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Math Master Arsenal 🚀

Extensive Formula Deep Dive: All 13 Chapters for Ultimate 80/80 Prep

🔗 Unit 1: Relations & Functions

Relations and Functions

Types of Relations

For a relation $R$ in a set $A$:

Reflexive: $(a, a) \in R$ for all $a \in A$.
Symmetric: $(a, b) \in R \implies (b, a) \in R$ for all $a, b \in A$.
Transitive: $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$ for all $a, b, c \in A$.
Equivalence Relation: A relation that is reflexive, symmetric, and transitive.
Equivalence Class [a]: Set of all elements related to $a$.
Number of relations from set $A$ (m elements) to $B$ (n elements) = $2^{mn}$.

Types of Functions

One-One (Injective): $f(x_1) = f(x_2) \implies x_1 = x_2$ for all $x_1, x_2 \in X$.
Onto (Surjective): Every element $y \in Y$ has a pre-image in $X$ i.e., $f(X) = Y$ (Range = Codomain).
Bijective: A function which is both one-one and onto.
Number of functions from $A$(m) to $B$(n) = $n^m$.
Number of One-One functions from $A$(m) to $B$(n) = $^nP_m$ (if $n \ge m$), else $0$.
Number of Bijections from $A$(n) to $A$(n) = $n!$

Composition & Invertible Functions

Composition: $g \circ f(x) = g(f(x))$. It is defined if Range of $f \subseteq$ Domain of $g$.
Associative: $h \circ (g \circ f) = (h \circ g) \circ f$.
Inverse Function: A function is invertible if and only if it is a bijection.
If $f \circ g = I_Y$ and $g \circ f = I_X$, then $g = f^{-1}$.
Property: $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

Inverse Trigonometric Functions

Domain & Principal Value Branches

\begin{array}{|c|c|c|} \hline \text{Function} & \text{Domain} & \text{Principal Value Range} \\ \hline y = \sin^{-1} x & [-1, 1] & [-\pi/2, \pi/2] \\ y = \cos^{-1} x & [-1, 1] & [0, \pi] \\ y = \tan^{-1} x & \mathbb{R} & (-\pi/2, \pi/2) \\ y = \cot^{-1} x & \mathbb{R} & (0, \pi) \\ y = \sec^{-1} x & \mathbb{R} - (-1, 1) & [0, \pi] - \{\pi/2\} \\ y = \text{cosec}^{-1} x & \mathbb{R} - (-1, 1) & [-\pi/2, \pi/2] - \{0\} \\ \hline \end{array}

Negative Angle Properties

\sin^{-1}(-x) = -\sin^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x, \quad \text{cosec}^{-1}(-x) = -\text{cosec}^{-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

Reciprocal & Complementary Properties

\sin^{-1}(1/x) = \text{cosec}^{-1}x \quad(|x| \ge 1) $$ $$ \cos^{-1}(1/x) = \sec^{-1}x \quad(|x| \ge 1) $$ $$ \tan^{-1}(1/x) = \cot^{-1}x \quad(x > 0) \quad \text{and} \quad -\pi + \cot^{-1}x \quad(x < 0) $$ $$ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}, \quad \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}, \quad \sec^{-1}x + \text{cosec}^{-1}x = \frac{\pi}{2}

Trigonometric Conversions (Important for Integrals/Derivatives)

\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) $$ $$ 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)

📊 Unit 2: Algebra

Matrices

Matrix Types & Operations

Scalar Multiplication: $k[a_{ij}] = [k \cdot a_{ij}]$
Multiplication: Possible if columns of $A$ = rows of $B$. ($A_{m \times n} \times B_{n \times p} = AB_{m \times p}$). Multiplication is generally not commutative ($AB \neq BA$).

Transpose of a Matrix

(A^T)^T = A $$ $$ (A \pm B)^T = A^T \pm B^T $$ $$ (kA)^T = k A^T $$ $$ (AB)^T = B^T A^T \quad \text{(Reversal Law)}

Symmetric & Skew Symmetric Matrices

Symmetric Matrix: $A^T = A \implies a_{ij} = a_{ji}$
Skew-Symmetric Matrix: $A^T = -A \implies a_{ij} = -a_{ji}$. All diagonal elements are zero.
Every square matrix can be expressed uniquely as sum of symmetric and skew-symmetric:

A = \underbrace{\frac{1}{2}(A + A^T)}_{\text{Symmetric}} + \underbrace{\frac{1}{2}(A - A^T)}_{\text{Skew-Symmetric}}

Determinants

Properties of Determinants

$|A| = |A^T|$
If any two rows/cols are interchanged, sign of $|A|$ changes.
If any two rows/cols are identical or proportional, $|A| = 0$.
$|kA| = k^n|A|$ where $n$ is order of matrix.
$|AB| = |A||B|$

Area of a Triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$: $$ \Delta = \frac{1}{2} \left| \begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{matrix} \right| $$

Minors, Cofactors & Adjoint

Cofactor: $A_{ij} = (-1)^{i+j} M_{ij}$.
Sum of product of elements of a row with their own cofactors gives determinant: $\sum a_{ij}A_{ij} = |A|$.
Sum of product with cofactors of OTHER row gives zero: $\sum a_{1j}A_{2j} = 0$.
Adjoint properties:

A(\text{adj } A) = (\text{adj } A)A = |A|I $$ $$ |\text{adj } A| = |A|^{n-1} $$ $$ \text{adj}(AB) = (\text{adj } B)(\text{adj } A) $$ $$ |\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}

Inverse & System of Equations

A^{-1} = \frac{1}{|A|} (\text{adj } A) $$ $$ (AB)^{-1} = B^{-1} A^{-1} \quad ; \quad (A^{-1})^{-1} = A \quad ; \quad (A^T)^{-1} = (A^{-1})^T

Solving linear equations ($AX=B$):

If $|A| \neq 0$: Consistent, unique solution $X = A^{-1}B$.
If $|A| = 0$ AND $(\text{adj } A)B = 0$: May be consistent (infinite solutions) or inconsistent.
If $|A| = 0$ AND $(\text{adj } A)B \neq 0$: Inconsistent (No solution).

∫ Unit 3: Calculus

Continuity & Differentiability

Continuity

A function $f(x)$ is continuous at $x = c$ if:

\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)

Every differentiable function is continuous, but converse is not always true (e.g., $f(x) = |x|$ at $x=0$).

Rules of Differentiation

\text{Product Rule:} \quad \frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx} $$ $$ \text{Quotient Rule:} \quad \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} $$ $$ \text{Chain Rule:} \quad \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}

Standard Derivatives

\frac{d}{dx}(x^n) = nx^{n-1}, \quad \frac{d}{dx}(e^x) = e^x, \quad \frac{d}{dx}(a^x) = a^x \ln a, \quad \frac{d}{dx}(\ln x) = \frac{1}{x} $$ $$ \frac{d}{dx}(\sin x) = \cos x, \quad \frac{d}{dx}(\cos x) = -\sin x $$ $$ \frac{d}{dx}(\tan x) = \sec^2 x, \quad \frac{d}{dx}(\cot x) = -\text{cosec}^2 x $$ $$ \frac{d}{dx}(\sec x) = \sec x \tan x, \quad \frac{d}{dx}(\text{cosec } x) = -\text{cosec } x \cot x $$ $$ \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}(\cos^{-1} x) = \frac{-1}{\sqrt{1-x^2}} $$ $$ \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}, \quad \frac{d}{dx}(\cot^{-1} x) = \frac{-1}{1+x^2} $$ $$ \frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}, \quad \frac{d}{dx}(\text{cosec}^{-1} x) = \frac{-1}{|x|\sqrt{x^2-1}}

Parametric & Logarithmic Diff.

Parametric Form ($x = f(t), y = g(t)$): $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
Logarithmic Diff. ($y = [f(x)]^{g(x)}$): Take $\log_e$ on both sides: $\ln y = g(x) \ln(f(x))$, then differentiate implicitly.

Application of Derivatives

Rate of Change

If $y = f(x)$, then $\frac{dy}{dx}$ represents rate of change of $y$ w.r.t $x$.

Increasing & Decreasing Functions

$f(x)$ is Strictly Increasing on interval $(a,b)$ if $f'(x) > 0$.
$f(x)$ is Strictly Decreasing on interval $(a,b)$ if $f'(x) < 0$.

Maxima and Minima

Critical Points: Find points where $f'(x) = 0$ or $f'(x)$ is undefined.
First Derivative Test: For a critical point $c$:
- If $f'(x)$ changes sign from $+$ to $-$ as $x$ increases through $c \implies$ Local Maxima.
- If $f'(x)$ changes sign from $-$ to $+$ $\implies$ Local Minima.
Second Derivative Test: At $f'(c) = 0$:
- $f''(c) < 0 \implies$ Local Maxima.
- $f''(c) > 0 \implies$ Local Minima.
- $f''(c) = 0 \implies$ Test fails, use First Derivative Test.
Absolute Max/Min: Find values at all critical points in $[a,b]$ AND at endpoints $f(a), f(b)$. The greatest is absolute max.

Integrals

Standard Integral Formulae

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1), \quad \int \frac{1}{x} \, dx = \ln|x| + C $$ $$ \int e^x \, dx = e^x + C, \quad \int a^x \, dx = \frac{a^x}{\ln a} + C $$ $$ \int \sin x \, dx = -\cos x + C, \quad \int \cos x \, dx = \sin x + C $$ $$ \int \sec^2 x \, dx = \tan x + C, \quad \int \text{cosec}^2 x \, dx = -\cot x + C $$ $$ \int \tan x \, dx = -\ln|\cos x| + C = \ln|\sec x| + C $$ $$ \int \cot x \, dx = \ln|\sin x| + C $$ $$ \int \sec x \, dx = \ln|\sec x + \tan x| + C, \quad \int \text{cosec } x \, dx = \ln|\text{cosec } x - \cot x| + C

Integrals of Special Functions

\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C $$ $$ \int \frac{dx}{a^2 - x^2} = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C $$ $$ \int \frac{dx}{x^2 + a^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C $$ $$ \int \frac{dx}{\sqrt{x^2 - a^2}} = \ln|x + \sqrt{x^2 - a^2}| + C $$ $$ \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln|x + \sqrt{x^2 + a^2}| + C $$ $$ \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C $$ $$ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C $$ $$ \int \sqrt{x^2 + a^2} \, dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\ln|x + \sqrt{x^2 + a^2}| + C $$ $$ \int \sqrt{x^2 - a^2} \, dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\ln|x + \sqrt{x^2 - a^2}| + C

Integration by Parts & e^x trick

\int u \cdot v \, dx = u \int v \, dx - \int \left(\frac{du}{dx} \int v \, dx\right) dx

Choose 1st function ($u$) using word ILATE (Inverse, Logarithmic, Algebraic, Trig, Expo).

\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C

Properties of Definite Integrals

P0: $ \int_a^b f(x) dx = \int_a^b f(t) dt $ P1: $ \int_a^b f(x) dx = -\int_b^a f(x) dx $ P2: $ \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx $ (useful for modulus functions) P3: $ \int_a^b f(x) dx = \int_a^b f(a+b-x) dx $ P4 (King's Rule VVI): $ \int_0^a f(x) dx = \int_0^a f(a-x) dx $ P6: $ \int_0^{2a} f(x) dx = \begin{cases} 2\int_0^a f(x) dx & \text{if } f(2a-x)=f(x) \\ 0 & \text{if } f(2a-x)=-f(x) \end{cases} $ P7 (Even/Odd Rule): $ \int_{-a}^a f(x) dx = \begin{cases} 2\int_0^a f(x) dx & \text{if } f(-x)=f(x) \text{ (Even)} \\ 0 & \text{if } f(-x)=-f(x) \text{ (Odd)} \end{cases} $

Application of Integrals

Area Under Curve

Area bounded by y=f(x), x-axis, and x=a, x=b: $ A = \int_a^b |y| \, dx = \int_a^b |f(x)| \, dx $
Area bounded by x=g(y), y-axis, and y=c, y=d: $ A = \int_c^d |x| \, dy = \int_c^d |g(y)| \, dy $
Area between two curves $y=f(x)$ and $y=g(x)$ (where $f(x) \ge g(x)$):
$ A = \int_a^b [f(x) - g(x)] \, dx $

Differential Equations

Order & Degree

Order: Order of highest order derivative occurring in DE.

Degree: Highest power of highest order derivative (Provided DE is polynomial in its derivatives. If derivative is stuck inside sine/log, degree is not defined).

Methods to Solve First Order DE

Variable Separable: Convert to form $f(y)dy = g(x)dx$, then integrate directly.
Homogeneous DE: If degree of all terms is same. Substitute $y = vx$. Then $\frac{dy}{dx} = v + x\frac{dv}{dx}$. Converts it to Variable Separable.
Linear Differential Equation (LDE): Form: $\frac{dy}{dx} + Py = Q$ (where P, Q are functions of x only or constants).

Integrating Factor (I.F.) = $\displaystyle e^{\int P \, dx}$ Solution is: $\quad y \cdot (\text{I.F.}) = \int [ Q \cdot (\text{I.F.}) ] dx + C$

LDE Form 2: $\frac{dx}{dy} + Px = Q$ (where P, Q are functions of y).
IF = $e^{\int P \, dy}$. Solution: $x \cdot (\text{IF}) = \int [Q \cdot (\text{IF})] dy + C$.

🧭 Unit 4: Vectors & 3D Geometry

Vector Algebra

Basic Operations & Unit Vector

For vector $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$: Magnitude: $$ |\vec{a}| = \sqrt{x^2 + y^2 + z^2} $$ Unit Vector: $$ \hat{a} = \frac{\vec{a}}{|\vec{a}|} $$ Direction Cosines: $$ l = \frac{x}{|\vec{a}|}, \, m = \frac{y}{|\vec{a}|}, \, n = \frac{z}{|\vec{a}|} \implies l^2+m^2+n^2=1 $$

Dot Product (Scalar Product)

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta $$ $$ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 $$ Projection of $\vec{a}$ on $\vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} $ Perpendicular Condition: $$ \vec{a} \cdot \vec{b} = 0

Cross Product (Vector Product)

|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin\theta $$ $$ \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} $$ Area of Triangle = $\frac{1}{2}|\vec{a} \times \vec{b}|$ Area of Parallelogram = $|\vec{a} \times \vec{b}|$ Parallel/Collinear Condition: $$ \vec{a} \times \vec{b} = \vec{0} \quad \text{or} \quad \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}

Three Dimensional Geometry

Equation of a Line

Passing through a point $\vec{a}$ and parallel to vector $\vec{b}$:

Vector Form: $$ \vec{r} = \vec{a} + \lambda \vec{b} $$ Cartesian Form: $$ \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} $$ *(where a,b,c are direction ratios of parallel vector)*

Passing through two points $\vec{a}$ and $\vec{b}$:

Vector Form: $$ \vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a}) $$ Cartesian Form: $$ \frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1} $$

Angles and Distances (Lines)

Angle between two lines: $\cos\theta = \left|\frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right|$
Shortest Distance between Skew Lines ($\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$):

d = \left| \frac{(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})}{|\vec{b_1} \times \vec{b_2}|} \right|

Distance between Parallel Lines ($\vec{r} = \vec{a_1} + \lambda\vec{b}$ and $\vec{r} = \vec{a_2} + \mu\vec{b}$):

d = \left| \frac{\vec{b} \times (\vec{a_2} - \vec{a_1})}{|\vec{b}|} \right|

📈 Unit 5: Linear Programming

Linear Programming

Linear Programming (Corner Point Method)

Formulate objective function $Z = ax + by$ (Maximize/Minimize).
Write down all linear constraints (e.g., $x+y \le 50$, $x \ge 0$, $y \ge 0$).
Graph the lines and identify the Feasible Region (common shaded region).
Find coordinates of all Corner Points of the feasible region.
Evaluate $Z$ at each corner point.
- If feasible region is Bounded: The max/min value among these is the final answer.
- If Unbounded: For maximum value $M$, draw open half-plane $ax+by > M$. If no common point with feasible region, $M$ is max. Else, no max value. (Same logic for min value $m$ using $ax+by < m$).

🎲 Unit 6: Probability

Probability

Conditional Probability

Probability of event A, given event B has already occurred:

P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad (P(B) > 0)

Multiplication Rule & Independence

Multiplication Theorem: $P(A \cap B) = P(A) \cdot P(B|A)$
Independent Events: Occurrence of one does not affect the other.
- $P(A|B) = P(A)$
- $P(A \cap B) = P(A) \cdot P(B)$

Total Probability & Bayes' Theorem

Theorem of Total Probability: If $E_1, E_2, ... E_n$ are mutually exclusive and exhaustive events, and $A$ is any event, then: $P(A) = P(E_1)P(A|E_1) + P(E_2)P(A|E_2) + \dots + P(E_n)P(A|E_n)$
Bayes' Theorem: Helps find the probability of a "cause" given an "effect/result" has happened. $P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum_{j=1}^{n} P(E_j)P(A|E_j)}$

Random Variable & Probability Distribution

Valid Distribution: $\sum P(X) = 1$ and $0 \le P(X) \le 1$.
Mean / Expectation $\mu$: $E(X) = \sum_{i=1}^{n} x_i p_i$
Variance $\sigma^2$: $\text{Var}(X) = \sum_{i=1}^{n} x_i^2 p_i - [E(X)]^2$
Standard Deviation $\sigma$: $\sqrt{\text{Var}(X)}$