Maths Maha Revision Maths Bihar - Bihar Board Mathematics

Number Systems and Algebraic Foundations

This summary covers essential concepts from natural numbers to polynomials, including definitions, properties, and exam-focused distinctions. It also addresses coordinate geometry basics and angle types, with explicit rules, methods, and formulae as directly stated.

Types of Numbers: Definitions and Properties

Numbers are classified based on their properties and representations. Understanding these distinctions is crucial for identifying number types in questions and avoiding common errors.

• Natural Numbers ($\mathbb{N}$): Counting numbers starting from $1$ ($1, 2, 3, \ldots$). Zero is not included.
• Whole Numbers ($\mathbb{W}$): All natural numbers plus zero ($0, 1, 2, 3, \ldots$).…

Number Systems and Algebraic Foundations

This summary covers essential concepts from natural numbers to polynomials, including definitions, properties, and exam-focused distinctions. It also addresses coordinate geometry basics and angle types, with explicit rules, methods, and formulae as directly stated.

Types of Numbers: Definitions and Properties

Numbers are classified based on their properties and representations. Understanding these distinctions is crucial for identifying number types in questions and avoiding common errors.

• Natural Numbers ($\mathbb{N}$): Counting numbers starting from $1$ ($1, 2, 3, \ldots$). Zero is not included.
• Whole Numbers ($\mathbb{W}$): All natural numbers plus zero ($0, 1, 2, 3, \ldots$). Negative numbers are not included.
• Integers ($\mathbb{Z}$ or $i$): All whole numbers and their negatives ($\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$).
• Rational Numbers: Numbers expressible as $p/q$ where $p$ and $q$ are integers and $q \neq 0$.
• Irrational Numbers: Numbers that cannot be written as $p/q$; their decimal expansions are non-terminating and non-repeating (e.g., $\sqrt{2}$, $\pi$).
• Key distinctions:
  • All natural numbers are whole numbers, but not all whole numbers are natural numbers (since $0$ is not natural).
  • All whole numbers are integers, but not all integers are whole numbers (since negatives are not whole).
  • $-1$ is not a whole number.
  • $0$ is a whole number and a rational number ($0/1$, $0/10$, etc.).
  • $0/0$ is undefined.

Rational and Irrational Numbers: Decimal Expansions

Decimal expansions help distinguish between rational and irrational numbers. Recognizing patterns in decimals is essential for classification.

• Terminating Decimals: Decimals that end after a finite number of digits (e.g., $0.5$, $0.71$). These are rational numbers.
• Non-Terminating, Repeating Decimals: Decimals with a repeating pattern (e.g., $0.\overline{3}$, $0.\overline{52}$). These are rational numbers.
• Non-Terminating, Non-Repeating Decimals: Decimals with no repeating pattern (e.g., digits after $\pi$). These are irrational numbers.
• Conversion to Fraction:
  • For $0.71$, write as $71/100$.
  • For $0.\overline{3}$, write as $3/9 = 1/3$.
  • For $0.\overline{52}$, write as $52/99$.
• Rule: The number of repeating digits determines the number of $9$s in the denominator.

Rationalization of Denominators

Rationalization is the process of removing irrational numbers from the denominator. The transcript provides explicit steps for both single and binomial denominators.

• Single Surd Denominator: Multiply numerator and denominator by the same surd.
  • Example: $\frac{1}{\sqrt{3}} \to \frac{\sqrt{3}}{3}$
• Binomial Surd Denominator: Multiply numerator and denominator by the conjugate (change the sign between terms).
  • Example: $\frac{1}{\sqrt{5}-2} \to \frac{\sqrt{5}+2}{(\sqrt{5})^2 - (2)^2} = \frac{\sqrt{5}+2}{5-4} = \sqrt{5}+2$
• Special Case: If the denominator contains $\sqrt{-3}$, use $i$ (iota), where $i^2 = -1$.
  • $\sqrt{-3} = \sqrt{3} \cdot i$
  • $(\sqrt{-3})^2 = -3$
• Key Rule: For $a-b$ in the denominator, multiply by $a+b$; for $a+b$, multiply by $a-b$.

Laws of Exponents

Exponent rules are essential for simplifying expressions and solving equations. The transcript provides explicit rules and examples.

• Multiplication (Same Base): Add exponents: $a^m \times a^n = a^{m+n}$
• Division (Same Base): Subtract exponents: $a^m / a^n = a^{m-n}$
• Negative Exponent: $a^{-n} = 1/a^n$
• Power of a Power: $(a^m)^n = a^{mn}$
• Product to a Power: $(ab)^n = a^n b^n$
• Quotient to a Power: $(a/b)^n = a^n / b^n$
• Zero Exponent: $a^0 = 1$ (for $a \neq 0$)
• Special Cases:
  • $1^n = 1$ for any natural number $n$
  • $0^n = 0$ for any natural number $n$
  • $0^0$ is undefined

Polynomials: Definition and Types

A polynomial is an algebraic expression with variables raised to non-negative integer powers and real coefficients. The transcript details criteria for identifying polynomials and their types.

• Polynomial: An expression where each term has a variable with a non-negative integer exponent and real coefficients.
  • Example: $3x^2 - 5x + 2$
• Not a Polynomial: If the variable is in the denominator or under a root (e.g., $1/x$, $\sqrt{x}$).
• Degree of a Polynomial: The highest power of the variable.
• Types by Degree:
  • Linear Polynomial: Degree $1$ (e.g., $3x+2$)
  • Quadratic Polynomial: Degree $2$ (e.g., $x^2-5x+2$)
  • Cubic Polynomial: Degree $3$ (e.g., $3x^3-5x^2+6x-9$)
  • Quartic Polynomial: Degree $4$
• Constant Polynomial: Degree $0$
• Key Rule: If the variable's exponent is negative or fractional, it is not a polynomial.

Zeros of a Polynomial

The zero (root) of a polynomial is the value of the variable that makes the polynomial equal to zero. The transcript provides the method for finding zeros.

• Finding Zeros:
  1. Set the polynomial equal to zero.
  2. Solve for the variable.
• Examples:
  • For $3x-6=0$, $x=2$ is the zero.
  • For $5x-10=0$, $x=2$ is the zero.
  • For $x^2-9=0$, $x=3$ and $x=-3$ are zeros.
• Number of Zeros:
  • Linear polynomial: $1$ zero
  • Quadratic polynomial: up to $2$ zeros
  • Cubic polynomial: up to $3$ zeros
  • Degree $n$ polynomial: up to $n$ zeros

Remainder Theorem

The remainder theorem provides a shortcut for finding the remainder when a polynomial is divided by a linear divisor.

• Method:
  1. Let $P(x)$ be the polynomial.
  2. For divisor $x-a$, set $x=a$.
  3. Substitute $x=a$ into $P(x)$; the result is the remainder.
• Example: For $P(x) = 3x^2-5x+2$ divided by $x-2$:
  • Substitute $x=2$: $3(2)^2 - 5(2) + 2 = 12 - 10 + 2 = 4$
  • Remainder is $4$.

Coordinate Geometry: Cartesian Plane and Coordinates

The Cartesian plane is divided by the $x$-axis (horizontal) and $y$-axis (vertical) into four quadrants. Points are represented by ordered pairs.

• Axes:
  • $x$-axis: Horizontal (abscissa)
  • $y$-axis: Vertical (ordinate)
• Origin: Intersection point of $x$ and $y$ axes; coordinates $(0,0)$.
• Quadrants: Numbered counterclockwise from the top right.
  • First: $(+,+)$
  • Second: $(-,+)$
  • Third: $(-,-)$
  • Fourth: $(+,-)$
• Coordinates:
  • On $x$-axis: $(x,0)$
  • On $y$-axis: $(0,y)$
• Terminology:
  • First value: Abscissa (x-coordinate)
  • Second value: Ordinate (y-coordinate)

Types of Angles

Angles are classified based on their measures. The transcript provides explicit ranges and definitions.

• Acute Angle: Greater than $0^\circ$ and less than $90^\circ$
• Right Angle: Exactly $90^\circ$
• Obtuse Angle: Greater than $90^\circ$ and less than $180^\circ$
• Straight Angle: Exactly $180^\circ$
• Reflex Angle: Greater than $180^\circ$ and less than $360^\circ$
• Complete Angle: Exactly $360^\circ$

Complementary and Supplementary Angles

Complementary and supplementary angles are pairs of angles with specific sum properties.

• Complementary Angles: Two angles whose sum is $90^\circ$.
  • To find the complement of $\theta$, compute $90^\circ - \theta$.
  • Example: Complement of $30^\circ$ is $60^\circ$.
• Supplementary Angles: Two angles whose sum is $180^\circ$.
  • To find the supplement of $\theta$, compute $180^\circ - \theta$.
  • Example: Supplement of $70^\circ$ is $110^\circ$.

Properties of Lines and Angles

When lines intersect or are parallel, specific angle relationships arise. The transcript details these with explicit rules.

• Angles on a Straight Line: Sum to $180^\circ$.
• Linear Pair: Two adjacent angles on a straight line; their sum is $180^\circ$.
• Vertically Opposite Angles: When two lines intersect, opposite angles are equal.
• Parallel Lines Cut by a Transversal:
  • Corresponding Angles: Equal; same relative position.
  • Alternate Interior Angles: Equal; on opposite sides of the transversal, inside the lines.
  • Co-interior (Consecutive Interior) Angles: Sum to $180^\circ$; on the same side of the transversal, inside the lines.

Key Formulas and Results

The transcript provides several direct formulas and results for quick recall.

• Square of Numbers Near 100:
  • For $102^2$:
    • $102$ is $2$ more than $100$.
    • $2^2 = 4$ (write as $04$ for two digits).
    • $102 + 2 = 104$.
    • Final answer: $10404$.
  • General method: For $(100 + a)^2$, compute $a^2$ (two digits), $100 + a + a$, and combine.
• If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$ (direct formula).
• Conversion of Repeating Decimals to Fractions:
  • $0.\overline{3} = 3/9 = 1/3$
  • $0.\overline{52} = 52/99$

These rules, methods, and formulae are directly stated and should be memorized for objective-type questions. Practice is emphasized for mastery.