Mathematics for Class X

1. Understanding Real Numbers and Their Properties

Since, which is an odd number as "a" is given to be a odd number.

Real Numbers Defined. Real numbers encompass both rational and irrational numbers, forming the foundation of much of mathematics. Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. Irrational numbers, on the other hand, cannot be expressed in this form; they have non-repeating, non-terminating decimal expansions.

Odd and Even Integers. The text explores properties of odd and even integers, demonstrating that the sum or product of such numbers follows specific rules. For instance, the sum of two odd numbers is always even, while the product of two consecutive positive integers is always divisible by 2. These properties are fundamental in number theory and are used in various proofs.

Divisibility Rules. The document also touches upon divisibility rules, such as proving that the product of three consecutive positive integers is divisible by 6. This involves considering different cases based on the form of the integer (e.g., 6q, 6q+1, etc.) and showing that in each case, the product is indeed divisible by 6. These divisibility rules are essential for simplifying calculations and understanding number relationships.

2. Mastering Euclidean Division and its Applications

By division lemma there exists integers q and r such that a = 6q+r where 0≤r<6.

Euclid's Division Lemma. This lemma states that for any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b. This lemma is a cornerstone in number theory and is used to prove various properties of integers.

HCF Calculation. The Euclidean division algorithm, derived from the division lemma, provides a systematic method for finding the highest common factor (HCF) of two positive integers. The process involves repeatedly applying the division lemma until the remainder is zero; the divisor at this stage is the HCF. For example, finding the HCF of 135 and 225 involves steps like 225 = 135 × 1 + 90, followed by 135 = 90 × 1 + 45, and so on, until the remainder is zero.

Linear Combinations. The HCF of two integers can be expressed as a linear combination of those integers. This means that if d is the HCF of a and b, then there exist integers x and y such that d = ax + by. This concept is crucial in solving Diophantine equations and understanding the structure of integers.

3. Prime Factorization: The Foundation of Number Theory

Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Fundamental Theorem of Arithmetic. This theorem asserts that every composite number can be uniquely expressed as a product of prime numbers, disregarding the order of the factors. This uniqueness is a fundamental property that underpins many results in number theory.

Expressing Integers as Products. The document provides examples of expressing integers as products of their prime factors. For instance, 420 = 2 × 2 × 3 × 5 × 7. This process involves systematically dividing the number by prime numbers until only prime factors remain.

Composite Numbers. The text explains why certain expressions, such as 7 × 11 × 13 + 13, are composite numbers. By factoring out common terms and demonstrating that the expression has factors other than 1 and itself, it's shown that these expressions are indeed composite.

4. HCF and LCM: Essential Tools for Number Manipulation

The largest positive integer which divides two or more integers without any remainder is called Highest Common Factor (HCF) or Greatest Common Divisor or Greatest Common Factor (GCF).

Definitions. The highest common factor (HCF) of two or more integers is the largest positive integer that divides all the integers without leaving a remainder. The least common multiple (LCM) is the smallest positive integer that is divisible by all the given integers.

Prime Factorization Method. The HCF and LCM can be found using the prime factorization method. This involves expressing each number as a product of its prime factors and then identifying the common factors (for HCF) or all factors with the highest powers (for LCM). For example, to find the HCF of 32 and 54, we have 32 = 2^5 and 54 = 2 × 3^3, so the HCF is 2.

Euclid's Algorithm. Euclid's division algorithm provides an efficient method for finding the HCF of two integers. This involves repeatedly applying the division lemma until the remainder is zero; the divisor at this stage is the HCF. This method is particularly useful for large numbers where prime factorization may be cumbersome.

5. Irrational Numbers: Proofs and Properties

Let assume that is rational.

Defining Irrationality. Irrational numbers cannot be expressed as a simple fraction p/q, where p and q are integers. Their decimal expansions are non-repeating and non-terminating. Common examples include √2, √3, and π.

Proof by Contradiction. The document demonstrates how to prove that certain numbers are irrational using proof by contradiction. This involves assuming the number is rational, then deriving a contradiction, thereby proving the initial assumption false. For example, to prove √2 is irrational, assume √2 = p/q, square both sides, and show that this leads to a contradiction regarding the nature of p and q.

Irrationality of Expressions. The text extends the concept to proving the irrationality of expressions involving irrational numbers, such as 1/√2 or 6 + √2. These proofs often involve algebraic manipulation to isolate the irrational number and then applying a similar contradiction argument.

6. Decimal Expansions: Terminating and Non-Terminating

If any number ends with the digit 0, it should be divisible by 10 or in other words its prime factorization must include primes 2 and 5 both as 10 = 2 ×5.

Terminating Decimals. A rational number p/q has a terminating decimal expansion if the prime factorization of q is of the form 2^m × 5^n, where m and n are non-negative integers. This means the denominator can only have 2 and 5 as prime factors.

Non-Terminating Repeating Decimals. If the prime factorization of q contains factors other than 2 or 5, then the decimal expansion of p/q is non-terminating and repeating (recurring). This is because the division process will continue indefinitely, with a repeating pattern of digits.

Converting Rational Numbers. The document provides examples of converting rational numbers to decimal expansions by writing their denominators in the form 2^m × 5^n. This involves multiplying the numerator and denominator by appropriate factors to achieve the desired form.

7. Polynomials: Zeros and Coefficients

In a polynomial the relations hold are as follows:sum of zeroes is equal to product of zeroes is equal to .

Zeros of Polynomials. A zero of a polynomial f(x) is a value of x for which f(x) = 0. For a quadratic polynomial ax^2 + bx + c, the zeros are the solutions to the equation ax^2 + bx + c = 0.

Relationship Between Zeros and Coefficients. There are specific relationships between the zeros (α and β) and the coefficients of a quadratic polynomial ax^2 + bx + c:

• Sum of zeros: α + β = -b/a
• Product of zeros: αβ = c/a

Forming Polynomials from Zeros. Given the zeros of a quadratic polynomial, one can form the polynomial using the formula: f(x) = k{x^2 - (sum of zeros)x + product of the zeros}, where k is a constant. This allows for the construction of polynomials with specific properties.

8. Division Algorithm for Polynomials

By applying division algorithm, we have: = ( )( )

Division Algorithm. For any two polynomials f(x) and g(x) (where g(x) ≠ 0), there exist polynomials q(x) (quotient) and r(x) (remainder) such that f(x) = g(x)q(x) + r(x), where r(x) = 0 or degree(r(x)) < degree(g(x)). This algorithm is fundamental in polynomial algebra.

Applying the Algorithm. The document provides examples of applying the division algorithm to find the quotient and remainder when dividing one polynomial by another. This involves long division of polynomials, similar to long division with numbers.

Finding Zeros. If a polynomial f(x) is divided by (x - a) and the remainder is zero, then a is a zero of f(x). This concept is used to find all zeros of a polynomial when some zeros are already known.

9. Quadratic Equations: Factorization Techniques

For zeros of f(x), f(x) = 0

Factorization. Factorization is a method to solve quadratic equations by expressing the quadratic polynomial as a product of two linear factors. This involves splitting the middle term of the quadratic equation in such a way that the equation can be easily factored.

Splitting the Middle Term. The document provides numerous examples of solving quadratic equations by splitting the middle term. This involves finding two numbers whose sum is equal to the coefficient of the middle term and whose product is equal to the product of the coefficient of the first term and the constant term.

Finding Roots. Once the quadratic equation is factored, the roots can be found by setting each factor equal to zero and solving for x. For example, if (x - a)(x - b) = 0, then x = a or x = b.

10. Nature of Roots: Discriminant Analysis

For a number to be divisible by 6, it should be divisible by 2 and 3 both,

Discriminant. For a quadratic equation ax^2 + bx + c = 0, the discriminant (D) is given by D = b^2 - 4ac. The discriminant determines the nature of the roots of the quadratic equation.

Types of Roots. The nature of the roots depends on the value of the discriminant:

• If D > 0, the roots are real and unequal.
• If D = 0, the roots are real and equal.
• If D < 0, the roots are not real (complex).

Applying Discriminant. The document provides examples of using the discriminant to determine whether quadratic equations have real roots and, if so, whether the roots are equal or unequal. This involves calculating the discriminant and then applying the rules above.

11. Solving Real-World Problems with Quadratic Equations

A quadratic equation when sum and product of its zeros is given by: , where k is a constant

Forming Quadratic Equations. Many real-world problems can be modeled using quadratic equations. These problems often involve finding unknown quantities that satisfy certain conditions, such as the product of consecutive integers or the dimensions of a rectangle.

Problem-Solving Strategies. The document provides examples of forming quadratic equations from word problems and then solving them using factorization or other methods. This involves carefully translating the problem into mathematical terms and then applying algebraic techniques to find the solution.

Interpreting Solutions. Once the quadratic equation is solved, it's important to interpret the solutions in the context of the original problem. This may involve discarding negative or non-sensical solutions and ensuring that the remaining solutions make sense in the given situation.

12. Coordinate Geometry: Distance, Section, and Midpoint Formulas

linear form:657a + 306b = 9The above equation have many solutions, one of them is a = -15, b = 22i.e. 9 = 657(-15) + 306(22)

Distance Formula. The distance between two points (x1, y1) and (x2, y2) in a coordinate plane is given by the formula: √((x2 - x1)^2 + (y2 - y1)^2). This formula is derived from the Pythagorean theorem and is used to calculate the length of a line segment.

Section Formula. The coordinates of a point P(x, y) that divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio m:n are given by:

• x = (mx2 + nx1) / (m + n)
• y = (my2 + ny1) / (m + n)
  This formula is used to find the coordinates of a point that divides a line segment in a specific ratio.

Midpoint Formula. The midpoint of a line segment joining A(x1, y1) and B(x2, y2) has coordinates:

• x = (x1 + x2) / 2
• y = (y1 + y2) / 2
  This is a special case of the section formula where the ratio is 1:1.

Last updated: April 3, 2025