Aspire Academy

Menu
BUY COURSES.
REGISTER/ LOGIN

Class 10 Maths

By admin Categories: Class 10
Wishlist

What Will You Learn?

  • Master your learning with fun.

Course Content

Class 10 Maths Video Lectures

  • Chapter 1 – Real Numbers Lecture – Natural Numbers
    31:15
  • Chapter 1 – Real Numbers Lecture – Euclid’s Division Lemma
    30:49
  • Chapter 1 – Real Numbers Lecture – Finding HCF using Euclid’s algorithm
    27:07
  • Chapter 1 – Real Numbers Lecture 4 – Fundamental Theorem of Arithmetic
    19:47
  • Chapter 1 – Real Numbers Lecture 5 – Prime numbers and their uses
    34:50
  • Chapter 1- Real Numbers Lecture 6 – Theorems of Integers
    25:37
  • Chapter 1- Real Numbers Lecture 7
    30:57
  • Chapter 1 – Real Numbers Lecture 8
    13:46
  • Chapter 2 – Polynomials Lecture 1
    33:41
  • Chapter 2- Polynomials Lecture 2
    31:07
  • Chapter 2 – Polynomials Lecture 3
    32:47
  • Chapter 2 – Polynomials Lecture 4
    31:36
  • Chapter 3 – Pair of Linear Equations in Two Variables Lecture 1
    34:59
  • Chapter 3 – Pair of Linear Equations in Two Variables Lecture 2
    30:05
  • Chapter 3 – Pair of Linear Equations in Two Variables Lecture 3
    35:17
  • Chapter 3 – Pair of Linear Equations in Two Variables Lecture 4
    34:38
  • Chapter 3 – Pair of Linear Equations in Two Variables Lecture 5
    27:09
  • Chapter 3 – Pair of Linear Equations in Two Variables Lecture 6
    32:36
  • Chapter 3 – Pair of Linear Equations in Two Variables Lecture 7
    35:58
  • Chapter 4 – Quadratic Equations – Lecture 1
    01:05:25
  • Chapter 4 – Quadratic Equations Lecture 2
    45:26
  • Chapter 4 – Quadratic Equations Lecture 3
    34:09
  • Chapter 4- Quadratic Equations Lecture 4
    30:03
  • Chapter 5 – Arithmetic Progressions Lecture 1
    35:31
  • Chapter 5- Arithmetic Progressions Lecture 2
    30:12
  • Chapter 5- Arithmetic Progressions Lecture 3
    32:34
  • Chapter 6 – Triangles Lecture 1
    38:41
  • Chapter 6 – Triangles Lecture 2
    31:49
  • Chapter 6 – Triangles Lecture 3
    34:23
  • Chapter 6 – Triangles Lecture 4
    33:25
  • Chapter 6 – Triangles Lecture 5
    00:00
  • Chapter 6 – Triangles Lecture 6
    31:20
  • Chapter 6 – Triangles Lecture 7
    34:03
  • Chapter 6 – Triangles Lecture 8
    38:24
  • Chapter 6 – Triangles Lecture 9
    00:00
  • Chapter 6 – Triangles Lecture 10
    00:00
  • Chapter 7- Co-Ordinate Geometry Lecture 1
    32:20
  • Chapter 7- Co-Ordinate Geometry Lecture 2
    00:00
  • Chapter 7- Co-Ordinate Geometry Lecture 3
    24:43
  • Chapter 7- Co-Ordinate Geometry Lecture 4
    28:24
  • Chapter 7- Co-Ordinate Geometry Lecture 5
    32:12
  • Chapter 7- Co-Ordinate Geometry Lecture 6
    37:49
  • Class 10 Maths: Chapter 8 – Introduction to Trigonometry
    31:04
  • Class 10 Maths Chapter 8: Introduction and Trigonometric ratios
    33:31
  • Class 10 Maths: Chapter 8 – Trigonometric ratios of some angles
    00:00
  • Class 10 Maths Chapter 8: Trigonometric ratios for Perpendicular Angles
    34:10
  • Class 10 Maths: Trigonometry – Trigonometric Ratios of Specific Angles Continued
    00:00
  • Class 10 Maths: Chapter 8- Trigonometric Ratios of Complimentary Angles
    00:00
  • Class 10 Maths: Chapter 8 – Trigonometric Ratios of Complimentary Angles Part 2
    30:59
  • Class 10 Maths: Chapter 8 – Trigonometric Identities
    28:38
  • Class 10 Maths: Chapter 8- Trigonometric Identities Part 2
    29:21
  • Class 10 Maths Chapter 8: Trigonometry Theorems Continued
    34:52
  • Class 10 Maths Chapter 8: Further Trigonometry Lectures Continued
    24:39
  • Class 10 Chapter 8- Trigonometry Revision
    36:23
  • Class 10 Maths: Chapter 9 – Applications of Trigonometry
    37:00
  • Class 10 Maths:Chapter 9- Applications of Trigonometry Part 2
    34:21
  • Class 10 Chapter 9 – Applications of Trigonometry Lecture 3
    30:11
  • Class 10 Chapter 9 – Applications of Trigonometry Lecture 4
    33:54
  • Chapter 10 – Circles Lecture 1
    37:58
  • Chapter 10 – Circles Lecture 2
    36:07
  • Chapter 10 – Circles Lecture 3
    30:38
  • Chapter 10 – Circles Lecture 4
    28:16
  • Chapter 12 Areas related to circles Lecture 1
    35:12
  • Chapter 12 Areas related to circles Lecture 2
    22:13
  • Chapter 12 Areas related to circles Lecture 3
    35:04
  • Chapter 12 Areas related to circles Lecture 4
    31:35
  • Chapter 12 Areas related to circles Lecture 5
    31:39
  • Chapter 12 Areas related to circles Lecture 6
    34:00
  • Chapter 12 Areas related to circles Lecture 7
    34:55
  • Chapter 12 Areas related to circles Lecture 8
    38:37
  • Chapter15 – Probability Lecture 1
    31:33
  • Chapter 15 – Probability Lecture 2
    30:55
  • Class 10_Maths_Chapter 8 – Trigonometry Special Class
    00:00
  • Class 10 Maths: Surface Area & Volume 1
    00:00
  • Class 10 Maths: Surface Area & Volume 2
    00:00
  • Class 10 Maths: Surface Area & Volume 3
    00:00
  • Class 10 Maths: Surface Area and Volume 4
    00:00
  • Class 10 Maths: Surface Area & Volume 5
    00:00
  • Class 10 Maths: Surface Area & Volume 6
    00:00
  • Class 10 Maths Chapter15 – Probability Lecture 1
    00:00
  • Class 10 Maths Chapter 15 – Probability Lecture 2
    00:00

Assamese Medium Lectures

Exam notes on Real Numbers
Exam notes on the NCERT Class 10 Mathematics chapter on "Real Numbers": **Introduction to Real Numbers:** - Real numbers include all rational and irrational numbers. - Rational numbers can be expressed as fractions (p/q), where p and q are integers and q is not equal to 0. - Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions. **Euclid's Division Lemma:** - For any two positive integers 'a' and 'b,' there exist unique integers 'q' and 'r' such that a = bq + r, where 0 ≤ r < b. - 'q' is called the quotient, and 'r' is the remainder. **Fundamental Theorem of Arithmetic:** - Every composite number can be expressed as a unique product of prime numbers (prime factorization). - Prime factorization helps in finding the LCM and HCF of numbers. **Proof of the Irrationality of √2:** - Assume that √2 is rational (can be expressed as p/q, where p and q are coprime). - Square both sides: 2 = (p²/q²). - Therefore, p² = 2q², implying that p² is even. - If p² is even, p must also be even (since the square of an odd number is odd). - Let p = 2k, where k is an integer. - Substituting in the equation: (2k)² = 4k² = 2q². - Dividing by 2: 2k² = q². - This implies that q² is also even, making q even. - But if both p and q are even, they are not coprime, contradicting the assumption. - Hence, the assumption that √2 is rational is false, proving it is irrational. **Decimal Representation of Rational Numbers:** - Rational numbers can have terminating or repeating (recurring) decimal expansions. - Terminating decimals have a finite number of decimal places (e.g., 0.125). - Repeating decimals have a repeating pattern of digits (e.g., 0.333... or 0.625625625...). **Division of Polynomials:** - Division of polynomials is similar to long division with numbers. - The divisor is called the "divisor," the polynomial being divided is the "dividend," and the result is the "quotient." **Finding HCF and LCM of Algebraic Expressions:** - The HCF (Highest Common Factor) is the greatest common factor of the coefficients of the terms in an algebraic expression. - The LCM (Least Common Multiple) is the product of the highest power of each prime factor in the expression. **Euclid's Algorithm for Finding HCF:** 1. Divide the larger number by the smaller number. 2. Replace the larger number with the remainder. 3. Repeat the process until the remainder becomes 0. 4. The divisor at this step is the HCF of the two numbers. **Rationalization:** - Rationalization is the process of converting an irrational number in the denominator of a fraction into a rational number. - For example, to rationalize √3 in the denominator, multiply by (√3)/(√3). **Summary:** - Real numbers encompass rational and irrational numbers. - Euclid's Division Lemma helps in understanding the relationship between numbers. - The Fundamental Theorem of Arithmetic emphasizes prime factorization. - The irrationality of √2 was proved by contradiction. - Rational numbers can have terminating or repeating decimal expansions. - Division of polynomials is performed using long division. - HCF and LCM are important in algebraic expressions. - Euclid's algorithm is used to find the HCF. - Rationalization is the process of converting irrational numbers in denominators. These notes provide a comprehensive overview of the "Real Numbers" chapter in NCERT Class 10 Mathematics and cover key concepts and proofs.

MCQs on Real Numbers
Here are some multiple-choice questions (MCQs) with answers based on the NCERT Class 10 Mathematics chapter on "Real Numbers": **1.** Which of the following numbers is not a real number? a. √5 b. -3/4 c. 0.123456... d. i (imaginary unit) **Answer:** d. i (imaginary unit) **2.** If p and q are two prime numbers, what is the HCF of p and q? a. 0 b. 1 c. p d. q **Answer:** b. 1 **3.** What is the prime factorization of the number 72? a. 2² × 3² b. 2³ × 3 c. 2⁴ × 3 d. 3⁴ × 2 **Answer:** b. 2³ × 3 **4.** Which of the following statements about irrational numbers is true? a. Irrational numbers can always be expressed as fractions. b. The decimal representation of an irrational number is always repeating. c. The square root of any positive integer is irrational. d. The product of two irrational numbers is always irrational. **Answer:** c. The square root of any positive integer is irrational. **5.** What is the decimal representation of the fraction 7/8? a. 0.875 b. 0.8750 c. 0.87500 d. 0.875000 **Answer:** a. 0.875 **6.** Which of the following is an example of a terminating decimal? a. 0.333... b. 0.123456789... c. 0.75 d. 0.625625625... **Answer:** c. 0.75 **7.** What is the HCF of 18 and 24? a. 2 b. 3 c. 6 d. 12 **Answer:** c. 6 **8.** The LCM of two numbers is 72, and their HCF is 8. If one of the numbers is 24, what is the other number? a. 18 b. 36 c. 48 d. 72 **Answer:** c. 48 **9.** Which of the following numbers is both a real number and a rational number? a. √2 b. -5 c. 0.75 d. π (pi) **Answer:** c. 0.75 **10.** If a = 5 and b = 15, what is the HCF of a and b? a. 3 b. 5 c. 15 d. 75 **Answer:** b. 5 These MCQs cover various concepts from the "Real Numbers" chapter of NCERT Class 10 Mathematics and can be used for practice and self-assessment.

Exam Notes on Polynomials
Here are exam notes on the NCERT Class 10 Mathematics chapter on "Polynomials": **Introduction to Polynomials:** - A polynomial is an algebraic expression with one or more terms, each consisting of a coefficient and a variable raised to a non-negative integer exponent. - Examples of polynomials include 2x² - 3x + 5 and 4y⁴ + 2y³ - y² + 7. **Degree of a Polynomial:** - The degree of a polynomial is the highest power of the variable in the polynomial. - For example, in the polynomial 3x⁴ - 2x³ + 7x² - 5x + 1, the degree is 4. **Types of Polynomials:** - Monomial: A polynomial with a single term, e.g., 2x³. - Binomial: A polynomial with two terms, e.g., 3x² + 4y. - Trinomial: A polynomial with three terms, e.g., 5a² - 2ab + 7b². - Polynomial: A polynomial with more than three terms, e.g., 4x⁴ - 3x³ + 2x² - x + 1. **Addition and Subtraction of Polynomials:** - To add or subtract polynomials, combine like terms (terms with the same variable and exponent). - For example, to add (3x² + 2x - 5) and (4x² - 3x + 7), we get 7x² - x + 2. **Multiplication of Polynomials:** - To multiply two polynomials, use the distributive property and combine like terms. - For example, to multiply (2x + 3) and (4x - 5), we get 8x² + 2x - 15. **Multiplying a Polynomial by a Monomial:** - Multiply each term of the polynomial by the monomial. - For example, to multiply 3x² - 2x + 5 by 2, we get 6x² - 4x + 10. **Polynomial Identities:** - Polynomial identities are equations that hold true for all values of the variables. - Examples include (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². **Factorization of Polynomials:** - Factorization involves expressing a polynomial as a product of two or more polynomials. - For example, x² - 4 can be factorized as (x + 2)(x - 2). **Division Algorithm for Polynomials:** - The division of one polynomial by another is carried out using the long division method, similar to division of numbers. - The quotient is the result, and the remainder is the remainder left after division. - For example, (x³ + 2x² - 3x + 1) ÷ (x - 1) results in a quotient of x² + 3x + 2 and a remainder of 3. **Zeroes of a Polynomial:** - Zeroes of a polynomial are values of the variable that make the polynomial equal to zero. - For example, the polynomial 2x² - 5x + 3 has zeroes x = 1 and x = 3/2. **Factor Theorem and Remainder Theorem:** - The Factor Theorem states that if (x - a) is a factor of a polynomial, then the polynomial is zero when x = a. - The Remainder Theorem states that if a polynomial f(x) is divided by (x - a), the remainder is f(a). **Synthetic Division:** - Synthetic division is a method for polynomial division when dividing by a linear factor (x - a). - It is particularly useful for finding zeroes of polynomials. **Summary:** - Polynomials are algebraic expressions with terms consisting of coefficients and variables raised to non-negative integer exponents. - The degree of a polynomial is the highest power of the variable. - Polynomials can be added, subtracted, and multiplied using various methods. - Factorization is the process of expressing a polynomial as a product of polynomials. - The Division Algorithm allows for the division of one polynomial by another. - Zeroes of a polynomial are values of the variable that make the polynomial zero. - The Factor Theorem and Remainder Theorem provide insights into polynomial factorization and division. - Synthetic division is a useful method for polynomial division. These notes provide a comprehensive overview of the "Polynomials" chapter in NCERT Class 10 Mathematics, covering key concepts and techniques for working with polynomials.

MCQs on Polynomials
Here are some multiple-choice questions (MCQs) with answers based on the NCERT Class 10 Mathematics chapter on "Polynomials": **1.** What is the degree of the polynomial 3x³ - 5x² + 2x - 7? a. 3 b. 2 c. 1 d. 0 **Answer:** a. 3 **2.** Which of the following is a monomial? a. 2x² - 3x + 1 b. 5y⁴ c. 3a² - 2b d. x⁴ + 2x³ - x² + 3x **Answer:** b. 5y⁴ **3.** What is the result of subtracting the polynomials (2x² + 3x - 5) and (4x² - 2x + 7)? a. 6x⁴ + x² - 12 b. -2x⁴ + 5x² - 2x - 12 c. -2x⁴ + 5x² + x - 12 d. 6x⁴ - x² + 12 **Answer:** c. -2x⁴ + 5x² + x - 12 **4.** Which of the following is a binomial? a. 4x³ - 2x² + 7x - 1 b. 5x⁵ c. 2y⁴ + 3y² - y d. x⁶ - 2x³ + 1 **Answer:** c. 2y⁴ + 3y² - y **5.** What is the product of the polynomials (x - 3) and (x + 5)? a. x² - 8x - 15 b. x² - 2x - 15 c. x² + 2x - 15 d. x² + 8x - 15 **Answer:** a. x² - 8x - 15 **6.** Which of the following is NOT a factor of the polynomial x³ - 8? a. x - 2 b. x + 2 c. x - 1 d. x + 1 **Answer:** b. x + 2 **7.** What is the degree of the constant polynomial 7? a. 0 b. 1 c. -7 d. It has no degree. **Answer:** a. 0 **8.** When the polynomial x⁴ - 5x³ + 3x² - 2x + k is divided by (x - 2), the remainder is 7. What is the value of 'k'? a. 2 b. -3 c. 7 d. 0 **Answer:** c. 7 **9.** What is the degree of the zero polynomial, denoted as '0'? a. 0 b. 1 c. It has no degree. d. It cannot be determined. **Answer:** c. It has no degree. **10.** What is the factor theorem used for in polynomials? a. To find the degree of a polynomial b. To find the quotient of polynomial division c. To determine the remainder in polynomial division d. To identify factors of a polynomial **Answer:** d. To identify factors of a polynomial These MCQs cover various concepts from the "Polynomials" chapter of NCERT Class 10 Mathematics and can be used for practice and self-assessment.

Exam notes on NCERT Class 10 chapter on Linear Equations in Two Variables
Here are exam notes on the NCERT Class 10 Mathematics chapter on "Linear Equations in Two Variables": **Introduction to Linear Equations in Two Variables:** - A linear equation in two variables, x and y, is an equation that can be written in the form ax + by = c, where a, b, and c are constants, and a and b are not both zero. - The solutions to a linear equation in two variables form a straight line on the coordinate plane. **Graphical Representation of Linear Equations:** - To graphically represent a linear equation, one can create a table of values or use the intercept method. - The x-intercept is the point where the line crosses the x-axis (y = 0), and the y-intercept is where the line crosses the y-axis (x = 0). **Slope-Intercept Form:** - The slope-intercept form of a linear equation is y = mx + b, where 'm' is the slope (rise over run) and 'b' is the y-intercept (where the line crosses the y-axis). - The slope indicates the steepness of the line, with positive values indicating an upward slope and negative values indicating a downward slope. **Finding Slope and Y-Intercept:** - To find the slope (m) between two points (x₁, y₁) and (x₂, y₂), use the formula: m = (y₂ - y₁) / (x₂ - x₁). - To find the y-intercept, set x = 0 in the equation and solve for y. **Forms of Linear Equations:** - Standard Form: ax + by = c, where a, b, and c are integers, and a and b have no common factors other than 1. - Slope-Intercept Form: y = mx + b. - Point-Slope Form: (y - y₁) = m(x - x₁), where (x₁, y₁) is a point on the line, and 'm' is the slope. **Parallel and Perpendicular Lines:** - Lines with the same slope are parallel to each other and will never intersect. - Lines with slopes that are negative reciprocals of each other (i.e., m₁ * m₂ = -1) are perpendicular and intersect at a right angle. **Solution of Linear Equations:** - A solution to a linear equation in two variables is a pair of values (x, y) that satisfies the equation. - Graphically, the solution is the point where the line representing the equation intersects the coordinate axes. **Simultaneous Linear Equations:** - A system of two simultaneous linear equations consists of two linear equations with two variables. - The solution to the system is the point (x, y) that satisfies both equations simultaneously. **Methods for Solving Simultaneous Linear Equations:** 1. **Graphical Method:** Plot both equations on the same graph and find the point of intersection. 2. **Substitution Method:** Solve one equation for one variable and substitute it into the other equation. 3. **Elimination Method:** Add or subtract equations to eliminate one variable and solve for the other. **Consistency of Linear Systems:** - A system of linear equations can be consistent (having one or more solutions) or inconsistent (having no solutions). - Consistent systems can be independent (infinitely many solutions) or dependent (unique solution). **Summary:** - Linear equations in two variables are represented as ax + by = c. - They can be graphically represented as straight lines on the coordinate plane. - The slope-intercept form is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. - Solutions to linear equations satisfy the equations and can be found graphically or algebraically. - Systems of simultaneous linear equations can be solved using graphical, substitution, or elimination methods. - The consistency of linear systems determines if they have solutions and, if so, whether they are unique or infinite. These notes provide a comprehensive overview of the "Linear Equations in Two Variables" chapter in NCERT Class 10 Mathematics, covering key concepts and methods for solving linear equations and systems.

MCQs on NCERT Class 10 chapter on Linear Equations in Two Variables
Here are some multiple-choice questions (MCQs) with answers based on the NCERT Class 10 Mathematics chapter on "Linear Equations in Two Variables": **1.** What is the standard form of a linear equation in two variables? a. y = mx + b b. ax + by = c c. y = ax + b d. ax + by = 0 **Answer:** b. ax + by = c **2.** Which of the following equations represents a horizontal line? a. y = 2x + 3 b. x = 4 c. 3x - 2y = 6 d. 2x + 3y = 5 **Answer:** b. x = 4 **3.** What is the slope of a line parallel to the x-axis? a. 0 b. 1 c. Undefined d. Any real number **Answer:** a. 0 **4.** If a line has a positive slope, which direction does it go in when graphed? a. Upward b. Downward c. Horizontal d. Vertical **Answer:** a. Upward **5.** What is the slope of a line that is perpendicular to a horizontal line? a. 0 b. 1 c. Undefined d. Any real number **Answer:** c. Undefined **6.** Which method involves substituting one equation into another to solve a system of linear equations? a. Graphical method b. Substitution method c. Elimination method d. Interception method **Answer:** b. Substitution method **7.** What is the solution to the system of linear equations: 2x + 3y = 8 4x + 6y = 16 a. (2, 2) b. (4, 0) c. (0, 4) d. (1, 2) **Answer:** a. (2, 2) **8.** In a consistent system of linear equations, how many solutions are possible? a. None b. One c. Infinitely many d. Two **Answer:** c. Infinitely many **9.** Which of the following systems of linear equations is inconsistent? a. 2x - 3y = 5 4x - 6y = 10 b. 3x + 2y = 7 6x + 4y = 14 c. 5x + 3y = 11 2x - 6y = 14 d. 2x + 3y = 8 4x + 6y = 16 **Answer:** c. 5x + 3y = 11 and 2x - 6y = 14 **10.** Which form of a linear equation is useful for finding the slope and y-intercept? a. Standard form b. Slope-intercept form c. Point-slope form d. Interception form **Answer:** b. Slope-intercept form These MCQs cover various concepts from the "Linear Equations in Two Variables" chapter of NCERT Class 10 Mathematics and can be used for practice and self-assessment.

Exam notes on NCERT Class 10 chapter on Quadratic Equations
Here are detailed exam notes on the NCERT Class 10 Mathematics chapter on "Quadratic Equations": **Introduction to Quadratic Equations:** - A quadratic equation is a polynomial equation of the second degree (degree 2). - It can be written in the standard form: ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to 0. **Solutions of Quadratic Equations:** - Quadratic equations have two solutions, which can be real or complex numbers. - The solutions are typically found using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). **Discriminant:** - The discriminant (D) of a quadratic equation is given by D = b² - 4ac. - The discriminant helps determine the nature of the solutions: - If D > 0, two distinct real solutions exist. - If D = 0, there is one real solution (a repeated root). - If D 0. - One real root (repeated) if D = 0. - Two complex conjugate roots if D < 0. **Factorization of Quadratic Equations:** - Quadratic equations can often be factorized to find their solutions. - For example, x² - 5x + 6 = 0 can be factored as (x - 2)(x - 3) = 0, leading to solutions x = 2 and x = 3. **Sum and Product of Roots:** - If α and β are the roots of a quadratic equation ax² + bx + c = 0, then: - α + β = -b/a (sum of roots) - αβ = c/a (product of roots) **Nature of Quadratic Graphs:** - The graph of a quadratic equation ax² + bx + c = 0 is a parabola. - If 'a' is positive, the parabola opens upward and has a minimum point. - If 'a' is negative, the parabola opens downward and has a maximum point. **Vertex Form of Quadratic Equations:** - A quadratic equation in vertex form is written as y = a(x - h)² + k, where (h, k) is the vertex of the parabola. **Completing the Square:** - Completing the square is a method to convert a quadratic equation into vertex form. - It involves adding and subtracting terms to create a perfect square trinomial. **Word Problems Involving Quadratic Equations:** - Quadratic equations are used to solve real-life problems involving areas, distances, and other quadratic relationships. **Summary:** - Quadratic equations are polynomial equations of the second degree. - They can be solved using the quadratic formula or by factoring. - The discriminant helps determine the nature of the roots (real, repeated, or complex). - Quadratic graphs are parabolas with vertex points. - Completing the square is a method for rewriting quadratic equations. - Quadratic equations are used to solve various word problems. These notes provide a comprehensive overview of the "Quadratic Equations" chapter in NCERT Class 10 Mathematics, covering key concepts and methods for solving and understanding quadratic equations.

MCQs on NCERT Class 10 chapter on Quadratic Equations
Here are some multiple-choice questions (MCQs) with answers based on the NCERT Class 10 Mathematics chapter on "Quadratic Equations": **1.** Which of the following equations is a quadratic equation? a. 3x - 7 = 0 b. 2x + 4y = 8 c. x² + 5x - 6 = 0 d. 4x + 3 = 2x - 1 **Answer:** c. x² + 5x - 6 = 0 **2.** What is the discriminant of the quadratic equation 2x² - 3x + 1 = 0? a. -5 b. 1 c. 9 d. 5 **Answer:** c. 9 **3.** How many real solutions does a quadratic equation with a negative discriminant have? a. 0 b. 1 c. 2 d. It depends on the coefficients. **Answer:** a. 0 **4.** Which of the following is the quadratic formula used to find the solutions of a quadratic equation ax² + bx + c = 0? a. x = (-b ± √b² - 4ac) / 2a b. x = -b ± √(b² + 4ac) / 2a c. x = (-b ± √b² + 4ac) / 2a d. x = -b ± 4a / √(b² - c) **Answer:** b. x = -b ± √(b² + 4ac) / 2a **5.** If a quadratic equation has one real solution, what can be said about its discriminant? a. D > 0 b. D = 0 c. D < 0 d. D can be positive, negative, or zero. **Answer:** b. D = 0 **6.** What is the sum of the roots of the quadratic equation x² - 6x + 8 = 0? a. 2 b. 3 c. 4 d. 6 **Answer:** b. 3 **7.** If the product of the roots of a quadratic equation is 5, what can be said about the coefficient of the linear term in the equation? a. It is equal to 5. b. It is equal to -5. c. It cannot be determined. d. It is irrelevant to the product of roots. **Answer:** b. It is equal to -5. **8.** In the quadratic equation x² - 9x + k = 0, the value of 'k' that makes the equation have one real solution is: a. 4 b. 9 c. 12 d. 6 **Answer:** b. 9 **9.** What is the vertex form of the quadratic equation y = x² - 4x + 3? a. y = (x - 2)² + 1 b. y = x² - 4x + 4 c. y = (x + 2)² - 1 d. y = (x - 2)² - 1 **Answer:** d. y = (x - 2)² - 1 **10.** Completing the square is a method used to: a. Factor quadratic equations. b. Solve linear equations. c. Convert quadratic equations into vertex form. d. Find the discriminant of quadratic equations. **Answer:** c. Convert quadratic equations into vertex form. These MCQs cover various concepts from the "Quadratic Equations" chapter of NCERT Class 10 Mathematics and can be used for practice and self-assessment.

Exam notes on NCERT Class 10 chapter on Arithmetic Progression
Here are detailed exam notes on the NCERT Class 10 Mathematics chapter on "Arithmetic Progression" (AP): **Introduction to Arithmetic Progression (AP):** - An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the "common difference" and is denoted by 'd.' **General Form of an AP:** - The general form of an arithmetic progression is: a, a + d, a + 2d, a + 3d, ... **Terms in an AP:** - The nth term of an AP is given by the formula: aₙ = a + (n - 1)d. - Here, 'a' is the first term, 'd' is the common difference, 'n' is the term number, and 'aₙ' is the nth term. **Sum of First 'n' Terms (Arithmetic Series):** - The sum of the first 'n' terms of an AP, denoted by 'Sₙ,' is given by the formula: Sₙ = (n/2)[2a + (n - 1)d]. - This formula is useful for finding the sum of an arithmetic series. **Sum of an AP:** - An infinite AP has a sum only if its common difference 'd' is not equal to zero. In that case, the sum is given by: S = a / (1 - r), where 'r' is the common ratio ('r = 1 + d'). **nth Term from the End:** - To find the nth term from the end of an AP, use the formula: aₙ = l - (n - 1)d, where 'l' is the last term. **Sum of 'n' Terms from the End:** - The sum of the last 'n' terms of an AP, denoted by 'Sₙ', is given by: Sₙ = (n/2)[2l - (n - 1)d], where 'l' is the last term. **Applications of AP:** - Arithmetic progressions are used in various real-life scenarios, such as calculating monthly installment payments, calculating the average of a series, and finding missing terms in a sequence. **Summary:** - An arithmetic progression is a sequence of numbers with a constant difference between consecutive terms. - The nth term of an AP is given by aₙ = a + (n - 1)d. - The sum of the first 'n' terms of an AP is given by Sₙ = (n/2)[2a + (n - 1)d]. - The sum of an infinite AP with a non-zero common difference is given by S = a / (1 - r). - Formulas for finding nth terms from the end and sums of 'n' terms from the end exist. - APs have practical applications in various fields. These notes provide a comprehensive overview of the "Arithmetic Progression" chapter in NCERT Class 10 Mathematics, covering key concepts and formulas related to APs and their applications.

MCQs on NCERT Class 10 chapter on Arithmetic Progression
Here are some multiple-choice questions (MCQs) with answers based on the NCERT Class 10 Mathematics chapter on "Arithmetic Progression" (AP): **1.** What is an arithmetic progression (AP)? a. A sequence of numbers with no common difference. b. A sequence of numbers with a constant difference between consecutive terms. c. A sequence of numbers with a constant ratio between consecutive terms. d. A sequence of prime numbers. **Answer:** b. A sequence of numbers with a constant difference between consecutive terms. **2.** In an arithmetic progression, what is the common difference? a. The ratio of the first term to the second term. b. The ratio of the second term to the first term. c. The difference between any two consecutive terms. d. The sum of the terms. **Answer:** c. The difference between any two consecutive terms. **3.** If the first term of an AP is 'a' and the common difference is 'd', what is the nth term of the AP? a. a + d b. a - d c. a × d d. a + (n - 1)d **Answer:** d. a + (n - 1)d **4.** What is the formula for finding the sum of the first 'n' terms of an arithmetic progression? a. Sₙ = (n/2)[2a + (n - 1)d] b. Sₙ = (n/2)(a + d) c. Sₙ = (n/2)(a + nd) d. Sₙ = (n/2)(2a + d) **Answer:** a. Sₙ = (n/2)[2a + (n - 1)d] **5.** If 'a' is the first term, 'l' is the last term, and 'n' is the number of terms in an AP, what is the sum of all terms in the AP? a. S = n(a + l) b. S = n(a - l) c. S = (n/2)(a + l) d. S = (n/2)(2a + l) **Answer:** c. S = (n/2)(a + l) **6.** What is the common difference in the arithmetic progression 3, 6, 9, 12, ...? a. 1 b. 2 c. 3 d. 4 **Answer:** c. 3 **7.** In an arithmetic progression, if the first term is 5 and the common difference is -2, what is the 6th term? a. -7 b. -5 c. 7 d. 5 **Answer:** a. -7 **8.** If the sum of the first 4 terms of an arithmetic progression is 24 and the common difference is 3, what is the first term? a. 2 b. 3 c. 4 d. 5 **Answer:** b. 3 **9.** What is the sum of the first 10 positive even numbers in an arithmetic progression? a. 50 b. 100 c. 110 d. 120 **Answer:** d. 120 **10.** If the sum of the first 'n' terms of an arithmetic progression is 42, and the first term 'a' is 3, what is the common difference 'd'? a. 4 b. 3 c. 2 d. 1 **Answer:** c. 2 These MCQs cover various concepts from the "Arithmetic Progression" chapter of NCERT Class 10 Mathematics and can be used for practice and self-assessment.

Exam notes on NCERT Class 10 maths chapter on Co-ordination Geometry
Some Exam notes on NCERT Class 10 Mathematics Chapter - "Coordinate Geometry": **Chapter Overview:** Coordinate Geometry is a branch of mathematics that combines algebra and geometry. It deals with the study of geometric shapes using the coordinate plane. In this chapter, students learn about the Cartesian coordinate system, plotting points, finding distances between points, and equations of lines and curves. **Introduction to Coordinate Geometry:** - Coordinate geometry is the study of geometry using a coordinate system. - The Cartesian coordinate system consists of a horizontal x-axis and a vertical y-axis, which intersect at the origin (0,0). - Points on the plane are represented as ordered pairs (x, y), where 'x' is the distance along the x-axis and 'y' is the distance along the y-axis. - The point of intersection of the axes is called the origin (0,0). **Plotting Points:** - To plot a point on the coordinate plane, locate its x-coordinate along the x-axis and its y-coordinate along the y-axis, then mark the point of intersection. - For example, to plot the point (3, 4), move 3 units to the right along the x-axis and 4 units up along the y-axis, and mark the point of intersection. **Distance Formula:** - The distance between two points (x₁, y₁) and (x₂, y₂) can be calculated using the distance formula: $$sqrt{(x₂ - x₁)² + (y₂ - y₁)²}$$ **Section Formula:** - The coordinates of the point that divides a line segment joining two points in a given ratio can be found using the section formula. - If a line segment joining points A(x₁, y₁) and B(x₂, y₂) is divided by a point P(x, y) in the ratio m:n, then the coordinates of point P are given by: $$(x, y) = left(frac{mx₂ + nx₁}{m + n}, frac{my₂ + ny₁}{m + n}right)$$ **Slope of a Line:** - The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by: $$text{Slope (m)} = frac{y₂ - y₁}{x₂ - x₁}$$ - The slope of a line is a measure of its steepness or incline. **Equation of a Line:** - The equation of a line can be written in slope-intercept form as: y = mx + c, where 'm' is the slope, and 'c' is the y-intercept (the point where the line crosses the y-axis). - Alternatively, the equation of a line can be expressed as Ax + By + C = 0, where A, B, and C are constants. **Parallel and Perpendicular Lines:** - Parallel lines have the same slope, and perpendicular lines have negative reciprocal slopes (the product of their slopes equals -1). **Summary:** - Coordinate Geometry combines algebra and geometry to study geometric shapes on the coordinate plane. - Points are represented as ordered pairs (x, y) in the Cartesian coordinate system. - The distance formula calculates the distance between two points. - The section formula finds the coordinates of a point dividing a line segment in a given ratio. - The slope of a line measures its steepness, and the slope-intercept form of a line is y = mx + c. - Parallel lines have the same slope, and perpendicular lines have negative reciprocal slopes. These notes cover the essential concepts and formulas from the "Coordinate Geometry" chapter in NCERT Class 10 Mathematics. Studying and practicing problems related to these concepts will help you excel in this chapter.

Exam notes for the NCERT Class 10 Maths Chapter on Triangles
Exam notes for NCERT Class 10 Maths Chapter on Triangles: **Chapter 6: Triangles** **Introduction:** - Triangles are one of the fundamental shapes in geometry. - This chapter explores various properties, theorems, and concepts related to triangles. **Key Concepts:** **1. Classification of Triangles:** - Based on the measure of angles: - Acute-angled triangle: All angles are less than 90 degrees. - Right-angled triangle: One angle is 90 degrees. - Obtuse-angled triangle: One angle is greater than 90 degrees. **2. Properties of Triangles:** - Sum of angles in a triangle is always 180 degrees (Angle Sum Property). - In an equilateral triangle, all sides and angles are equal. - In an isosceles triangle, two sides and two angles are equal. - In a scalene triangle, all sides and angles are unequal. **3. Pythagoras Theorem:** - In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. - c² = a² + b² (where c is the hypotenuse, and a and b are the other two sides). **4. Congruence of Triangles:** - Two triangles are congruent if their corresponding sides and angles are equal. - The criteria for congruence include SSS, SAS, ASA, RHS, and HL. **5. Basic Proportionality Theorem (Thales Theorem):** - If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. - Also known as Thales' Theorem. **6. Similarity of Triangles:** - Two triangles are similar if their corresponding angles are equal. - The criteria for similarity include AA, SSS, and SAS. **7. Pythagoras Theorem Converse:** - If in a triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, it is a right-angled triangle. **8. Areas of Similar Triangles:** - The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. **9. Applications of Similar Triangles:** - Height and distance problems often involve the use of similar triangles. **10. Circles and Triangles:** - An angle formed by a chord in a circle is half the angle formed by the same chord at the center. **11. Theorems and Constructions:** - The chapter covers various theorems such as the Angle Bisector Theorem and the Basic Proportionality Theorem. - Construction problems related to triangles are also included. **Summary:** - Triangles are classified based on angles and sides. - Pythagoras Theorem, congruence, similarity, and theorems related to triangles are essential concepts. - Similar triangles have proportional sides and areas. - Circles and triangles have specific angle relationships. - Constructions related to triangles are a part of the chapter. **Practice Problems:** 1. Prove the Pythagoras Theorem using similar triangles. 2. Construct an equilateral triangle given its side length. 3. Determine the length of an altitude in a right-angled triangle. 4. Prove the Angle Bisector Theorem. 5. Solve height and distance problems involving triangles and circles.

Exam notes for the NCERT Class 10 Maths Chapter on Trigonometry
Exam notes for NCERT Class 10 Maths Chapter on Trigonometry: **Chapter 8: Trigonometry** **Introduction:** - Trigonometry is a branch of mathematics that deals with the study of relationships between the angles and sides of triangles. **Key Concepts:** **1. Trigonometric Ratios:** - In a right-angled triangle, the three primary trigonometric ratios are defined as follows: - Sine (sin θ) = Opposite side / Hypotenuse - Cosine (cos θ) = Adjacent side / Hypotenuse - Tangent (tan θ) = Opposite side / Adjacent side **2. Trigonometric Identities:** - The Pythagorean identities: sin² θ + cos² θ = 1 and 1 + tan² θ = sec² θ. - Reciprocal identities: csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ. **3. Trigonometric Ratios of Specific Angles:** - Special angles (0°, 30°, 45°, 60°, and 90°) have specific trigonometric ratios. **4. Complementary Angles:** - Two angles are complementary if their sum is 90 degrees. - The trigonometric ratios of complementary angles are complementary to each other. **5. Heights and Distances:** - Trigonometry is used to solve problems related to heights and distances, such as finding the height of a building or the distance between two objects. **6. Introduction to Trigonometric Identities:** - Trigonometric identities are equations that involve trigonometric ratios and are true for all values of the variables. **7. Applications of Trigonometry:** - Trigonometry is applied in various fields, including physics, engineering, navigation, and astronomy. **8. Trigonometric Tables:** - Trigonometric values for standard angles can be tabulated and used for calculations. **Summary:** - Trigonometry deals with the study of angles and sides of triangles. - Sine, cosine, and tangent are the primary trigonometric ratios. - Trigonometric identities are equations involving trigonometric ratios. - Heights and distances can be determined using trigonometry. - Trigonometry has practical applications in different fields. **Practice Problems:** 1. Calculate the values of sin 30°, cos 60°, and tan 45°. 2. Find the length of the shadow of a 10-meter pole when the angle of elevation of the sun is 30°. 3. Prove the Pythagorean identity sin² θ + cos² θ = 1 using a right-angled triangle. 4. Determine the distance between two ships when the angle of elevation to the top of the other ship's mast is given. 5. Solve problems involving trigonometric identities, such as simplifying expressions using them.

Exam notes for the NCERT Class 10 Maths Chapter on Application of Trigonometry
Exam notes for NCERT Class 10 Maths Chapter on "Application of Trigonometry": **Chapter 9: Application of Trigonometry** **Introduction:** - This chapter focuses on practical applications of trigonometry in various real-life situations. **Key Concepts:** **1. Heights and Distances:** - Trigonometry is used to calculate the heights and distances of objects or locations when the angle of elevation or depression is known. - Angle of Elevation: The angle formed by the line of sight above the horizontal when an observer looks up. - Angle of Depression: The angle formed by the line of sight below the horizontal when an observer looks down. **2. Solving Problems Involving Heights and Distances:** - To find the height of an object or the distance between two objects, trigonometric ratios (sine, cosine, or tangent) are used in conjunction with the given angle of elevation or depression. **3. Angle of Elevation and Angle of Depression Problems:** - Examples include determining the height of a building, the depth of a well, or the distance between two objects. - The concept of similar triangles is often applied in these problems. **4. Line of Sight:** - Trigonometry helps in calculating the angle of elevation or depression for line of sight in various scenarios, such as surveying, navigation, and construction. **5. Use of Trigonometric Tables and Calculators:** - Trigonometric values of standard angles are often used from tables or calculators for solving real-life problems. **Summary:** - This chapter deals with the practical application of trigonometry in determining heights and distances. - It involves solving problems related to the angle of elevation and angle of depression. - Trigonometric ratios are used to find unknown quantities. - Similar triangles play a crucial role in solving these problems. **Practice Problems:** 1. Calculate the height of a flagpole when the angle of elevation from a certain distance is given. 2. Determine the distance between two buildings when the angles of elevation from a point between them are provided. 3. Find the depth of a river when the angle of depression from a bridge is given. 4. Solve problems involving the angle of elevation and depression in different scenarios. 5. Use trigonometric ratios to find heights or distances in real-life situations.

Exam notes for the NCERT Class 10 Maths Chapter on Probability
Exam notes for NCERT Class 10 Maths Chapter on "Probability": **Chapter 15: Probability** **Introduction:** - Probability is a branch of mathematics that deals with the likelihood of events occurring. - It is used to quantify uncertainty and randomness. **Key Concepts:** **1. Random Experiments:** - A random experiment is an experiment whose outcome cannot be predicted with certainty. - Examples include tossing a coin, rolling a die, or drawing a card from a deck. **2. Sample Space and Events:** - The sample space (S) is the set of all possible outcomes of a random experiment. - An event (E) is a subset of the sample space, consisting of certain outcomes. **3. Probability of an Event:** - Probability (P) of an event E is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. - P(E) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes) **4. Probability Scale:** - Probabilities range from 0 (impossible event) to 1 (certain event). **5. Complementary Events:** - The probability of an event E occurring is denoted as P(E), and the probability of the event not occurring is denoted as P(not E) or P(E'). - P(E) + P(E') = 1 **6. Addition Theorem of Probability:** - For two mutually exclusive events (events that cannot occur simultaneously), the probability of either event happening is the sum of their individual probabilities. - P(A or B) = P(A) + P(B) **7. Multiplication Theorem of Probability:** - For two independent events (events that do not affect each other's outcomes), the probability of both events happening is the product of their individual probabilities. - P(A and B) = P(A) * P(B) **8. Conditional Probability:** - Conditional probability is the probability of an event occurring given that another event has already occurred. - P(A|B) = P(A and B) / P(B) **9. Probability of an Event Not Happening:** - P(not A) = 1 - P(A) **Summary:** - Probability measures the likelihood of events occurring in random experiments. - It is expressed as a ratio of favorable outcomes to total possible outcomes. - The addition and multiplication theorems help in calculating probabilities of compound events. - Conditional probability is used when events are dependent. **Practice Problems:** 1. Calculate the probability of getting a head when tossing a fair coin. 2. Find the probability of drawing an Ace from a standard deck of 52 cards. 3. Determine the probability of rolling a prime number on a fair six-sided die. 4. Solve problems involving conditional probability. 5. Calculate the probability of events not happening using complementary probabilities.

Exam notes for the NCERT Class 10 Maths Chapter on Surface Area and Volume
Exam notes for NCERT Class 10 Maths Chapter on "Surface Area and Volume": **Chapter 13: Surface Areas and Volumes** **Introduction:** - This chapter deals with the calculation of surface areas and volumes of various 3D shapes such as cubes, cuboids, cylinders, cones, and spheres. **Key Concepts:** **1. Surface Area of Solids:** - The surface area of a solid is the total area of its outer surfaces. - Different types of solids have different formulas for calculating their surface areas. **2. Volume of Solids:** - The volume of a solid is the amount of space occupied by it. - Different types of solids have different formulas for calculating their volumes. **3. Surface Area and Volume of Cubes and Cuboids:** - For a cube, the surface area is 6a^2, and the volume is a^3 (where 'a' is the length of the side). - For a cuboid with length (l), width (b), and height (h), the surface area is 2(lb + bh + hl), and the volume is lbh. **4. Surface Area and Volume of Cylinders:** - For a cylinder with radius (r) and height (h), the curved surface area is 2πrh, the total surface area is 2πrh + 2πr^2, and the volume is πr^2h. **5. Surface Area and Volume of Cones:** - For a cone with radius (r) and slant height (l), the curved surface area is πrl, and the total surface area is πrl + πr^2. The volume is (1/3)πr^2h. **6. Surface Area and Volume of Spheres:** - For a sphere with radius (r), the surface area is 4πr^2, and the volume is (4/3)πr^3. **7. Frustum of a Cone:** - A frustum of a cone is the portion of a cone that remains after cutting off the top by a plane parallel to the base. - Formulas are used to calculate its surface area and volume. **8. Hemisphere:** - A hemisphere is half of a sphere. It has its own surface area and volume formulas. **Summary:** - This chapter covers the calculation of surface areas and volumes of various 3D shapes. - Formulas are provided for cubes, cuboids, cylinders, cones, spheres, frustums, and hemispheres. - The understanding of these formulas and their application is crucial in solving real-world problems. **Practice Problems:** 1. Calculate the surface area and volume of a cube with a side length of 5 cm. 2. Find the curved surface area and total surface area of a cylinder with a radius of 3 cm and height of 8 cm. 3. Determine the volume of a cone with a radius of 6 cm and slant height of 10 cm. 4. Calculate the surface area of a hemisphere with a radius of 7 cm. 5. Solve problems involving frustums of cones.

Exam notes for the NCERT Class 10 Maths Chapter on Area Related To Circles
Exam notes for NCERT Class 10 Maths Chapter on "Areas Related to Circles": **Chapter 12: Areas Related to Circles** **Introduction:** - This chapter deals with the calculation of areas related to circles and sectors. **Key Concepts:** **1. Area of a Circle:** - The area (A) of a circle with radius (r) is given by A = πr^2. **2. Circumference of a Circle:** - The circumference (C) of a circle with radius (r) is given by C = 2πr. **3. Area of a Sector:** - A sector is a portion of a circle enclosed by two radii and an arc. Its area is calculated as (θ/360) * πr^2, where θ is the angle (in degrees) at the center of the circle. **4. Length of an Arc:** - The length (L) of an arc in a circle with radius (r) and subtending angle θ (in degrees) at the center is given by L = (θ/360) * 2πr. **5. Perimeter of a Sector:** - The perimeter of a sector is the sum of the lengths of its arc and the two radii. **6. Areas of Combinations of Figures:** - Problems involve finding the areas of shapes formed by combining circles, semicircles, and rectangles. **Summary:** - This chapter focuses on the calculation of areas related to circles, sectors, and arcs. - Understanding the formulas for the area of a circle and related concepts like circumference, sector, and arc length is crucial. - The chapter also covers practical applications of these concepts. **Practice Problems:** 1. Calculate the area of a circle with a radius of 5 cm. 2. Find the circumference of a circle with a diameter of 14 cm. 3. Determine the area of a sector with a radius of 8 cm and a central angle of 60 degrees. 4. Calculate the length of an arc in a circle with a radius of 10 cm, subtending an angle of 45 degrees. 5. Find the area of a region enclosed between two concentric circles with radii 6 cm and 9 cm. 6. Solve problems involving combinations of circles, semicircles, and rectangles in practical scenarios.

Exam Notes on Class 10 Euclid’s Division Lemma
### Class 10 Mathematics Notes: Euclid's Division Lemma **Chapter 1: Real Numbers** **Euclid's Division Lemma:** Euclid's Division Lemma states that given two positive integers "a" and "b," there exist unique integers "q" and "r" satisfying the equation: [ a = bq + r ] where "a" and "b" are given positive integers, while "q" is the quotient and "r" is the remainder such that ( 0 leq r < b ). #### Key Points: 1. **Integers:** "a" and "b" are the numbers being divided. 2. **Quotient:** "q" is the result of the division. 3. **Remainder:** "r" should be a non-negative integer and smaller than "b". #### Applications of Euclid's Division Lemma: 1. **GCD (Greatest Common Divisor):** Euclid's Division Lemma is instrumental in finding the GCD of two positive integers. [ GCD(a, b) = GCD(b, r) ] When "a" is divided by "b," if "r" equals zero, "b" is the GCD of "a" and "b". If "r" is not zero, then the GCD of "a" and "b" is the same as the GCD of "b" and "r". 2. **Proving the Fundamental Theorem of Arithmetic:** The lemma plays a pivotal role in establishing the Fundamental Theorem of Arithmetic by helping demonstrate the existence and uniqueness of prime factorization. 3. **Solving Diophantine Equations:** Euclid's Lemma can be employed to solve linear Diophantine equations, which are equations that seek integer solutions. #### Example Problems: 1. **GCD Calculation:** Find the GCD of 56 and 98 using Euclid’s Division Lemma. Applying the lemma, divide 98 by 56 to get the remainder. [ 98 = 56 times 1 + 42 ] Now, GCD(98, 56) = GCD(56, 42) Repeat the process: [ 56 = 42 times 1 + 14 ] So, GCD(56, 42) = GCD(42, 14) [ 42 = 14 times 3 + 0 ] When the remainder is zero, the divisor at this stage (which is 14) is the GCD of 56 and 98. 2. **Diophantine Equation:** Solve for (x) and (y) in the equation ( 37x + 43y = 1 ) By applying Euclid's Lemma and using back substitution, solutions to such equations can be determined, albeit with additional steps and manipulation. #### Exam Tips: - Understand the application of Euclid's Division Lemma in finding GCD, as this is frequently asked in the exam. - Practice different types of problems where the lemma is applied to various contexts - like Diophantine equations. - Be mindful of the condition that the remainder "r" must be non-negative and less than "b". - Pay attention to problems involving consecutive divisors and remainders, ensuring that you follow each step accurately to arrive at the GCD. Remember, practice and understanding of concepts are crucial to applying Euclid's Division Lemma accurately in solving problems.

Exam Notes On Geometrical Meaning of the Zeroes of a Polynomial
### Class 10 Mathematics Exam Notes: Geometrical Meaning of the Zeroes of a Polynomial **Chapter 2: Polynomials** #### Understanding Zeroes of a Polynomial - **Definition**: The zeroes of a polynomial are the values of (x) for which the polynomial (P(x)) equals zero. Mathematically, (P(a) = 0), where (a) is a zero of the polynomial (P(x)). - **Graph**: The graphical representation of a polynomial equation is a curve. The points where the curve intersects the x-axis are the zeroes of the polynomial. #### Key Concepts 1. **Linear Polynomial** ((ax + b)) - Degree: 1 - Zeroes: One zero - Graph: A straight line - Intersection with X-Axis: One point Example: (P(x) = 2x + 3) To find the zero, set (P(x) = 0) and solve for (x). 2. **Quadratic Polynomial** ((ax^2 + bx + c)) - Degree: 2 - Zeroes: Zero, one, or two zeroes - Graph: A parabola - Intersection with X-Axis: Up to two points Example: (P(x) = x^2 - 3x + 2) Use factoring, the quadratic formula, or completing the square to find zeroes. 3. **Cubic Polynomial** ((ax^3 + bx^2 + cx + d)) - Degree: 3 - Zeroes: Up to three - Graph: Can have various shapes with possible inflection points Example: (P(x) = x^3 - 6x^2 + 11x - 6) #### Graphical Representation and Zeroes - The **graph** of a polynomial function of degree (n) will cut or touch the x-axis at at most (n) points. - The **number of times** the graph of a polynomial cuts or touches the x-axis at a point is called the multiplicity of the zero. #### Key Takeaways for Exams - **Graph-Intercept**: Always remember that the zeroes of a polynomial are the x-coordinates of the points where the graph of the polynomial intersects the x-axis. - **Factoring**: Be adept at factoring polynomials to find their zeroes quickly. - **Graph Analysis**: Practice analyzing graphs to deduce the zeroes and potentially the factors of the polynomial. - **Zeroes Count**: Remember that a polynomial of degree (n) will have at most (n) zeroes. #### Example Problem 1. **Zeroes Identification from Graph**: If you are given a graph, identify the points where the curve intersects the x-axis and read off the zeroes. 2. **Graph Sketching**: If you are given a polynomial, find its zeroes and then sketch the graph ensuring it passes through the points on the x-axis corresponding to the zeroes. #### Practice and Understanding - Solve different types of problems which involve finding the zeroes of a polynomial and interpreting the geometrical meaning of the zeroes. - Practice sketching graphs of polynomials, ensuring they pass through the appropriate x-intercepts. - Practice questions where you interpret the number of zeroes based on the graph of a polynomial. Understand and recall these key points to enhance your problem-solving skills related to polynomials, their zeroes, and their graphical representations during the exam.

Relationship between Zeroes and Coefficients of a Polynomial
### Exam Notes: Relationship between Zeroes and Coefficients of a Polynomial **Class 10 Mathematics** **Chapter 2: Polynomials** Understanding the relationship between the zeroes and the coefficients of a polynomial can provide valuable insights into its structure and behavior without having to fully factor or graph it. #### Key Concepts 1. **Linear Polynomial:** - General Form: (ax + b = 0) - Zero ((alpha)): (alpha = -b/a) 2. **Quadratic Polynomial:** - General Form: (ax^2 + bx + c = 0) - Zeroes ((alpha) and (beta)): - Sum of zeroes ((alpha + beta)): (-b/a) (Sum of the roots = -Coefficient of (x)/Coefficient of (x^2)) - Product of zeroes ((alphabeta)): (c/a) (Product of the roots = Constant term/Coefficient of (x^2)) 3. **Cubic Polynomial:** - General Form: (ax^3 + bx^2 + cx + d = 0) - Zeroes ((alpha), (beta), and (gamma)): - Sum of zeroes ((alpha + beta + gamma)): (-b/a) - Sum of the product of zeroes taken two at a time: (c/a) - Product of zeroes ((alphabetagamma)): (-d/a) #### Note the Patterns - The sign alternates starting with negative in the sum of zeroes. - Coefficients and constant terms are always divided by the coefficient of the highest degree term. #### Example Questions and Tips for Exam 1. **Finding Zeroes from Coefficients:** - Given a polynomial and a zero, use the relationship to find other zeroes. - E.g., If one zero of (3x^2 - kx + 12 = 0) is 3, find the value of k and the other zero. 2. **Creating Polynomials:** - Formulate a polynomial when the relationship between its zeroes and coefficients is provided. - E.g., Form a quadratic polynomial having zeroes 4 and -3. 3. **Analyzing Polynomials:** - Determine possible expressions for a polynomial given specific relationships among its zeroes. - E.g., If two zeroes of the cubic polynomial (kx^3 - 3x^2 + x + 1 = 0) are 0 and 1, find the value of k. #### Exam Tips - **Understand Relationships:** Grasp the concept behind the relationship between zeroes and coefficients instead of just memorizing them. - **Sign Consistency:** Be mindful of the signs (positive or negative) in the relationships. - **Practical Application:** Practice applying these relationships in various types of problems, like finding the zeroes of the polynomial or constructing a polynomial with given zeroes. - **Verification:** If the zeroes are provided, substitute them back into the polynomial to ensure they satisfy the equation. - **Factoring and Zeroes:** Sometimes factoring the polynomial can provide a quick check or insight into its zeroes. Being thorough with these relationships and practicing various types of problems that utilize them will ensure a strong understanding and capability to solve problems in the exam effectively. Remember that sometimes these relationships can provide a shortcut to answering questions about a polynomial without having to fully factor it.

Exam Notes on Algebraic Methods of Solving a Pair of Linear Equations
### Exam Notes: Algebraic Methods of Solving a Pair of Linear Equations **Class 10 Mathematics** **Chapter: Pair of Linear Equations in Two Variables** A pair of linear equations in two variables (also known as simultaneous equations) can be solved algebraically using various methods. The primary algebraic methods include: #### 1. Substitution Method - **Procedure:** - Solve one of the equations for one variable. - Substitute this expression in the other equation. - Solve the resulting equation. - Sub back to get the value of the other variable. - **Usage:** - Best used when one of the variables can be easily isolated. - **Exam Tips:** - Ensure all equations are simplified before starting. - Validate the solution by substituting back into the original equations. #### 2. Elimination Method - **Procedure:** - Make the coefficients of one of the variables the same in both equations (if not already). - Add or subtract the equations to eliminate one variable, making it possible to find the value of the remaining variable. - Substitute the found value back into one of the original equations to find the other variable. - **Usage:** - Effective when the coefficients of one of the variables are the same or can easily be made the same. - **Exam Tips:** - Ensure the signs are correct when adding or subtracting. - Always check the solution in both original equations. #### 3. Cross-Multiplication Method - **Procedure:** - Given (a_1x + b_1y + c_1 = 0) and (a_2x + b_2y + c_2 = 0), solve for (x) and (y) by: [ x = frac{{b_1c_2 - b_2c_1}}{{a_1b_2 - a_2b_1}}, quad y = frac{{c_1a_2 - c_2a_1}}{{a_1b_2 - a_2b_1}} ] - **Usage:** - Useful when variables do not have the same coefficients. - **Exam Tips:** - Be cautious with the signs to avoid calculation errors. - Validate your solution. #### Application Questions - **Word Problems:** Translate the information into two linear equations and solve using a suitable method. - **Equations with Parameters:** Sometimes parameters are included, which may result in no solution, a unique solution, or infinitely many solutions, depending on their value. #### Special Cases - **Parallel Lines (No Solution):** When two lines are parallel, there's no common point, indicating no solution. - **Coincident Lines (Infinitely Many Solutions):** When two equations represent the same line, there are infinitely many solutions. - **Intersecting Lines (Unique Solution):** When two lines intersect, there's a unique solution representing the point of intersection. #### Exam Tips - **Selection of Method:** Choose the method that seems most straightforward based on the given equations. - **Validation:** Always validate solutions in both original equations to ensure accuracy. - **Parameter Evaluation:** For equations with parameters, discuss the various possibilities of their values. Understanding and practicing each method with different types of equations will help you quickly identify and apply the most efficient solution method in the exam. Remember to review the procedure for each method and understand the contexts in which they're most useful to enhance speed and accuracy during the exam.

Exam Notes on Equations Reducible to a Pair of Linear Equations in Two Variables
### Exam Notes: Equations Reducible to a Pair of Linear Equations in Two Variables **Class 10 Mathematics** **Chapter: Pair of Linear Equations in Two Variables** Certain types of non-linear equations, or equations that do not initially appear linear, can be transformed into a pair of linear equations, which can then be solved using familiar methods (e.g., substitution, elimination, or cross-multiplication). #### Key Concepts 1. **Identifying Reducible Equations:** - Typically involves expressions where variables are under radicals or within trigonometric functions. - May contain terms where variables are multiplied together. 2. **Transformation:** - Replace combinations of variables with a single variable to make the equation appear linear. - E.g., if an equation involves (sqrt{x}) and (sqrt{y}), let (p = sqrt{x}) and (q = sqrt{y}), and modify the equation accordingly. #### Transformative Substitutions 1. **Radical Equations:** - Example Form: ( sqrt{x} + sqrt{y} = a ) - Substitution: Let (p = sqrt{x}) and (q = sqrt{y}). 2. **Reciprocal Equations:** - Example Form: ( frac{a}{x} + frac{b}{y} = c ) - Substitution: Let (p = 1/x) and (q = 1/y). 3. **Trigonometric Equations:** - Example Form: ( asin x + bcos y = c ) - Substitution: Expressions involving trigonometric identities might be simplified by suitable substitutions. #### Solving Steps 1. **Perform Substitutions:** Replace complex terms with simpler variables. 2. **Form Linear Equations:** Use substituted variables to formulate two linear equations. 3. **Solve the Linear System:** Use a suitable method (substitution, elimination, or cross-multiplication) to find the values of the new variables. 4. **Back-Substitute:** Replace the newfound variable values into the original substitutions to find the values of the original variables. #### Example Questions - **Quadratic Equations:** Sometimes, a pair of equations might involve a quadratic and a linear equation in two variables. - **Mixed Types:** Often, the pair of equations may involve radicals and simple linear expressions, requiring apt substitutions. #### Exam Tips - **Method Selection:** Upon substitution, decide which method of solving linear equations is most efficient. - **Accuracy in Substitution:** Ensure substitutions and back-substitutions are done accurately to prevent errors. - **Check Solutions:** Verify obtained values by substituting them back into the original equations. #### Application Questions - **Word Problems:** Be adept at converting word problems into mathematical models, which may involve reducible equations. - **Geometry Problems:** Some problems might involve geometric figures, and relations among their elements might yield reducible equations. #### Key Takeaway - **Understand Transformations:** Have a clear understanding of different substitutions and how they simplify equations. - **Practice Different Types:** Engage in varied types of problems to enhance adaptability during the exam. Understanding and practicing the method of reducing non-linear equations to linear ones and solving them will bolster your problem-solving skills and enable you to tackle a wide array of problems in the exam effectively. Remember to always validate your solutions in the original equations to ensure accuracy.

Exam Notes on Solutions of Quadratic Equations by Factorization
### Exam Notes: Solutions of Quadratic Equations by Factorization **Class 10 Mathematics** **Chapter: Quadratic Equations** Quadratic equations are of the form ( ax^2 + bx + c = 0 ), where (a), (b), and (c) are constants, and (a neq 0). One method to find the roots (solutions) of such equations is the Factorization Method. #### Key Concepts 1. **Quadratic Equations:** - General form: ( ax^2 + bx + c = 0 ) - Roots: The values of (x) that satisfy the equation. 2. **Zero of a Polynomial:** - A number (k) is a zero of a polynomial (p(x)) if (p(k) = 0). 3. **Factorization:** - Expressing an expression as a product of its factors. #### Factorization Method ##### Step-by-Step Process: 1. **Expressing in Standard Form:** - Ensure the quadratic equation is in the standard form ( ax^2 + bx + c = 0 ). 2. **Expressing the Middle Term:** - Factorize (ac), if required, into two parts such that their sum is (b). - Modify the equation: ( ax^2 + mx + nx + c = 0 ) where (m) and (n) are the two parts. 3. **Factoring:** - Factorize by grouping the terms: ( a(x^2 + frac{m}{a}x) + (x^2 + frac{n}{a}x) = 0 ). - Or, if it's a perfect square, recognize and express it as ((px + q)^2 = 0). 4. **Finding the Roots:** - Apply the Zero-Product Property: If a product of two factors is zero, at least one of the factors must be zero. - Equate each factor to zero and solve for (x) in each case. #### Examples - **Simple Quadratic:** (x^2 - 5x + 6 = 0) - Factoring: ((x - 2)(x - 3) = 0) - Roots: (x = 2, 3) - **Coefficient of x^2 ≠ 1:** (6x^2 - 5x - 6 = 0) - Middle Term Split: (6x^2 - 6x + x - 6 = 0) - Factoring: (6x(x - 1) + 1(x - 1) = 0) - Roots: (x = 1, -frac{1}{6}) #### Special Cases - **Perfect Square:** ( (x + a)^2 = 0 ) - Root: (x = -a) - **Complex Roots:** Sometimes factorization may not be possible with real numbers, indicating complex roots. #### Exam Tips - **Factorization Skills:** Practice different types of factoring techniques. - **Verification:** Always verify by substituting the roots back into the original equation. - **No Real Roots:** If factorization is not possible, it may indicate that the equation has no real roots. - **Word Problems:** Translate word problems into quadratic equations and then solve by factorization. #### Practice Questions - **Factoring Practice:** Engage with various factoring techniques. - **Real-Life Applications:** Practice solving word problems like projectile motion, area problems, and more. #### Key Takeaways - **Ensure Factoring Accuracy:** The correct splitting of the middle term is crucial. - **Root Verification:** Always substitute back to validate the roots. - **Complex Roots:** Understand that not all quadratic equations will have real roots. By effectively understanding the factorization method and practicing various types of quadratic equations, you will be able to adeptly find the roots in the examination, ensuring accuracy and speed. Always remember to verify your solutions by substituting them back into the original equations.

Exam Notes: Finding the nth Term of an Arithmetic Progression (AP)
### Exam Notes: Finding the nth Term of an Arithmetic Progression (AP) **Class 10 Mathematics** **Chapter: Arithmetic Progressions** Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always constant. This constant difference could be positive, negative, or zero. #### Key Concepts 1. **Arithmetic Progression (AP):** - Defined by: (a, a + d, a + 2d, a + 3d, ldots ) - Where (a) is the first term and (d) is the common difference. 2. **Common Difference:** - (d = a_2 - a_1 = a_3 - a_2 = ldots = a_{n} - a_{n-1}) #### Finding the nth Term (General Term) of an AP The general or nth term ((a_n)) of an AP is given by: [ a_n = a + (n - 1)d ] ##### Explanation: - (a): First term of the AP. - (d): Common difference, found by subtracting the preceding term from the succeeding term. - (n): Position of the term in the AP. #### Application and Understanding 1. **Determining Any Term:** - Utilize the formula to determine any specific term of an AP without listing all preceding terms. 2. **Identifying Terms:** - You can identify if a number is part of an AP and determine its position by rearranging the formula. #### Exam Tips - **Deriving the Formula:** Ensure you understand how the formula is derived as it reinforces understanding. - **Negative Common Difference:** Be cautious with negative common differences as it means the AP is decreasing. - **Validation:** After calculating the nth term, verify by checking with a few terms if feasible. #### Sample Problems 1. **Find the nth term:** - Given an AP and a certain value of (n), compute (a_n). 2. **Identify Position:** - Given a term, find its position (n) in the AP. #### Examples - **Example 1:** Find the 10th term of the AP: 3, 7, 11, 15, … - Here, (a = 3) and (d = 4) - (a_{10} = 3 + (10 - 1)(4) = 39) - **Example 2:** Which term of the AP: 4, 2, 0, -2, … is -26? - (a = 4) and (d = -2) - (a_n = -26) - Using the formula: -26 = 4 + (n - 1)(-2) - Solving, (n = 15) #### Possible Missteps - **Neglecting “n-1”:** Sometimes students mistakenly use (n) instead of (n-1) in the formula. - **Incorrect Common Difference:** Ensure you subtract in the correct order to obtain the common difference. #### Practice Questions - **Varied APs:** Engage with APs of varied nature: increasing, decreasing, fractional, etc. - **Solve for Different Variables:** Sometimes you may need to find (d) or (a) given other variables. #### Conclusion - **Understand the Concept:** Rather than memorizing the formula, ensure you comprehend its derivation and usage. - **Practice:** Engage in a variety of problems to solidify your understanding. Mastering the concept of finding the nth term in an AP equips you to navigate through various problems in examinations, especially ones that require formulating and solving arithmetic sequences. Remember to validate your solutions and practice with a multitude of problems to enhance your problem-solving skills.

Exam Notes: Pythagoras Theorem
### Exam Notes: Pythagoras Theorem **Class 10 Mathematics** **Chapter: Triangles** The Pythagorean Theorem is a fundamental principle in geometry, discovered by the Greek mathematician Pythagoras, that describes a special relationship between the three sides of a right-angled triangle. #### Key Concepts 1. **Definition of Pythagoras Theorem:** - In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. [ c^2 = a^2 + b^2 ] 2. **Identifying the Sides:** - (c): Hypotenuse (longest side). - (a) and (b): Other two sides (perpendicular and base). #### Application - **Distance Between Two Points:** - Derive the distance formula using the Pythagorean theorem. - **Height of a Triangle:** - If the base and hypotenuse are known, you can find the height of the triangle. #### Proof of Pythagoras Theorem - **Geometric Proof:** Using similar triangles or by rearranging squares and rectangles. - **Algebraic Proof:** Leveraging algebraic equations and known geometric properties. Note: Understanding one proof deeply is often more valuable than knowing multiple proofs superficially. #### Converse of Pythagoras Theorem - Statement: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right-angled. - Mathematically: If ( c^2 = a^2 + b^2 ), then the triangle is right-angled at the vertex joining the sides (a) and (b). #### Exam Tips - **Diagram Drawing:** Always draw a labeled diagram to identify sides correctly. - **Unit Consistency:** Ensure all the side lengths are in the same units before applying the theorem. - **Accuracy:** Double-check calculations and substitutions to avoid silly mistakes. #### Example Problems - **Find the Hypotenuse:** Given the other two sides. - **Find the Other Side:** Given the hypotenuse and one side. - **Verify the Right Angle:** Determine whether a triangle is right-angled by verifying with the theorem. #### Examples - **Example 1:** Find the length of the hypotenuse in a triangle with other sides 3 cm and 4 cm. - ( c^2 = 3^2 + 4^2 ) - ( c = 5 ) cm - **Example 2:** Determine if the triangle with sides 5 cm, 12 cm, and 13 cm is a right-angle triangle. - Check: ( 13^2 = 5^2 + 12^2 ) - Verified: Triangle is right-angled. #### Practice Questions - **Application Questions:** Questions where the theorem must be applied indirectly (like in a word problem or a problem involving diagrams). - **Theoretical Questions:** Explaining the theorem, proofs, or related concepts in your own words. #### Summary and Key Takeaways - **Theoretical Understanding:** Ensure you understand the theorem and its converse, not just memorizing it. - **Problem-Solving:** Apply the theorem in various contexts, such as solving for different sides and identifying right-angled triangles. - **Proof:** Be well-versed with at least one proof of the theorem. The Pythagorean Theorem and its applications are crucial for solving various geometric problems in examinations. Ensure you have a robust understanding of the theorem, its converse, and its practical applications, and do remember to practice a variety of problems to solidify your understanding.

Exam Notes: Area of a Quadrilateral
### Exam Notes: Area of a Quadrilateral **Class: Variable (Generally applicable)** Quadrilaterals are four-sided polygons. While there are several types of quadrilaterals, we will focus on general methods of finding areas. Specific methods might apply to special quadrilaterals like squares, rectangles, parallelograms, etc. #### Key Concepts: 1. **Basic Formulae for Special Quadrilaterals:** - **Rectangle:** Area = Length × Breadth - **Square:** Area = Side × Side - **Parallelogram:** Area = Base × Height (Perpendicular height) - **Rhombus:** Area = (Diagonal1 × Diagonal2)/2 - **Trapezium:** Area = (1/2) × (Sum of parallel sides) × Height 2. **Using Triangles:** - Often, an arbitrary quadrilateral can be divided into two triangles. If the area of these triangles can be found, they can be added together to find the area of the quadrilateral. 3. **Using Diagonals and Height:** - Sometimes, using a diagonal to divide the quadrilateral and then applying respective formulas for each section can be useful. #### Detailed Concepts: - **Triangle Area using Sides (Heron's Formula):** If the sides of a triangle are (a), (b), and (c), and (s) is the semi-perimeter: [ s = frac{a + b + c}{2} ] [ text{Area} = sqrt{s(s-a)(s-b)(s-c)} ] - **Coordinates Geometry Method:** If the vertices of the quadrilateral are given, applying the Shoelace Theorem (or Shoelace Formula) is effective. If vertices are (A(x_1, y_1), B(x_2, y_2), C(x_3, y_3),) and (D(x_4, y_4)): [ text{Area} = frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - x_2y_1 - x_3y_2 - x_4y_3 - x_1y_4| ] #### Exam Tips: - **Choosing the Right Method:** Always opt for the method that seems simplest based on the given information. - **Accuracy in Calculations:** Ensure to double-check your calculations, especially in methods involving multiple steps. - **Diagram Construction:** Drawing an accurately labeled diagram will assist in choosing the method and ensuring no mistakes in the identification of dimensions or vertices. #### Examples: 1. **Find the Area of a Quadrilateral Given Sides:** - Break the figure into triangles, use appropriate methods for the triangles, and add the areas. 2. **Given Vertices:** - Apply the Shoelace Formula for a direct solution. #### Practice Questions: - **Geometric Application:** Find areas by forming quadrilaterals in combined figures or intersecting figures. - **Word Problems:** Apply area concepts in practical scenarios or problems where dimensions need to be extracted from given information. #### Summary: - **Understand Multiple Methods:** Be familiar with different approaches to find areas to be adaptable during exams. - **Application Practice:** Engage with problems that require application of area finding in diverse contexts. - **Conceptual Clarity:** Have clarity on the derivations and applications of each formula and method. Quadrilaterals offer a rich area (pun intended) for problem-solving in exams due to their varied nature and applicability of diverse methods based on available data. A comprehensive understanding of the principles, formulas, and application methods related to finding the area of quadrilaterals is key to scoring well in such problems.

Exam Notes on Area Of A Triangle
### Exam Notes: Area of a Triangle Understanding how to calculate the area of a triangle is fundamental in geometry. There are several methods, and the application of each method depends on the given parameters within the question. #### Key Concepts 1. **Basic Formula** - ( text{Area} = frac{1}{2} times text{Base} times text{Height} ) 2. **Using Three Sides (Heron’s Formula)** - ( s = frac{a + b + c}{2} ) - ( text{Area} = sqrt{s(s-a)(s-b)(s-c)} ) 3. **Using Trigonometry** - ( text{Area} = frac{1}{2} times a times b times sin(C) ) - This method is beneficial when two sides and the included angle is given. #### Detailed Concepts - **Equilateral Triangle** - ( text{Area} = frac{sqrt{3}}{4} times text{Side}^2 ) - **Isosceles Triangle** - ( text{Area} = frac{b}{4}sqrt{4a^2 - b^2} ) - Where ( a ) is the equal side length and ( b ) is the base. - **Using Coordinates of Vertices** - If vertices A(x1, y1), B(x2, y2), and C(x3, y3) are given: ( text{Area} = frac{1}{2} |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| ) #### Exam Tips - **Selecting the Right Method** - Always analyze what information is given and choose the method accordingly. - **Accurate Calculations** - Be especially careful with calculations under the square root in Heron's formula. - **Understand Variations** - Be clear with different formulae for varied types of triangles like isosceles, equilateral, etc. #### Examples - **Example 1:** If given the base and height: - Simply use ( text{Area} = frac{1}{2} times text{Base} times text{Height} ) - **Example 2:** If given all three sides, employ Heron’s Formula. #### Practice Questions - **Application-Based Problems:** - Calculating the area when points are given on a graph or part of a larger geometric figure. - **Problem Solving:** - Engage with word problems where you extract relevant information to calculate the area. #### Summary and Key Takeaways - **Multiple Methods:** Be aware of different approaches to calculate the area and choose wisely based on given data. - **Application:** Be adept at applying these formulas in various problems and real-world applications. - **Diagrammatic Representation:** Always sketch a labeled diagram to visualize the information provided. Understanding the different methods to calculate the area of a triangle and when to apply them is crucial for solving geometry problems efficiently in the exam. Ensure to practice multiple problems and understand the conceptual framework behind each formula to perform effectively in the exam.

Exam Notes On Trigonometric Ratios of Some Specific Angles
### Exam Notes: Trigonometric Ratios of Some Specific Angles **Key Concepts** 1. **Angles & Their Ratios:** - **0° and 90°:** Sine 0° = 0, Cosine 0° = 1, and Tangent 0° = 0; while Sine 90° = 1, Cosine 90° = 0, and Tangent 90° is undefined. - **30° (or π/6) and 60° (or π/3):** Common angles with standard trigonometric ratios (see below). 2. **Quadrants:** - **1st Quadrant (0° to 90°):** All trigonometric ratios are positive. - **2nd Quadrant (90° to 180°):** Sine is positive. - **3rd Quadrant (180° to 270°):** Tangent is positive. - **4th Quadrant (270° to 360°):** Cosine is positive. **Specific Ratios** - **30°:** - (sin 30° = frac{1}{2}, cos 30° = frac{sqrt{3}}{2}, tan 30° = frac{1}{sqrt{3}}) - (csc 30° = 2, sec 30° = frac{2}{sqrt{3}}, cot 30° = sqrt{3}) - **45°:** - (sin 45° = frac{1}{sqrt{2}}, cos 45° = frac{1}{sqrt{2}}, tan 45° = 1) - (csc 45° = sqrt{2}, sec 45° = sqrt{2}, cot 45° = 1) - **60°:** - (sin 60° = frac{sqrt{3}}{2}, cos 60° = frac{1}{2}, tan 60° = sqrt{3}) - (csc 60° = frac{2}{sqrt{3}}, sec 60° = 2, cot 60° = frac{1}{sqrt{3}}) **Exam Tips** - **Memorizing Values:** - It’s crucial to remember the standard values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90° as they are frequently utilized. - **Sine & Cosine Graphs:** - Understanding the sine and cosine wave graphs and their properties will help identify values and apply transformations. - **Applying Pythagorean Identity:** - Utilizing (sin^2 theta + cos^2 theta = 1) to derive other values as needed. - **Use of Calculators:** - If calculators are allowed, ensure you understand how to toggle between degrees and radians and use inverse functions. **Examples & Application** - **Solving Trigonometric Equations:** - Applying known ratios to solve for unknown variables within trigonometric functions. - **Evaluating Expressions:** - Evaluating expressions like ( sin 2theta ) and ( tan (90° - theta) ) using known angles and their respective trigonometric values. **Practice Questions** - **Ratios:** - Compute ratios for compound angles, like (sin (45° + 30°)) or (cos (90° - 60°)), using sum and difference of angle formulas. - **Applications:** - Solve problems involving right-angled triangles and their applications in real-world scenarios using trigonometry. **Summary** In trigonometry, understanding and memorizing the basic trigonometric ratios for specific angles is essential as it lays the foundation for solving problems involving angles and right-angled triangles. Focus on understanding the relationships between the ratios, and remember to apply the appropriate formulas and identities, particularly in contexts involving different quadrants and compound angles, to be successful in exam scenarios.

Study Notes on Class 10 Maths: Pair of Linear Equations in Two Variables
Study Notes on Class 10 Maths: Pair of Linear Equations in Two Variables --- Chapter Overview This chapter deals with linear equations in two variables, their graphical representation, methods to solve these equations, and their applications in real-life scenarios. --- 1. Introduction to Linear Equations in Two Variables A linear equation in two variables is of the form: [ ax + by + c = 0 ] where (a), (b), and (c) are real numbers, and (a) and (b) are not both zero. Examples: 1. ( 2x + 3y = 5 ) 2. ( x - y = 2 ) 3. ( 4x + 5y + 6 = 0 ) Key Points: - Standard Form: The equation is usually written as ( ax + by = c ). - Variables: (x) and (y) are the variables. - Solution: Any ordered pair ((x, y)) that satisfies the equation. --- 2. Graphical Method of Solution Graphical Representation: - Each linear equation in two variables represents a straight line in the Cartesian plane. - The solution of the pair of linear equations is the point of intersection of these lines. Steps to Plot: 1. Rewrite the equation in slope-intercept form ( y = mx + c ). 2. Identify the slope (m) and y-intercept (c). 3. Plot the y-intercept. 4. Use the slope to find another point on the line. 5. Draw the line through the points. Example: For ( 2x + 3y = 5 ): 1. Rewrite as ( y = -frac{2}{3}x + frac{5}{3} ). 2. Plot ( (0, frac{5}{3}) ). 3. Use slope ( -frac{2}{3} ) to find another point. 4. Draw the line. Intersection Points: - If the lines intersect at a point, that point is the solution. - If the lines are parallel, there is no solution. - If the lines coincide, there are infinitely many solutions. --- 3. Algebraic Methods of Solution Three main methods to solve the pair of linear equations: a. Substitution Method 1. Solve one equation for one variable. 2. Substitute this value in the other equation. 3. Solve for the second variable. 4. Substitute back to find the first variable. Example: Solve ( x + y = 7 ) and ( x - y = 3 ): 1. From ( x - y = 3 ), ( x = y + 3 ). 2. Substitute ( x = y + 3 ) in ( x + y = 7 ): ( (y + 3) + y = 7 ) ⟹ ( 2y = 4 ) ⟹ ( y = 2 ). 3. Substitute ( y = 2 ) in ( x = y + 3 ): ( x = 2 + 3 ) ⟹ ( x = 5 ). Solution: ( x = 5, y = 2 ). b. Elimination Method 1. Multiply equations to align coefficients. 2. Add or subtract equations to eliminate one variable. 3. Solve the resulting equation. 4. Substitute back to find the other variable. Example: Solve ( 3x + 2y = 11 ) and ( 2x + 3y = 4 ): 1. Multiply first equation by 3, second by 2: ( 9x + 6y = 33 ) and ( 4x + 6y = 8 ). 2. Subtract second from first: ( (9x + 6y) - (4x + 6y) = 33 - 8 ) ⟹ ( 5x = 25 ) ⟹ ( x = 5 ). 3. Substitute ( x = 5 ) in ( 3x + 2y = 11 ): ( 3(5) + 2y = 11 ) ⟹ ( 15 + 2y = 11 ) ⟹ ( 2y = -4 ) ⟹ ( y = -2 ). Solution: ( x = 5, y = -2 ). c. Cross Multiplication Method For equations ( a1x + b1y + c1 = 0 ) and ( a2x + b2y + c2 = 0 ): 1. Use formula: [ x = frac{b1c2 - b2c1}{a1b2 - a2b1}, quad y = frac{c1a2 - c2a1}{a1b2 - a2b1} ] 2. Check if ( a1b2 - a2b1 neq 0 ) (unique solution). 3. If ( a1b2 - a2b1 = 0 ): - If ( b1c2 - b2c1 = 0 ), infinitely many solutions. - Otherwise, no solution. Example: Solve ( 2x - 3y + 4 = 0 ) and ( x + y - 3 = 0 ): 1. Rewrite as ( 2x - 3y = -4 ) and ( x + y = 3 ). 2. Use cross multiplication: [ x = frac{(-3)(-3) - 1(-4)}{2(1) - (-3)(1)} = frac{9 + 4}{2 + 3} = frac{13}{5} = 2.6 ] [ y = frac{2(-3) - (-4)(1)}{2(1) - (-3)(1)} = frac{-6 + 4}{2 + 3} = frac{-2}{5} = -0.4 ] Solution: ( x = 2.6, y = -0.4 ). --- 4. Applications of Linear Equations Word Problems: 1. Number Problems: - Example: Sum of two numbers is 9, and their difference is 3. Find the numbers. - Solution: Let ( x ) and ( y ) be the numbers. [ x + y = 9 ] [ x - y = 3 ] Solve using substitution or elimination. 2. Age Problems: - Example: The sum of ages of two persons is 50, and one is 10 years older than the other. Find their ages. - Solution: Let ( x ) and ( y ) be their ages. [ x + y = 50 ] [ x - y = 10 ] Solve using substitution or elimination. 3. Cost and Revenue Problems: - Example: Cost of 3 pens and 4 pencils is ₹50, and the cost of 5 pens and 6 pencils is ₹78. Find the cost of each pen and pencil. - Solution: Let ( x ) be the cost of a pen, and ( y ) be the cost of a pencil. [ 3x + 4y = 50 ] [ 5x + 6y = 78 ] Solve using elimination or cross multiplication. --- 5. Special Cases of Linear Equations 1. No Solution (Parallel Lines): - When the lines are parallel, there is no solution. - Condition: ( frac{a1}{a2} = frac{b1}{b2} neq frac{c1}{c2} ). 2. Infinite Solutions (Coincident Lines): - When the lines coincide, there are infinitely many solutions. - Condition: ( frac{a1}{a2} = frac{b1}{b2} = frac{c1}{c2} ). 3. Unique Solution: - When the lines intersect at one point. - Condition: ( frac{a1}{a2} neq frac{b1}{b2} ). --- 6. Conclusion Understanding and solving pairs of linear equations in two variables is essential for mathematical proficiency in higher studies and practical applications. Mastery of graphical, substitution, elimination, and cross multiplication methods ensures a robust grasp of the concepts and their real-life utility. --- These notes provide a comprehensive overview of the topic, facilitating an in-depth understanding and ability to solve related problems efficiently.

Looking for the best coaching experience in the Northeast? Join our Aspire Academy and unlock your full potential. Our experienced teachers and personalized approach will help you achieve academic success and personal growth like never before. Don’t miss out on this opportunity to excel!

Quick Links

Social Icons

Facebook Youtube Instagram

Fill your information

We will contact you