Class 11 Mathematics — Complete Formula & Concept Guide

A comprehensive, authoritative reference guide for Class 11 Mathematics (NCERT & CBSE syllabus). Covers all 15 chapters including Sets, Relations & Functions, Trigonometry, Complex Numbers, Inequalities, Permutations & Combinations, Binomial Theorem, Sequences & Series, Straight Lines, Conic Sections, 3D Geometry, Limits & Derivatives, Mathematical Reasoning, Statistics, and Probability with detailed formulas, LaTeX equations, and interactive SVG diagrams.

1. Sets — Formulas & Fundamental Concepts

A set is a well-defined collection of distinct objects.

Venn Diagram — Union (A ∪ B) & Intersection (A ∩ B)

1.1 Key Formulas, Definitions & Examples

1️⃣ Empty Set ($\emptyset$):
A set containing no elements.
$\emptyset = \{ \}$

(e.g., a set with no members)

2️⃣ Complement of a Set ($A'$):
All elements in Universal set $U$ that do not belong to $A$.
$A' = U - A$

Example: If $U = \{1,2,3,4,5\}$ and $A = \{1,2\}$, then $A' = \{3,4,5\}$

3️⃣ Union of Sets ($A \cup B$):
All elements that are in $A$, or $B$, or both.
$A \cup B = \{x : x \in A \text{ or } x \in B\}$

Example: $A = \{1,2,3\}, B = \{3,4,5\} \implies A \cup B = \{1,2,3,4,5\}$

4️⃣ Intersection of Sets ($A \cap B$):
All elements common to both $A$ and $B$.
$A \cap B = \{x : x \in A \text{ and } x \in B\}$

Example: $A = \{1,2,3\}, B = \{3,4,5\} \implies A \cap B = \{3\}$

5️⃣ Difference of Sets ($A - B$):
Elements belonging to $A$ but not to $B$.
$A - B = \{x : x \in A \text{ and } x \notin B\}$

Example: $A = \{1,2,3\}, B = \{3,4,5\} \implies A - B = \{1,2\}$

6️⃣ Subset ($A \subseteq B$):
Every element of $A$ is also an element of $B$.
$A \subseteq B \iff (\forall x \in A \implies x \in B)$

Example: If $A = \{1,2\}$ and $B = \{1,2,3\}$, then $A \subseteq B$

7️⃣ Power Set ($P(A)$):
The set of all subsets of $A$.
$\text{Formula: } n(P(A)) = 2^{n(A)}$

Example: If $A = \{a,b\}$, then $P(A) = \{\emptyset, \{a\}, \{b\}, \{a,b\}\}$ and $n(P(A)) = 2^2 = 4$

8️⃣ Cardinality Formula for Two Sets:
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

Example: If $n(A) = 10, n(B) = 15, n(A \cap B) = 5 \implies n(A \cup B) = 10 + 15 - 5 = 20$

9️⃣ Cardinality Formula for Three Sets:
$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$

Example: If $n(A) = 5, n(B) = 6, n(C) = 7, n(A \cap B) = 2, n(B \cap C) = 3, n(C \cap A) = 2, n(A \cap B \cap C) = 1$:
$n(A \cup B \cup C) = 5 + 6 + 7 - 2 - 3 - 2 + 1 = 12$

🔟 De Morgan's Laws:
$(A \cup B)' = A' \cap B' \quad \text{and} \quad (A \cap B)' = A' \cup B'$

Example: If $U = \{1,2,3,4\}, A = \{1,2\}, B = \{2,3\}$:
$(A \cup B)' = \{4\}$ and $(A \cap B)' = \{1,3,4\}$

1.2 Symbols Used in Set Theory

Symbol 🔤	Meaning 📋
$\cup$	Union
$\cap$	Intersection
$\subseteq$	Subset
$\subset$	Proper Subset
$\supset$	Superset
$\emptyset$	Empty / Null Set
$U$	Universal Set
$'$	Complement

1.3 Additional Solved Examples

🔹 Example 1: $A = \{1,2,3\}$ and $B = \{3,4,5\}$:

👉 $A \cup B = \{1,2,3,4,5\}$

👉 $A \cap B = \{3\}$

👉 $A - B = \{1,2\}$

🔹 Example 2: Universal set $U = \{1,2,3,4,5,6,7,8,9\}$ and $A = \{2,4,6,8\}$:

👉 $A' = \{1,3,5,7,9\}$

🔹 Example 3: If $n(A) = 10, n(B) = 15,$ and $n(A \cap B) = 5$:

👉 $n(A \cup B) = 10 + 15 - 5 = 20$

🔹 Example 4: If $n(A) = 3$, number of elements in power set $P(A)$:

👉 $n(P(A)) = 2^3 = 8$

2. Relations & Functions — Comprehensive Guide

A relation is a subset of the Cartesian product of two sets. A function is a special relation where each input has exactly one output.

Function Mapping Diagram (A → B)

2.1 Relations — 11 Key Types & Examples

1️⃣ Definition of Relation & Cartesian Product:
For non-empty sets $A$ and $B$, Cartesian product $A \times B = \{(a,b) \mid a \in A, b \in B\}$.

👉 Example 1: If $A = \{1,2\}$ and $B = \{3,4\}$:
Step 1: Pair each element of $A$ with every element of $B$.
Step 2: Ordered pairs: $(1,3), (1,4), (2,3), (2,4)$.
Step 3: Thus $A \times B = \{(1,3), (1,4), (2,3), (2,4)\}$.

2️⃣ Reflexive Relation:
A relation $R$ on set $A$ is reflexive if $(a,a) \in R$ for every $a \in A$.

👉 Example 2: $A = \{1,2\}$ and $R = \{(1,1), (2,2)\}$. Since $(1,1)$ and $(2,2)$ are present, $R$ is Reflexive.

3️⃣ Symmetric Relation:
A relation $R$ is symmetric if $(a,b) \in R \implies (b,a) \in R$.

👉 Example 3: $A = \{1,2\}$ and $R = \{(1,2), (2,1)\}$. Since $(1,2)$ and $(2,1)$ both exist, $R$ is Symmetric.

4️⃣ Transitive Relation:
A relation $R$ is transitive if $(a,b) \in R$ and $(b,c) \in R \implies (a,c) \in R$.

👉 Example 4: $A = \{1,2,3\}$ and $R = \{(1,2), (2,3), (1,3)\}$. Since $(1,2)$ and $(2,3)$ imply $(1,3)$ which is present, $R$ is Transitive.

5️⃣ Universal Relation:
Where $R = A \times A$ (contains all possible ordered pairs).

👉 Example 5: For $A = \{1,2\}$, $R = \{(1,1), (1,2), (2,1), (2,2)\}$ is Universal.

6️⃣ Empty Relation:
Where $R = \emptyset$ (contains no ordered pairs).

👉 Example 6: For $A = \{1,2,3\}$, $R = \emptyset$ is an Empty Relation.

7️⃣ One-One Relation:
Each element of domain maps to a distinct element in codomain.

👉 Example 7: $A = \{1,2,3\}, B = \{4,5,6\}, R = \{(1,4), (2,5), (3,6)\}$.

8️⃣ Functional Relation:
Each element in domain has exactly one image.

👉 Example 8: $A = \{a,b\}, B = \{1,2,3\}, R = \{(a,2), (b,3)\}$.

9️⃣ Transitive Property Rule:
$(a,b) \in R \land (b,c) \in R \implies (a,c) \in R$.

👉 Example 9: For $R = \{(1,2), (2,3), (1,3)\}$, Transitive condition holds.

🔟 Cartesian Product:
Set of all ordered pairs $(a,b)$ where $a \in A, b \in B$.

👉 Example 10: $A=\{1,2\}, B=\{3,4\} \implies A \times B = \{(1,3), (1,4), (2,3), (2,4)\}$.

1️⃣1️⃣ Equivalence Relation:
A relation that is Reflexive, Symmetric, AND Transitive.

👉 Example 11: $A = \{1,2,3\}$ and $R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$.
Reflexive: $(1,1), (2,2), (3,3)$ present ✅.
Symmetric: $(1,2)$ and $(2,1)$ present ✅.
Transitive: $(1,2)$ and $(2,1) \implies (1,1)$ present ✅.
Thus, $R$ is an Equivalence Relation.

2.2 Functions — Types & Range

1️⃣ Function Definition:
A relation $f: A \to B$ is a function if every element in $A$ has a unique image in $B$.

👉 Example 1: If $A=\{1,2\}, B=\{3,4\}$, $A \times B = \{(1,3), (1,4), (2,3), (2,4)\}$ is not a function because 1 maps to both 3 and 4.

2️⃣ One-to-One (Injective) Function:
Distinct elements in $A$ map to distinct elements in $B$. $f(a_1) = f(a_2) \implies a_1 = a_2$.

👉 Example 2: $f(1) = 3, f(2) = 4$ is One-to-One.

3️⃣ Onto (Surjective) Function:
Every element in $B$ is the image of at least one element in $A$ ($\text{Range} = \text{Codomain}$).

👉 Example 3: For $A=\{1,2\}, B=\{3,4\}$, $f(1)=3, f(2)=4$ is Onto.

4️⃣ Bijective Function:
A function that is both One-to-One and Onto.

👉 Example 4: $f(1) = 3, f(2) = 4$ is Bijective.

5️⃣ Range of a Function:
The set of all actual output values in $B$ produced by $f(x)$.

👉 Example 5: For $A=\{1,2\}, B=\{3,4,5\}$ with $f(1)=3, f(2)=4$, $\text{Range} = \{3,4\}$.

3. Trigonometric Functions — Master Formula Guide

Trigonometry analyzes relationships between triangle side lengths and angles as well as circular functions.

Unit Circle & Quadrants Overview

3.1 Radian & Degree Measure Conversions

$1 \text{ rad} = \frac{180^\circ}{\pi} \approx 57^\circ 17' 45''$
$1^\circ = \frac{\pi}{180} \text{ rad} \approx 0.017453 \text{ rad}$
$1' = \frac{\pi}{180 \times 60} \text{ rad} \approx 0.000291 \text{ rad}$
$1'' = \frac{\pi}{180 \times 3600} \text{ rad} \approx 0.000005 \text{ rad}$

3.2 Fundamental Ratios & Reciprocals

In a right-angled triangle: $\text{Opposite} = p, \text{Adjacent} = b, \text{Hypotenuse} = h$

$h^2 = p^2 + b^2, \quad p^2 = h^2 - b^2, \quad b^2 = h^2 - p^2$

Ratio	Definition	Reciprocal
$\sin\theta$	$\frac{p}{h}$	$\csc\theta = \frac{1}{\sin\theta} = \frac{h}{p}$
$\cos\theta$	$\frac{b}{h}$	$\sec\theta = \frac{1}{\cos\theta} = \frac{h}{b}$
$\tan\theta$	$\frac{p}{b}$	$\cot\theta = \frac{1}{\tan\theta} = \frac{b}{p}$

Quotient Identities:

$\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}$

3.3 Standard Values Chart

Angle ($\theta$)	0°	30°	45°	60°	90°
Sine ($\sin\theta$)	0	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	1
Cosine ($\cos\theta$)	1	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	0
Tangent ($\tan\theta$)	0	$\frac{1}{\sqrt{3}}$	1	$\sqrt{3}$	$\infty$
Cotangent ($\cot\theta$)	$\infty$	$\sqrt{3}$	1	$\frac{1}{\sqrt{3}}$	0
Secant ($\sec\theta$)	1	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	2	$\infty$
Cosecant ($\csc\theta$)	$\infty$	2	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	1

3.4 Pythagorean, Even-Odd & Quadrant Identities

Pythagorean Identities:

$\sin^2\theta + \cos^2\theta = 1, \quad \sec^2\theta - \tan^2\theta = 1, \quad \csc^2\theta - \cot^2\theta = 1$

Even-Odd Identities:

$\sin(-\theta) = -\sin\theta, \quad \cos(-\theta) = \cos\theta, \quad \tan(-\theta) = -\tan\theta$ $\csc(-\theta) = -\csc\theta, \quad \sec(-\theta) = \sec\theta, \quad \cot(-\theta) = -\cot\theta$

Co-Ratio Transformations:

Angle $\theta$	sin	cos	tan	cot	sec	csc
$90^\circ - \theta$	$\cos\theta$	$\sin\theta$	$\cot\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$
$90^\circ + \theta$	$\cos\theta$	$-\sin\theta$	$-\cot\theta$	$-\tan\theta$	$-\csc\theta$	$\sec\theta$
$180^\circ - \theta$	$\sin\theta$	$-\cos\theta$	$-\tan\theta$	$-\cot\theta$	$-\sec\theta$	$\csc\theta$
$180^\circ + \theta$	$-\sin\theta$	$-\cos\theta$	$\tan\theta$	$\cot\theta$	$-\sec\theta$	$-\csc\theta$

3.5 Compound, Double & Multiple Angle Formulas

Addition & Subtraction Formulas:

$\sin(x+y) = \sin x \cos y + \cos x \sin y, \quad \sin(x-y) = \sin x \cos y - \cos x \sin y$ $\cos(x+y) = \cos x \cos y - \sin x \sin y, \quad \cos(x-y) = \cos x \cos y + \sin x \sin y$ $\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}, \quad \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$

Double Angle Formulas:

$\sin 2\theta = 2\sin\theta\cos\theta = \frac{2\tan\theta}{1+\tan^2\theta}$ $\cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1 = \frac{1-\tan^2\theta}{1+\tan^2\theta}$ $\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}, \quad \cot 2\theta = \frac{\cot^2\theta - 1}{2\cot\theta}$

Multiple Angle Formulas:

$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$ $\sin 4\theta = 4\sin\theta\cos\theta - 8\sin^3\theta\cos\theta$ $\sin 5\theta = 5\sin\theta - 20\sin^3\theta + 16\sin^5\theta$ $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ $\cos 4\theta = 8\cos^4\theta - 8\cos^2\theta + 1$ $\cos 5\theta = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta$ $\tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$

3.6 Laws of Triangles, Euler's Formulas & General Solutions

Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$

Area of Triangle: $\text{Area} = \frac{1}{2} ab \sin C$

Euler's Exponential Formulas:

$e^{i\theta} = \cos\theta + i\sin\theta, \quad \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}, \quad \cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$

General Solutions of Trig Equations:

$\sin\theta = 0 \implies \theta = n\pi, \quad \cos\theta = 0 \implies \theta = (2n+1)\frac{\pi}{2}$ $\sin\theta = a \implies \theta = n\pi + (-1)^n \arcsin(a)$ $\cos\theta = a \implies \theta = 2n\pi \pm \arccos(a)$ $\tan\theta = a \implies \theta = n\pi + \arctan(a)$

4. Complex Numbers — Exhaustive Reference

A complex number is expressed as $z = a + bi$ where $a$ is the real part, $b$ is the imaginary part, and $i = \sqrt{-1}$.

Argand Plane — Modulus & Argument Visualization

4.1 All 17 Formulas, Identities & Examples

1️⃣ Standard Form:
$z = a + bi \quad (i = \sqrt{-1}, \, i^2 = -1)$

👉 Example 1: $z = 3 + 4i$. Real part $a = 3$, Imaginary part $b = 4$.

2️⃣ Addition of Complex Numbers:
$(a + bi) + (c + di) = (a + c) + (b + d)i$

👉 Example 2: $(3 + 4i) + (1 + 2i) = 4 + 6i$

3️⃣ Multiplication:
$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

👉 Example 3: $(3 + 4i)(1 + 2i) = (3 - 8) + (6 + 4)i = -5 + 10i$

4️⃣ Conjugate ($\bar{z}$):
$\bar{z} = a - bi$

👉 Example 4: For $z = 3 + 4i$, $\bar{z} = 3 - 4i$.

5️⃣ Division:
$\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

👉 Example 5: $\frac{3 + 4i}{1 + 2i} = \frac{11 - 2i}{5} = \frac{11}{5} - \frac{2}{5}i$

6️⃣ Modulus ($|z|$):
$|z| = r = \sqrt{a^2 + b^2}$

👉 Example 6: $z = 3 + 4i \implies |z| = \sqrt{3^2 + 4^2} = 5$

7️⃣ Argument ($\theta$):
$\theta = \tan^{-1}\left(\frac{b}{a}\right)$

👉 Example 7: $z = 1 + \sqrt{3}i \implies \theta = \frac{\pi}{3}$

8️⃣ Polar Form:
$z = r (\cos\theta + i \sin\theta)$

9️⃣ Exponential Form:
$z = r e^{i\theta}$

🔟 Real and Imaginary Part Formulas:
$\text{Re}(z) = \frac{z + \bar{z}}{2}, \quad \text{Im}(z) = \frac{z - \bar{z}}{2i}$

1️⃣1️⃣ Cartesian to Polar Conversion:
$r = \sqrt{a^2 + b^2}, \quad \theta = \arctan(b/a)$

1️⃣2️⃣ Polar to Cartesian Conversion:
$a = r \cos\theta, \quad b = r \sin\theta$

1️⃣3️⃣ De Moivre's Theorem:
$(\cos\theta + i \sin\theta)^n = \cos(n\theta) + i \sin(n\theta)$

1️⃣4️⃣ Roots of Complex Numbers:
$z^{1/n} = r^{1/n} \left[ \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right]$

1️⃣5️⃣ Trigonometric Form:
$z = |z| (\cos\theta + i \sin\theta)$

1️⃣6️⃣ Cube Formula:
$(a + bi)^3 = (a^3 - 3ab^2) + (3a^2b - b^3)i$

1️⃣7️⃣ Cube Roots of Unity:
$1, \quad \omega = \frac{-1 + i\sqrt{3}}{2}, \quad \omega^2 = \frac{-1 - i\sqrt{3}}{2}$ $\text{Properties: } 1 + \omega + \omega^2 = 0, \quad \omega^3 = 1$

5. Quadratic Equations & Linear Inequalities — Complete Guide

Parabolic Graph of Quadratic Function ($y = ax^2 + bx + c$)

5.1 Quadratic Equations

1️⃣ Standard Form:
$ax^2 + bx + c = 0 \quad (a \neq 0)$

2️⃣ Quadratic Formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3️⃣ Discriminant ($\Delta$) & Nature of Roots:
$\Delta = b^2 - 4ac$

🔹 If $\Delta > 0$: Two distinct real roots.
🔹 If $\Delta = 0$: Two equal real roots ($x = -b / 2a$).
🔹 If $\Delta < 0$: Two complex conjugate roots ($x = \frac{-b \pm i\sqrt{4ac - b^2}}{2a}$).

4️⃣ Methods of Solving:
👉 Factorization
👉 Completing the Square
👉 Quadratic Formula

5️⃣ Example 1: $x^2 - 5x + 6 = 0$
Step 1: Split middle term $-5x = -2x - 3x$
Step 2: $(x - 2)(x - 3) = 0$
Step 3: $x = 2$ or $x = 3$

6️⃣ Example 2: $2x^2 + 3x - 2 = 0$
Step 1: $\Delta = 3^2 - 4(2)(-2) = 25$
Step 2: $x = \frac{-3 \pm 5}{4}$
Step 3: $x = 1/2$ and $x = -2$

5.2 Linear Inequalities

1️⃣ Forms:
$ax + b < c, \quad ax + b > c, \quad ax + b \le c, \quad ax + b \ge c$

2️⃣ Example 1: $2x + 3 \le 7$
Subtract 3 $\implies 2x \le 4 \implies x \le 2$ or $x \in (-\infty, 2]$

3️⃣ Example 2: $3x - 5 > 10$
Add 5 $\implies 3x > 15 \implies x > 5$ or $x \in (5, \infty)$

4️⃣ Example 3: $5x + 4 \ge 9$
Subtract 4 $\implies 5x \ge 5 \implies x \ge 1$ or $x \in [1, \infty)$

6. Permutations & Combinations — Complete Formulas

Permutations involve ordering objects where sequence matters. Combinations involve selecting objects where order does not matter.

6.1 Key Principles & Formulas

1️⃣ Factorial Notation ($n!$):
$n! = n \times (n-1) \times \dots \times 1, \quad 0! = 1$

2️⃣ Permutations ($P(n,r)$):
Arrangements where order is significant.
$P(n,r) = \frac{n!}{(n-r)!}, \quad P(n) = n!$

🔹 Circular Permutations: $(n-1)!$
🔹 Permutations with repetitions: $\frac{n!}{p! \, q! \, r!}$

3️⃣ Combinations ($C(n,r)$):
Selections where order is irrelevant.
$C(n,r) = \frac{n!}{r!(n-r)!}$

🔹 Identity: $C(n,r) = C(n, n-r)$
🔹 Pascal's Identity: $C(n,r) + C(n,r-1) = C(n+1,r)$

Feature	Permutations	Combinations
Focus	Arrangement	Selection
Order Sensitivity	Order Matters	Order Does Not Matter
Formula	$P(n,r) = \frac{n!}{(n-r)!}$	$C(n,r) = \frac{n!}{r!(n-r)!}$

4️⃣ Solved Comparison Example:
Choose 3 people out of 5:

👉 Permutations (Seating / Positions):
$P(5,3) = \frac{5!}{2!} = 60 \text{ ways}$
👉 Combinations (Committee / Team):
$C(5,3) = \frac{5!}{3!2!} = 10 \text{ ways}$

7. Binomial Theorem & Key Mathematical Identities

The Binomial Theorem provides an algebraic expansion of powers of a binomial expression.

7.1 Binomial Expansion Formulas

1️⃣ Binomial Expansion Formula:
$(a+b)^n = \sum_{r=0}^n ^{n}C_{r} a^{n-r} b^r = ^{n}C_{0}a^n + ^{n}C_{1}a^{n-1}b + ^{n}C_{2}a^{n-2}b^2 + \dots + ^{n}C_{n}b^n$

2️⃣ General Term ($T_{r+1}$):
$T_{r+1} = ^{n}C_{r} \, a^{n-r} \, b^r$

3️⃣ Middle Term Rules:
🔹 If $n$ is Even: Single middle term $\implies T_{(n/2) + 1}$
🔹 If $n$ is Odd: Two middle terms $\implies T_{(n+1)/2}$ and $T_{(n+3)/2}$

4️⃣ Sum of Coefficients:
$^{n}C_{0} + ^{n}C_{1} + ^{n}C_{2} + \dots + ^{n}C_{n} = 2^n$ $^{n}C_{0} + ^{n}C_{2} + ^{n}C_{4} + \dots = 2^{n-1}$

5️⃣ Inequality & Mathematical Identities:
🔹 Triangle Inequality: $|a + b| \le |a| + |b|$
🔹 Simple Interest: $SI = \frac{P \times R \times T}{100}$
🔹 Pyramid Volume: $V = \frac{1}{3} B h$

8. Sequence & Series — AP, GP & Summation Formulas

A sequence is an ordered set of numbers governed by a rule. A series is the sum of terms of a sequence.

8.1 Arithmetic & Geometric Progression Formulas

1️⃣ Arithmetic Progression (AP):
$\text{General Form: } a, \, a+d, \, a+2d, \, \dots$ $n\text{-th term: } a_n = a_1 + (n-1)d$ $\text{Sum of } n \text{ terms: } S_n = \frac{n}{2} [2a_1 + (n-1)d] = \frac{n}{2} [a_1 + a_n]$

👉 Example (AP): $a_1 = 2, d = 3, n = 5$
$S_5 = \frac{5}{2} [2(2) + (5-1)3] = \frac{5}{2} [4 + 12] = 40$

2️⃣ Geometric Progression (GP):
$\text{General Form: } a, \, ar, \, ar^2, \, \dots$ $n\text{-th term: } a_n = a_1 r^{n-1}$ $\text{Sum of } n \text{ terms: } S_n = \frac{a_1 (1 - r^n)}{1 - r} \quad (r \neq 1)$

👉 Example (GP 4 terms): $a_1 = 3, r = 2, n = 4$
$S_4 = \frac{3(1 - 2^4)}{1 - 2} = \frac{3(-15)}{-1} = 45$

3️⃣ Infinite GP Sum:
For $|r| < 1$:
$S_\infty = \frac{a_1}{1 - r}$

👉 Example (Infinite GP): $a_1 = 5, r = 1/2$
$S_\infty = \frac{5}{1 - 1/2} = 10$

9. Straight Lines — 16 Forms, Formulas & Examples

Coordinate geometry representations, slopes, forms, and distances between lines and points in 2D space.

Straight Lines — Slope & Point of Intersection

9.1 All 16 Forms, Formulas & Solved Examples

1️⃣ General Form:
$Ax + By + C = 0 \implies \text{Slope } m = -\frac{A}{B}$

👉 Example 1: $2x + 3y - 5 = 0$.

2️⃣ Slope of Line:

👉 Example 2: For $4x + 5y + 7 = 0$, slope $m = -4/5$.

3️⃣ Two Point Form:
$(y - y_1) = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$

👉 Example 3: Line through $(2,3)$ & $(4,7)$ gives $2x - y - 1 = 0$.

4️⃣ Slope-Point Form:
$(y - y_1) = m(x - x_1)$

👉 Example 4: $m = 3$, $(1,2) \implies 3x - y - 1 = 0$.

5️⃣ Intercept Form:
$\frac{x}{a} + \frac{y}{b} = 1$

👉 Example 5: $a = 3, b = 4 \implies 4x + 3y = 12$.

6️⃣ Angle Between Two Lines:
$\tan\theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$

👉 Example 6: $m_1 = 1, m_2 = 2 \implies \theta = \arctan(1/3)$.

7️⃣ Distance from Point to Line:
$d = \frac{|A x_1 + B y_1 + C|}{\sqrt{A^2 + B^2}}$

👉 Example 7: Point $(2,3)$ to line $3x + 4y - 5 = 0$ gives $d = 2.6$.

8️⃣ Distance Between Parallel Lines:
$d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}$

👉 Example 8: Lines $3x+4y-5=0$ and $3x+4y+7=0$ give $d = 2.4$.

9️⃣ Concurrent Lines:
Lines passing through a common point.

👉 Example 9: $x+y-5=0, 2x-y+1=0, x-3y+7=0$.

🔟 Normal Form:
$x \cos\theta + y \sin\theta = p$

1️⃣1️⃣ Perpendicular Lines Condition:
$m_1 \times m_2 = -1$

1️⃣2️⃣ Line Through Point:
$y - y_1 = m(x - x_1)$

1️⃣3️⃣ Parallel Lines Condition:
$m_1 = m_2$

1️⃣4️⃣ Slope-Intercept Form:
$y = mx + c$

1️⃣5️⃣ Point of Intersection:
System of lines solution. $x+y=5, x-y=1 \implies (3,2)$.

1️⃣6️⃣ Section Formula:
$\left( \frac{m x_2 + n x_1}{m + n}, \, \frac{m y_2 + n y_1}{m + n} \right)$

10. Conic Sections — Circles, Parabolas, Ellipses & Hyperbolas

Curves formed by the intersection of a plane with a double-napped cone.

Conic Shapes Overview

10.1 Formulas, Equations & Examples

1️⃣ Circle:
$(x-h)^2 + (y-k)^2 = r^2$

👉 Example 1: Center $(2,3)$, radius $r = 5 \implies (x-2)^2 + (y-3)^2 = 25$.

2️⃣ Parabola:
Standard form: $y^2 = 4ax$

👉 Example 2: Vertex $(0,0)$, Focus $(2,0) \implies y^2 = 8x$.

4 Parabola Orientations:
🔹 $y^2 = 4ax$ (Opens Right)
🔹 $y^2 = -4ax$ (Opens Left)
🔹 $x^2 = 4ay$ (Opens Up)
🔹 $x^2 = -4ay$ (Opens Down)

3️⃣ Ellipse:
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

👉 Example 3: $a = 5, b = 3 \implies \frac{x^2}{25} + \frac{y^2}{9} = 1$.

4️⃣ Hyperbola:
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

👉 Example 4: $a = 4, b = 3 \implies \frac{x^2}{16} - \frac{y^2}{9} = 1$.

5️⃣ Focus & Directrix Formulas:
🔹 Parabola $y^2=4ax \implies$ Focus $(a,0)$, Directrix $x = -a$
🔹 Ellipse & Hyperbola Latus Rectum = $\frac{2b^2}{a}$
🔹 Shifted Vertex Example: $(y-2)^2 = 8(x+3) \implies$ Vertex $(-3,2)$.

11. Three-Dimensional Geometry — 19 Essential Formulas

Cartesian coordinates, lines, planes, direction cosines, and distances in 3D space.

3D Coordinate Frame (X, Y, Z Axes)

11.1 All 19 Formulas in 3D Geometry

1️⃣ 3D Distance Formula:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

2️⃣ Midpoint Formula:
$M = \left( \frac{x_1 + x_2}{2}, \, \frac{y_1 + y_2}{2}, \, \frac{z_1 + z_2}{2} \right)$

3️⃣ Distance from Point to Axes:
🔹 From X-axis $= \sqrt{y^2 + z^2}$
🔹 From Y-axis $= \sqrt{x^2 + z^2}$
🔹 From Z-axis $= \sqrt{x^2 + y^2}$

4️⃣ Line Equations (Vector & Cartesian):
$\text{Vector: } \vec{r} = \vec{a} + \lambda \vec{b}$ $\text{Cartesian: } \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$

5️⃣ Angle Between Two Lines:
$\cos\theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$

6️⃣ Plane General Equation:
$Ax + By + Cz + D = 0$

7️⃣ Plane Vector Equation:
$\vec{r} \cdot \vec{n} = d$

8️⃣ Distance from Point to Plane:
$D = \frac{|A x_1 + B y_1 + C z_1 + D|}{\sqrt{A^2 + B^2 + C^2}}$

9️⃣ Angle Between Two Planes:
$\cos\theta = \frac{A_1 A_2 + B_1 B_2 + C_1 C_2}{\sqrt{A_1^2 + B_1^2 + C_1^2} \sqrt{A_2^2 + B_2^2 + C_2^2}}$

🔟 Angle Between Line and Plane:
$\sin\theta = \frac{Aa + Bb + Cc}{\sqrt{A^2 + B^2 + C^2} \sqrt{a^2 + b^2 + c^2}}$

1️⃣1️⃣ Line-Plane Intersection: Solve for parameter $\lambda$.

1️⃣2️⃣ Shortest Distance Between Skew Lines:
$D = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|}$

1️⃣3️⃣ Plane Normal Form: $l x + m y + n z = p$

1️⃣4️⃣ Perpendicular Distance from Origin to Plane.

1️⃣5️⃣ Foot of Perpendicular from Point to Plane.

1️⃣6️⃣ Distance from Line to Parallel Plane.

1️⃣7️⃣ Perpendicular Planes Condition: $A_1 A_2 + B_1 B_2 + C_1 C_2 = 0$

1️⃣8️⃣ 3D Cone Equation: $z^2 = x^2 + y^2$

1️⃣9️⃣ Vertical Cross-Section of Cone: $y^2 = 4ax$

12. Limits & Derivatives — Foundations of Calculus

Calculus studies continuous change through limits and derivatives representing rates of change.

Derivative — Slope of Tangent Line ($\tan\theta = f'(x)$)

12.1 Limits Formulas

1️⃣ Algebraic Limits:
$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}, \quad \lim_{x \to 0} \frac{(1+x)^n - 1}{x} = n$

2️⃣ Trigonometric Limits:
$\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{\tan x}{x} = 1, \quad \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$

3️⃣ Exponential & Logarithmic Limits:
$\lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$

12.2 Derivative Rules (All 13 Types)

1️⃣ First Principle:
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

2️⃣ Power Rule: $\frac{d}{dx}(x^n) = n x^{n-1}$

3️⃣ Product Rule: $\frac{d}{dx} [f \cdot g] = f' g + f g'$

4️⃣ Quotient Rule: $\frac{d}{dx} [f / g] = \frac{f' g - f g'}{g^2}$

5️⃣ Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

6️⃣ Trigonometric Derivatives:
$\frac{d}{dx}(\sin x) = \cos x, \quad \frac{d}{dx}(\cos x) = -\sin x, \quad \frac{d}{dx}(\tan x) = \sec^2 x$

7️⃣ Inverse Trig Derivatives:
$\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$

13. Mathematical Reasoning — Statements & Logic

Deductive reasoning, logical connectives, truth tables, contrapositives, and methods of proof.

13.1 Logical Rules & Definitions

1️⃣ Statement Definition:
A sentence that is either strictly true or strictly false, but not both.

2️⃣ Negation ($\sim p$):
The denial of statement $p$.

3️⃣ Connectives:
🔹 AND ($\wedge$): True only if both components are true.
🔹 OR ($\vee$): True if at least one component is true.

4️⃣ Conditional Statements ($p \implies q$):
🔹 Contrapositive: $\sim q \implies \sim p$
🔹 Converse: $q \implies p$

5️⃣ Methods of Proof:
Direct Proof, Proof by Contradiction, Proof by Contrapositive.

14. Statistics — 18 Master Formulas

Measures of central tendency (Mean, Median, Mode) and dispersion (Mean Deviation, Standard Deviation, Variance).

Histogram & Normal Distribution Curve

14.1 All 18 Statistical Formulas

1️⃣ Class Width: $\frac{\text{Max} - \text{Min}}{\text{Classes}}$

2️⃣ Class Midpoint: $\frac{\text{Lower Limit} + \text{Upper Limit}}{2}$

3️⃣ Mean ($\bar{x}$): $\bar{x} = \frac{\sum x}{n}$ or $\frac{\sum fx}{\sum f}$

4️⃣ Median ($M$): $M = L + \left(\frac{N/2 - CF}{f}\right) \times h$

5️⃣ Mode ($Z$): $Z = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$

6️⃣ Mean Deviation: $MD = \frac{\sum |x - \bar{x}|}{n}$

7️⃣ Standard Deviation ($\sigma$): $\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$

8️⃣ Variance: $\sigma^2$

9️⃣ Coefficient of Variation: $CV = \left(\frac{\sigma}{\bar{x}}\right) \times 100\%$

🔟 Combined Mean: $\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$

1️⃣1️⃣ Range: $\text{Max} - \text{Min}$

1️⃣2️⃣ Quartile Deviation: $QD = \frac{Q_3 - Q_1}{2}$

1️⃣3️⃣ Coefficient of QD: $\frac{Q_3 - Q_1}{Q_3 + Q_1}$

1️⃣4️⃣ Coefficient of MD: $\frac{MD}{\bar{x}}$

1️⃣5️⃣ Coefficient of SD: $\frac{\sigma}{\bar{x}}$

1️⃣6️⃣ Assumed Mean Method: $\bar{x} = a + \frac{\sum fd}{\sum f}$

1️⃣7️⃣ Step Deviation Method: $\bar{x} = a + \left(\frac{\sum fu}{\sum f}\right) h$

1️⃣8️⃣ Empirical Relationship: $\text{Mode} = 3 \, \text{Median} - 2 \, \text{Mean}$

15. Probability — 15 Master Formulas

Quantifying uncertainty, conditional events, Bayes' theorem, and probability distributions.

Probability Tree Diagram (Bayes' Rule Visualizer)

15.1 All 15 Probability Formulas

1️⃣ Classical Probability: $P(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$, $0 \le P(E) \le 1$

2️⃣ Complementary Probability: $P(E') = 1 - P(E)$

3️⃣ Addition Law: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

4️⃣ Multiplication Law: $P(A \cap B) = P(A) \cdot P(B|A)$

5️⃣ Conditional Probability: $P(B|A) = \frac{P(A \cap B)}{P(A)}$

6️⃣ Binomial Distribution: $P(X = r) = ^{n}C_{r} \, p^r \, (1-p)^{n-r}$

7️⃣ Expected Value: $E(X) = \mu = \sum x_i P(x_i)$

8️⃣ Variance & SD: $V(X) = E(X^2) - [E(X)]^2$, $\sigma = \sqrt{V(X)}$

9️⃣ Permutations & Combinations in Probability.

🔟 Generalized Probability Rules.

1️⃣1️⃣ Law of Total Probability: $P(B) = \sum P(A_i) P(B|A_i)$

1️⃣2️⃣ Bayes' Theorem: $P(A_i | B) = \frac{P(B | A_i) P(A_i)}{\sum P(B | A_k) P(A_k)}$

1️⃣3️⃣ Continuous Distribution Probability: $P(a \le X \le b) = \int_a^b f(x) \, dx$

1️⃣4️⃣ Continuous Mean: $\mu = \int_{-\infty}^{\infty} x f(x) \, dx$

1️⃣5️⃣ Probabilistic Modelling Applications.

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Angle \(\theta\)	sin	cos	tan	cot	sec	csc
\(90^\circ - \theta\)	\(\cos\theta\)	\(\sin\theta\)	\(\cot\theta\)	\(\tan\theta\)	\(\csc\theta\)	\(\sec\theta\)
\(90^\circ + \theta\)	\(\cos\theta\)	\(-\sin\theta\)	\(-\cot\theta\)	\(-\tan\theta\)	\(-\csc\theta\)	\(\sec\theta\)
\(180^\circ - \theta\)	\(\sin\theta\)	\(-\cos\theta\)	\(-\tan\theta\)	\(-\cot\theta\)	\(-\sec\theta\)	\(\csc\theta\)
\(180^\circ + \theta\)	\(-\sin\theta\)	\(-\cos\theta\)	\(\tan\theta\)	\(\cot\theta\)	\(-\sec\theta\)	\(-\csc\theta\)