Math Maha Marathon UP Exam - Class 10 UP Board Mathematics

Real Numbers, Polynomials, and Pair of Linear Equations in Two Variables

Real Numbers: Types and Properties

Real numbers consist of both rational and irrational numbers. Rational numbers can be written in the form $p/q$ where $p$ and $q$ are integers and $q \neq 0$. Irrational numbers cannot be written in this form, and typically include numbers under a positive square root that are not perfect squares.

• Rational Numbers: All integers, positive, negative, and zero, are rational. All natural numbers and whole numbers are included.
• Irrational Numbers: Numbers like $\sqrt{2}$, $\sqrt{5}$, etc., where the value under the…

Real Numbers, Polynomials, and Pair of Linear Equations in Two Variables

Real Numbers: Types and Properties

Real numbers consist of both rational and irrational numbers. Rational numbers can be written in the form $p/q$ where $p$ and $q$ are integers and $q \neq 0$. Irrational numbers cannot be written in this form, and typically include numbers under a positive square root that are not perfect squares.

• Rational Numbers: All integers, positive, negative, and zero, are rational. All natural numbers and whole numbers are included.
• Irrational Numbers: Numbers like $\sqrt{2}$, $\sqrt{5}$, etc., where the value under the root is not a perfect square.
• Real Number Definition: The set of real numbers is the collection of all rational and irrational numbers.

Prime, Composite, and Co-prime Numbers

Prime numbers are those with exactly two factors: 1 and the number itself. Composite numbers have more than two factors. Co-prime numbers are pairs of numbers whose only common factor is 1.

• Prime Numbers: Examples include 2, 3, 5, 7, 11.
• Composite Numbers: Examples include 4, 6, 8, 9, 12.
• Co-prime Numbers: Two numbers like 8 and 9, whose only common factor is 1.

Fundamental Theorem of Arithmetic and Prime Factorization

The fundamental theorem of arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way (apart from the order of the factors). Prime factorization is the process of breaking a number down into its prime factors.

• Prime Factorization Example: $336 = 2^4 \times 3^1 \times 7^1$
• Application: Used to find HCF and LCM of numbers.

HCF and LCM: Calculation and Formula

HCF (Highest Common Factor) and LCM (Lowest Common Multiple) can be found using the prime factorization method. The product of HCF and LCM of two numbers equals the product of the numbers themselves.

• Formula: For numbers $a$ and $b$, $a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$
• HCF: Product of common prime factors with the lowest powers.
• LCM: Product of all prime factors with the highest powers.

Exam-Focused Examples and Questions

Several exam-style questions are directly solved using the above concepts. For instance, finding the prime factorization of $156$ yields $2^2 \times 3 \times 13$. For $96$ and $404$, their HCF is $4$ and LCM is $9696$. For $15$ and $25$ with HCF $5$, LCM is $75$.

• Prime Factorization: $140 = 2^2 \times 5 \times 7$
• HCF Example: $306$ and $657$ have HCF $9$.
• Co-prime Pair Example: $18$ and $25$ are co-prime.

Properties of Rational and Irrational Numbers

The sum of a rational and an irrational number is always irrational. The product or quotient (except by zero) of a rational and an irrational number is also irrational.

• Rule: Rational $+$ Irrational $=$ Irrational
• Rule: Rational $\times$ Irrational $=$ Irrational (except when rational is zero)

Proving Irrationality

To prove a number is irrational, assume it is rational, express it as $p/q$, rearrange to isolate the irrational part, and show a contradiction arises because the irrational part cannot be written as a rational number.

• Example: To prove $3 + 2\sqrt{5}$ is irrational:
  • Assume $3 + 2\sqrt{5} = p/q$
  • Rearranged: $2\sqrt{5} = p/q - 3$
  • $p/q - 3$ is rational, but $2\sqrt{5}$ is irrational, leading to contradiction.

Polynomials: Definition, Types, and Properties

A polynomial is an algebraic expression where the powers of the variables are always whole numbers. The variables are called indeterminates or "chars," and the constants are called coefficients.

• Polynomial Definition: Expressions where variable powers are whole numbers (e.g., $x^2$, $y^3$).
• Types by Number of Terms:
  • Monomial: One term
  • Binomial: Two terms
  • Trinomial: Three terms
• Types by Degree:
  • Linear: Degree 1
  • Quadratic: Degree 2
  • Cubic: Degree 3

Degree, Terms, and Coefficients

The degree of a polynomial is the highest power of the variable. Each term is separated by $+$ or $-$ signs. The coefficient is the number multiplying the variable.

• Degree: Highest exponent in the polynomial.
• Terms: Each part separated by $+$ or $-$.
• Coefficient: Number attached to the variable, including its sign.

Value and Zeroes of a Polynomial

The value of a polynomial at a given variable is found by substituting the value. The zero of a polynomial is the value of the variable for which the polynomial equals zero.

• Zero of Polynomial: If $P(x) = 0$ for some $x$, then $x$ is a zero.
• Linear Polynomial Zero: Set $P(x) = 0$ and solve for $x$.
• Quadratic Polynomial Zero: Factorize and set each factor to zero.

Geometric Interpretation of Zeroes

The graph of a polynomial cuts the $x$-axis at points corresponding to its zeroes. The number of times the graph cuts the $x$-axis equals the maximum number of zeroes.

• Linear Polynomial: Cuts $x$-axis at one point (maximum one zero).
• Quadratic Polynomial: Parabola cuts $x$-axis at up to two points (maximum two zeroes).
• Cubic Polynomial: Wave cuts $x$-axis at up to three points (maximum three zeroes).

Relationship Between Zeroes and Coefficients in Quadratic Polynomials

For a quadratic polynomial $ax^2 + bx + c$, if $\alpha$ and $\beta$ are its zeroes:

• Sum of Zeroes: $\alpha + \beta = -b/a$
• Product of Zeroes: $\alpha \beta = c/a$
• Constructing Quadratic Polynomial: $P(x) = k[x^2 - (\text{sum of zeroes})x + (\text{product of zeroes})]$, where $k$ is any real number (usually $1$).

Pair of Linear Equations in Two Variables

A pair of linear equations in two variables is two equations of the form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, where $a_1, b_1, c_1, a_2, b_2, c_2$ are real numbers and $a_1, b_1$ and $a_2, b_2$ are not both zero.

• Form: $ax + by + c = 0$
• Solution: The values of $x$ and $y$ that satisfy both equations.

Methods to Solve Pair of Linear Equations

Three methods are included: Graphical, Substitution, and Elimination.

• Graphical Method: Plot both equations on a graph. The intersection point(s) give the solution(s).
• Substitution Method: Solve one equation for one variable, substitute into the other, and solve.
• Elimination Method: Multiply equations to equalize coefficients, add or subtract to eliminate one variable, solve for the other.

Nature of Solutions and Graphical Interpretation

The nature of solutions depends on the relationship between the coefficients:

• Unique Solution (Intersecting Lines): $a_1/a_2 \neq b_1/b_2$

• No Solution (Parallel Lines): $a_1/a_2 = b_1/b_2 \neq c_1/c_2$

• Infinite Solutions (Coincident Lines): $a_1/a_2 = b_1/b_2 = c_1/c_2$

• Consistent System: At least one solution exists (unique or infinite).

• Inconsistent System: No solution exists.

Exam-Style Application and Problem Types

Problems include finding HCF/LCM, proving irrationality, constructing polynomials from zeroes, and solving pairs of linear equations using all three methods. Graphical questions require plotting and interpreting intersection points. Substitution and elimination methods are used for word problems involving ages, prices, and other relationships.

• Key Formula: $a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$

• Quadratic Zeroes: $\alpha + \beta = -b/a$, $\alpha \beta = c/a$

• Linear Equation Solution Types: Unique, none, infinite, based on coefficient ratios.

• For graphical method, plot two points for each equation, draw lines, and find intersection.

• For substitution, express one variable in terms of the other, substitute, and solve.

• For elimination, equalize coefficients, add/subtract, and solve.

• Always check the nature of the solution before solving: intersecting, parallel, or coincident lines.