TS SSC | Basics In Maths

Chapter 3 Linear Equations in two varaibles

Chapter 3 Pair of Linear Equations in Two Variables | NCERT Solutions

Chapter 3: Pair of Linear Equations in Two Variables | NCERT Solutions A pair of linear equations is the backbone of algebra and forms the foundation for many advanced mathematical concepts. This chapter delves into the pair of linear equations in two variables. This topic holds significant importance in the curriculum and has extensive real-world applications. Understanding the intricacies of solving these equations will enhance our analytical skills and enable us to confidently tackle complex mathematical problems Pair of Linear Equations in Two Variables Linear equations in two variables: An equation of the form ax + by + c = o where a, b, c are real numbers and (a2 + b2 ≠ 0) is called a linear equation in two variables. Two linear equations in two variables of the same type are called a pair of linear equations in two variables. a1x + b1y + c1 = 0 (a12 + b12 ≠ 0), a2x + b2 y + c2 = 0 (a22 + b22≠0); a1, a2, b1, b2, c1, c2 are real numbers A pair of two linear equations can be solved by four methods Graphical method Substitution method Elimination method Cross multiplication method Visit my YouTube channel: Click on the logo.

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Chapter 2 Polynomials Answers | NCERT Solutions

TS SSC / Nayini Satyanarayana Reddy

Chapter 2 Polynomials Answers | NCERT Solutions Chapter 2 Polynomials Solution polynomials are polynomials that describe the roots or solutions of a given polynomial. When you have a polynomial, finding polynomials solutions is often a crucial step in solving problems in various fields, including mathematics, physics, engineering, and economics. For example, consider the polynomial 𝑓(𝑥)=𝑎𝑥2+𝑏𝑥+𝑐=0 where a, b, and c are constants and x is the variable. The solutions to this quadratic polynomial, also known as the roots of the equation, can be found using various methods such as factoring, completing the square, or using the quadratic formula. The solution polynomial in this case would be a polynomial representing these roots. In general, the solution polynomial can have multiple terms depending on the original polynomial equation’s degree and its roots’ multiplicities. Solution polynomials are important because they provide a concise way to represent the solutions of polynomial equations, making it easier to analyze and understand the behavior of the equations in various contexts. They are beneficial in fields like control theory, where understanding the roots of polynomial equations helps in designing stable systems. When teaching polynomial solutions to students, it’s important to start with the basics and gradually increase the complexity as they become more comfortable with the concepts. Here’s a suggested approach: Chapter 2 Polynomials Introduction: Define what a polynomial is: an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Give examples of polynomials and non-polynomials. Introduce key terms like degree, leading coefficient, and standard form. Degree of Polynomials: Explain how to determine the degree of a polynomial. Practice identifying the degree of various polynomials. Operations with Polynomials: Addition and subtraction: Teach how to add and subtract polynomials by combining like terms. Multiplication: Show how to multiply polynomials using the distributive property and FOIL method. Division: Introduce polynomial long division and synthetic division for dividing polynomials Factoring Polynomials: Teach students various factoring techniques such as: GCF (Greatest Common Factor) factoring Factoring by grouping Difference of squares Factoring trinomials (e.g., using the AC method or trial and error) Emphasize the importance of factoring in finding polynomial solutions. Finding Solutions (Roots) of Polynomials: Introduce the concept of roots or zeros of polynomials. Explain how to find roots graphically and algebraically. EXERCISE 2.1 The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case. Solution: (i) From the graph, the graph of the polynomial is parallel to X – axis. It does not cut X – axis at any point ∴ the polynomial has no zeroes (ii) From the graph, the polynomial graph cuts the X – axis at only one point. ∴ the polynomial has one zero (iii) From the graph, the polynomial graph cuts the X – axis at three points. ∴ the polynomial has three zeroes (iv) From the graph, the polynomial graph cuts the X – axis at two points. ∴ the polynomial has two zeroes. (v) From the graph, the polynomial graph cuts the X – axis at four points. ∴ the polynomial has four zeroes (vi) From the graph, the polynomial graph cuts the X – axis at three points. ∴ the polynomial has three zeroes. Teach methods such as: Factoring to find roots Using the quadratic formula for quadratic polynomials Synthetic Division and the Remainder Theorem Rational root theorem and synthetic division for higher-degree polynomials Provide plenty of practice problems for students to apply these methods. Graphing Polynomials: Show how to graph polynomial functions using their roots and leading coefficients. Discuss end behavior and how it relates to the degree and leading coefficient of the polynomial. Real-World Applications: Illustrate how polynomials are used in various real-world scenarios such as finance, physics, and engineering. Provide examples and problems related to these applications to demonstrate the practical significance of polynomial solutions. Review and Practice: Regularly review previous topics and provide ample opportunities for students to practice solving problems involving polynomials. Offer additional resources such as worksheets, online practice problems, or interactive activities to reinforce learning. By following these steps and adjusting the pace and depth of instruction based on students’ comprehension and progress, you can effectively teach polynomial solutions to your students. EXERCISE 2.2 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) x2 – 2x – 8 (ii) 4s2 – 4s + 1 (iii) 6×2 – 3 – 7x (iv) 4u2 + 8u (v) t2 – 15 (vi) 3×2 – x – 4 Solution: (i) Given polynomial is x2 – 2x – 8 Let P (x) = x2 – 2x – 8 For the zeroes of the polynomial P (x) = 0 ⟹ x2 – 2x – 8 = 0 x2 – 4x + 2x – 8 = 0 x (x – 4) + 2(x – 4) = 0 (x – 4) (x + 2) = 0 x – 4 = 0 or x + 2 = 0 x = 4 or x = – 2 (ii) Given polynomial is 4s2 – 4s + 1 Let P (s) = 4s2 – 4s + 1 For the zeroes of the polynomial P (s) = 0 ⟹ 4s2 – 4s + 1 = 0 s2 – 2s – 2s + 1 = 0 s (2s – 1) – 1(2s – 1) = 0 (2s – 1) (2s– 1) = 0 2s – 1 = 0 or 2s – 1= 0 s = or s = let α = and β = (iii) Given polynomial is 6×2 – 3 – 7x Let P (x) = 6×2 – 7x – 3 For the zeroes of the polynomial P (x) = 0 ⟹ 6×2 – 7x – 3 = 0 6×2 – 9x + 2x – 3 = 0 3x (2x – 3) + 1(2x – 3) =

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Chapter 1 Real Numbers Answers | NCERT Solutions

TS SSC / Nayini Satyanarayana Reddy

Chapter 1 Real Numbers Answers | NCERT Solutions Chapter 1 Real numbers are a fundamental concept in mathematics that includes all rational and irrational numbers. Here’s a brief overview: Chapter 1 Real Numbers Definition: Real numbers encompass all numbers that can be located on the number line. They include rational numbers, which can be expressed as a fraction of integers, and irrational numbers, which cannot be expressed as fractions and have non-repeating, non-terminating decimal representations. Representation: Real numbers can be represented in various forms, including decimal notation, fraction notation, or as roots of algebraic equations. Properties: Closure under addition and multiplication: The sum or product of any two real numbers is also a real number. Commutativity and associativity: The addition and multiplication of real numbers are commutative and associative. Distributive property: For any real numbers a, b, and c, a(b + c) = ab + ac. Order property: Real numbers can be compared using the less than (<) or greater than (>) symbols, and they obey the trichotomy law, which states that for any two real numbers a and b, exactly one of the following holds: a < b, a = b, or a > b. Density property: Between any two distinct real numbers, there exists another real number. This property illustrates the continuous nature of the real number line. Operations: Arithmetic operations such as addition, subtraction, multiplication, and division are defined for real numbers, and they follow the usual rules of arithmetic. Solutions: Real numbers are often the solutions to various mathematical equations and inequalities. For example, in algebraic equations, real numbers are the values of the variable that make the equation true. Understanding real numbers and their properties is crucial in various fields of mathematics, including algebra, calculus, and analysis, as well as in many scientific and engineering applications. NCERT Solutions for Class 10 Mathematics Chapter 1 Real Numbers: Exercise 1.1: In this exercise, students will learn about Euclid’s division lemma, the Fundamental Theorem of Arithmetic, and the concept of the highest common factor (HCF). They’ll solve problems related to the division algorithm and the uniqueness of quotients and remainders. Exercise 1.2: Students will understand the concept of irrational numbers and the decimal representation of rational and irrational numbers. Problems involve proving that given numbers are irrational, expressing irrational numbers in decimal form, and finding rational approximations of irrational numbers. Exercise 1.3: This exercise focuses on operations on real numbers, such as addition, subtraction, multiplication, and division. Students will learn about the properties of rational numbers under these operations. Exercise 1.4: Here, students will explore rationalizing factors, rationalizing the denominator, and simplifying expressions involving surds. Exercise 1.5: The concept of laws of exponents for real numbers is discussed here. Students will learn about the laws of exponents and how they are applied to real numbers. Miscellaneous Exercises: These exercises cover a mix of topics from the chapter, including finding the HCF and LCM of given numbers, verifying the irrationality of numbers, simplifying surds, and applying the laws of exponents. Each exercise includes a variety of problems to ensure that students grasp the fundamental concepts of real numbers thoroughly. The solutions provide step-by-step explanations to aid in understanding and learning. Visit my YouTube channel: Click on the Logo

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TS 10th class maths concept and formulae

TS SSC / Nayini Satyanarayana Reddy

10th class math concepts and formulae 10th class maths concept 10th Class Maths: Are you a student gearing up for your 10th-grade mathematics exam? Perhaps you’re feeling a bit overwhelmed by the myriad of concepts and formulas you need to grasp. Fear not! In this blog post, we’ll break down some of the fundamental concepts and essential formulas you need to know to conquer your 10th-grade math with confidence. 10th Class Maths—Understanding the Basics Before diving into the specific formulas, it’s crucial to have a solid understanding of the basic concepts that form the foundation of 10th-grade mathematics. These include: Number Systems: Understanding real numbers, rational and irrational numbers, integers, and their properties is essential. Algebra: Knowing how to solve linear and quadratic equations, understanding polynomials, and mastering factorization techniques. 10th class maths concept Geometry: Familiarize yourself with concepts such as lines, angles, triangles, circles, and their properties. Trigonometry: Grasping trigonometric ratios, trigonometric identities, and their applications in solving triangles. Mensuration: Learning formulas for calculating the area, perimeter, and volume of various geometric shapes like rectangles, squares, circles, cylinders, and cones. Statistics and Probability: Understanding measures of central tendency, dispersion, and basic probability concepts. Essential Formulas Now, let’s delve into some of the key formulas you’ll encounter in your 10th-grade maths syllabus: Tips for Success Practice Regularly: Mathematics is a subject that requires consistent practice. Solve a variety of problems regularly to reinforce your understanding of concepts and formulas. Understand the Logic: Don’t just memorize formulas; understand the logic and reasoning behind them. This will help you apply them effectively in different problem-solving scenarios. Seek Clarification: If you encounter difficulties or have doubts about any concept or formula, don’t hesitate to seek help from your teachers, classmates, or online resources. X CLASS MATHS CONCEPTS AND FORMULAE BY SATYAM 2024 Create Summary Sheets: Summarize key formulas and concepts in a concise format for quick revision before exams. TS 10th class maths concept and formulae PPT Stay Calm and Confident: Approach your maths exams with a calm and confident mindset. Believe in your abilities and tackle each problem systematically. Conclusion Mastering 10th-grade mathematics requires a combination of understanding fundamental concepts and memorizing essential formulas. By familiarizing yourself with the concepts discussed in this blog post and practicing diligently, you’ll be well-equipped to tackle any maths problem that comes your way. Remember, consistency and perseverance are the keys to success in mathematics! Visit my YouTube channel: Click on the Logo

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TS 10th class maths model papers Get Ready for Success

Qestion Papers, TS SSC / Nayini Satyanarayana Reddy

TS 10th Class Maths Model Papers Get Ready for Success “TS 10th Class Maths Model Paper with Solutions” is not just a static document but a dynamic tool for learning and growth. Its carefully structured format encourages active engagement, enabling students to immerse themselves in the world of mathematics and cultivate a deeper appreciation for the subject. This model paper is designed to spark curiosity and encourage exploration, with each question serving as a gateway to new insights and discoveries. Whether grappling with algebraic equations, geometric theorems, or statistical analyses, students are challenged to think critically, reason logically, and apply their knowledge in innovative ways. Furthermore, the inclusion of detailed solutions fosters a collaborative learning environment, where students can learn from their mistakes, seek clarification, and refine their problem-solving techniques. By embracing a growth mindset and embracing the process of learning from both successes and setbacks, students can unleash their full potential and achieve academic excellence. As students journey through “TS 10th Class Maths Model Paper with Solutions 2024,” they not only enhance their mathematical proficiency but also cultivate essential life skills such as perseverance, resilience, and adaptability. Armed with this holistic approach to education, students are empowered to thrive not only in the examination hall but also in the ever-evolving landscape of the 21st century. Maths Model Papers—How to Prepare? Preparing for the TS 10th Class Math examination in 2024 requires a structured approach and effective study strategies. Here’s a guide to help you prepare: Understand the Syllabus: Familiarize yourself with the syllabus prescribed by the Telangana State Board for Class 10 Mathematics. Make sure you know all the topics and subtopics that are going to be covered in the exam. TS 10th Class Maths Model Paper – 1 2024 PDF Book Organize Your Study Material: Gather all the necessary textbooks, notes, and reference materials. Organize them according to the syllabus to make it easier to access and study. Create a Study Schedule: Develop a study timetable that allocates sufficient time for each topic based on its weightage in the exam and your understanding of the subject. Make sure to include regular breaks to avoid burnout. 10th class maths model papers Conceptual Understanding: Focus on understanding the concepts rather than rote memorization. Mathematics is a subject that requires a strong foundation of concepts. If you’re struggling with any concept, don’t hesitate to ask your teachers or seek online resources for clarification. Practice Regularly: Mathematics is all about practice. Solve as many problems as you can from different sources, including textbooks, previous year question papers, and sample papers. Practice different types of problems to get a thorough understanding of the concepts. Time Management: Practice solving problems within the stipulated time to improve your time management skills. This will be crucial during the exam when you have to answer questions within a specified time frame. Revise Regularly: Schedule regular revision sessions to reinforce your learning and to ensure that you retain the information for a longer duration. Use techniques like flashcards, summarization, and teaching others to aid in revision. TS 10th Class Maths Model Paper – 2 2024 PDF Book Mock Tests: Take mock tests to simulate exam-like conditions. This will help you familiarize yourself with the exam pattern, manage exam stress, and identify your strengths and weaknesses. Stay Healthy: Ensure you get adequate sleep, eat healthily, and exercise regularly. A healthy body leads to a healthy mind, which is essential for effective studying. Stay Positive: Maintain a positive attitude towards your studies. Believe in yourself and your abilities. Visualize success and stay motivated throughout your preparation journey. Remember, consistency and dedication are key to succeeding in any examination. Good luck with your preparations! maths model papers Practicing : Familiarity with Exam Pattern: By practicing model question papers, students become familiar with the exam pattern, format, and types of questions typically asked in the TS 10 maths exam. Time Management: Solving model question papers helps students improve their time management skills by simulating exam conditions and practicing efficient strategies for completing the paper within the allotted time. Identifying Strengths and Weaknesses: Analyzing responses to model papers enables students to identify their strengths and weaknesses in different topics. This allows them to focus their revision efforts on areas where they need improvement. TS 10th Class Maths Model Paper – 3 2024 PDF Book maths model papers Application of Concepts: Model question papers provide opportunities for students to apply the mathematical concepts they’ve learned in real-world contexts, helping them deepen their understanding and retention of the subject matter. Building Confidence: Regular practice with model papers boosts students’ confidence levels as they become more familiar with the types of questions asked and improve their ability to tackle them effectively. TS 10th Class Maths Model Paper – 4 2024 PDF Book Assessment and Feedback: After completing model papers, students can assess their performance and seek feedback from teachers or mentors to further refine their problem-solving skills and address any misconceptions. In summary, practicing TS 10 maths model question papers is an essential part of exam preparation as it enhances students’ understanding, performance, and confidence levels, ultimately leading to better outcomes in the actual exam. maths model papers Visit my YouTube channel: Click on the Logo

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TS 10th Real Numbers Previous Question Papers – Solutions

Education, TS SSC / Nayini Satyanarayana Reddy

Real Numbers Real Numbers 1. Insert 4 rational numbers between 1 and without using formula Sol: we have to insert 4 rational numbers between 1 and and 1 4 +1 = 5 1 can be written as Lcm of (4, 5) = 20 2. The prime factorisation of a natural number (n) is 23 × 32 × 52 × 7. How many consecutive zeros will it have at the end of it? Justify your answer. Sol: n = 23 × 32 × 52 × 7 = 2 × 22 × 32 × 52 × 7 = 2 × 32 × 7 (52 × 22) = 2 × 32 × 7 × (10)2 = 12600 ∴ n has 2 consecutive zeros at the end 3. Find the value of Sol: 4. Write any two irrational numbers between 3 and 4 Sol: 5. Find the value of 6. Find the HCF and LCM of 90, 144 by using the prime factorisation method Sol: 7. Is rational or irrational? justify your answer Sol: 8. Expand Sol: 9. Find the HCF of 24 and 33 by using division method Sol: 33 > 24 33 = 24 × 1 + 9 24 = 9 × 2 + 6 9 = 6 × 1 + 3 6 = 3 × 2 + 0 ∴ HCF of 24 and 33 = 3 10. Find the value of Sol: 11. Ramu says, “if = 0, the value of x is 0”. Do you agree with him? Give reason. Sol: Real Numbers 1. Write any three numbers of two digits. Find the LCM and HCF for the above numbers by the prime factorisation method. Sol: let 10, 12, 16 be three two-digit numbers Prime factarisation of 10 = 2 × 5 = 21× 51 Prime factarisation of 12 = 2 × 2× 3 = 22 × 3 Prime factarisation of 16 = 2 × 2× 2× 2 = 24 HCF of 10, 12 and 16 = 21 = 2 LCM of 10, 12 and 16 = 24 × 3× 5 = 8 × 3× 5 = 120 2. Give an example for each of the following (i) The product of two irrational numbers is a rational number (ii) The product of two irrational numbers is an irrational number Sol: 3. State with reasons which of the following are rational numbers and which are irrational numbers 4. Expand Visit My Youtube Channel: Click on below logo

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Pair Of linear equations in two variables Feature Image

TS 10th Chapter 4 Pair of Linear Equations in Two Variables

Education, TS SSC / Nayini Satyanarayana Reddy

TS 10th Chapter 4 Pair of Linear Equations in Two Variables TS 10th Chapter 4 Pair of Linear Equations in Two Variables, Parallel, coincident, and intersecting lines. Dependent and independents lines An equation of the form ax + by + c = 0 where a, b, c are real numbers and a2 + b2 ≠ 0 is called a linear equation in two variables x and y. Two linear equations in the same variables are called a pair of linear equations in two variables a1x + b1y + c1 = 0; a2x + b2y + c2 = 0 are the pair of linear equations in two variables in x and y. Solution of Pair of linear equations in two variables: For linear equations in two variables, there are infinitely many solutions. Ex: x + y = 10 x =7, y = 3; x = 6, y = 4; x = 1, y = 9; x =2, y = 8; x=3, y = 7 like we have infinitely many solutions. ⇰ For finding exact values of x and y we have to know two linear equations. ⇰ A pair linear equations in two variables solved by four methods Graphical method Substitution method Elimination method Cross – Multiplication method Solving the pair of linear equations in two variables by using graphical Method: 1. 2x + y −5 = 0, 3x – 2y − 4 = 0 After plotting the points in the above tables in Cartesian plane, we observe that two straight lines intersect at the point (2, 1) There is only one solution for this pair of linear equations in two variables. These equations are known as consistent pair of linear equations and they have a unique solution. 2. 2x – 3y = 15; 4x – 6y = 9 After plotting the points in the above tables in Cartesian plane, we observe that two straight lines are parallel There is no solution for this pair of linear equations in two variables. These equations are known as inconsistent pair of linear equations and they have no solution. 3. 3x + 4y = 2; 6x + 8y = 4 After plotting the points in the above tables in Cartesian plane, we observe that two straight lines are coincide There are infinitely many solutions for this pair of linear equations in two variables. These equations are known as consistent pair of linear equations and they have infinitely solution. Consistent and inconsistent: If the system of equations has a solution, then they are consistent. If the system of equations has no solution, then they are inconsistent. The relationship between coefficients and the nature of the equation system: Examples: 1. Draw the graph of the following pair of linear equations in two variables and find their solution from the graph 3x – 2y = 2 and 2x + y = 6 The two lines intersect at the point (2, 2) ∴ solutions is x = 2, y = 2 2. Draw the graph of the following pair of linear equations in two variables and find their solution from the graph x – 2y = –1 and 2x – y – 4 = 0 The two lines intersect at the point (3, 2) ∴ solutions is x = 3, y = 2 3. Represent the solution of linear equation graphically x – 2y = –3 and 2x + y = 4 The two lines intersect at the point (1, 2) ∴ solutions is x = 1, y = 2 Word problems: 1. Neha went to a sale to purchase some points and skirts. When her friend asked her how many of each she had bought, she answered “The number of skirts is two less than twice the number of points purchased. Also, the number of number of skirts is four less than four times the number of pants purchased”. Help her friend to find how many pants and skirts Neha bought. Sol: Let the number of points = x and the number of skirts = y the number of skirts is two less than twice the number points purchased ⟹ y = 2x – 2 the number of number of skirts is four less than four times the number of pants purchased ⟹ y = 4x – 4 The two lines intersect at the point (1, 0) ∴ solution is x = 1, y = 2 No. of Points = 1 and no. of skirts = 0 2. 10 students of a class X took part in a Mathematics quiz. If the no. of girls is 4 more than the no. of boys, then find the no, of boys and no. of girls who took part in the quiz. Sol: let the no. of boys = x No. of girls = y Total no. of students = 10 ⟹ x + y = 10 The no. of girls is 4 more than the no. of boys ⟹ y = x + 4 ⟹ x – y = – 4 The two lines intersect at the point (3, 7) ∴ solution is x = 3, y = 7 No. of boys = 3 and no. of girls = 7 3. Half the perimeter of a rectangular garden, whose length is 4m more than its width is 36 m. Find the dimensions of the garden. Sol: let the width of garden = x m. Length of garden = y m. Length of garden is 4m mote than its width ⟹ y = x + 4 x – y = – 4 half of the perimeter of rectangular garden is 36 m. ⟹ x + y = 36 ⟹ x – y = – 4 The two lines intersect at the point (16, 20) ∴ solution is x = 16, y = 20 Width of garden = 16 m and Length of garden = 20 m 4. The area of a rectangle gets reduced by 80

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TS 10th class maths concept (E/M)

Education, TS SSC / Nayini Satyanarayana Reddy

TS 10th class maths concept 10th Class Maths – Studying mathematics successfully means that children take responsibility for their own learning and learn to apply the concepts to solve problems. This note is designed by the ‘Basic In Maths’ team. These notes are to help students fall in love with mathematics and overcome fear. 1. REAL NUMBERS • Rational number: A number, which is written in the form of p/q, where p and q are integers, q is not equal to zero, is called a rational number. It is denoted by Q. • Irrational numbers:- the number, which is not rational, is called an irrational number. It is denoted by Q’ or S. • Euclid division lemma:- For any positive integers a and b, then q, r are integers exists uniquely satisfying the rules a = bq + r, 0 ≤ r < b. • Prime number:- A number that has only two factors, 1 and itself, is called a prime number. (2, 3, 5, 7 …. Etc.) • Composite number:- the number that has more than two factors is called a composite number. (4, 6, 8, 9, 10,… etc.) • Co-prime numbers:- Two numbers are said to be co-prime numbers if they have no common factor except 1. [Ex: (1, 2), (3, 4), (4, 7)…etc.] • To find HCF, LCM by using prime factorisation method: H. C.F= product of the smallest power of each common prime factor of given numbers. L.C.M = product of the greatest power of each prime factor of the given numbers. In p/q, if the prime factorisation of q is in the form 2m 5n, then p/q is a terminating decimal. Otherwise, non terminating repeating decimal. Decimal numbers with a finite number of digits are called terminating. Decimal numbers with an infinite number of digits are called non-terminating decimals. In a decimal, a digit or sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non-terminating repeating decimals. • ‘p’ is a prime number and ‘a’ is a positive integer, if p divides a2, then p divides a. • If ax = N then x = (i) log (xy) = log(x) + log(y) (ii) log (x/y) = log( x) – log( y) (iii) log (xm ) = m log (x) Pair of Linear Equations In two Variables Concept and Solutions: Click Here TS 10th class maths concept FOR MORE CONCEPT, click here for pdf file Visit My YouTube Channel: Click on the Logo

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TS 10th Class Maths Concept (T/M)

Education, TS SSC / Nayini Satyanarayana Reddy

10 వ తరగతి గణితం నోట్స్ 10 వ తరగతి గణితం 10 వ తరగతి గణిత శాస్త్రాన్ని అధ్యయనం చేయడం అంటే, పిల్లలు తమ స్వంత అభ్యాసానికి బాధ్యత వహిస్తారు మరియు సమస్యలను పరిష్కరించడానికి భావనలను వర్తింపజేయడం నేర్చుకుంటారు. ఈ విషయం . ఈ గమనికలు విద్యార్థులకు గణితంను ఇస్టపడేలా మరియు భయాన్ని అధిగమించడానికి సహాయపడతాయి. 1. వాస్తవ సంఖ్యలు మనం ముందు తరగతులలో వివిధ రకాలైన సంఖ్యలను గురించి తెలుసుకున్నాము .అంటే సహజ సంఖ్యలు, పూర్ణాంకాలు, పూర్ణ సంఖ్యలు, కరణీయ , అకరణీయ సంఖ్యలను గురించి నేర్చుకున్నాము . అకరణీయ సంఖ్యలు : p,q లు పూర్ణ సంఖ్య లై , q ≠ 0 అయిన సందర్భం లో రూపం లో రాయగల సంఖ్య లను అకరణీయ సంఖ్యలు అంటారు . దీనిని Q తో సూచిస్తారు . ఉదా :- మొదలగునవి. కరణీయ సంఖ్యలు : రూపం లో రాయలేని సంఖ్యలను కరణీయ సంఖ్యలు అంటారు . దీనిని QI లేదా S తో సూచిస్తారు . ఉదా :- మొదలగునవి. వాస్తవ సంఖ్యలు : అకరణీయ , కరణీయ సంఖ్యల సమూహాన్ని వాస్తవ సంఖ్యలు అంటారు . కింది పటములో మనం వీటిని చూడ వచ్చు. భాగహార శేష నిధి : a, b అనే ధన పూర్ణాంకాలు ఇచ్చినప్పుడు a = b q + r, 0≤ r <b అయ్యే విధంగా ఏకైక జత పూర్ణాంకాలు q ,r లు వ్యవస్తితం అవుతాయి. ఇది అందరికి తెలిసినప్పటికీ యూక్లిడ్ పుస్తకాల సంకలనం లోని 7 వ పుస్తకం లో మొట్టమొదటగా నమోదు చేయడం జరిగింది. ఈ భాగహార శేషనిధి మీద యూక్లిడ్ భాగహార శేష నిధి ఆధారపడి ఉంది. యూక్లిడ్ భాగహార శేషనిధి కేవలం ధన పూర్ణ సంఖ్యల పైనే నిర్వచించ బడినా , దానిని అన్ని శూన్యేతర పూర్ణ సంఖ్యలకు అనువర్తింప చేయవచ్చు . యూక్లిడ్ భాగహార శేషనిధి ఉపయోగించి గ . సా . భా ను కనుక్కోవడం : రెండు ధన పూర్ణ సంఖ్యల సామాన్య కారాణాంకాలలోని అతి పెద్ద కారణాo న్కాన్ని గ .సా. భా అంటారు . ఉదా:- 9 , 24 ల గ . సా .భా కనుక్కోవడం దీనిని 24 = 9×2 + 18 గా రాయవచ్చు 9 , 24 కన్నా పెద్దది కావున 24 ను 9 చే భాగిస్తే శేషం 6 వస్తుంది పై దానిలో ని భాజకం 9 మరియు 6 పై యూక్లిడ్ న్యాయాన్ని అనువర్తింప చేయగా 9 = 6 ×1 + 3 గా రాయవచ్చు పై దానిలో ని భాజకం 6 మరియు శేషం 3 పై యూక్లిడ్ న్యాయాన్ని అనువర్తింప చేయగా దానిని 6 = 3 ×2 + 0 గా రాయవచ్చు పై దాని లో శేషం సున్నా వచ్చింది కావున 9 , 24 ల గ . సా .భా 3 అవుతుంది. ప్రాథమిక అంకగణిత సిద్ధాంతం : ప్రతి సంయుక్త సఖ్యను ప్రదానానంకముల లబ్దంగా రాయవచ్చు మరియు ప్రధాన కారణాంకాల క్రమం ఏదైనప్పటికీ ఈ కారణాంకాల లబ్దం ఏకైకం . ఒక సంయుక్త సంఖ్య x ను x = p 1 p 2 ….p n అని రాయవచ్చు . దీనిలో p 1 , p 2, …., p n ఆరోహణ క్రమం లో రాయబడిన ప్రధానాంకాలు , అంటే p 1≤ p 2 ≤….≤ p n. ఈ సందర్భం లో ఒకే రకమైన ప్రదానంకములు వాడినచో వాటిని ప్రధానాంకాల ఘా తాoకాలుగా రాస్తాము . ఒకసారి మనం ఈ సంఖ్యలు ఆరోహణ క్రమంలో ఉన్నాయని భావిస్తే . అప్పుడు లబ్దం ఏకైకం . ఉదా :- 360 = 3×3×2× 2 × 2 × 5 = 32 × 23 × 5 ప్రధాన కారణాంకాల లబ్ద పద్ధతి ద్వారా గా. సా . భా మరియు కా . సా . గు కనుక్కోవడం; 9 , 24 ల గ . సా .భా మరియు కా. సా . గు. కనుక్కోవడం 9 యొక్క ప్రధాన కారణాంకాలు = 3 × 3 = 32 24 యొక్క ప్రధాన కారణాంకాలు = 2 × 2 ×2 × 3 = 23 ×31 9 , 24 ల గ . సా .భా = 31 = 3 ( సంక్యల యొక్క సామాన్య కారణాంకంల కనిష్ఠ ఘాతాల లబ్ధం ) 9 , 24 ల కా. సా . గు.= 32× 23 = 9×8 = 72 (సంఖ్యల యొక్క కారణాంకంల గరిష్ఠ ఘాతాల లబ్ధం) అకరణీయ సంఖ్యలు మరియు వాటి దశాంశ రూపాలు : x అనేది ఒక అకరణీయ సంఖ్య మరియు దీని ధశాంశ రూపం ఒక అంతమయ్యే దశాంశము ,అయినప్పుడు x ను p, q లు పరస్పర ప్రధా నాంకములు అయివున్న p /q రూపం లో వ్యక్త పరచవచ్చు . మరియు q యొక్క ప్రధాన కారాణాంకాల లబ్దం 2m 5 n అగును , n ,m లు ఋణేతర పూర్ణ సంఖ్యలు . పై దాని విపర్యయం ఇలా ఉంటెుంది • n ,m లు ఋణేతర పూర్ణ సంఖ్యలు మరియు q యొక్క ప్రధాన కారాణాంకాల లబ్దం 2m 5 n కలిగినటువంటి అకరణీయ సంఖ్య x = p /q అయిన, xయొక్క ధశాంశ రూపం ఒక అంతమయ్యే దశాంశము అగును , • n ,m లు ఋణేతర పూర్ణ సంఖ్యలు మరియు q యొక్క ప్రధాన కారాణాంకాల లబ్దం 2m 5 n రూపంలో లేకుంటే , అకరణీయ సంఖ్య x = p /q అయిన, xయొక్క ధశాంశ రూపం ఒక అంతంకాని దశాంశము అగును. ఉదా :- కరణీయ సంఖ్యలు :- • p, q లు కరణీయ సంఖ్యలు మరయు q ≠ 0 అయిన p /q రూపం లో రాయలేని సంఖ్యలను కరణీయ సంఖ్యలు అంటారు . • ప్రతీ కరణీయ సంఖ్య ధశాంశ రూపం ఒక అంతంకాని దశాంశము అగును. ప్రవచనం: p అనేది ఒక ప్రధాన సంఖ్య మరియు a ఒక ధనపూర్ణ సంఖ్య అయితే “ a2 ను p నిశ్శేషంగా భాగిస్తే a ను p నిశ్శేషంగాభాగిస్తుంది. ఘాతాలు : • a n ను ఘాతాంక రూపం అంటాము. a ను భూమి అని , n ను ఘాతము అని అంటారు. (i) (ii) (iii) ( am)n = amn (iv) a0 = 1 సంవర్గమానాలు:- x మరియు aలు ధనపూర్ణసంఖ్యలై a ≠1 అయివుండి ax = n అయిన x = అగును. 2. సమితులు • గణిత పరిశోధనలలో సమితి వాదాన్ని ‘ జార్జి కాంటర్’ అభివృద్ధి పరిచారు. సమితి: సునిర్విచిత వస్తువుల సముదాయాన్ని సమితి అంటారు. • సునిర్విచితం అనగా : 1 . సమితిలోని వస్తువులన్నిటికి ఒకే విధమైన సామాన్య పోలిక లేదా ధర్మం కలిగి ఉండాలి . 2 . ఏదైనా ఓకే సమితికి చెందినది, లేనిది నిర్దారించే టట్లు ఉండాలి. • సమితి పేరును ఇంగ్లీష్ వర్ణమాల లోని పెద్ద అక్షరాలతో సూచిస్తారు. ఉదాహరణకు A, B, … మొదలగునవి. • ఏదైనా ఓకే వస్తువు ఒక సమితికి చెందితే దాన్ని వస్తువులు/ మూలకాలు అంటారు . చెందినది (belongs to) అని తెలపటానికి మనం ∈ గుర్తు తో సూచిస్తాము.సమితికి చెందినది అయితే దానిని ∉ చే సూచిస్తాము. • జాబితా రూపం లేదా రోస్టర్ రూపం : సమితికి చెందిన మూలకాలన్నిటిని ‘కామ’ (,) తో వేరు చేసి ప్లవర్ బ్రాకెట్ { } లో ఉంచితే వచ్చే రూపాన్ని జాబితా రూపం లేదా రోస్టర్ రూపం అంటారు. ఉదా :- A = {1, 2, 3, 4}, B = { a, e, I, o, u}. • సమితి నిర్మాణ రూపం లేదా లాక్షణిక రూపం : సమితి లోని మూలకాన్ని x ( లేక y, z మొదలగు ఏవైన గుర్తులు ) గా సూచించి , x ప్రక్కన : లేదా / (colon ) ఉంచి ఆ సమితి కి చెందిన మూలకాల యొక్క లక్షణాలు లేదా ధర్మాలను రాసి ప్లవర్ బ్రాకెట్ { } ఉంచితే వచ్చే రూపాన్ని సమితి నిర్మాణ రూపం లేదా లాక్షణిక రూపం అంటారు . : లేదా / గుర్తులను such that అని చదువుతాము . ఉదా :- A = { x/ x ఒక సరి సంఖ్య మరియి x ∈N }, B = { y : y ఒక ప్రధాన సంఖ్య మరియు x < 10 }. సమితులు – రకాలు

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