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Class 12 Mathematics — Complete Formulas & Master Repository
Complete, 100% exhaustive formula guide for Class 12 Mathematics (NCERT, CBSE & State Board Syllabus). Includes Relations & Functions, Inverse Trigonometry, Matrices, Determinants, Calculus (Limits, Continuity, Derivatives, Integrals, Differential Equations), Vector Algebra, 3D Geometry, Linear Programming, Probability, Mensuration (2D & 3D Shapes), and Statistics.
1. Relations and Functions
Concepts of relations on sets, types of relations, classification of functions, composition of functions, invertible functions, and binary operations.
1.1 Relations and Types
Definition: A relation \(R\) from a non-empty set \(A\) to \(B\) is a subset of \(A \times B\) (\(R \subseteq A \times B\)). If \((a, b) \in R\), we write \(a R b\).
Types of Relations on Set \(A\):
🔹 Empty Relation: \(R = \emptyset \subset A \times A\)
🔹 Universal Relation: \(R = A \times A\)
🔹 Identity Relation: \(I_A = \{(a, a) : a \in A\}\)
🔹 Reflexive Relation: If \((a, a) \in R\) for every \(a \in A\).
🔹 Symmetric Relation: If \((a, b) \in R \implies (b, a) \in R\) for all \(a, b \in A\).
🔹 Transitive Relation: If \((a, b) \in R\) and \((b, c) \in R \implies (a, c) \in R\).
🔹 Equivalence Relation: A relation that is Reflexive, Symmetric, and Transitive simultaneously.
🔹 Empty Relation: \(R = \emptyset \subset A \times A\)
🔹 Universal Relation: \(R = A \times A\)
🔹 Identity Relation: \(I_A = \{(a, a) : a \in A\}\)
🔹 Reflexive Relation: If \((a, a) \in R\) for every \(a \in A\).
🔹 Symmetric Relation: If \((a, b) \in R \implies (b, a) \in R\) for all \(a, b \in A\).
🔹 Transitive Relation: If \((a, b) \in R\) and \((b, c) \in R \implies (a, c) \in R\).
🔹 Equivalence Relation: A relation that is Reflexive, Symmetric, and Transitive simultaneously.
1.2 Functions and Types
A function \(f: A \to B\) is a rule assigning each element of domain \(A\) to a unique element in codomain \(B\). The set of actual output values \(f(A)\) is called the Range.
| Type of Function | Mathematical Definition | Example |
|---|---|---|
| One-One (Injective) | \(f(x_1) = f(x_2) \implies x_1 = x_2\) | \(f(x) = 2x + 3\) |
| Many-One | Distinct domain elements map to the same image. | \(f(x) = x^2\) |
| Onto (Surjective) | Range = Codomain (Every element in B has a pre-image in A). | \(f(x) = x^3\) |
| Into Function | At least one element in B has no pre-image in A. | \(f: \mathbb{R} \to \mathbb{R}, f(x) = x^2\) |
| Bijective | Both One-One and Onto. | \(f(x) = 3x - 5\) |
| Even Function | \(f(-x) = f(x)\) (Symmetric about y-axis) | \(f(x) = \cos x, x^2\) |
| Odd Function | \(f(-x) = -f(x)\) (Symmetric about origin) | \(f(x) = \sin x, x^3\) |
1️⃣ Composition of Functions:
If \(f: A \to B\) and \(g: B \to C\), the composite function \((g \circ f): A \to C\) is defined as: $(g \circ f)(x) = g(f(x)), \quad \forall x \in A$ Note: Generally, \(f \circ g \neq g \circ f\).
If \(f: A \to B\) and \(g: B \to C\), the composite function \((g \circ f): A \to C\) is defined as: $(g \circ f)(x) = g(f(x)), \quad \forall x \in A$ Note: Generally, \(f \circ g \neq g \circ f\).
2️⃣ Inverse Function:
A function \(f: A \to B\) is invertible if and only if \(f\) is Bijective. There exists a unique function \(g: B \to A\) such that: $f(x) = y \iff g(y) = x \implies f^{-1}(y) = x$ Properties: \((f^{-1})^{-1} = f\), and \((g \circ f)^{-1} = f^{-1} \circ g^{-1}\).
A function \(f: A \to B\) is invertible if and only if \(f\) is Bijective. There exists a unique function \(g: B \to A\) such that: $f(x) = y \iff g(y) = x \implies f^{-1}(y) = x$ Properties: \((f^{-1})^{-1} = f\), and \((g \circ f)^{-1} = f^{-1} \circ g^{-1}\).
3️⃣ Binary Operations:
A binary operation \(*\) on set \(A\) is a function \(*: A \times A \to A\).
🔹 Commutative: \(a * b = b * a\)
🔹 Associative: \((a * b) * c = a * (b * c)\)
🔹 Identity Element: \(a * e = e * a = a\)
🔹 Inverse Element: \(a * b = b * a = e \implies b = a^{-1}\)
A binary operation \(*\) on set \(A\) is a function \(*: A \times A \to A\).
🔹 Commutative: \(a * b = b * a\)
🔹 Associative: \((a * b) * c = a * (b * c)\)
🔹 Identity Element: \(a * e = e * a = a\)
🔹 Inverse Element: \(a * b = b * a = e \implies b = a^{-1}\)
2. Inverse Trigonometric Functions
Principal Value Branches, Domains, Ranges, and Master Transformation Formulas for Inverse Trigonometric Functions.
2.1 Domain & Principal Value Branch Table
| Function \(y = f(x)\) | Domain | Principal Value Branch / Range |
|---|---|---|
| \(y = \sin^{-1} x\) | \([-1, 1]\) | \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) |
| \(y = \cos^{-1} x\) | \([-1, 1]\) | \([0, \pi]\) |
| \(y = \tan^{-1} x\) | \(\mathbb{R}\) | \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) |
| \(y = \cot^{-1} x\) | \(\mathbb{R}\) | \((0, \pi)\) |
| \(y = \sec^{-1} x\) | \(\mathbb{R} - (-1, 1)\) | \([0, \pi] - \left\{\frac{\pi}{2}\right\}\) |
| \(y = \csc^{-1} x\) | \(\mathbb{R} - (-1, 1)\) | \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\) |
2.2 Essential Identities & Formulas
1️⃣ Negative Angle Identities:
$\sin^{-1}(-x) = -\sin^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x, \quad \csc^{-1}(-x) = -\csc^{-1}x$ $\cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \cot^{-1}(-x) = \pi - \cot^{-1}x, \quad \sec^{-1}(-x) = \pi - \sec^{-1}x$
$\sin^{-1}(-x) = -\sin^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x, \quad \csc^{-1}(-x) = -\csc^{-1}x$ $\cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \cot^{-1}(-x) = \pi - \cot^{-1}x, \quad \sec^{-1}(-x) = \pi - \sec^{-1}x$
2️⃣ Complementary Identities:
$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \quad (|x| \le 1)$ $\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \quad (x \in \mathbb{R})$ $\sec^{-1}x + \csc^{-1}x = \frac{\pi}{2} \quad (|x| \ge 1)$
$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \quad (|x| \le 1)$ $\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \quad (x \in \mathbb{R})$ $\sec^{-1}x + \csc^{-1}x = \frac{\pi}{2} \quad (|x| \ge 1)$
3️⃣ Reciprocal Identities:
$\sin^{-1}\left(\frac{1}{x}\right) = \csc^{-1}x, \quad \cos^{-1}\left(\frac{1}{x}\right) = \sec^{-1}x, \quad \tan^{-1}\left(\frac{1}{x}\right) = \cot^{-1}x \quad (x > 0)$
$\sin^{-1}\left(\frac{1}{x}\right) = \csc^{-1}x, \quad \cos^{-1}\left(\frac{1}{x}\right) = \sec^{-1}x, \quad \tan^{-1}\left(\frac{1}{x}\right) = \cot^{-1}x \quad (x > 0)$
4️⃣ \(\tan^{-1}\) Addition & Subtraction Formulas:
$\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \quad (xy < 1)$ $\tan^{-1}x - \tan^{-1}y = \tan^{-1}\left(\frac{x-y}{1+xy}\right) \quad (xy > -1)$
$\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \quad (xy < 1)$ $\tan^{-1}x - \tan^{-1}y = \tan^{-1}\left(\frac{x-y}{1+xy}\right) \quad (xy > -1)$
5️⃣ Multiple Angle Transformations:
$2\tan^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \tan^{-1}\left(\frac{2x}{1-x^2}\right)$ $\sin^{-1}x \pm \sin^{-1}y = \sin^{-1}\left(x\sqrt{1-y^2} \pm y\sqrt{1-x^2}\right)$ $\cos^{-1}x \pm \cos^{-1}y = \cos^{-1}\left(xy \mp \sqrt{1-x^2}\sqrt{1-y^2}\right)$
$2\tan^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \tan^{-1}\left(\frac{2x}{1-x^2}\right)$ $\sin^{-1}x \pm \sin^{-1}y = \sin^{-1}\left(x\sqrt{1-y^2} \pm y\sqrt{1-x^2}\right)$ $\cos^{-1}x \pm \cos^{-1}y = \cos^{-1}\left(xy \mp \sqrt{1-x^2}\sqrt{1-y^2}\right)$
3. Matrices
Rectangular arrays of numbers/functions, types of matrices, matrix algebra, transpose, symmetric, skew-symmetric matrices, and inverse matrix.
3.1 Types & Matrix Operations
1️⃣ Order of Matrix: \(m \times n\) (\(m\) rows, \(n\) columns).
Types: Row matrix (\(1 \times n\)), Column matrix (\(m \times 1\)), Zero matrix (\(O\)), Square matrix (\(n \times n\)), Diagonal, Scalar, Identity matrix (\(I_n\)).
Types: Row matrix (\(1 \times n\)), Column matrix (\(m \times 1\)), Zero matrix (\(O\)), Square matrix (\(n \times n\)), Diagonal, Scalar, Identity matrix (\(I_n\)).
2️⃣ Addition & Subtraction:
Defined for matrices of identical order: $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \pm \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}\pm b_{11} & a_{12}\pm b_{12} \\ a_{21}\pm b_{21} & a_{22}\pm b_{22} \end{pmatrix}$
Defined for matrices of identical order: $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \pm \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}\pm b_{11} & a_{12}\pm b_{12} \\ a_{21}\pm b_{21} & a_{22}\pm b_{22} \end{pmatrix}$
3️⃣ Matrix Multiplication (Row-by-Column Rule):
If \(A\) has order \(m \times n\) and \(B\) has order \(n \times p\), product \(AB\) has order \(m \times p\): $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix}$ Note: Matrix multiplication is non-commutative in general (\(AB \neq BA\)).
If \(A\) has order \(m \times n\) and \(B\) has order \(n \times p\), product \(AB\) has order \(m \times p\): $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix}$ Note: Matrix multiplication is non-commutative in general (\(AB \neq BA\)).
4️⃣ Transpose of Matrix (\(A^T\) or \(A'\)):
Interchanging rows and columns.
🔹 Properties: \((A^T)^T = A, \quad (kA)^T = k A^T, \quad (A+B)^T = A^T + B^T, \quad (AB)^T = B^T A^T\)
🔹 Symmetric Matrix: \(A^T = A\)
🔹 Skew-Symmetric Matrix: \(A^T = -A\) (Main diagonal elements are zero).
🔹 Any square matrix can be expressed as: $A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$
Interchanging rows and columns.
🔹 Properties: \((A^T)^T = A, \quad (kA)^T = k A^T, \quad (A+B)^T = A^T + B^T, \quad (AB)^T = B^T A^T\)
🔹 Symmetric Matrix: \(A^T = A\)
🔹 Skew-Symmetric Matrix: \(A^T = -A\) (Main diagonal elements are zero).
🔹 Any square matrix can be expressed as: $A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$
4. Determinants
Scalar values associated with square matrices, properties of determinants, minors, cofactors, adjoint, inverse matrix, and solving system of linear equations.
4.1 Evaluation & Properties
1️⃣ Evaluation of $2 \times 2$ and $3 \times 3$ Determinants:
$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$ $\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)$
$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$ $\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)$
2️⃣ Key Properties:
🔹 Interchanging rows and columns leaves the value unchanged (\(|A^T| = |A|\)).
🔹 Interchanging any two rows or columns reverses the sign.
🔹 If any two rows/columns are identical, the determinant is \(0\).
🔹 Multiplying a row/column by scalar \(k\) multiplies determinant by \(k\) (\(|kA| = k^n |A|\)).
🔹 Property: \(|AB| = |A| \cdot |B|\).
🔹 Interchanging rows and columns leaves the value unchanged (\(|A^T| = |A|\)).
🔹 Interchanging any two rows or columns reverses the sign.
🔹 If any two rows/columns are identical, the determinant is \(0\).
🔹 Multiplying a row/column by scalar \(k\) multiplies determinant by \(k\) (\(|kA| = k^n |A|\)).
🔹 Property: \(|AB| = |A| \cdot |B|\).
3️⃣ Area of Triangle:
For vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\): $\Delta = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
For vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\): $\Delta = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
4️⃣ Adjoint & Inverse of Matrix:
🔹 Minor \(M_{ij}\), Cofactor \(A_{ij} = (-1)^{i+j} M_{ij}\)
🔹 Adjoint (\(\text{adj}\,A\)): Transpose of cofactor matrix.
🔹 Inverse Matrix Formula: $A^{-1} = \frac{\text{adj}(A)}{|A|}, \quad (|A| \neq 0)$ 🔹 Key result: \(A \cdot (\text{adj}\,A) = |A| I_n\), and \(|\text{adj}\,A| = |A|^{n-1}\).
🔹 Minor \(M_{ij}\), Cofactor \(A_{ij} = (-1)^{i+j} M_{ij}\)
🔹 Adjoint (\(\text{adj}\,A\)): Transpose of cofactor matrix.
🔹 Inverse Matrix Formula: $A^{-1} = \frac{\text{adj}(A)}{|A|}, \quad (|A| \neq 0)$ 🔹 Key result: \(A \cdot (\text{adj}\,A) = |A| I_n\), and \(|\text{adj}\,A| = |A|^{n-1}\).
5️⃣ System of Linear Equations (Matrix Method):
\(AX = B \implies X = A^{-1} B\).
🔹 If \(|A| \neq 0 \implies\) Unique solution (Consistent system).
🔹 If \(|A| = 0\) and \((\text{adj}\,A)B \neq 0 \implies\) No solution (Inconsistent system).
🔹 If \(|A| = 0\) and \((\text{adj}\,A)B = 0 \implies\) Infinitely many solutions.
\(AX = B \implies X = A^{-1} B\).
🔹 If \(|A| \neq 0 \implies\) Unique solution (Consistent system).
🔹 If \(|A| = 0\) and \((\text{adj}\,A)B \neq 0 \implies\) No solution (Inconsistent system).
🔹 If \(|A| = 0\) and \((\text{adj}\,A)B = 0 \implies\) Infinitely many solutions.
5. Calculus — Continuity and Differentiability
5.1 Essential Limits & Continuity
Condition for Continuity at \(x = a\):
$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \quad (\text{LHL} = \text{RHL} = f(a))$
Standard Limits:
$\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{\tan x}{x} = 1, \quad \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$
$\lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a, \quad \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$
5.2 Master Derivatives Table
| Function \(f(x)\) | Derivative \(\frac{d}{dx}f(x)\) | Function \(f(x)\) | Derivative \(\frac{d}{dx}f(x)\) |
|---|---|---|---|
| Constant \(c\) | \(0\) | \(x^n\) | \(n x^{n-1}\) |
| \(e^x\) | \(e^x\) | \(a^x\) | \(a^x \ln a\) |
| \(\ln x\) | \(\frac{1}{x}\) | \(\log_a x\) | \(\frac{1}{x \ln a}\) |
| \(\sin x\) | \(\cos x\) | \(\cos x\) | \(-\sin x\) |
| \(\tan x\) | \(\sec^2 x\) | \(\cot x\) | \(-\csc^2 x\) |
| \(\sec x\) | \(\sec x \tan x\) | \(\csc x\) | \(-\csc x \cot x\) |
| \(\sin^{-1} x\) | \(\frac{1}{\sqrt{1-x^2}}\) | \(\cos^{-1} x\) | \(-\frac{1}{\sqrt{1-x^2}}\) |
| \(\tan^{-1} x\) | \(\frac{1}{1+x^2}\) | \(\cot^{-1} x\) | \(-\frac{1}{1+x^2}\) |
| \(\sec^{-1} x\) | \(\frac{1}{|x|\sqrt{x^2-1}}\) | \(\csc^{-1} x\) | \(-\frac{1}{|x|\sqrt{x^2-1}}\) |
Differentiation Rules:
🔹 Product Rule: \(\frac{d}{dx}[u \cdot v] = u \frac{dv}{dx} + v \frac{du}{dx}\)
🔹 Quotient Rule: \(\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\)
🔹 Chain Rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)
🔹 Product Rule: \(\frac{d}{dx}[u \cdot v] = u \frac{dv}{dx} + v \frac{du}{dx}\)
🔹 Quotient Rule: \(\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\)
🔹 Chain Rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)
Mean Value Theorems:
🔹 Rolle's Theorem: If \(f\) continuous on \([a,b]\), differentiable on \((a,b)\) and \(f(a)=f(b)\), then there exists \(c \in (a,b)\) such that \(f'(c) = 0\).
🔹 Lagrange's Mean Value Theorem (LMVT): There exists \(c \in (a,b)\) such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$
🔹 Rolle's Theorem: If \(f\) continuous on \([a,b]\), differentiable on \((a,b)\) and \(f(a)=f(b)\), then there exists \(c \in (a,b)\) such that \(f'(c) = 0\).
🔹 Lagrange's Mean Value Theorem (LMVT): There exists \(c \in (a,b)\) such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$
6. Applications of Derivatives
6.1 Rates, Tangents & Maxima/Minima
🔹 Increasing/Decreasing: Strictly Increasing if \(f'(x) > 0\), Strictly Decreasing if \(f'(x) < 0\).
🔹 Tangents & Normals: Slope \(m = f'(x_1)\)
Tangent: \(y - y_1 = m(x - x_1)\), Normal: \(y - y_1 = -\frac{1}{m}(x - x_1)\)
Tangent: \(y - y_1 = m(x - x_1)\), Normal: \(y - y_1 = -\frac{1}{m}(x - x_1)\)
🔹 Second Derivative Test for Maxima/Minima: At critical point where \(f'(c) = 0\):
• If \(f''(c) < 0 \implies\) Local Maximum at \(x = c\)
• If \(f''(c) > 0 \implies\) Local Minimum at \(x = c\)
• If \(f''(c) < 0 \implies\) Local Maximum at \(x = c\)
• If \(f''(c) > 0 \implies\) Local Minimum at \(x = c\)
7. Indefinite & Definite Integration
7.1 Standard Integrals Table
| Integral \(\int f(x) dx\) | Formula (+ C) | Integral \(\int f(x) dx\) | Formula (+ C) |
|---|---|---|---|
| \(\int x^n dx\) | \(\frac{x^{n+1}}{n+1} \quad (n \neq -1)\) | \(\int \frac{1}{x} dx\) | \(\ln|x|\) |
| \(\int e^x dx\) | \(e^x\) | \(\int a^x dx\) | \(\frac{a^x}{\ln a}\) |
| \(\int \sin x dx\) | \(-\cos x\) | \(\int \cos x dx\) | \(\sin x\) |
| \(\int \sec^2 x dx\) | \(\tan x\) | \(\int \csc^2 x dx\) | \(-\cot x\) |
| \(\int \sec x \tan x dx\) | \(\sec x\) | \(\int \csc x \cot x dx\) | \(-\csc x\) |
| \(\int \tan x dx\) | \(\ln|\sec x|\) | \(\int \cot x dx\) | \(\ln|\sin x|\) |
Special Integrals:
$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right), \quad \int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right|$ $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right), \quad \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln\left|x + \sqrt{x^2 + a^2}\right|$
$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right), \quad \int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right|$ $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right), \quad \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln\left|x + \sqrt{x^2 + a^2}\right|$
Integration by Parts (ILATE Rule):
$\int u \cdot v \, dx = u \int v dx - \int \left[ \frac{du}{dx} \cdot \int v dx \right] dx$
$\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$
Properties of Definite Integrals:
\(\int_0^a f(x) dx = \int_0^a f(a - x) dx\) (King's Property)
\(\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx\) (If Even), \(0\) (If Odd)
\(\int_0^a f(x) dx = \int_0^a f(a - x) dx\) (King's Property)
\(\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx\) (If Even), \(0\) (If Odd)
8. Applications of Integrals (Area Bounded by Curves)
8.1 Area Bounded Formulas
Area under curve \(y = f(x)\) from \(x=a\) to \(x=b\): \(A = \int_a^b f(x) dx\)
Area between two curves \(y=f(x)\) and \(y=g(x)\): \(A = \int_a^b [f(x) - g(x)] dx\)
Area between two curves \(y=f(x)\) and \(y=g(x)\): \(A = \int_a^b [f(x) - g(x)] dx\)
9. Differential Equations
9.1 Linear Differential Equations
Standard Form: \(\frac{dy}{dx} + P(x) y = Q(x)\)
Integrating Factor: \(\text{I.F.} = e^{\int P(x) dx}\)
General Solution: \(y \cdot (\text{I.F.}) = \int [Q(x) \cdot (\text{I.F.})] dx + C\)
Integrating Factor: \(\text{I.F.} = e^{\int P(x) dx}\)
General Solution: \(y \cdot (\text{I.F.}) = \int [Q(x) \cdot (\text{I.F.})] dx + C\)
10. Vector Algebra
10.1 Key Vector Formulas
🔹 Dot Product: \(\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3\)
🔹 Cross Product: \(\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\)
🔹 Scalar Triple Product: \([\vec{a} \, \vec{b} \, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})\)
🔹 Cross Product: \(\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\)
🔹 Scalar Triple Product: \([\vec{a} \, \vec{b} \, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})\)
11. Three-Dimensional Geometry
11.1 Line & Plane Formulas
🔹 Vector Line: \(\vec{r} = \vec{a} + \lambda \vec{b}\)
🔹 Cartesian Line: \(\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}\)
🔹 Shortest Distance: \(d = \left| \frac{(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)}{|\vec{b}_1 \times \vec{b}_2|} \right|\)
🔹 Cartesian Line: \(\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}\)
🔹 Shortest Distance: \(d = \left| \frac{(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)}{|\vec{b}_1 \times \vec{b}_2|} \right|\)
12. Linear Programming (LPP)
12.1 Objective Function & Corner Point Method
Objective Function \(Z = ax + by\)
Constraints: \(ax + by \le c\), \(x \ge 0, y \ge 0\).
Constraints: \(ax + by \le c\), \(x \ge 0, y \ge 0\).
13. Probability
13.1 Bayes' Theorem & Distributions
🔹 Conditional Probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
🔹 Bayes' Theorem: \(P(A_i | B) = \frac{P(A_i) P(B | A_i)}{\sum P(A_j) P(B | A_j)}\)
🔹 Binomial Distribution: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)
🔹 Bayes' Theorem: \(P(A_i | B) = \frac{P(A_i) P(B | A_i)}{\sum P(A_j) P(B | A_j)}\)
🔹 Binomial Distribution: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)
14. Mensuration & Geometry (2D & 3D Shapes)
14.1 2D & 3D Master Table
| Shape | Surface Area | Volume / Perimeter |
|---|---|---|
| Cube | \(6a^2\) | \(a^3\) |
| Cuboid | \(2(lb + bh + hl)\) | \(l \times b \times h\) |
| Cylinder | \(2\pi r(h+r)\) | \(\pi r^2 h\) |
| Sphere | \(4\pi r^2\) | \(\frac{4}{3}\pi r^3\) |
| Cone | \(\pi r (r+l)\) | \(\frac{1}{3}\pi r^2 h\) |
15. Statistics & Distributions
15.1 Statistical Formulas
🔹 Mean \(\bar{x} = \frac{\sum x_i}{n}\)
🔹 Standard Deviation \(\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}\)
🔹 Coefficient of Variation \(CV = \frac{\sigma}{\bar{x}} \times 100\%\)
🔹 Standard Deviation \(\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}\)
🔹 Coefficient of Variation \(CV = \frac{\sigma}{\bar{x}} \times 100\%\)
