Class 9 Mathematics — Complete Formula & Concept Guide

Animated Diagram: Real Numbers Hierarchy

1.1 Natural Numbers (\(N\))

Definition: Counting numbers starting from 1 and extending to infinity (\(\infty\)).

Set Notation: \(N = \{1, 2, 3, 4, 5, \dots\}\)

Key Property: The smallest natural number is 1. Zero and negative numbers are excluded.

1.2 Whole Numbers (\(W\))

Definition: The collection of all natural numbers along with zero (0).

Set Notation: \(W = \{0, 1, 2, 3, 4, \dots\}\)

Key Property: Every natural number is a whole number (\(N \subset W\)), but 0 is a whole number that is not a natural number.

1.3 Integers (\(Z\))

Definition: The set of all whole numbers along with negative natural numbers.

Set Notation: \(Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}\)

• Positive Integers (\(Z^+\)): \(1, 2, 3, 4, \dots\)
• Negative Integers (\(Z^-\)): \(-1, -2, -3, -4, \dots\)
• Zero (0): Neither positive nor negative.

1.4 Rational Numbers (\(Q\))

Definition: Any number that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).

Examples: \(\frac{1}{2}, \frac{3}{4}, -\frac{2}{5}, 5 = \frac{5}{1}, 0 = \frac{0}{1}\)

Decimal Expansion: Either Terminating or Non-terminating Recurring (Repeating).

1.5 Irrational Numbers (\(Q'\))

Definition: Numbers that cannot be written in the form \(\frac{p}{q}\) where \(p, q \in Z, q \neq 0\).

Examples: \(\sqrt{2}, \sqrt{3}, \sqrt{5}, \pi, 0.1010010001\dots\)

Decimal Expansion: Always Non-terminating Non-repeating.

1.6 Real Numbers (\(R\))

Definition: The combined set of all rational (\(Q\)) and irrational (\(Q'\)) numbers.

Relation: \(R = Q \cup Q'\)

Key Property: Every real number corresponds to a unique point on the number line, and every point on the line represents a unique real number.

1.7 Types of Decimal Expansions

• 1. Terminating Decimal: The remainder becomes zero after finite steps. Example: \(\frac{1}{2} = 0.5\), \(\frac{7}{8} = 0.875\). (Denominator has prime factors only of form \(2^n \cdot 5^m\)).
• 2. Non-terminating Recurring Decimal: The remainder never becomes zero, and a block of digits repeats infinitely. Example: \(\frac{1}{3} = 0.333\dots = 0.\bar{3}\).
• 3. Non-terminating Non-recurring Decimal: The remainder never vanishes and digits show no repeating pattern. Example: \(\sqrt{2} = 1.41421356\dots\), \(\pi = 3.14159265\dots\).

1.8 Converting Non-Terminating Recurring Decimal to \(\frac{p}{q}\) Form

Step-by-Step Procedure:

1. Let \(x\) equal the repeating decimal: \(x = 0.\bar{6} = 0.666\dots\) (Eq. 1)
2. Multiply both sides by \(10^n\) where \(n\) is the number of repeating digits (for 1 digit, multiply by 10): \(10x = 6.666\dots\) (Eq. 2)
3. Subtract Eq. 1 from Eq. 2: \(10x - x = 6.666\dots - 0.666\dots \implies 9x = 6\)
4. Solve for \(x\): \(x = \frac{6}{9} = \frac{2}{3}\)

1.9 Representation of Real Numbers on the Number Line

Every real number is represented by a unique point on the real number line. Points to the right of the origin \(O(0)\) are positive, while points to the left are negative.

1.10 Process of Successive Magnification

Visualizing the position of a real number with a terminating/non-terminating decimal (e.g., \(3.765\)) using a magnifying lens step by step:

• Locate between integers 3 and 4; divide into 10 equal sub-intervals.
• Zoom into interval \([3.7, 3.8]\); divide into 10 equal sub-parts to find \([3.76, 3.77]\).
• Zoom into \([3.76, 3.77]\) to locate the exact point \(3.765\).

1.11 Representing Irrationals (\(\sqrt{2}, \sqrt{3}, \sqrt{5}\)) on Number Line

Using Pythagorean Theorem:

To locate \(\sqrt{n}\), first construct \(\sqrt{n-1}\) geometrically.

• For \(\sqrt{2}\): Mark unit length \(OA = 1\). Draw perpendicular \(AB = 1\). Hypotenuse \(OB = \sqrt{1^2 + 1^2} = \sqrt{2}\). Draw arc with radius \(OB\) onto the number line.
• For \(\sqrt{3}\): Draw perpendicular \(BC = 1\) on \(OB\). Hypotenuse \(OC = \sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{3}\).

1.12 Operations on Real Numbers

• Sum or difference of a rational and an irrational number is irrational. (e.g., \(3 + \sqrt{2}\))
• Product or quotient of a non-zero rational and an irrational number is irrational. (e.g., \(2\sqrt{3}\))
• Sum, difference, product, or quotient of two irrational numbers may be rational or irrational. (e.g., \(\sqrt{2} + (-\sqrt{2}) = 0\), which is rational).

1.13 Rationalization of the Denominator

Process of converting an irrational denominator into a rational number by multiplying numerator and denominator by a rationalizing factor:

$\begin{aligned} \frac{1}{\sqrt{a}} &= \frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a} \\ \frac{1}{\sqrt{a} + \sqrt{b}} &= \frac{1}{\sqrt{a} + \sqrt{b}} \times \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b} \\ \frac{1}{a + b\sqrt{c}} &= \frac{1}{a + b\sqrt{c}} \times \frac{a - b\sqrt{c}}{a - b\sqrt{c}} = \frac{a - b\sqrt{c}}{a^2 - b^2 c} \end{aligned}$

1.14 Laws of Exponents for Real Numbers

For positive real bases \(a, b > 0\) and rational exponents \(m, n\):

$\begin{aligned} a^m \cdot a^n &= a^{m+n} \\ (a^m)^n &= a^{m \cdot n} \\ \frac{a^m}{a^n} &= a^{m-n} \quad (a \neq 0) \\ a^m \cdot b^m &= (a \cdot b)^m \\ \left(\frac{a}{b}\right)^m &= \frac{a^m}{b^m} \\ a^0 &= 1 \quad (a \neq 0) \\ a^{-n} &= \frac{1}{a^n}, \quad a^{1/n} = \sqrt[n]{a} \end{aligned}$

1.15 Radicals and Surds

If \(a > 0\) is a rational number and \(\sqrt[n]{a}\) is irrational, it is called a Surd of order \(n\). Radical Identities:

$\begin{aligned} \sqrt{ab} &= \sqrt{a} \cdot \sqrt{b} \\ \sqrt{\frac{a}{b}} &= \frac{\sqrt{a}}{\sqrt{b}} \\ (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) &= a - b \\ (a + \sqrt{b})(a - \sqrt{b}) &= a^2 - b \\ (\sqrt{a} + \sqrt{b})^2 &= a + 2\sqrt{ab} + b \end{aligned}$

1.16 Density Property of Rational Numbers

Property: Between any two distinct rational numbers \(a\) and \(b\), there exist infinitely many rational and irrational numbers.

• Midpoint rational number \(= \frac{a + b}{2}\).
• To insert \(n\) rational numbers: Convert to equivalent fractions with denominator \((n+1)\).
• Geometric mean for irrational: \(\sqrt{ab}\) (provided \(ab\) is not a perfect square).

Solved Example 1.1: Rationalizing Denominator

Question: Rationalize the denominator of \(\frac{1}{7+3\sqrt{2}}\).

Solution: Multiply numerator and denominator by conjugate \((7-3\sqrt{2})\):

$\frac{1}{7+3\sqrt{2}} \times \frac{7-3\sqrt{2}}{7-3\sqrt{2}} = \frac{7-3\sqrt{2}}{7^2 - (3\sqrt{2})^2} = \frac{7-3\sqrt{2}}{49 - 18} = \frac{7-3\sqrt{2}}{31}$

2.1 Standard Form of Polynomial in One Variable

\(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\)

where \(a_n, a_{n-1}, \dots, a_0\) are real constants (\(a_n \neq 0\)) and exponent \(n\) is a non-negative integer \(\{0, 1, 2, 3, \dots\}\).

2.2 Degree of a Polynomial

The highest power of variable \(x\) in a polynomial is called its degree.

• Constant Polynomial: Degree = 0 (e.g., \(P(x) = 5 = 5x^0\))
• Zero Polynomial: \(P(x) = 0\) (Degree is Undefined)
• Linear Polynomial: Degree = 1 (e.g., \(ax + b, a \neq 0\))
• Quadratic Polynomial: Degree = 2 (e.g., \(ax^2 + bx + c, a \neq 0\))
• Cubic Polynomial: Degree = 3 (e.g., \(ax^3 + bx^2 + cx + d, a \neq 0\))
• Biquadratic Polynomial: Degree = 4

2.3 Classification by Number of Terms

• Monomial: 1 term only (e.g., \(2x, -5x^2\))
• Binomial: 2 terms (e.g., \(x + 1, 3x^2 - 4\))
• Trinomial: 3 terms (e.g., \(x^2 + 2x + 5\))

2.4 Zeroes of a Polynomial

A real number \(k\) is called a zero of polynomial \(P(x)\) if \(P(k) = 0\).

A polynomial of degree \(n\) has at most \(n\) real zeroes.

2.5 Remainder Theorem

If \(P(x)\) is a polynomial of degree \(\ge 1\) and is divided by linear polynomial \((x - a)\), the remainder is \(R = P(a)\).

2.6 Factor Theorem

Linear term \((x - a)\) is a factor of polynomial \(P(x)\) if and only if \(P(a) = 0\).

2.7 All 10 Algebraic Identities

$\begin{aligned} 1. \quad (a + b)^2 &= a^2 + 2ab + b^2 \\ 2. \quad (a - b)^2 &= a^2 - 2ab + b^2 \\ 3. \quad a^2 - b^2 &= (a - b)(a + b) \\ 4. \quad (x + a)(x + b) &= x^2 + (a + b)x + ab \\ 5. \quad (a + b + c)^2 &= a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ 6. \quad (a + b)^3 &= a^3 + b^3 + 3ab(a + b) = a^3 + 3a^2b + 3ab^2 + b^3 \\ 7. \quad (a - b)^3 &= a^3 - b^3 - 3ab(a - b) = a^3 - 3a^2b + 3ab^2 - b^3 \\ 8. \quad a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \\ 9. \quad a^3 - b^3 &= (a - b)(a^2 + ab + b^2) \\ 10. \quad a^3 + b^3 + c^3 - 3abc &= (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \end{aligned}$

Special Corollary: If \(a + b + c = 0\), then \(a^3 + b^3 + c^3 = 3abc\).

Solved Example 2.1: Factorization using Identity

Question: Factorize \(8x^3 + 27y^3 + z^3 - 18xyz\).

Solution: Rewrite as \((2x)^3 + (3y)^3 + z^3 - 3(2x)(3y)(z)\).

Applying Identity 10:

$= (2x + 3y + z)(4x^2 + 9y^2 + z^2 - 6xy - 3yz - 2zx)$

Animated Diagram: Cartesian Plane & Point P(x, y)

3.1 Axes & Origin

X-axis: The horizontal line in the Cartesian plane.

Y-axis: The vertical line in the Cartesian plane.

Origin (\(O\)): The point of intersection of X-axis and Y-axis with coordinates \(O(0,0)\).

3.2 Abscissa & Ordinate

For any point \(P(x, y)\):

• Abscissa (x-coordinate): Perpendicular distance from Y-axis.
• Ordinate (y-coordinate): Perpendicular distance from X-axis.
• Points on Axes: Any point on X-axis has ordinate 0, i.e., \((x, 0)\). Any point on Y-axis has abscissa 0, i.e., \((0, y)\).

3.3 Quadrant Signs

$\begin{aligned} \text{Quadrant I:} &\quad (+, +) \quad (x > 0, y > 0) \\ \text{Quadrant II:} &\quad (-, +) \quad (x < 0, y > 0) \\ \text{Quadrant III:} &\quad (-, -) \quad (x < 0, y < 0) \\ \text{Quadrant IV:} &\quad (+, -) \quad (x > 0, y < 0) \end{aligned}$

3.4 Distance & Midpoint Formulas

Distance between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\):

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Distance of \(P(x,y)\) from origin \(O(0,0)\): \(OP = \sqrt{x^2 + y^2}\)

Coordinates of midpoint of segment \(AB\): \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

4.1 Solutions of a Linear Equation

Standard Form: \(ax + by + c = 0\)

Number of Solutions: A linear equation in two variables has infinitely many solutions.

Each solution \((x, y)\) corresponds to a point lying on the straight line representing the graph of the equation.

4.2 Graph of a Linear Equation

The graph of every linear equation in two variables is always a straight line.

• Equation of line passing through origin: \(y = mx\) (where constant \(c = 0\))
• Equation of X-axis: \(y = 0\)
• Equation of Y-axis: \(x = 0\)

4.3 Equations of Lines Parallel to Coordinate Axes

$\begin{aligned} x = a \quad &\longrightarrow \text{ Line parallel to Y-axis at distance } a \\ y = b \quad &\longrightarrow \text{ Line parallel to X-axis at distance } b \end{aligned}$

4.4 Slope-Intercept Form

$y = mx + c$

where slope \(m = -\frac{a}{b}\) and Y-intercept \(c = -\frac{c}{b}\).

5.1 Euclid's 5 Postulates

1. A straight line may be drawn from any one point to any other point.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any center and any radius.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles (\(180^\circ\)), the two lines if produced indefinitely meet on that side.

5.2 Types of Angles

• Acute Angle: \(0^\circ < \theta < 90^\circ\)
• Right Angle: \(\theta = 90^\circ\)
• Obtuse Angle: \(90^\circ < \theta < 180^\circ\)
• Straight Angle: \(\theta = 180^\circ\)
• Reflex Angle: \(180^\circ < \theta < 360^\circ\)
• Complete Angle: \(\theta = 360^\circ\)

5.3 Angle Pairs & Axioms

• Complementary Angles: Sum of two angles \(= 90^\circ\) (e.g., \(30^\circ\) and \(60^\circ\))
• Supplementary Angles: Sum of two angles \(= 180^\circ\) (e.g., \(110^\circ\) and \(70^\circ\))
• Linear Pair Axiom: If a ray stands on a line, then the sum of two adjacent angles formed is \(180^\circ\). (\(\angle 1 + \angle 2 = 180^\circ\))
• Vertically Opposite Angles: Vertically opposite angles formed when two lines intersect are equal. (\(\angle 1 = \angle 3, \angle 2 = \angle 4\))

5.4 Transversal & Parallel Lines

When a transversal intersects two parallel lines \(l \parallel m\):

• Corresponding Angles: Are equal (\(\angle 1 = \angle 5, \angle 2 = \angle 6\))
• Alternate Interior Angles: Are equal (\(\angle 3 = \angle 5, \angle 4 = \angle 6\))
• Co-interior Angles: Are supplementary (\(\angle 3 + \angle 6 = 180^\circ, \angle 4 + \angle 5 = 180^\circ\))

5.5 Angle Sum Property of a Triangle

1. Sum of all interior angles of a triangle is always \(180^\circ\):

$\angle A + \angle B + \angle C = 180^\circ$

2. Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of its two opposite interior angles:

$\angle \text{ext} = \angle A + \angle B$

6.1 5 Criteria for Congruence

1. SAS (Side-Angle-Side): Two sides and the included angle are equal.
2. ASA (Angle-Side-Angle): Two angles and the included side are equal.
3. AAS (Angle-Angle-Side): Two angles and any one corresponding side are equal.
4. SSS (Side-Side-Side): All three corresponding sides are equal.
5. RHS (Right angle-Hypotenuse-Side): Hypotenuse and one side of right triangles are equal.

6.2 Properties of Isosceles Triangles

• Isosceles Triangle Theorem: Angles opposite to equal sides are equal. (If \(AB = AC\), then \(\angle B = \angle C\))
• Converse: Sides opposite to equal angles are equal. (If \(\angle B = \angle C\), then \(AB = AC\))

6.3 Inequalities in a Triangle

• Sum of any two sides of a triangle is strictly greater than the third side: \(a + b > c\)
• In any triangle, the side opposite to the larger angle is longer.
• Angle opposite to the longer side is larger.

7.1 6 Trigonometric Ratios in a Right Triangle

For angle \(\theta\) in a right triangle (Perpendicular = \(P\), Base = \(B\), Hypotenuse = \(H\)):

$\begin{aligned} \sin\theta &= \frac{P}{H}, &\quad \csc\theta &= \frac{H}{P} \\[0.5em] \cos\theta &= \frac{B}{H}, &\quad \sec\theta &= \frac{H}{B} \\[0.5em] \tan\theta &= \frac{P}{B}, &\quad \cot\theta &= \frac{B}{P} \end{aligned}$

7.2 Reciprocal & Quotient Relations

$\begin{aligned} \csc\theta &= \frac{1}{\sin\theta}, &\quad \sec\theta &= \frac{1}{\cos\theta}, &\quad \cot\theta &= \frac{1}{\tan\theta} \\[0.5em] \tan\theta &= \frac{\sin\theta}{\cos\theta}, &\quad \cot\theta &= \frac{\cos\theta}{\sin\theta} \end{aligned}$

7.3 Fundamental Trigonometric Identities

$\begin{aligned} 1. \quad &\sin^2\theta + \cos^2\theta = 1 \implies \sin^2\theta = 1 - \cos^2\theta, \quad \cos^2\theta = 1 - \sin^2\theta \\[0.5em] 2. \quad &1 + \tan^2\theta = \sec^2\theta \implies \sec^2\theta - \tan^2\theta = 1, \quad \tan^2\theta = \sec^2\theta - 1 \\[0.5em] 3. \quad &1 + \cot^2\theta = \csc^2\theta \implies \csc^2\theta - \cot^2\theta = 1, \quad \cot^2\theta = \csc^2\theta - 1 \end{aligned}$

7.4 Complete Trigonometric Table for Standard Angles

Ratio \(\theta\)	\(0^\circ\)(0 rad)	\(30^\circ\)(\(\frac{\pi}{6}\))	\(45^\circ\)(\(\frac{\pi}{4}\))	\(60^\circ\)(\(\frac{\pi}{3}\))	\(90^\circ\)(\(\frac{\pi}{2}\))
\(\sin\theta\)	\(0\)	\(\frac{1}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{\sqrt{3}}{2}\)	\(1\)
\(\cos\theta\)	\(1\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{1}{2}\)	\(0\)
\(\tan\theta\)	\(0\)	\(\frac{1}{\sqrt{3}}\)	\(1\)	\(\sqrt{3}\)	Undefined (\(\infty\))
\(\cot\theta\)	Undefined (\(\infty\))	\(\sqrt{3}\)	\(1\)	\(\frac{1}{\sqrt{3}}\)	\(0\)
\(\sec\theta\)	\(1\)	\(\frac{2}{\sqrt{3}}\)	\(\sqrt{2}\)	\(2\)	Undefined (\(\infty\))
\(\csc\theta\)	Undefined (\(\infty\))	\(2\)	\(\sqrt{2}\)	\(\frac{2}{\sqrt{3}}\)	\(1\)

7.5 Complementary Angle Ratios

$\begin{aligned} \sin(90^\circ - \theta) &= \cos\theta, &\quad \cos(90^\circ - \theta) &= \sin\theta \\[0.5em] \tan(90^\circ - \theta) &= \cot\theta, &\quad \cot(90^\circ - \theta) &= \tan\theta \\[0.5em] \sec(90^\circ - \theta) &= \csc\theta, &\quad \csc(90^\circ - \theta) &= \sec\theta \end{aligned}$

8.1 Angle Sum Property of a Quadrilateral

The sum of all four interior angles of a quadrilateral is always \(360^\circ\):

$\angle A + \angle B + \angle C + \angle D = 360^\circ$

8.2 Properties of Parallelograms

• Opposite sides are equal and parallel (\(AB \parallel CD, AB = CD\)).
• Opposite angles are equal (\(\angle A = \angle C, \angle B = \angle D\)).
• Diagonals bisect each other.
• Special Types: Rectangle (all angles \(90^\circ\)), Rhombus (all sides equal, diagonals bisect at \(90^\circ\)), Square (Rectangle + Rhombus).

8.3 Mid-point Theorem

Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it.

$DE \parallel BC \quad \text{and} \quad DE = \frac{1}{2} BC$

8.4 Key Theorems on Circles

• Perpendicular from Center: Perpendicular from center to a chord bisects the chord.
• Equal Chords: Equal chords subtend equal angles at the center and are equidistant from the center.
• Angle Subtended at Center: Angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining circumference. (\(\angle AOB = 2 \angle ACB\))
• Angles in Same Segment: Angles in the same segment of a circle are equal.
• Angle in a Semicircle: Angle in a semicircle is a right angle (\(90^\circ\)).

8.5 Cyclic Quadrilaterals

A quadrilateral whose all four vertices lie on a circle is called a Cyclic Quadrilateral.

Property: Opposite angles of a cyclic quadrilateral are supplementary (sum \(= 180^\circ\)):

$\angle A + \angle C = 180^\circ, \quad \angle B + \angle D = 180^\circ$

9.1 Heron's Formula

For a triangle with side lengths \(a, b, c\):

1. Semi-perimeter (\(s\)):

$s = \frac{a + b + c}{2}$

2. Area of Triangle:

$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$

9.2 Formulas for Special Triangles

$\begin{aligned} \text{Right Triangle:} &\quad \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \\[0.5em] \text{Equilateral Triangle (side } a\text{):} &\quad \text{Area} = \frac{\sqrt{3}}{4} a^2, \quad \text{height } h = \frac{\sqrt{3}}{2} a \\[0.5em] \text{Isosceles Triangle (sides } a, a, b\text{):} &\quad \text{Area} = \frac{b}{4} \sqrt{4a^2 - b^2} \end{aligned}$

9.3 Applications in Finding Areas of Quadrilaterals

To find the area of an irregular quadrilateral, divide it into two triangles by drawing a diagonal, calculate the area of both triangles using Heron's formula, and add them together:

$\text{Area of Quad } ABCD = \text{Area}(\Delta ABC) + \text{Area}(\Delta ADC)$

10.1 Cuboid & Cube

1. Cuboid (length \(l\), breadth \(b\), height \(h\)):

$\begin{aligned} \text{Lateral Surface Area (LSA / Area of 4 walls)} &= 2h(l + b) \\[0.5em] \text{Total Surface Area (TSA)} &= 2(lb + bh + hl) \\[0.5em] \text{Volume} &= l \cdot b \cdot h \\[0.5em] \text{Diagonal} &= \sqrt{l^2 + b^2 + h^2} \end{aligned}$

2. Cube (edge \(a\)):

$\begin{aligned} \text{Lateral Surface Area (LSA)} &= 4a^2 \\[0.5em] \text{Total Surface Area (TSA)} &= 6a^2 \\[0.5em] \text{Volume} &= a^3 \\[0.5em] \text{Diagonal} &= a\sqrt{3} \end{aligned}$

10.2 Right Circular Cylinder (radius \(r\), height \(h\))

$\begin{aligned} \text{Curved Surface Area (CSA)} &= 2\pi r h \\[0.5em] \text{Total Surface Area (TSA)} &= 2\pi r h + 2\pi r^2 = 2\pi r(r + h) \\[0.5em] \text{Volume} &= \pi r^2 h \end{aligned}$

10.3 Right Circular Cone (radius \(r\), height \(h\), slant height \(l\))

$\begin{aligned} \text{Slant Height (} l\text{)} &= \sqrt{r^2 + h^2} \\[0.5em] \text{Curved Surface Area (CSA)} &= \pi r l \\[0.5em] \text{Total Surface Area (TSA)} &= \pi r l + \pi r^2 = \pi r(l + r) \\[0.5em] \text{Volume} &= \frac{1}{3}\pi r^2 h \end{aligned}$

10.4 Sphere & Solid Hemisphere (radius \(r\))

1. Sphere:

$\begin{aligned} \text{Surface Area} &= 4\pi r^2 \\[0.5em] \text{Volume} &= \frac{4}{3}\pi r^3 \end{aligned}$

2. Solid Hemisphere:

$\begin{aligned} \text{Curved Surface Area (CSA)} &= 2\pi r^2 \\[0.5em] \text{Total Surface Area (TSA)} &= 3\pi r^2 \\[0.5em] \text{Volume} &= \frac{2}{3}\pi r^3 \end{aligned}$

11.1 Data Collection & Concepts

• Primary Data: Information collected directly by the investigator for a specific purpose.
• Secondary Data: Information gathered from pre-existing sources (newspapers, records, reports).
• Range: Difference between maximum and minimum values (\(\text{Range} = \text{Max} - \text{Min}\)).
• Class Mark: \(\frac{\text{Upper Limit} + \text{Lower Limit}}{2}\)

11.2 Measures of Central Tendency

1. Mean (\(\bar{x}\)):

$\bar{x} = \frac{\sum x_i}{n} = \frac{\sum f_i x_i}{\sum f_i}$

2. Median: Arranging data in ascending order:

$\text{Median} = \begin{cases} \left(\frac{n+1}{2}\right)\text{th term} & \text{if } n \text{ is odd} \\[0.5em] \frac{\left(\frac{n}{2}\right)\text{th term} + \left(\frac{n}{2} + 1\right)\text{th term}}{2} & \text{if } n \text{ is even} \end{cases}$

3. Mode: The observation value with maximum frequency in the data.

$\text{Mode} \approx 3(\text{Median}) - 2(\text{Mean})$

11.3 Probability (Empirical / Experimental)

The probability of an event \(E\), denoted \(P(E)\):

$P(E) = \frac{\text{Number of trials in which event } E \text{ occurred}}{\text{Total number of trials}}$

Fundamental Axioms of Probability:

• Probability of any event always lies between 0 and 1 inclusive: \(0 \le P(E) \le 1\)
• Impossible Event: \(P(E) = 0\) (e.g., getting a 7 on a standard 6-sided die)
• Sure / Certain Event: \(P(E) = 1\)
• Complementary Event Rule: Sum of probability of occurrence \(P(E)\) and non-occurrence \(P(\bar{E})\) is 1: $P(E) + P(\bar{E}) = 1 \implies P(\bar{E}) = 1 - P(E)$