pressure. Studies have shown that meditating for just 10-20 minutes per day can have significant positive impacts on both mental and physical health.
“To estimate the charge induced on each one of the two identical Styrofoam (o...VanshPatil7
This document is a certificate certifying that Vansh Patil of class 12th at SNBP International School completed a physics project on estimating the charge induced on two identical Styrofoam balls suspended vertically using Coulomb's Law, in partial fulfillment of a CBSE exam. The student thanks their physics teachers Miss Luna and Miss Ruchita for guidance. The project report includes an introduction to transformers, the theory behind them, sample circuit diagrams, observation tables showing measurements taken, results discussing relationships between voltage ratios and coils, and applications of transformers such as in voltage regulators.
Computer science class 12 project on Super Market BillingHarsh Kumar
This document is Harsh Kumar's final project report for the subject of Computer Science in Class XII. It details the development of a "Super Market Billing" software project under the guidance of his teacher, Mr. Manoj Kumar Singh. The report contains documentation of the project's features, code, and output. Harsh Kumar declares that all coding was the result of his personal efforts for his school's CBSE examination.
Rectifier class 12th physics investigatory projectndaashishk7781
This document is a physics project submitted by Ashish Kumar to his teacher, Mr. C.S. Jha, on the topic of rectifiers. It describes constructing a full wave bridge rectifier to convert alternating current (AC) to direct current (DC). The project aims to understand rectification and explain center tapped and bridge full wave rectification. It details the circuit components used, including a transformer, diodes, capacitors and resistor. The document explains how the full wave bridge rectifier works during each half cycle to allow current flow in one direction only, producing a pulsating DC output that is filtered by the capacitors. Testing showed the rectifier produced a 12V DC current.
Maths Class 12 Probability Project PresentationAaditya Pandey
The document discusses the concept of probability. It defines probability as the likelihood of an event occurring based on the number of possible outcomes. It provides an example of calculating the probability of picking a red ball from a basket containing balls of different colors. The document then discusses key terms related to probability like sample space, sample point, events, mutually exclusive events, and exhaustive events. It also explains the concepts of conditional probability and Bayes' theorem along with examples. It discusses the multiplication theorem of probability and the concept of independent events.
Computer Science Investigatory Project Class 12Self-employed
The document describes a project report submitted by Rahul Kushwaha on a railway ticket reservation system. It includes certificates from the guide and examiner approving the report. The report contains sections describing the header files used, files generated, the working of the program, the coding, output screens, and conclusion. It was submitted for a computer science class and thanks the guide, principal, parents and classmates for their support.
CLASS 12 PHYSICS PROJECT - Measuring current using halfwave rectifierMathesh T
The document describes an experiment to measure current using a half-wave rectifier. A student named Mathesh from VELS Vidhyashram School designed the experiment for their class 12 physics project. The experiment involves connecting a diode, capacitor, voltmeter and resistance box in a circuit to form a half-wave rectifier. By measuring the voltage output across different resistances and using Ohm's law, the average current of 0.264 Amperes was calculated. The summary concludes that a half-wave rectifier is rarely used in practice due to its high ripple factor but is cheap and simple to construct.
This document describes a chemistry student's school project on detecting food adulterants. It includes a certificate signed by the teacher, acknowledgements, introduction on food adulteration and laws, objectives, theory on common adulterants and tests to detect them in samples. The experiments describe tests to detect adulterants in oils/fats, sugar, chili powder, turmeric and pepper. Observations of sample tests show no adulterants detected. The conclusion stresses the importance of selecting non-adulterated food for health.
This document discusses functions and their properties. It defines a function as a special relation where each first element is paired with exactly one second element. Functions are represented as sets of ordered pairs. The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values. Functions can be represented graphically and through equations, and can be transformed through shifts, reflections, and stretching/shrinking. Common function families include linear, quadratic, exponential, and trigonometric functions.
The document discusses quadratic functions and their graphs. It defines quadratic functions as functions of the form f(x)=ax^2+bx+c, where a is not equal to 0. The graph of a quadratic function is a parabola with certain characteristics: it is symmetrical about an axis of symmetry and has a vertex which is either a maximum or minimum point. The axis of symmetry is the line x=0 for functions of the form f(x)=ax^2 and the vertex is at (0,0). For functions of the form f(x)=ax^2+k, the graph is a translation of f(x)=ax^2, so the vertex is (0,k) and the
K-Notes are concise study materials intended for quick revision near the end of preparation for exams like GATE. Each K-Note covers the concepts from a subject in 40 pages or less. They are useful for final preparation and travel. Students should use K-Notes in the last 2 months before the exam, practicing questions after reviewing each note. The document then provides a summary of key concepts in linear algebra and matrices, including matrix properties, operations, inverses, and systems of linear equations.
This document provides definitions and formulas for vector algebra and calculus. It includes:
- Definitions of orthonormal vectors, scalar and vector products, equations of lines and planes
- Formulas for vector multiplication, scalar and vector triple products, and non-orthogonal bases
- Notation for vector and scalar functions, and identities for gradient, divergence, curl, and Laplacian operators
- Formulas for gradient, divergence, curl in Cartesian, cylindrical and spherical coordinates
- Definitions of eigenvalues/eigenvectors, and formulas for matrix operations like transpose, inverse, and determinant
- Theorems of Gauss, Stokes, and Green relating integrals over volumes, surfaces and curves
This document discusses algebraic functions, including polynomial and rational functions. Polynomial functions are functions of the form y = p(x) = a0 + a1x + a2x2 + ... + anxn, where ai ∈ R and an ≠ 0. Rational functions are functions of the form y = R(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions. The document outlines how to analyze the domain, intercepts, symmetries, asymptotes, and graph of algebraic functions. It provides examples of discussing these "aids to graphing" and sketching the graphs of specific rational functions.
1) Functions relate inputs to outputs through ordered pairs where each input maps to exactly one output. The domain is the set of inputs and the range is the set of outputs.
2) There are different types of functions including linear, quadratic, and composition functions. A linear function's graph is a straight line while a quadratic function's graph is a parabola.
3) Composition functions combine other functions where the output of one becomes the input of another. Together functions provide a powerful modeling tool used across many fields including medicine.
This document provides an overview of some basic mathematics concepts for machine learning, including:
1. Probability theory - definitions of probability, joint and conditional probability, Bayes' rule, expectations.
2. Linear algebra - definitions of vectors, matrices, matrix multiplication and properties, inverses, eigenvalues.
3. Differentiation - definitions of the derivative, gradient, maxima/minima, approximations, the chain rule.
A New Approach on the Log - Convex Orderings and Integral inequalities of the...inventionjournals
In this paper, we introduce a new approach on the convex orderings and integral inequalities of the convex orderings of the triangular fuzzy random variables. Based on these orderings, some theorems and integral inequalities are established.
This document discusses eigenvalues and diagonalization of matrices. Some key points:
- Eigenvalues are values that satisfy the equation AX = λX, where X is a non-zero eigenvector. The eigenspace associated with λ contains all eigenvectors for that eigenvalue.
- The characteristic polynomial of a matrix A is defined as det(xI - A). The eigenvalues of A are the roots of its characteristic polynomial.
- Similar matrices have the same eigenvalues, determinant, rank, trace, and characteristic polynomial. Two matrices are similar if one can be obtained from the other by conjugation via an invertible matrix.
- The trace of a matrix is the sum of its diagonal entries
The document defines and provides properties of various mathematical functions including:
- Relations and sets including Cartesian products and relations.
- Functions including domain, co-domain, range, and the number of possible functions between sets.
- Types of functions such as polynomial, algebraic, transcendental, rational, exponential, logarithmic, and absolute value functions.
- Graphs of important functions are shown such as 1/x, sinx, logx, |x|, [x], and their key properties are described.
This document discusses various topics related to piecewise functions and rational functions:
- It defines piecewise functions and provides examples of evaluating piecewise functions at given values.
- It introduces rational functions as functions of the form p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not equal to zero. It discusses representing rational functions in different forms.
- It explains how to identify restrictions or extraneous roots of rational functions by setting the denominator equal to zero. It also discusses how to determine the domain of a rational function based on its restrictions.
- Finally, it defines vertical and horizontal asymptotes of rational functions. It provides
1. The document outlines various formulas and concepts related to trigonometric, exponential, and calculus functions including differentiation, integration, asymptotes, derivatives, inverse functions, and volumes of revolution.
2. Formulas are provided for trigonometric functions, exponential growth and decay, and the definitions of continuity, derivatives, and inverse functions.
3. Theorems and properties are described for mean value theorem, L'Hopital's rule, fundamental theorem of calculus, and volumes generated by revolving regions about axes.
The document provides an overview of Calculus I taught by Professor Matthew Leingang at New York University. It outlines key topics that will be covered in the course, including different classes of functions, transformations of functions, and compositions of functions. The first assignments are due on January 31 and February 2, with first recitations on February 3. The document uses examples to illustrate concepts like linear functions, other polynomial functions, and trigonometric functions. It also explains how vertical and horizontal shifts can transform the graph of a function.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
This document provides information about Baraka Loibanguti, who is the author of an advanced mathematics book. It includes his contact information and some notes about copyright and permissions. The document then begins discussing functions, including definitions of domain, range, and different types of functions like linear, quadratic, cubic, and polynomial functions. It provides examples of how to graph different types of functions by creating tables of values or using intercepts.
Finding the opening of the parabola, vertex, axis of symmetry, y-intercept, x- intercept, domain, range, and the minimum/maximum value including the illustration of the graph
105
The number of possible functions g: A → A is 10^10 = 105 since |A| = 10.
For gof = f to hold, g must be the identity function. There is only one identity function.
Hence, the number of possible functions g is 105.
The answer is A.
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Similar to CBSE Class 12 Mathematics formulas (20)
PNS Ghazi was a Pakistani submarine that played a key role in the 1971 Indo-Pakistani War. Ghazi's mission was to locate and sink the Indian aircraft carrier INS Vikrant and to mine India's eastern seaboard. On December 3rd, 1971, Ghazi was reported sunk off the coast of Visakhapatnam, India, with all 93 crew on board. The exact circumstances of Ghazi's sinking remain unclear, with competing claims from the Indian and Pakistani navies. Ghazi's loss contributed to tilting the war in favor of India.
This document discusses key aspects of population in India including:
- Population size, distribution, and growth rates based on census data
- Characteristics like age composition, sex ratio, literacy rates, and occupational structure
- Important processes that influence population change like birth rates, death rates, and migration patterns
- Government policies aim to promote planned parenthood, education, healthcare, and delayed marriage to influence population growth.
The document summarizes the plot of Sir Arthur Conan Doyle's short story "The Tragedy of Birlstone". It describes the setting of Birlstone village and the manor house where John Douglas lived with his wife. On the night of January 6th, John Douglas was found murdered. The local doctor investigated and questioned the other resident, Cecil Baker, about the events of that night. Several theories were proposed about how the killer may have entered the locked manor house and committed the crime. The summary leaves it to the reader to consider whether Douglas committed suicide, was murdered by an outside intruder, or was killed by someone inside the house.
Guys download this ppt if you want to know more about bermuda triangle. Its from Main Course Book of class 9th. NCERT textbook. Its also useful for users studying in Kendriya Vidhyalaya like me .
Weather refers to current atmospheric conditions like temperature, wind, and precipitation, while climate describes average weather patterns over 30+ years. India's climate is influenced by its latitude near the Tropic of Cancer, the Himalayan mountains, ocean currents, and monsoon winds. The monsoon season from June to September brings most of India's annual rainfall, though it can vary significantly between wet and dry periods. India experiences four main seasons - a cold winter, hot summer, advancing monsoon rains, and retreating monsoon transition.
The document provides information on the classification of living organisms. It discusses the need for classification due to the huge diversity of life. It explains the levels of classification from kingdom down to species. The five kingdom system of Whittaker is described, including the kingdoms of Monera, Protista, Fungi, Plantae, and Animalia. Characteristics of each kingdom are provided. The classification of plants and animals is then outlined down to class levels. Finally, scientific naming conventions are explained.
The document summarizes the status of women in 19th century India and various social reform movements that helped improve their conditions. It discusses that women faced severe discrimination and were treated poorly. It describes various discriminatory social practices like sati system, child marriage, untouchability etc. It then discusses the role of social reformers like Raja Ram Mohan Roy, Ishwarchandra Vidyasagar, Pandita Ramabai, Tarabai Shinde and others who fought to reform practices and promote women's rights and education through establishing schools and writing texts. Their efforts along with other reform movements helped raise awareness and change the social status of women over time.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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2. 1. Relations and
Functions
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
Empty relation is the relation R in X given by R = X × X.
Universal relation is the relation R in X given by R = X × X.
Reflexive relation R in X is a relation with (a, a) R a X.
Symmetric relation R in X is a relation satisfying (a, b) R implies (b, a) R.
Transitive relation R in X is a relation satisfying (a, b) R and (b, c) R
implies that (a, c) R.
Equivalence relation R in X is a relation which is reflexive, symmetric and
transitive.
Equivalence class [a] containing a X for an equivalence relation R in X is
the subset of X containing all elements b related toa.
A function f : X Y is one-one (or injective) if
f (x ) = f (x ) x = x x , x X.
1 2 1 2 1 2
A function f : X Y is onto (or surjective) if given any y Y, x X such
that f (x) = y.
A function f : X Y is one-one and onto (or bijective), if f is both one-one
and onto.
The composition of functions f : A B and g : B C is the function
gof : A C given by gof (x) = g(f (x)) x A.
A function f : X Y is invertible if g : Y X such that gof = IX and
fog = IY.
A function f : X Y is invertible if and only if f is one-one and onto.
◆ Given a finite set X, a function f : X X is one-one (respectively onto) if and only if f
is onto (respectively one-one). This is the characteristic property of a finite set. This
is not true for infinite set.
◆ A binary operation on a set A is a function from A × A to A.
◆ An element e X is the identity element for binary operation : X × X X, if a e
= a = e a a X.
◆ An element a X is invertible for binary operation : X × X X, if there exists b
X such that a b = e = b a where, e is the identity for the binary operation .
The element b is called inverse of a and is denoted by a–1.
◆ An operation on X is commutative if a b = b a a, b in X.
◆ An operation on X is associative if (a b) c = a (b c) a, b, c in X.
3. 2.INVERSE TRIGONOMETRIC
FUNCTIONS
◆ The domains and ranges (principal value branches) of inverse trigonometric
functionsare given in the followingtable:
Functions Domain Range
(Principal Value Branches)
y = sin–1 x [–1, 1]
2
,
2
y = cos–1 x [–1, 1] [0, ]
y = cosec–1 x R – (–1,1)
2
,
2
– {0}
y = sec–1 x R – (–1, 1)
[0, ] – { }
2
y = tan–1 x R
,
2 2
y = cot–1 x R (0, )
◆ sin–1x should not be confused with (sin x)–1. In fact (sin x)–1 =
1
sinx
and
similarlyforother trigonometricfunctions.
◆ The value of an inverse trigonometric functions which lies in its principal
value branch is called the principal value of that inverse trigonometric
functions.
For suitable values of domain, wehave
◆
◆
y = sin–1 x x = sin y
sin (sin–1 x) = x
◆
◆
x = sin y y = sin–1 x
sin–1 (sin x) = x
◆
1
sin–1 = cosec–1 x ◆ cos–1 (–x) = – cos–1 x
◆ cos–1
x
1
x
= sec–1
x ◆ cot–1
(–x) = – cot–1
x
◆
1
x
tan–1
= cot–1
x ◆ sec–1
(–x) = – sec–1
x
4. sin–1 (–x) = – sin–1 x tan–1 (–x) = – tan–1 x
◆
◆ 2
tan–1 x + cot–1 x =
◆
◆ cosec–1 (–x) = –cosec–1 x
◆ tan–1x + tan–1y = tan–1
x y
1 xy ◆ 2
sin–1 x + cos–1 x =
◆ tan–1
x + tan–1
y = tan–1
x y
1 xy ◆ 2
cosec–1
x + sec–1
x =
◆ 2tan–1 x = sin–1
2x
1 x2 = cos–1
1 x2
1 x2 ◆ 2tan–1x = tan–1
2
2x
1 x
x y
xy
◆ tan–1x + tan–1y = tan–1
1 , xy>1; x, y> 0
5. ◆ A matrix is an ordered rectangular array of numbers or functions.
◆ A matrix having m rows and n columns is called a matrix of order m × n.
◆ [aij]m × 1 is a column matrix.
◆ [aij]1 × n is a row matrix.
◆ An m × n matrix is a square matrix if m = n.
◆ A = [aij]m × m is a diagonal matrix if aij = 0, when i j.
3.Matrices
A = [aij]n × n is a scalar matrix if aij = 0, when i j, aij = k, (k is some
constant), when i = j.
A = [aij]n × n is an identity matrix, if aij = 1, when i = j, aij = 0, when i j. A
zero matrix has all its elements as zero.
A = [aij] = [bij] = B if (i) A and B are of same order, (ii) aij = bij for all
possible values of i and j.
kA = k[aij]m × n =[k(aij)]m × n
– A = (–1)A
A – B = A + (–1) B
A + B = B + A
(A + B) + C = A + (B + C), where A, B and C are of same order.
k(A + B) = kA + kB, where A and B are of same order, k is constant.
(k + l ) A = kA + lA, where k and l are constant.
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆ If A = [aij]m× n and B = [bjk]n × p , thenAB = C = [cik]m × p, where cik ∑ aij bjk
j1
(i) A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC
◆
◆
◆
◆
◆
◆
If A = [a ] , then A
or AT = [a ]
ij m × n ji n × m
(i) (
A
)
= A, (ii) (kA)= kA
,(iii) (A + B
)
= A
+ B
, (iv) (AB)= B
A
A is a symmetric
matrix if A
=A.
A is a skew symmetric matrix if A
= – A.
Any square matrix can be represented as the sum of a symmetric and a
skew symmetric matrix.
Elementary operations of a matrix are asfollows:
(i) Ri
Rj
or Ci
Cj
(ii) Ri kRi or Ci kCi
(iii) Ri Ri + kRj or Ci Ci + kCj
If A and B are two square matrices such that AB = BA = I, then B is the
inverse matrix of A and is denoted by A–1 and A is the inverse ofB.
Inverse of a square matrix, if it exists, is unique.
◆
◆
◆
6. 4.Determinants
◆ Determinant of a matrix A = [a11]1× 1 is given by | a11| = a11
a a
◆ Determinant of a matrix A
a11
21 22
a12
is given by
21 22
a11 a12
A
a a
= a a – a a
11 22 12 21
◆ Determinantof a matrix A a2 2
a3
a1 b1 c1
b2
b3 c3
1
c is given by (expandingalongR )
2 2 2 2
3 3 3 3 3 3
b c a c a b
2 2 2 1
b c 1
a 1
a
3 3 3
c b
a b c
a1 b1 c1
A a b c a 2
b 2
c
For any square matrix A, the |A| satisfy following properties.
◆ |A|= |A|, where A
= transpose of A.
◆ If we interchange any two rows (or columns), then sign of determinant
changes.
◆ If any two rows or any two columns are identical or proportional, then value
of determinant iszero.
◆ If we multiplyeach elementof a row or a column of a determinantby constant
k, then value of determinant is multiplied by k.
◆ Multiplying a determinant by k means multiply elements of only one row
(or one column) by k.
◆If A [a ] , then k .A k3
A
◆
ij 33
◆ If elements of a row or a column in a determinant can be expressed as sum
of two or more elements, then the given determinant can be expressed as
sum of two or more determinants.
If to each element of a row or a column of a determinant the equimultiples of
corresponding elements of other rows or columns are added, then value of
determinant remains same.
7. ◆ Area of a triangle with vertices (x1
, y1
), (x2
, y2
) and (x3
, y3
) is given by
2 2
2
x3 y3
x1 y1 1
1
x y 1
1
Minor of an element aij of the determinant of matrix A is the determinant
ij
obtained by deleting ith
row and jth
column and denoted by M .
Cofactor of a of given by A = (– 1)i + j
M
ij ij ij
◆
◆
◆ Valueof determinantof a matrixAis obtainedby sum of productof elements
of a row (or a column) with corresponding cofactors. Forexample,
A = a11 A11 + a12 A12 + a13 A13.
◆ If elements of one row (or column) are multiplied with cofactors of elements
of any other row (or column), then their sum is zero. For example, a11 A21 + a12
A + a A = 0
◆ a a
22 13 23
a11 a12 a13
,
If A a
21 22 23
a31 a32 a33
12 22
A11 A21 A31
32
A13 A23 A33
ij
then adj A A A A , where A is
cofactor of aij
◆
◆
◆
A (adj A) = (adj A) A = |A| I, where A is square matrix of order n.
A square matrix A is said to be singular or non-singular according as
|A| = 0 or |A| 0.
If AB = BA = I, where B is square matrix, then B is called inverse of A.
Also A–1 = B or B–1 = A and hence (A–1)–1 = A.
A square matrixAhas inverse if and only ifA is non-singular.
A–1
1
(adj A)
A
◆
◆
◆ If a1 x + b1 y + c1 z = d1 a2 x
+ b2 y + c2 z = d2 a3 x +
b3 y + c3 z = d3,
b3
then these equations can be written as A X = B, where
a1 b1 c1 x d1
b c ,X= y and B= d
A a 2 2
z
2
a3 c3
2
d3
8. ◆ Unique solution of equation AX = B is given by X = A–1 B, where A 0.
◆ A system of equation is consistent or inconsistent according as its solution
exists or not.
◆ For a square matrixA in matrix equationAX = B
(i) |A| 0, there exists unique solution
(ii) |A| = 0 and (adj A) B 0, then there exists no solution
(iii)|A| = 0 and (adj A) B = 0, then system may or may not be consistent.
9. .
5.Continuity and
Differentiability
◆ A real valued function is continuous at a point in its domain if the limit of the
function at that point equals the value of the function at that point. A function
is continuous if it is continuous on the whole of its domin.
◆ Sum, difference, product and quotient of continuous functions are continuous.
i.e., if f and g are continuous functions,then
(f ± g) (x) = f (x) ± g (x) is continuous.
(f . g) (x) = f (x) . g (x) is continuous.
g(x)
f
(x)
f (x)
(wherever g (x) 0) is continuous.
g
◆ Every differentiable function is continuous, but the converse is not true.
◆ Chain rule is rule to differentiate composites of functions. If f = v o u, t = u (x)
and if both
dt
and
dv
existthen
dx dt
df
dv
dt
dx dt dx
◆ Following are some of the standard derivatives (in appropriate domains):
11. Alternatively, if f (x) 0 for each x in (a, b)
(b) decreasing on (a,b) if
x < x in (a, b) f (x ) f (x ) for all x , x (a, b).
1 2 1 2 1 2
(c) constant in (a, b), if f (x) = c for all x (a, b), where c is a constant.
◆ The equation of the tangent at (x0
, y0
) to the curve y = f (x) is given by
y y dy
dx
0 0
(x0 ,y0 )
(x x )
◆ dx 0 0
dy
If does not exist at the point ( x , y ) , then the tangent at this point is
parallel to the y-axis and its equation is x = x0.
◆ 0
0
dxx x
If tangent to a curve y = f (x) at x = x is parallel to x-axis, then
dy
0.
◆ Equation of the normal to the curve y = f (x) at a point ( x0 , y0 ) is given by
1
0 0
dx (x , y )
y y0
dy
(x x0 )
◆ dx 0 0 0
dy
If at the point ( x , y ) is zero, then equation of the normal is x = x.
dx 0 0
◆ If
dy
at the point ( x , y ) does not exist, then the normal is parallel tox-axis
and its equation is y = y0.
◆ Let y = f (x), x be a small increment in x and y be the increment in y
corresponding to the increment in x, i.e., y = f (x + x) – f (x). Then dy
givenby
dy f (x)dxor dy
dy
x .
◆
dx
isa good approximationofy when dx x is relativelysmalland we denote
it by dy y.
A point c in the domain of a function f at which either f (c)= 0 or f is not
differentiable is called a critical point of f.
12. ◆ First Derivative Test Let f be a function defined on an open interval I. Let
f be continuous at a critical point c in I. Then
(ii)
(iii)
(i) If f (x)changes sign from positive to negative as x increases through c,
i.e., if f (x)> 0 at every point sufficiently close to and to the left of c,
and f (x) < 0 at every point sufficiently close to and to the right of c,
then c is a point of local maxima.
If f (x)changes sign from negative to positive as x increases through c,
i.e., if f (x)< 0 at every point sufficiently close to and to the left of c,
and f (x) > 0 at every point sufficiently close to and to the right of c,
then c is a point of local minima.
If f (x)does not change sign as x increases through c, then c is neither a
point of local maxima nor a point of local minima. Infact, such a point
is called point of inflexion.
◆
◆
Second Derivative Test Let f be a function defined on an interval I and
c I. Let f be twice differentiable at c. Then
(i) x = c is a point of local maxima if f (c)= 0 and f (c) < 0
The values f (c) is local maximum value of f .
(ii) x = c is a point of local minima if f (c)= 0 and f (c) > 0
In this case, f (c) is local minimum value of f.
(iii) The test fails if f (c)= 0 and f (c) = 0.
In this case, we go back to the first derivative test and find whether c is
a pointof maxima,minimaor a pointof inflexion.
Working rulefor findingabsolutemaxima and/orabsoluteminima
Step 1: Find all critical points of f in the interval, i.e., find points x where
either f (x)= 0 or f is not differentiable.
Step 2:Take the end points of theinterval.
Step 3:At all these points (listed in Step 1 and 2), calculate the values of f .
Step 4: Identify the maximum and minimum values of f out of the values
calculated in Step 3. This maximum value will be the absolute maximum
value of f and the minimum value will be the absolute minimum value of f .
14. Some standard integrals
(i) n
x dx
xn1
n1
C , n – 1. Particularly, dx x C
(ii) (iii)
(iv) (v)
sin x dx – cosx C
cosec2
x dx – cot x C
(vi)
(vii)
cos x dx sin x C
sec2
x dx tan x C
sec x tan x dx secx C
cosec x cot x dx – cosec x C (viii)
1 x2
dx
sin1
x C
(ix)
1 x2
dx
cos1
x C (x)
1 x2
dx
tan 1
x C
(xi)
1 x2
dx
cot 1
x C (xii) ex
dx ex
C
(xiii)
x ax
loga
C
a dx (xiv) sec 1
x C
(xv)
dx
cosec 1
x C (xvi)
dx
x x2
1
1
x
dx log| x | C
x x2
1
Integration by partial fractions
Recall that a rational function is ratio of two polynomials of the form P(x) ,
Q(x)
where P(x) and Q (x) are polynomials in x and Q (x) 0. If degree of the
polynomial P (x) is greater than the degree of the polynomial Q (x), then we
may divide P (x) by Q (x) so that
P(x)
T (x)
P1(x)
, where T(x) is a
Q(x) Q(x)
polynomial in x and degree of P1 (x) is less than the degree of Q(x). T(x)
being polynomial can be easily integrated.
P (x)
1
Q(x)
can be integratedby
15. expressing
P1(x)
Q(x)
as the sum of partial fractions of the following type:
1. =
A B
x a x b
, a b
2. 2
px q
(x a) (x b)
px q
(x a)
= 2
A B
x a (x a)
3. =
A
B
C
x a x b x c
4. =
5.
px2
qx r
(x a) (x b) (x c)
px2
qx r
(x a)2
(x b)
px2
qx r
(x a) (x2
bx c)
=
A
B
C
x a (x a)2
x b
A
Bx +C
x a x2
bx c
where x2 + bx + c can not be factorised further.
Integration by substitution
A change in the variableof integrationoften reduces an integralto one of the
fundamentalintegrals. The method in which we change the variable to some
othervariableiscalledthemethod ofsubstitution.Whentheintegrandinvolves
some trigonometricfunctions, we use some well known identitiesto find the
integrals. Using substitution technique, we obtain the following standard
integrals.
(ii) cot x dx log sin x C
(i)
(iii)
(iv)
tan x dx log secx C
sec x dx log secx tan x C
cosecx dx log cosecx cot x C
Integrals of some special functions
(i) log
dx 1 x a
C
x2
a2
2a x a
(ii) log
dx 1 a x
C
a2
x2
2a a x
(iii)
x
a
1
tan C
dx 1
x2
a2
a
16. (iv)
x2
a2
dx
log x x2
a2
C (v)
a
a2
x2
dx
sin 1 x
C
(vi)
x2
a2
dx
log |x x2
a2
| C
Integration by parts
1
For given functions f and f , we have
2
, i.e., the
integral of the product of two functions = first function × integral of the
second function – integral of {differential coefficient of the first function ×
integral of the second function}. Care must be taken in choosing the first
function and the second function. Obviously, we must take that function as
the second function whose integral is well known to us.
ex
[ f (x) f
(x)] dx ex
f (x) dx C
Some special types of integrals
2 2 2 2 2 2
x a2
x a dx x a log x x a C
2 2
(i)
(ii)
2 2 2 2 2 2
x a2
x a dx x a log x x a C
2 2
(iii) 2 2 2 2
x a2
1 x
a x dx a x sin C
2 2 a
(iv) Integrals of the types
ax2
bx c ax2
bx c
dx
or
dx
can be
transformed into standard form by expressing
b c b 2
c b2
x
ax2
+ bx + c = a x2
a x
a a 2a a 4a2
px qdx
or
px q dx
ax bx c
(v) Integrals of the types 2
ax2
bx c
can be
17. b
x
x
transformed into standard form byexpressing
px q A
d
(ax2
bx c) B A (2ax b) B , whereAand B are
dx
determined by comparing coefficients on both sides.
We have defined a
f (x) dx as the area of the region bounded by thecurve
y = f (x), a x b, the x-axis and the ordinates x = a and x = b. Let x be a
given point in [a, b]. Then a
f (x) dx represents the Area function A (x).
This concept of area functionleads to the FundamentalTheorems of Integral
Calculus.
First fundamental theorem of integral calculus
Let the area function be defined by A(x) = a
f (x) dx for all x a, where
the function f is assumed to be continuous on [a, b].ThenA
(x) = f (x) for all
x [a, b].
Second fundamental theorem of integral calculus
Let f be a continuous function of x defined on the closed interval [a, b] and
dx
d
let F be another functionsuch that F(x) f (x) for all x in the domain of
b b
a
a
f, then f (x) dx F(x) C F (b) F (a) .
This is called the definite integral of f over the range [a, b], where a and b
are called the limits of integration, a being the lower limit and b the
upperlimit.
18. 8.Application of Integrals
b b
d d
The area of the region bounded by the curve y = f (x), x-axis and the lines
x = a and x = b (b > a) is given by the formula: Area a
ydx a
f (x)dx .
The area of the region bounded by the curve x = (y), y-axis and the lines
y = c, y = d is given by the formula: Area c
xdy c
(y)dy .
The area of the region enclosed between two curves y = f (x), y = g (x) and
the lines x = a, x = b is given by theformula,
b
a
Area f (x) g(x) dx , where, f(x) g (x) in [a, b]
If f (x) g (x) in [a, c] and f (x) g (x) in [c, b], a < c < b, then
c b
a c
Area f (x) g(x) dx g(
x) f (x)dx.
19. dx dy
dx
g (x, y) where, f (x, y) and g(x, y) are homogenous
An equation involving derivatives of the dependent variable with respect to
independent variable (variables) is known as a differential equation.
Order of a differential equation is the order of the highest order derivative
occurringin the differentialequation.
Degree of a differential equation is defined if it is a polynomial equation in its
derivatives.
Degree (when defined) of a differential equation is the highest power (positive
integer only) of the highest order derivative in it.
A function which satisfies the given differential equation is called its solution.
The solution which contains as many arbitrary constants as the order of the
differential equation is called a general solution and the solution free from
arbitraryconstants is called particularsolution.
To form a differential equation from a given function we differentiate the
function successively as many times as the number of arbitrary constants in
the given function and then eliminate the arbitrary constants.
Variable separable method is used to solve such an equation in which variables
can be separated completely i.e. terms containing y should remain with dy
and terms containing x should remain withdx.
A differential equation which can be expressed in the form
dy
f (x, y) or
functions of degree zero is called a homogeneous differential equation.
A differentialequationofthe form
dy
+Py Q , where P and Q are constants
dx
or functions of x only is called a first order linear differential equation.
9. Differential Equations
20. 10.Vector Algebra
Positionvectorof a pointP(x, y, z) is given as ,andits
magnitude by x2
y2
z2 .
The scalar components of a vector are its direction ratios, and represent its
projections along the respective axes.
The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of
any vector are related as:
l
a
, m
b
, n
c
r r r
21. , then their cross product is
If is the angle between two vectors
given as
where nˆ is a unitvectorperpendicularto the planecontaining .Such
that form right handed system of coordinate axes.
If we have two vectors , given in component form as
and any scalar,
then = (a1 b1)iˆ (a2 b2 ) ˆj (a3 b3) kˆ;
= (a1 )iˆ (a2 ) ˆj (a3 )kˆ;
= a1b1 a2b2 a3b3 ;
and =
ˆj kˆ
iˆ
a1 b1 c1 .
a2 b2 c2
22. 12. Linear
Programming
A linear programming problemis one that is concerned with finding the optimal
value (maximum or minimum) of a linear function of several variables (called
objective function) subject to the conditions that the variables are
non-negative and satisfy a set of linear inequalities (called linear constraints).
Variables are sometimes called decision variables and are non-negative.
A few important linearprogramming problems are:
(i) Dietproblems
(ii) Manufacturingproblems
(iii) Transportationproblems
The common region determined by all the constraints including the non-negative
constraints x 0, y 0 of a linear programming problem is called the feasible
region (or solution region) for the problem.
Points within and on the boundary of the feasible region represent feasible
solutions of the constraints.
Any point outside the feasible region is an infeasible solution.
23. Any point in the feasible region that gives the optimal value (maximum or
minimum) of the objective function is called an optimal solution.
The following Theorems are fundamental in solving linear programming
problems:
Theorem 1 Let R be the feasible region (convex polygon) for a linear
programming problem and let Z = ax + by be the objective function. When Z
has an optimal value (maximum or minimum), where the variables x and y
are subject to constraints described by linear inequalities, this optimal value
must occur at a corner point (vertex) of the feasible region.
Theorem 2 Let R be the feasible region for a linear programming problem,
and let Z = ax + by be the objective function. If R is bounded, then the
objective function Z has both a maximum and a minimum value on R and
each of these occurs at a corner point (vertex) of R.
If the feasible region is unbounded, then a maximum or a minimum may not
exist. However, if it exists, it must occur at a corner point of R.
Corner point method for solving a linear programming problem. The method
comprisesof the following steps:
(i) Findthefeasibleregionofthelinearprogrammingproblemanddetermine
its corner points (vertices).
(ii) Evaluate the objective function Z = ax + by at each corner point. Let M
and m respectively be the largest and smallest values at thesepoints.
(iii) If thefeasibleregionisbounded,M andm respectivelyarethemaximum
and minimumvaluesof theobjectivefunction.
If the feasible region is unbounded, then
(i) M is the maximum value of the objective function, if the open half plane
determined by ax + by > M has no point in common with the feasible
region. Otherwise, the objective function has no maximum value.
(ii) m is the minimum value of the objective function, if the open half plane
determined by ax + by < m has no point in common with the feasible
region. Otherwise, the objective function has no minimum value.
If two corner points of the feasible region are both optimal solutions of the
same type, i.e., both produce the same maximum or minimum, then any point
on the line segment joining these two points is also an optimal solution of the
same type.
24. 13. Probability
The salient features of the chapter are –
Theconditionalprobabilityof an eventE, given theoccurrenceof the eventF
is given by P(E | F)
P(E F) , P(F) 0
P(F)
0 P (E|F) 1, P (E|F)= 1 – P (E|F)
P ((E F)|G) = P (E|G) + P (F|G) – P ((E F)|G)
P (E F) = P (E) P (F|E), P (E) 0
P (E F) = P (F) P (E|F), P (F) 0
If E and F are independent, then
P (E F) = P (E) P (F)
P (E|F) = P (E), P (F) 0
P (F|E) = P (F), P(E) 0
Theorem of total probability
Let {E1, E2, ...,En) be a partition of a sample space and suppose that each of
E1
, E2
, ..., En
has nonzero probability. Let A be any event associated with S,
then
P(A) = P(E1) P (A|E1) + P (E2) P (A|E2) + ... + P (En) P(A|En)
Bayes' theorem If E , E , ..., E are events which constitute a partition of
1 2 n
sample space S, i.e. E , E , ..., E are pairwise disjoint and E E ... E = S
1 2 n 1 2 n
andA be any event with nonzero probability, then
P(Ei )P(A|Ei )
i n
P(E | A)
P(E j )P(A|E j )
j1
A random variable is a real valued function whose domain is the sample
space of a random experiment.
The probability distribution ofa random variableX is the system of numbers
X : x1
P(X) : p1
x
2
p2
...
...
x
n
pn
n
where, pi 0, pi 1, i 1,2,...,n
i1
25. Let X be a random variable whose possible values x , x , x , ..., x occur with
1 2 3 n
1 2 3 n
probabilities p , p , p , ... p respectively. The mean of X, denoted by , is
n
the number xi pi .
i1
The mean of a randomvariableX is alsocalledtheexpectationof X, denoted
by E (X).
Let X be a random variable whose possible values x1, x2, ..., xn occur with
1 2 n
probabilities p(x ), p(x ), ..., p(x ) respectively.
Let = E(X) be the mean of X. The variance of X, denoted by Var (X) or
x
2
, is defined as
2
or equivalently = E (X –) 2
x
The non-negativenumber
is called the standard deviation of the random variable X.
Var (X) = E (X2) – [E(X)]2
Trials of a random experiment are called Bernoulli trials, if they satisfy the
following conditions:
(i) There should be a finite number of trials.
(ii) The trials should be independent.
(iii) Each trial has exactly two outcomes : success or failure.
(iv) The probability of success remains the same in eachtrial.
For Binomial distribution B (n, p), P (X = x) = nC q n–x px, x = 0, 1,..., n
x
(q = 1 – p)