Mathematics in Music is fascinating. The world of Korvais are intriguing and interesting. A short but extremely insightful introduction into this world has been shared in this presentation.
Chitravina Ravikiran- SeamLess korvais - Music Academy Lec-Dem - 2014sowmya acharya
This document discusses different types of seamless korvais in Carnatic music. It begins by defining korvais and describing different levels of complexity. It then discusses natural, seamless korvais that have elegant mathematical patterns not found in textbooks. Several examples of seamless korvais are provided, including those with progressive, dovetailing, boomerang, and 3-speed patterns. The document also discusses "keyless korvais" that have aesthetic simplicity despite lacking clear mathematical relationships. Finally, it explores extensions of seamless korvais concepts to other talas like Khanda Triputa and Mishra Chapu.
This document provides a review for algebra sections on various topics including:
1) Solving word problems involving equations with one unknown variable.
2) Finding sums, differences, and ratios of numbers.
3) Representing word problems using tables and equations.
4) Solving uniform motion problems using tables, diagrams, and the appropriate equation based on the type of motion described.
5) Graphing linear equations and finding slopes of lines from equations or two points.
6) Writing equations of lines in different forms given information like slopes, intercepts, or two points.
The document discusses algorithms for finding shortest paths between all pairs of vertices in a directed graph, including:
- Floyd-Warshall algorithm, which uses dynamic programming and matrix multiplication to compute the shortest paths matrix in O(V^3) time.
- Johnson's algorithm, which first reweights the graph to make all edge weights nonnegative, allowing it to use Dijkstra's algorithm repeatedly to solve the all-pairs shortest paths problem more efficiently for sparse graphs.
- Reweighting transforms the original graph in a way that preserves shortest path distances while ensuring nonnegative edge weights.
The document discusses sample questions from TCS recruitment tests. It provides details about the new test pattern introduced in 2010, including that it is computer-based, has 35 questions to be answered in 60 minutes, and excludes English questions. The questions are divided into four categories: quantitative ability, logical reasoning, analytical reasoning, and reading comprehension. Several sample quantitative questions are then presented covering topics like trigonometry, probability, linear equations, and more. Step-by-step solutions are provided for each sample question.
1. The document is a sample paper for a mathematics class consisting of 26 questions divided into 3 sections - A, B, and C. Section A has 6 one-mark questions, Section B has 13 four-mark questions, and Section C has 7 six-mark questions.
2. The paper tests topics like trigonometry, calculus, matrices, and probability. It involves evaluating expressions, solving equations, proving identities, finding integrals, and applying mathematical concepts to word problems.
3. Students are instructed to answer all questions, show working, and choose one alternative for internal options. Use of calculators is not allowed.
Chitravina Ravikiran- SeamLess korvais - Music Academy Lec-Dem - 2014sowmya acharya
This document discusses different types of seamless korvais in Carnatic music. It begins by defining korvais and describing different levels of complexity. It then discusses natural, seamless korvais that have elegant mathematical patterns not found in textbooks. Several examples of seamless korvais are provided, including those with progressive, dovetailing, boomerang, and 3-speed patterns. The document also discusses "keyless korvais" that have aesthetic simplicity despite lacking clear mathematical relationships. Finally, it explores extensions of seamless korvais concepts to other talas like Khanda Triputa and Mishra Chapu.
This document provides a review for algebra sections on various topics including:
1) Solving word problems involving equations with one unknown variable.
2) Finding sums, differences, and ratios of numbers.
3) Representing word problems using tables and equations.
4) Solving uniform motion problems using tables, diagrams, and the appropriate equation based on the type of motion described.
5) Graphing linear equations and finding slopes of lines from equations or two points.
6) Writing equations of lines in different forms given information like slopes, intercepts, or two points.
The document discusses algorithms for finding shortest paths between all pairs of vertices in a directed graph, including:
- Floyd-Warshall algorithm, which uses dynamic programming and matrix multiplication to compute the shortest paths matrix in O(V^3) time.
- Johnson's algorithm, which first reweights the graph to make all edge weights nonnegative, allowing it to use Dijkstra's algorithm repeatedly to solve the all-pairs shortest paths problem more efficiently for sparse graphs.
- Reweighting transforms the original graph in a way that preserves shortest path distances while ensuring nonnegative edge weights.
The document discusses sample questions from TCS recruitment tests. It provides details about the new test pattern introduced in 2010, including that it is computer-based, has 35 questions to be answered in 60 minutes, and excludes English questions. The questions are divided into four categories: quantitative ability, logical reasoning, analytical reasoning, and reading comprehension. Several sample quantitative questions are then presented covering topics like trigonometry, probability, linear equations, and more. Step-by-step solutions are provided for each sample question.
1. The document is a sample paper for a mathematics class consisting of 26 questions divided into 3 sections - A, B, and C. Section A has 6 one-mark questions, Section B has 13 four-mark questions, and Section C has 7 six-mark questions.
2. The paper tests topics like trigonometry, calculus, matrices, and probability. It involves evaluating expressions, solving equations, proving identities, finding integrals, and applying mathematical concepts to word problems.
3. Students are instructed to answer all questions, show working, and choose one alternative for internal options. Use of calculators is not allowed.
This document provides a blueprint for the topics and questions that will be covered on a 12th grade mathematics pre-board exam. It lists the topics that will be covered in short answer (SA), long answer (LA), and very short answer (VSA) questions. The topics include relations and functions, matrices, determinants, continuity and differentiability, applications of derivatives, integrals, differential equations, vectors, and three dimensional geometry. It provides the number of questions at each mark value (1, 4, 6) for each topic. In total there will be 10 VSA questions worth 1 mark each, 12 SA questions worth 4 marks each, and 7 LA questions worth 6 marks each, for a total of 100 marks
The pattern of question paper in the subject Mathematics has been changed in CBSE,India.I am uploading the paper with marking scheme so that students will be benefitted-Pratima Nayak,KVS
The document contains 20 math assignment questions covering various topics:
- Solving systems of linear equations using substitution, elimination, and cross multiplication methods
- Solving pairs of linear equations and finding values of variables
- Finding values that satisfy or cause certain properties in systems of linear equations
- Solving quadratic and cubic polynomial equations
- Finding quadratic polynomials based on properties of their zeros
- Solving geometry problems using concepts like midpoints, centroids, and collinearity
- Calculating probabilities of outcomes in experiments involving balls, cards, dice, and coins
- Solving quadratic equations by finding discriminants and values that produce equal roots
- Solving word problems involving rates, speeds, mixtures, and geometric concepts
The document defines key terms related to algebraic forms, including:
- Algebraic form, term, variable, coefficient, like term, unlike term
It provides examples of algebraic expressions and their terms. Operations covered include addition, subtraction, simplification of algebraic expressions, substitution, and finding the lowest common multiple and highest common factor of algebraic expressions.
The document is the marking scheme for a mathematics exam consisting of 26 questions divided into 3 sections. Section A has 6 one-mark questions, Section B has 13 four-mark questions, and Section C has 7 six-mark questions. For questions involving calculus, the marking scheme provides the full working and steps to arrive at the solution. For other questions it states the final answer or shows a short reasoning to justify the answer. The marking scheme also sometimes explains the concepts involved in the question to help examiners understand the approach and marking.
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
This document contains multiple questions related to programming, geometry, probability, and other topics. It provides the questions, solutions, and explanations for the following:
1) A question about the number of lines of code that can be written by a certain number of programmers in a given time period.
2) A geometry question about the minimum number of "1-sets" (points separated from others by a line) for 5 points in a plane.
3) A question about the number of times a particular number appears when labeling items with a numeric system of a certain base.
4) A question regarding the angle between the hands of a clock on the planet Oz, which has different timekeeping conventions than
The document is a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. All questions are compulsory. The paper tests concepts related to matrices, trigonometry, calculus, differential equations, and vectors. Internal choices are provided in some questions. Calculators are not permitted.
This document contains a test with 26 multiple choice questions covering topics in mathematics, geometry, operations, and data interpretation from graphs and tables. The questions range in complexity and cover finding ratios, remainders, number of solutions to equations, rates, paths on grids, areas after operations, and interpreting data from tables and graphs.
A multiple choice problem consists of a set of color classes P = {C1 , C2 , . . . , Cn }. Each color class Ci consists of a pair of objects typically a pair of points. Objective of such a problem, is to select one object from each color class such that certain optimality criteria is satisfied. One example of such problem is rainbow minmax gap problem(RMGP). In RMGP, given P, the objective is to select exactly one point from each color class, such that the maximum distance between a pair of consecutive selected points is minimized. This problem was studied by Consuegra and Narasimhan. We show that the problem is NP-hard. For our proof we also describe an auxiliary result on satisfiability. A 3-SAT formula is an LSAT formula if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. We show that the problem of deciding whether an LSAT formula is satisfiable or not is NP-complete. We also briefly describe some approximation results of some multiple choice problems.
This document provides a module on functions and simultaneous equations for Additional Mathematics Form 4 students in Terengganu, Malaysia. It contains 15 problems on functions and 12 problems on simultaneous equations to help students prepare for their SPM examinations. The module is published by the Terengganu Education Department and involves several teachers from technical and science schools in the state.
1. The document contains 35 math problems involving matrices, determinants, vectors, trigonometry, calculus and their applications.
2. Key concepts covered include finding the inverse, determinant and adjoint of matrices; evaluating integrals; solving differential equations; and proving geometric and trigonometric identities.
3. The problems range from straightforward calculations to proofs requiring the use of matrix, vector and calculus properties.
451 sample questions 2 with some ans by rajanjsanket8989
The document contains a collection of math and logic problems with their step-by-step solutions. The problems cover a range of topics including ratios, ages, probabilities, geometry, and more. For each problem, the relevant information is presented, the steps to solve are shown, and the final answer is provided. The level of detail in the solutions allows readers to understand the reasoning behind arriving at the correct answers.
This document contains 83 mathematics problems ranging from algebra, trigonometry, calculus, probability, and statistics. The problems cover a variety of concepts including limits, derivatives, relations, expansions, inequalities, and geometry. They involve finding values, proving identities, solving equations, evaluating expressions, and more. The level of difficulty ranges from straightforward to more complex problems requiring multiple steps.
This document contains a set of 47 sample questions that cover various mathematical concepts such as number theory, probability, algebra, and geometry. The questions are intended to help a test taker prepare for an assessment by learning relevant concepts rather than focusing on specific questions. Each question is multiple choice with 4 possible answer options.
- 1-2 significant figures are used for things like grades, ages described in decades, cooking measurements, distances, areas, weights, temperatures.
- 3 significant figures are used for more precise measurements like heights, biological works, accurate measurements with a ruler.
- 4 or more significant figures are used for things like trigonometric ratios, logarithms, and highly precise scientific works.
- In general, fewer significant figures are used for less precise measurements and descriptions, while more are used for highly accurate scientific or technical contexts.
The document discusses systems of linear equations in two variables. It explains that two lines can intersect in 0, 1, or infinitely many points, corresponding to no solutions, exactly one solution, or infinitely many solutions (dependent system) for a system of two linear equations. Several examples are worked through, including inconsistent, dependent, and consistent systems. Matrices are introduced as another method for solving systems, but it is noted they have limitations.
This document contains a sample question paper for Class XII Mathematics. It has 5 sections (A-E). Section A contains 18 multiple choice questions and 2 assertion-reason questions worth 1 mark each. Section B has 5 very short answer questions worth 2 marks each. Section C contains 6 short answer questions worth 3 marks each. Section D has 4 long answer questions worth 5 marks each. Section E contains 3 case study/passage based questions worth 4 marks each with internal subparts. The document provides sample questions on topics including trigonometry, calculus, matrices, probability, linear programming and more.
This document provides a blueprint for the topics and questions that will be covered on a 12th grade mathematics pre-board exam. It lists the topics that will be covered in short answer (SA), long answer (LA), and very short answer (VSA) questions. The topics include relations and functions, matrices, determinants, continuity and differentiability, applications of derivatives, integrals, differential equations, vectors, and three dimensional geometry. It provides the number of questions at each mark value (1, 4, 6) for each topic. In total there will be 10 VSA questions worth 1 mark each, 12 SA questions worth 4 marks each, and 7 LA questions worth 6 marks each, for a total of 100 marks
The pattern of question paper in the subject Mathematics has been changed in CBSE,India.I am uploading the paper with marking scheme so that students will be benefitted-Pratima Nayak,KVS
The document contains 20 math assignment questions covering various topics:
- Solving systems of linear equations using substitution, elimination, and cross multiplication methods
- Solving pairs of linear equations and finding values of variables
- Finding values that satisfy or cause certain properties in systems of linear equations
- Solving quadratic and cubic polynomial equations
- Finding quadratic polynomials based on properties of their zeros
- Solving geometry problems using concepts like midpoints, centroids, and collinearity
- Calculating probabilities of outcomes in experiments involving balls, cards, dice, and coins
- Solving quadratic equations by finding discriminants and values that produce equal roots
- Solving word problems involving rates, speeds, mixtures, and geometric concepts
The document defines key terms related to algebraic forms, including:
- Algebraic form, term, variable, coefficient, like term, unlike term
It provides examples of algebraic expressions and their terms. Operations covered include addition, subtraction, simplification of algebraic expressions, substitution, and finding the lowest common multiple and highest common factor of algebraic expressions.
The document is the marking scheme for a mathematics exam consisting of 26 questions divided into 3 sections. Section A has 6 one-mark questions, Section B has 13 four-mark questions, and Section C has 7 six-mark questions. For questions involving calculus, the marking scheme provides the full working and steps to arrive at the solution. For other questions it states the final answer or shows a short reasoning to justify the answer. The marking scheme also sometimes explains the concepts involved in the question to help examiners understand the approach and marking.
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
This document contains multiple questions related to programming, geometry, probability, and other topics. It provides the questions, solutions, and explanations for the following:
1) A question about the number of lines of code that can be written by a certain number of programmers in a given time period.
2) A geometry question about the minimum number of "1-sets" (points separated from others by a line) for 5 points in a plane.
3) A question about the number of times a particular number appears when labeling items with a numeric system of a certain base.
4) A question regarding the angle between the hands of a clock on the planet Oz, which has different timekeeping conventions than
The document is a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. All questions are compulsory. The paper tests concepts related to matrices, trigonometry, calculus, differential equations, and vectors. Internal choices are provided in some questions. Calculators are not permitted.
This document contains a test with 26 multiple choice questions covering topics in mathematics, geometry, operations, and data interpretation from graphs and tables. The questions range in complexity and cover finding ratios, remainders, number of solutions to equations, rates, paths on grids, areas after operations, and interpreting data from tables and graphs.
A multiple choice problem consists of a set of color classes P = {C1 , C2 , . . . , Cn }. Each color class Ci consists of a pair of objects typically a pair of points. Objective of such a problem, is to select one object from each color class such that certain optimality criteria is satisfied. One example of such problem is rainbow minmax gap problem(RMGP). In RMGP, given P, the objective is to select exactly one point from each color class, such that the maximum distance between a pair of consecutive selected points is minimized. This problem was studied by Consuegra and Narasimhan. We show that the problem is NP-hard. For our proof we also describe an auxiliary result on satisfiability. A 3-SAT formula is an LSAT formula if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. We show that the problem of deciding whether an LSAT formula is satisfiable or not is NP-complete. We also briefly describe some approximation results of some multiple choice problems.
This document provides a module on functions and simultaneous equations for Additional Mathematics Form 4 students in Terengganu, Malaysia. It contains 15 problems on functions and 12 problems on simultaneous equations to help students prepare for their SPM examinations. The module is published by the Terengganu Education Department and involves several teachers from technical and science schools in the state.
1. The document contains 35 math problems involving matrices, determinants, vectors, trigonometry, calculus and their applications.
2. Key concepts covered include finding the inverse, determinant and adjoint of matrices; evaluating integrals; solving differential equations; and proving geometric and trigonometric identities.
3. The problems range from straightforward calculations to proofs requiring the use of matrix, vector and calculus properties.
451 sample questions 2 with some ans by rajanjsanket8989
The document contains a collection of math and logic problems with their step-by-step solutions. The problems cover a range of topics including ratios, ages, probabilities, geometry, and more. For each problem, the relevant information is presented, the steps to solve are shown, and the final answer is provided. The level of detail in the solutions allows readers to understand the reasoning behind arriving at the correct answers.
This document contains 83 mathematics problems ranging from algebra, trigonometry, calculus, probability, and statistics. The problems cover a variety of concepts including limits, derivatives, relations, expansions, inequalities, and geometry. They involve finding values, proving identities, solving equations, evaluating expressions, and more. The level of difficulty ranges from straightforward to more complex problems requiring multiple steps.
This document contains a set of 47 sample questions that cover various mathematical concepts such as number theory, probability, algebra, and geometry. The questions are intended to help a test taker prepare for an assessment by learning relevant concepts rather than focusing on specific questions. Each question is multiple choice with 4 possible answer options.
- 1-2 significant figures are used for things like grades, ages described in decades, cooking measurements, distances, areas, weights, temperatures.
- 3 significant figures are used for more precise measurements like heights, biological works, accurate measurements with a ruler.
- 4 or more significant figures are used for things like trigonometric ratios, logarithms, and highly precise scientific works.
- In general, fewer significant figures are used for less precise measurements and descriptions, while more are used for highly accurate scientific or technical contexts.
The document discusses systems of linear equations in two variables. It explains that two lines can intersect in 0, 1, or infinitely many points, corresponding to no solutions, exactly one solution, or infinitely many solutions (dependent system) for a system of two linear equations. Several examples are worked through, including inconsistent, dependent, and consistent systems. Matrices are introduced as another method for solving systems, but it is noted they have limitations.
This document contains a sample question paper for Class XII Mathematics. It has 5 sections (A-E). Section A contains 18 multiple choice questions and 2 assertion-reason questions worth 1 mark each. Section B has 5 very short answer questions worth 2 marks each. Section C contains 6 short answer questions worth 3 marks each. Section D has 4 long answer questions worth 5 marks each. Section E contains 3 case study/passage based questions worth 4 marks each with internal subparts. The document provides sample questions on topics including trigonometry, calculus, matrices, probability, linear programming and more.
This document contains a 3-page excerpt from the textbook "Elementary Mathematics" by W W L Chen and X T Duong. The excerpt discusses basic algebra concepts including:
- The real number system and subsets like natural numbers, integers, rational numbers, irrational numbers
- Rules of arithmetic operations like addition, subtraction, multiplication, division
- Properties of square roots
- Distributive laws for multiplication
- Arithmetic of fractions including addition and subtraction of fractions.
This document provides an outline of topics in algebra including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples and explanations for each topic.
This document contains 10 math questions related to trigonometric functions, sets, relations and functions, complex numbers, sequences and series, straight lines, conic sections. The questions range from proving identities and equations to finding specific values based on given information. They require various trigonometric, algebraic and geometric problem solving skills at a higher-order thinking level.
This document provides a review of various algebra topics including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples to practice each topic.
MATHEMATICS BRIDGE COURSE (TEN DAYS PLANNER) (FOR CLASS XI STUDENTS GOING TO ...PinkySharma900491
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This document provides an overview of topics covered in intermediate algebra revision including: collecting like terms, multiplying terms, indices, expanding single and double brackets, substitution, solving equations, finding nth terms of sequences, simultaneous equations, inequalities, factorizing common factors and quadratics, solving quadratic equations, rearranging formulas, and graphing curves and lines. The document contains examples and practice problems for each topic.
This document provides a review for algebra sections on various topics including:
1) Solving word problems involving equations with one unknown variable.
2) Finding sums, differences, and ratios of numbers.
3) Representing word problems using tables and equations.
4) Solving uniform motion problems using tables, diagrams, and the appropriate equation based on the type of motion described.
5) Graphing linear equations and finding slopes of lines from equations or two points.
6) Writing equations of lines in different forms given information like slopes, intercepts, or two points.
This document provides solutions to exercises from a complex analysis homework assignment. It includes:
1) Computing powers and roots of complex numbers using polar forms and identities
2) Sketching and describing properties of sets involving complex quantities like openness, boundedness, and being a domain
3) Finding and sketching images of complex mappings
The document summarizes the history of solving cubic equations and the discovery of complex numbers. It describes how del Ferro, Tartaglia, and Bombelli contributed to solving cubic equations of the form x3 = bx + c. Bombelli realized that taking the cube root of a negative number results in an "imaginary" number containing the square root of -1. He used this insight to find all three cube roots of a complex number. Modern algebra approaches finding the cube roots of a complex number using polar form and trisecting angles. This allowed solving any cubic equation with three real or complex solutions. The document traces the conceptual breakthroughs that led to the acceptance and understanding of imaginary and complex numbers.
The document discusses algebra of radicals. It provides rules for simplifying expressions involving radicals, such as √x·y = √x·√y and √x·√x = x. An example problem is worked through step-by-step, simplifying the expression 3√3 * √2* 2 * √2 * √3 * √2. The concept of conjugates is also introduced, where the conjugate of x + y is x - y.
- The document discusses expanding expressions by removing brackets.
- It provides examples of expanding expressions like 4(a + 2), 3x(x + 5), and -2a^2(a^3 - 3b^2 + 4c).
- It also discusses common mistakes made when expanding expressions, like incorrectly distributing negative signs, and provides examples to watch out for such mistakes.
The document provides examples and explanations of adding, subtracting, multiplying, and expanding polynomials. It demonstrates multiplying polynomials using the FOIL (First, Outer, Inner, Last) method and provides examples of sum and difference of squares, square of a binomial, cube of a binomial, and multiplying three binomials. Common patterns that arise when multiplying polynomials are identified.
1. The radius of curvature at a point on a curve is defined as the reciprocal of the curvature at that point. It represents the radius of the circle that best approximates the curve near that point.
2. For the circle x^2 + y^2 = 25, the radius of curvature at any point is equal to the radius of the circle, which is 25.
3. For the curve xy = c^2, the radius of curvature at the point (c, c) is c.
The document provides an introduction to basic algebra and graphing concepts for two students, Kaya and Zoey. It covers one new idea (using variables to represent unknown numbers) and four things to learn: substitution, combining, solving, and graphing. It defines key terms and provides examples of substituting values for variables, combining algebraic expressions, solving equations, and graphing points and lines. Higher-level concepts like graphing parabolas, circles, and three-dimensional graphs are also introduced.
The document provides an overview of basic algebra concepts including:
- What algebra is and how it uses variables and letters to represent unknown quantities and write rules in a general way.
- Common algebra terms like variables, constants, expressions, equations, and their definitions.
- How to work with expressions by combining like terms, distributing, factoring, and other operations.
- Examples of monomials, binomials, trinomials and polynomials.
- How to solve basic equations by isolating the variable and using properties of equality.
- Practice questions covering topics like identifying coefficients, combining like/unlike terms, evaluating expressions, and solving equations.
This document contains 11 math problems involving algebra concepts such as factorizing, solving equations, expanding expressions, and finding simultaneous solutions. The key details are:
1) Factorize quadratics and solve equations for variables.
2) Find the area of a circle given the radius.
3) Simplify algebraic expressions and set up equations to solve word problems.
4) Solve simultaneous equations to find values of variables.
5) Expand expressions and set up and solve equations to solve word problems involving ages multiplying to a given value in the future.
The document discusses simplifying expressions involving radicals. It provides examples of simplifying expressions using properties of radicals, such as √x∙y = √x∙√y and √x∙√x = x. One example simplifies the expression 3√3√2∙2√2√3√2 to 36√2. Another example simplifies the expression √12(√3 + 3√2) to 12√6.
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2. Prelude
Numbers have fascinated man for millennia.
India’s contributions in this area is mammoth in
general.
It is therefore unsurprising that Indian rhythm has
led the way in world music when it comes to
musical mathematics.
Even between Indian’s two major classical
systems, Carnatic culture stands out for not just
rhythmic virtuosity but in its sophisticated
approach towards structured mathematical
patterns.
3. Korvai Types
Level I: Any number taken after appropriate units
after samam to end as required. Ex: (3+3, 5+3, 7)x3
after 1. My very first attempt at a korvai at age 5…
Level II: Taken from samam to end at samam with
1 or 2 karvais between patterns to fill out the
remaining units (in say 32/64/28/40 units in Adi
1/2 kalais, Mishra chapu/Khanda chapu etc)
Level III: Same as II but to end a few units after or
before samam.
Level IV: Same as I or II but with different gatis
thrown in.
All these can be termed as man-made korvais
4. Natural Korvais – seamless elegance
Seamless korvai – DEFINITION: Patterns (of usually two or more parts)
from samam to samam/landing point of song that do not have remainder
indivisible by 3 in talas or landings indivisible by 3.
In other words, these do not have remainder of any number of units not
divisible by 3 (like 2 or 4) which have to be patched up as 1 or 2 karvais it in
between patterns. These have a grace or sophistication in the numbers that
are obvious only when one is inspired.
Intellectually, they require multi-layered thinking rather than just
conventional approaches.
Some of them involve precise and logical patterns but not found in
mathematical text books.
I literally stumbled upon most them as some of them are not accessible
through intuitive methods.
A couple of them have been in vogue for decades – 6, 8, 10 (or 8+8+8) as
first part then 3x5, 2x5, (1x5) x3.
Typically, they are in one gati though there are exceptions (but overuse of
multiple gati will make it a different concept.)
5. Seamless korvais – amazing options
ADI 2 kalais = 64 units
Challenges: To get 3 khandams (3x5) in Part B, Part A has to be
49. Similarly, for 3x6, 3x7 or 3x9 in B, we need A to be 46, 43 or
37, none of which is divisible by 3. So, simple approaches will
not work.
1. Simple progressive: These are most obvious
types. Ex 1: 7+3 (karvais), 6+3….. 1+3 as first part (A)
and 5x3 as the second part (B). Ex 2: 7+2…0+2 as A
and B is 5x3 in tishra gati. (A can also be in
srotovaha yati)
2.Progressive with addition in multiple parts:
A= (2,3,4)+(2,3,4,5)+(2,3,4,5,6); B=7x3.
6. Seamless korvais – amazing options
3. Inverted progression in 3 parts:
A=3,3,3 + 5X1, B= 3,3+7X2, C= 3+9X3
Another example:
A=2,2,3 + 7x1; B=2,3 + 7x2; C=3 + 7x3(tishra gati)
4. Progressive in second part: A= 6, 6, 6; B = (3x9) +
(2x7) + (1x5). Impressive when B is rendered 3 times
with A alternating between the 9, 7 and 5s.
5. Progressive in each part:
A = (5x3karvais)+(5x2karvais)+5x1;
B= (6x3karvais)+(6x2karvais)+6x2
C = (7x3karvais)+(7x2karvais)+7x3
7. Seamless korvais – amazing options
5. 3-speed korvais (example for 4 after samam):
(A=7, 2+7, 4+7; B=9x3)x3 karvais;
(A=7, 2+7, 4+7; B=9x3)x2 karvais;
A=7, 2+7, 4+7; B=9x3
Another example, employing second part
progression also (samam to samam):
6+2, 5+2, 4+2, 3+2, (3x5)x3; 6+2, 5+2, 4+2, 3+2,
(2x7)x3; 6+2, 5+2, 4+2, 3+2, (1x9)x3
8. Seamless korvais – dovetailing patterns
The beauty of these are part of A will dovetail into B in a
seamless manner.
(a) G,R,S,, R,S,N,, - G,R,SND - GR,S,, RS,N,, - GR,SND -
GRS,, RSN,, - GRSND RSNDP SNDPM
(b) GR, S, N, S,,, R,,, - GRSND – R,SN, S,,, R,,, - GRSND –
SN, S,,, R,,, - GRSND RSNDP SNDPM
(c) G,,,,, R,,,,, G,, R,, S,, N,, D,, - G,,, R,,, G, R, S, N, D, -
G,R, GRSND RSNDP SNDPM
(d) G,,, R,,, S,,, N,,, D – GRSND – R,, S,, N,, D,, P –
RSNDP – S,N,D,P,D - GRSND RSNDP SNDPM
It would be obvious that some are 13+5, 13+5 and 13+(3 times
5) in various ways. If song starts after +6, various manifestations
of 15+5, 15+5 and 15+(3 times 5) can be created.
9. Seamless korvais – Boomerang patterns
Let’s look at the sequence of numbers: (a) 7, 12, 15, 16….
(b) 6, 10, 12, 12... What are the next numbers?
Typically, these are not part of general math textbooks and do
not make sense to most mathematicians. But they are fine
examples of how Carnatic music can transcend science and
math. Remarkably, the series will turn back on itself. I call these
Double layered progressive sequences which boomerang. The
first few numbers are formed using multiplication progression
in (a) are: 7x1, 6x2, 5x3, 4x4. Thus, the next few numbers are
15, 12 and 7. Similarly, in (b), they are 10 and 6.
An example of a korvai with this: A= 6x2, 5x3, 4x4; B = 7x3
Another ex: A= 7, 12, 15, 16, 15, 12. B= 3 mishrams C= 3x10
(which can be said as ta.. Ti.. Ki ta. Tom (to give an illusion of 7)
10. The concept of Keyless korvais
At times, one stumbles upon korvais with no apparent mathematical
relationship. These cannot be logically deciphered or developed by locking
on to their key (usually the average of their various parts/2nd repeat out of
3). Yet, these are elegant beyond words in their simplicity.
1. A 3-part Korvai in 3 speeds: The amazing aesthetics of
this is mind-boggling – simple when rendered but looks a
jungle of numbers when expressed as below!
A = (8+3)x3 + (1x5)x3
B = (6+3)x2 + (2x7) x 2
C = (4+3)x1 + (3x9) x1
2. A 3-part korvai over 2 cycles (128 units): A stunning set of
patterns found in nature.
A= [(5+2), (4+2), (3+2)] + (3x5);
B = [(5+2), (4+2), (3+2), (2+2)] + (3x7), C = [(5+2), (4+2),
(3+2), (2+2), (1+2)] + (3x9).
11. Keyless korvais extensions to other talas
Keyless methods give scope to execute amazing finishes in
seemingly impossible situations. For instance, a tala like
Khanda Triputa @ 8 units per beat (72 units) or Rupakam,
which is already divisible by 3, can hardly offer scope for a
samam to + 2 or + 4 finish… Let’s look at a couple of
aesthetic solutions.
1. Khanda triputa – samam to +2 (out of 8) in 2 cycles
A= [(5+2), (4+2), (3+2), (2+2)] + (3x5), B = [(5+2), (4+2), (3+2),
(2+2), (1+2)] + (3x8), C = [(5+2), (4+2), (3+2), (2+2), (1+2), (0+2)]
+ (3x11).
2. A 3-part Korvai in 3 speeds for same landing as above
A = (11+3)x3 + (1x5)x3 (Can be rendered as G, R, GRSN
DPD N,, - GRSND in a raga like Vachaspati)
B = (9+3)x2 + (2x7) x 2
C = (7+3)x1 + (3x9) x1
12. Keyless korvais extensions to other talas
3. Khanda triputa – samam to +3 (out of 8)
[A= 7+3 (karvais), 6+3…..1+3, 0+3 B= 7x3] (To be rendered
3 times or change B as 5x3, 7x3 and 9x3 each time etc).
4. Mishra Chapu: Samam to -1
[(5x4)+1]x3, [(4x4)+1]x3, [(3x4)+1]x3, [(2x4)+1]x3,
[(1x4)+1]x3 (for landings like Suvaasita nava javanti in Shri
matrubhootam)
5. Roopakam: Samam to +2
A= [(5+2), (4+2), (3+2)] + (3x5), B = [(5+2), (4+2), (3+2),
(2+2)] + (3x9), C = [(5+2), (4+2), (3+2), (2+2), (1+2)] +
(3x13).
(The 3x(5/9/13) can be rendered as just 3x5 all 3 times. Or as
3x9, 3x13, 3x17 etc.
13. Seamless korvais for other talas
ADI 1 kalai (32 units)
Most korvais in this smaller space require patch work. Some
of the most famous ones are even mathematically incorrect.
(ta, tom… taka tom.. Takita tom.. + 3x5).
1. Simple progressive: A few years ago, I had introduced
A = 2, 3, 4, 5; B = 6x3.
2.Single part apparently wrong but actually correct
korvai:
GR,-GRS,-GRSN,-GRSNP,-GRSNPG,-GRSNPGR
Typical hearing will make it seem like 1+2 karvais… 5+2
karvais and final phrase illogically being 7. In reality, it is
2+1, 3+1…6+1 ending in 7.
14. Seamless korvais for other talas… contd
ADI 1 kalais = 32 units
3. An elegant solution in 3 cycles for songs starting after 6
(34 units/cycle)
A = (3x5) x3; B= (2x6)x3; C = (1x7) x3
4. Several other progressive solutions work beautifully
for samam to songs starting after 6:
7+7 (karvais), 6+7….2+7 +1 (landing on the song)
The same one can be rendered with 6 karvais for songs
starting on samam.
5. A simple 3-speed solution for 6 after samam:
A = (6x3 + 5x3)x3; B = (6x3 + 5x3)x2
C = (6x3 + 5x3)x1
15. Seamless korvais for other talas… contd
ADI 1 kalais = 32 units
6. A progressive 3-speed korvai for 6 after samam:
(7+7+3; 5)x3 karvais; (GR,S,N, DP,D,N, S,, - GRSND)X3
(6+6+3; 5)x2 karvais;
5+5+3; 5,5,5
Roopakam from samam to +3
A= [6, (2+6), (4+6)] B = (5 x 4 karvais + 3x5)
C = [6, (2+6), (4+6)] D = (7 x 4 karvais + 3x7)
E= [6, (2+6), (4+6)] B = (9 x 4 karvais + 3x9)
Note: A, C and E can be any combination divisible by 12
16. Seamless korvais in other gatis
Just as many korvais for Adi can be extended to other talas, they can be extended
to other gatis too. For instance, Adi - Khanda gati (double speed) = 80 units
Eg: GR, SN, DP, DN, S,, - G, R, SND – RS, ND, PM, P D, N,,- R,S |
,NDP – SN, DP, MG, MP, D,, - G | ,R,SND – R,S,NDP – S,N,DPM ||
But there are highly interesting possibilities which are
original for this like the one I had presented in my solo
concert at the Academy 2-3 years ago: A = (4x5) + (3x7) +
(2x9); B= 5+7+9
There is a lovely possibility in 3 gatis:
G,R, SN, S,, - GRSND (tishram)
GR, SN, S,, - G, R, SND (Chaturashram)
GRSN, S,, - G,R,GRSND – R,S,RSNDP – S,N,SNDPM
17. Seamless korvais with other approaches
I had remarked in a mrdanga arangetram about how most of our music is
elementary arithmetic and why percussionists must focus on aesthetics once they
have got the patterns right. This got me into thinking about experimenting with
korvais that represent some other math concepts such as a couple below:
1. Fibonachi series: Leonardo of Pisa, known as Fibonacci in
1200 AD but attributed to a much earlier Indian mathematician
Pingala (450-200 BC). The series is any two initial numbers like 3,
4 which are added to get 7. Now, add the last two numbers (4+7)
to get 11 and so forth. A korvai in that sequence (in say, Kalyani):
A = G,, - R,,, - G,R,SND – GRSNDPMGRSN – DN,R,, GM,D,, MD,N,, B=
G,R,SND – R,S,NDP – D,P,MGR
2. A simple korvai using squares of numbers as first part
(3)2+(4)2+(5)2:
A= G,,R,,S,, - G,,, R,,, S,,, N,,, - G,,,, R,,,, S,,,, N,,,, D,,,,
B= 3 mishrams in tishra gati double speed.
18. Creating Seamless korvais
It now would be obvious that anyone can create seamless
korvais with the thinking and methods I have shared.
I have used mostly familiar sounding easy patterns to create
these, mainly with melodic aesthetics in mind.
I have shown only a few small samples here, even from the
ones I have discovered/presented.
Pure rhythmic seamless korvais can deal with typical patterns
suited for percussion.
This is a vast exciting new world with tremendous scope to
expand the horizons both melodically and rhythmically.
Each door I’ve opened leads to exhilarating worlds…
Happy exploring!!!