Formula Sheet
Every key formula of the course on one page, grouped by chapter, with a link back to the section that develops it.
Chapter 1 · Relations and Functions
Types of relation on a set A§ 1.1 →
Reflexive: $(a,a)\in R\ \forall a\in A$ · Symmetric: $(a,b)\in R\Rightarrow(b,a)\in R$ · Transitive: $(a,b),(b,c)\in R\Rightarrow(a,c)\in R$
Trivial relations§ 1.1 →
Empty: $R=\varnothing$; universal: $R=A\times A$
Equivalence class§ 1.1 →
$R$ reflexive, symmetric and transitive; $[a]=\{x\in A:(x,a)\in R\}$
Classes form a partition§ 1.1 →
For an equivalence relation $R$: $a\in[a]$; $[a]=[b]\iff(a,b)\in R$; either $[a]=[b]$ or $[a]\cap[b]=\varnothing$
Counting relations (JEE), $n(A)=n$§ 1.1 →
all $2^{n^2}$ · reflexive $2^{n^2-n}$ · symmetric $2^{n(n+1)/2}$ · reflexive and symmetric $2^{n(n-1)/2}$
One-one (injective)§ 1.2 →
$f:A\to B$ is one-one iff $f(x_1)=f(x_2)\Rightarrow x_1=x_2$ for all $x_1,x_2\in A$
Onto (surjective)§ 1.2 →
$f:A\to B$ is onto iff for every $y\in B$ there is $x\in A$ with $f(x)=y$, i.e. $f(A)=B$
Bijective§ 1.2 →
one-one and onto: each $y\in B$ has exactly one preimage
Finite sets§ 1.2 →
$n(A)\gt n(B)$: no one-one $f$; $n(A)\lt n(B)$: no onto $f$; $n(A)=n(B)$ finite: $f$ one-one $\iff$ $f$ onto
JEE: counting, composition, inverse§ 1.2 →
$n(A)=m$, $n(B)=n$: $n^m$ functions, $\tfrac{n!}{(n-m)!}$ one-one ($m\le n$); $(g\circ f)(x)=g(f(x))$; only bijections have $f^{-1}$
Chapter 2 · Inverse Trigonometric Functions
sin⁻¹, cos⁻¹, tan⁻¹: domains and branches§ 2.1 →
$\sin^{-1}:[-1,1]\to\left[-\tfrac\pi2,\tfrac\pi2\right]$; $\cos^{-1}:[-1,1]\to[0,\pi]$; $\tan^{-1}:\mathbb R\to\left(-\tfrac\pi2,\tfrac\pi2\right)$
cosec⁻¹, sec⁻¹, cot⁻¹: domains and branches§ 2.1 →
$\operatorname{cosec}^{-1}:\mathbb R-(-1,1)\to\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}$; $\sec^{-1}:\mathbb R-(-1,1)\to[0,\pi]-\left\{\tfrac\pi2\right\}$; $\cot^{-1}:\mathbb R\to(0,\pi)$
Meaning of sin⁻¹ x§ 2.1 →
$y=\sin^{-1}x\iff\sin y=x$ and $-\tfrac\pi2\le y\le\tfrac\pi2$ (similarly for the other five, with their own branches); $\sin^{-1}x\ne\tfrac1{\sin x}$
Cancellation (Theorem 2.1)§ 2.1 →
$\sin(\sin^{-1}x)=x$ for $x\in[-1,1]$; $\sin^{-1}(\sin y)=y$ only for $y\in\left[-\tfrac\pi2,\tfrac\pi2\right]$ (likewise $\cos$ on $[0,\pi]$, $\tan$ on $\left(-\tfrac\pi2,\tfrac\pi2\right)$)
JEE: elementary properties§ 2.1 →
$\sin^{-1}(-x)=-\sin^{-1}x$, $\cos^{-1}(-x)=\pi-\cos^{-1}x$ ($|x|\le1$); $\sin^{-1}x+\cos^{-1}x=\tfrac\pi2$ ($|x|\le1$); $\tan^{-1}x+\cot^{-1}x=\tfrac\pi2$ ($x\in\mathbb R$)
Graph of an inverse (Theorem 2.2)§ 2.2 →
$(a,b)$ on $y=f(x)$ $\iff$ $(b,a)$ on $y=f^{-1}(x)$: reflect in $y=x$ ($f$ one-one and onto)
sin⁻¹(sin θ) for any θ§ 2.2 →
$\sin^{-1}(\sin\theta)=(-1)^k(\theta-k\pi)$ for $k\pi-\tfrac\pi2\le\theta\le k\pi+\tfrac\pi2$; e.g. $\pi-\theta$ on $\left[\tfrac\pi2,\tfrac{3\pi}2\right]$
cos⁻¹(cos θ) for any θ§ 2.2 →
$\cos^{-1}(\cos\theta)=|\theta-2k\pi|$ for $(2k-1)\pi\le\theta\le(2k+1)\pi$; e.g. $2\pi-\theta$ on $[\pi,2\pi]$
tan⁻¹(tan θ) for any θ§ 2.2 →
$\tan^{-1}(\tan\theta)=\theta-k\pi$ for $k\pi-\tfrac\pi2\lt\theta\lt k\pi+\tfrac\pi2$
JEE: tan⁻¹ addition§ 2.2 →
$\tan^{-1}x+\tan^{-1}y=\tan^{-1}\dfrac{x+y}{1-xy}$ ($xy\lt1$); $2\tan^{-1}x=\tan^{-1}\dfrac{2x}{1-x^2}$ ($|x|\lt1$)
Chapter 3 · Matrices
Matrix notation§ 3.1 →
$A=[a_{ij}]_{m\times n}$: $m$ rows, $n$ columns, $mn$ entries; $a_{ij}$ = entry in row $i$, column $j$
Possible orders§ 3.1 →
$N$ entries: orders $m\times n$ with $mn=N$; number of orders = number of positive divisors of $N$
Special square matrices§ 3.1 →
diagonal: $a_{ij}=0$ for $i\ne j$; scalar: diagonal with $a_{ii}=k$; identity $I$: $k=1$; zero $O$: all entries $0$ (any order)
Equality§ 3.1 →
$A=B\iff$ same order and $a_{ij}=b_{ij}$ for all $i,j$
Addition and scalar multiples§ 3.2 →
same order: $A+B=[a_{ij}+b_{ij}]$, $A-B=[a_{ij}-b_{ij}]$; any $k\in\mathbb R$: $kA=[ka_{ij}]$
Product of matrices§ 3.2 →
$A_{m\times n}B_{n\times p}=C_{m\times p}$, $c_{ik}=\sum_{j=1}^{n}a_{ij}b_{jk}$; undefined if columns of $A$ ≠ rows of $B$
Laws of multiplication§ 3.2 →
$(AB)C=A(BC)$, $A(B+C)=AB+AC$, $(A+B)C=AC+BC$, $I_mA=A=AI_n$ (orders compatible)
Warnings§ 3.2 →
in general $AB\ne BA$; $AB=O\not\Rightarrow A=O$ or $B=O$; $AB=AC\not\Rightarrow B=C$; $(A+B)^2=A^2+AB+BA+B^2$
JEE: 2 × 2 relation§ 3.2 →
$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$: $A^2-(a+d)A+(ad-bc)I=O$; $\begin{bmatrix}1&1\\0&1\end{bmatrix}^n=\begin{bmatrix}1&n\\0&1\end{bmatrix}$
Transpose§ 3.3 →
$A=[a_{ij}]_{m\times n}\ \Rightarrow\ A'=[a_{ji}]_{n\times m}$ (also written $A^T$)
Properties of the transpose§ 3.3 →
$(A')'=A$, $(kA)'=kA'$, $(A+B)'=A'+B'$, $(AB)'=B'A'$ (when defined)
Symmetric / skew-symmetric§ 3.3 →
square $A$: symmetric $\iff A'=A$; skew-symmetric $\iff A'=-A$ (then every $a_{ii}=0$)
Decomposition (unique)§ 3.3 →
square $A$: $A=\tfrac12(A+A')+\tfrac12(A-A')$, symmetric + skew-symmetric
Inverse of a square matrix§ 3.4 →
$A$ square, $AB=BA=I\ \Rightarrow\ B=A^{-1}$ (unique if it exists)
Inverse of a product§ 3.4 →
$A,B$ invertible, same order: $(AB)^{-1}=B^{-1}A^{-1}$
Rules for inverses§ 3.4 →
$(A^{-1})^{-1}=A$, $(A')^{-1}=(A^{-1})'$, $(kA)^{-1}=\tfrac1kA^{-1}$ ($k\ne0$); a zero row or column ⇒ no inverse
Using an inverse§ 3.4 →
$A$ invertible: $AX=B\Rightarrow X=A^{-1}B$; $AB=AC\Rightarrow B=C$
JEE: inverse from a relation§ 3.4 →
$A^2-kA+lI=O$, $l\ne0$ $\Rightarrow$ $A^{-1}=\tfrac1l(kI-A)$
Chapter 4 · Determinants
Determinant of order 2§ 4.1 →
$\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$; order 1: $|[a]|=a$ (may be negative)
Determinant of order 3 (along row 1)§ 4.1 →
$|A|=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{12}(a_{21}a_{33}-a_{23}a_{31})+a_{13}(a_{21}a_{32}-a_{22}a_{31})$
Expansion along any row or column (Theorem 4.1)§ 4.1 →
Entry $a_{ij}$ carries the sign $(-1)^{i+j}$: $\begin{smallmatrix}+&-&+\\-&+&-\\+&-&+\end{smallmatrix}$; every row and every column gives the same $|A|$ — choose the one with most zeros.
Area of a triangle (Theorem 4.2)§ 4.1 →
$\Delta=\tfrac12\left|D\right|$, $D=\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}$ (take the absolute value; an unknown vertex usually gives two answers)
Collinearity and the line through two points§ 4.1 →
$(x_i,y_i)$ collinear $\iff D=0$; line through $(x_1,y_1)\ne(x_2,y_2)$: $\begin{vmatrix}x&y&1\\x_1&y_1&1\\x_2&y_2&1\end{vmatrix}=0$
JEE: properties of determinants§ 4.1 →
$|A'|=|A|$; swapping two rows changes the sign; equal or proportional rows give $0$; a row factor $k$ comes outside; $R_i\to R_i+kR_j$ leaves $|A|$ unchanged.
Minor and cofactor§ 4.2 →
$M_{ij}$ = determinant after deleting row $i$, column $j$; $A_{ij}=(-1)^{i+j}M_{ij}$
Expanding with cofactors (Theorem 4.3)§ 4.2 →
$a_{i1}A_{i1}+a_{i2}A_{i2}+a_{i3}A_{i3}=|A|$; $a_{i1}A_{k1}+a_{i2}A_{k2}+a_{i3}A_{k3}=0$ for $i\ne k$ (same for columns)
Adjoint§ 4.2 →
$\operatorname{adj}A=[A_{ij}]'$ (transpose of the cofactor matrix); $\operatorname{adj}\begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$
Adjoint identity (Theorem 4.4)§ 4.2 →
$A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|\,I$
Inverse by the adjoint (Theorems 4.5, 4.6)§ 4.2 →
$A^{-1}$ exists $\iff|A|\ne0$, and then $A^{-1}=\tfrac1{|A|}\operatorname{adj}A$; $|AB|=|A|\,|B|$; $|A^{-1}|=\tfrac1{|A|}$
JEE: |kA| and |adj A|§ 4.2 →
For order $n$: $|kA|=k^n|A|$; $|\operatorname{adj}A|=|A|^{n-1}$; $\operatorname{adj}(\operatorname{adj}A)=|A|^{n-2}A$ ($|A|\ne0$)
A system as AX = B§ 4.3 →
$a_ix+b_iy+c_iz=d_i\ (i=1,2,3)\iff AX=B$, $A=[\,\text{coefficients}\,]$, $X=\begin{bmatrix}x\\y\\z\end{bmatrix}$, $B=\begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix}$
Matrix method (Theorem 4.7)§ 4.3 →
If $|A|\ne0$: exactly one solution, $X=A^{-1}B=\tfrac1{|A|}(\operatorname{adj}A)B$
When |A| = 0 (Theorem 4.8)§ 4.3 →
$(\operatorname{adj}A)B\ne O\Rightarrow$ no solution; $(\operatorname{adj}A)B=O\Rightarrow$ infinitely many or none — solve directly to decide
Homogeneous system§ 4.3 →
$AX=O$ always has $X=O$; if $|A|\ne0$ it is the only solution
Two equations, two unknowns§ 4.3 →
$a_1b_2-a_2b_1\ne0$: lines cross (one solution); $=0$: parallel (none) or coincident (infinitely many)
Chapter 5 · Continuity and Differentiability
Continuity at a point§ 5.1 →
$f$ continuous at $c$ (in the domain) $\iff\lim\limits_{x\to c^-}f(x)=\lim\limits_{x\to c^+}f(x)=f(c)$
Continuity on $[a,b]$§ 5.1 →
continuous on $(a,b)$, and $\lim\limits_{x\to a^+}f(x)=f(a)$, $\lim\limits_{x\to b^-}f(x)=f(b)$
Algebra of continuous functions§ 5.1 →
$f,g$ continuous at $c$ $\Rightarrow$ $f\pm g$, $fg$, $\lambda f$ continuous at $c$; $\tfrac fg$ continuous at $c$ if $g(c)\ne0$
Composite of continuous functions§ 5.1 →
$g$ continuous at $c$ and $f$ continuous at $g(c)$ $\Rightarrow$ $f\circ g$ continuous at $c$
Standard functions§ 5.1 →
polynomials, $\sin x$, $\cos x$, $|x|$: continuous on $\mathbb R$; rational and $\tan x$ etc.: continuous on their domains; $[x]$: discontinuous exactly at the integers
Limits used for “find $k$”§ 5.1 →
$\lim\limits_{x\to0}\dfrac{\sin ax}{x}=a$, $\ \lim\limits_{x\to0}\dfrac{1-\cos ax}{x^2}=\dfrac{a^2}{2}$
Differentiability at $c$§ 5.2 →
$Lf'(c)=\lim\limits_{h\to0^-}\dfrac{f(c+h)-f(c)}{h}$, $Rf'(c)=\lim\limits_{h\to0^+}\dfrac{f(c+h)-f(c)}{h}$; differentiable $\iff$ both exist and are equal
Differentiable ⇒ continuous§ 5.2 →
$f$ differentiable at $c$ $\Rightarrow$ $f$ continuous at $c$ (not conversely: $|x|$ at $0$)
Chain rule§ 5.2 →
$\dfrac{d}{dx}v(u(x))=v'(u(x))\,u'(x)$, i.e. $\dfrac{dy}{dx}=\dfrac{dy}{dt}\cdot\dfrac{dt}{dx}$ ($u$ differentiable at $x$, $v$ at $u(x)$)
Implicit differentiation§ 5.2 →
$\dfrac{d}{dx}y^n=ny^{n-1}\dfrac{dy}{dx}$, $\ \dfrac{d}{dx}(xy)=y+x\dfrac{dy}{dx}$
$\sin^{-1}$, $\cos^{-1}$§ 5.2 →
$\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}$, $\ \dfrac{d}{dx}\cos^{-1}x=-\dfrac{1}{\sqrt{1-x^2}}$, $\ |x|\lt1$
$\tan^{-1}$, $\cot^{-1}$, $\sec^{-1}$, $\operatorname{cosec}^{-1}$§ 5.2 →
$\dfrac{d}{dx}\tan^{-1}x=\dfrac{1}{1+x^2}=-\dfrac{d}{dx}\cot^{-1}x$; $\ \dfrac{d}{dx}\sec^{-1}x=\dfrac{1}{|x|\sqrt{x^2-1}}=-\dfrac{d}{dx}\operatorname{cosec}^{-1}x$, $|x|\gt1$
$e^x$§ 5.3 →
$\dfrac{d}{dx}e^x=e^x$, $\ \dfrac{d}{dx}e^{u}=e^{u}\,\dfrac{du}{dx}$
$\log x$ ($=\log_ex$)§ 5.3 →
$\dfrac{d}{dx}\log x=\dfrac1x$ ($x\gt0$), $\ \dfrac{d}{dx}\log|x|=\dfrac1x$ ($x\ne0$), $\ \dfrac{d}{dx}\log u=\dfrac{u'}{u}$
Other bases§ 5.3 →
$\dfrac{d}{dx}a^x=a^x\log a$ ($a\gt0$); $\ \dfrac{d}{dx}\log_ax=\dfrac{1}{x\log a}$ ($a\gt0$, $a\ne1$, $x\gt0$)
Laws of logarithms§ 5.3 →
$\log xy=\log x+\log y$, $\log\tfrac xy=\log x-\log y$, $\log x^n=n\log x$ ($x,y\gt0$); $\log_bx=\tfrac{\log x}{\log b}$
Logarithmic differentiation§ 5.3 →
$y=u^v$ ($u\gt0$): $\log y=v\log u$, $\ \dfrac{dy}{dx}=u^v\left(v'\log u+\dfrac{v\,u'}{u}\right)$; e.g. $\dfrac{d}{dx}x^x=x^x(1+\log x)$
Parametric derivative§ 5.4 →
$x=f(t)$, $y=g(t)$: $\ \dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{g'(t)}{f'(t)}$, provided $f'(t)\ne0$
Parametric second derivative§ 5.4 →
$\dfrac{d^2y}{dx^2}=\dfrac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}$ $\ \left(\text{not }\dfrac{d^2y/dt^2}{d^2x/dt^2}\right)$
Standard parametric forms§ 5.4 →
circle $(a\cos t,a\sin t)$; ellipse $(a\cos t,b\sin t)$; parabola $y^2=4ax$: $(at^2,2at)$, $\tfrac{dy}{dx}=\tfrac1t$; cycloid $(a(t-\sin t),a(1-\cos t))$, $\tfrac{dy}{dx}=\cot\tfrac t2$
Second-order derivative§ 5.4 →
$y_2=\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)=f''(x)$
Useful relations§ 5.4 →
$y=A\cos kx+B\sin kx\Rightarrow y_2+k^2y=0$; $\ y=e^{kx}\Rightarrow y_2=k^2y$; $\ y=(\sin^{-1}x)^2\Rightarrow(1-x^2)y_2-xy_1=2$
JEE: Rolle and mean value theorems§ 5.4 →
$f$ continuous on $[a,b]$, differentiable on $(a,b)$: some $c\in(a,b)$ has $f'(c)=\dfrac{f(b)-f(a)}{b-a}$ (Rolle: if $f(a)=f(b)$, then $f'(c)=0$)
Chapter 6 · Application of Derivatives
Rate of change (Definition 6.1)§ 6.1 →
average rate $=\dfrac{f(x_0+h)-f(x_0)}{h}$; rate at $x_0$: $\left.\dfrac{dy}{dx}\right|_{x_0}=f'(x_0)$ ($f$ differentiable at $x_0$); units: units of $y$ per unit of $x$
Motion on a line§ 6.1 →
$v=\dfrac{ds}{dt}$, $a=\dfrac{dv}{dt}=\dfrac{d^2s}{dt^2}$; at rest when $v=0$
Marginal cost, revenue, profit§ 6.1 →
$\text{MC}=\dfrac{dC}{dx}$, $\text{MR}=\dfrac{dR}{dx}$, $\dfrac{dP}{dx}=\text{MR}-\text{MC}$
Linked rates (Theorem 6.1)§ 6.1 →
$y=f(x)$, $x=x(t)$ both differentiable: $\dfrac{dy}{dt}=\dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$; differentiate the relation first, substitute the instant's values last
Standard linked rates§ 6.1 →
circle $\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}$; sphere $\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}$, $\dfrac{dS}{dt}=8\pi r\dfrac{dr}{dt}$; cube $\dfrac{dV}{dt}=3a^2\dfrac{da}{dt}$; ladder $x\dfrac{dx}{dt}+y\dfrac{dy}{dt}=0$
Increasing / decreasing (Definition 6.3)§ 6.2 →
On an interval $I$: increasing if $x_1\lt x_2\Rightarrow f(x_1)\le f(x_2)$; strictly increasing if $\Rightarrow f(x_1)\lt f(x_2)$; (strictly) decreasing with $\ge$ ($\gt$)
Derivative test (Theorem 6.2)§ 6.2 →
$f$ continuous on $[a,b]$, differentiable on $(a,b)$: $f'\gt0$ on $(a,b)\Rightarrow$ strictly increasing on $[a,b]$; $f'\lt0\Rightarrow$ strictly decreasing; $f'=0\Rightarrow$ constant
Isolated zeros of f′§ 6.2 →
$f'\ge0$ with $f'=0$ only at finitely many points $\Rightarrow$ still strictly increasing (e.g. $x^3$); the converse of the test fails
Cubic increasing on ℝ§ 6.2 →
$f'(x)=ax^2+bx+c$, $a\gt0$: $f$ strictly increasing on $\mathbb R\iff b^2-4ac\le0$
Local extrema (Definition 6.5)§ 6.3 →
local max at $c$: $f(c)\ge f(x)$ for all $x\in(c-h,c+h)$, some $h\gt0$; local min: $f(c)\le f(x)$
Critical points (Theorem 6.3, Definition 6.6)§ 6.3 →
local extremum at interior $c$ and $f$ differentiable at $c$ $\Rightarrow f'(c)=0$; critical point: $f'(c)=0$ or $f'(c)$ does not exist
First derivative test (Theorem 6.4)§ 6.3 →
$f$ continuous at critical point $c$: $f'$ changes $+\to-$ ⇒ local max; $-\to+$ ⇒ local min; no change ⇒ neither (point of inflection if $f'(c)=0$)
Second derivative test (Theorem 6.5)§ 6.3 →
$f'(c)=0$, $f$ twice differentiable at $c$: $f''(c)\lt0$ ⇒ local max; $f''(c)\gt0$ ⇒ local min; $f''(c)=0$ ⇒ test fails
Absolute extrema exist (Theorem 6.6)§ 6.4 →
$f$ continuous on $[a,b]$ $\Rightarrow$ $f$ has an absolute maximum and an absolute minimum on $[a,b]$
Where to look (Theorem 6.7)§ 6.4 →
$f$ continuous on $[a,b]$: compare $f(a)$, $f(b)$ and $f(c)$ for every critical point $c\in(a,b)$; largest = absolute max, smallest = absolute min
One critical point (Theorem 6.8)§ 6.4 →
$f$ continuous on an interval with exactly one interior critical point $c$: local max at $c$ $\Rightarrow$ absolute max; local min $\Rightarrow$ absolute min
Open box from a sheet§ 6.4 →
$a\times b$ sheet, cut $x$: $V=x(a-2x)(b-2x)$, $0\lt x\lt\tfrac12\min(a,b)$
Best cylinders§ 6.4 →
fixed volume, least area: closed $h=2r$; open-top $h=r$
Chapter 7 · Integrals
Indefinite integral§ 7.1 →
$\int f(x)\,dx=F(x)+C\iff F'(x)=f(x)$ on an interval; any two antiderivatives differ by a constant
Powers, exponentials, log§ 7.1 →
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\ (n\ne-1)$; $\int\frac{dx}{x}=\log|x|+C$; $\int e^x\,dx=e^x+C$; $\int a^x\,dx=\frac{a^x}{\log a}+C\ (a\gt0,\ a\ne1)$
Trigonometric integrals§ 7.1 →
$\int\cos x\,dx=\sin x+C$; $\int\sin x\,dx=-\cos x+C$; $\int\sec^2x\,dx=\tan x+C$; $\int\operatorname{cosec}^2x\,dx=-\cot x+C$; $\int\sec x\tan x\,dx=\sec x+C$; $\int\operatorname{cosec}x\cot x\,dx=-\operatorname{cosec}x+C$
Inverse-trigonometric forms§ 7.1 →
$\int\frac{dx}{\sqrt{1-x^2}}=\sin^{-1}x+C\ (|x|\lt1)$; $\int\frac{dx}{1+x^2}=\tan^{-1}x+C$; $\int\frac{dx}{x\sqrt{x^2-1}}=\sec^{-1}x+C\ (x\gt1)$
Substitution§ 7.1 →
$u=g(x)$, $du=g'(x)\,dx$: $\int f(g(x))\,g'(x)\,dx=F(g(x))+C$ where $F'=f$; $\int f(ax+b)\,dx=\frac1aF(ax+b)+C\ (a\ne0)$
Logarithmic integrals§ 7.1 →
$\int\frac{f'(x)}{f(x)}\,dx=\log|f(x)|+C$; $\int\tan x\,dx=\log|\sec x|+C$; $\int\cot x\,dx=\log|\sin x|+C$; $\int\sec x\,dx=\log|\sec x+\tan x|+C$; $\int\operatorname{cosec}x\,dx=\log|\operatorname{cosec}x-\cot x|+C$
Special integrals (no root), a > 0§ 7.2 →
$\int\frac{dx}{x^2-a^2}=\frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C$; $\int\frac{dx}{a^2-x^2}=\frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+C$; $\int\frac{dx}{x^2+a^2}=\frac1a\tan^{-1}\frac xa+C$
Special integrals (with a root), a > 0§ 7.2 →
$\int\frac{dx}{\sqrt{x^2-a^2}}=\log\left|x+\sqrt{x^2-a^2}\right|+C\ (|x|\gt a)$; $\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac xa+C\ (|x|\lt a)$; $\int\frac{dx}{\sqrt{x^2+a^2}}=\log\left|x+\sqrt{x^2+a^2}\right|+C$
Quadratic denominators§ 7.2 →
$ax^2+bx+c=a\left[\left(x+\frac b{2a}\right)^2+\frac{4ac-b^2}{4a^2}\right]$; for $\frac{px+q}{ax^2+bx+c}$ write $px+q=A(2ax+b)+B$
Partial fraction forms§ 7.2 →
$\frac{px+q}{(x-a)(x-b)}=\frac A{x-a}+\frac B{x-b}$; $\frac{px+q}{(x-a)^2}=\frac A{x-a}+\frac B{(x-a)^2}$; $\frac{px^2+qx+r}{(x-a)(x^2+k)}=\frac A{x-a}+\frac{Bx+C}{x^2+k}$ ($k\gt0$); proper fractions only
Integration by parts§ 7.3 →
$\int u\,dv=uv-\int v\,du$; $\int uv\,dx=u\int v\,dx-\int\left(u'\int v\,dx\right)dx$; choose $u$ by ILATE ($u$ with continuous derivative)
The ex[f + f′] pattern§ 7.3 →
$\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C$
Cyclic integrals§ 7.3 →
$\int e^{ax}\sin bx\,dx=\frac{e^{ax}(a\sin bx-b\cos bx)}{a^2+b^2}+C$; $\int e^{ax}\cos bx\,dx=\frac{e^{ax}(a\cos bx+b\sin bx)}{a^2+b^2}+C$
√(a² − x²), a > 0§ 7.3 →
$\int\sqrt{a^2-x^2}\,dx=\frac x2\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac xa+C$ ($|x|\le a$)
√(x² ± a²), a > 0§ 7.3 →
$\int\sqrt{x^2\pm a^2}\,dx=\frac x2\sqrt{x^2\pm a^2}\pm\frac{a^2}{2}\log\left|x+\sqrt{x^2\pm a^2}\right|+C$ (same sign throughout; $|x|\ge a$ for the minus sign)
Fundamental theorems§ 7.4 →
$f$ continuous on $[a,b]$: $\frac{d}{dx}\int_a^xf(t)\,dt=f(x)$; $\int_a^bf(x)\,dx=\big[F(x)\big]_a^b=F(b)-F(a)$ for any $F$ with $F'=f$
Substitution with limits§ 7.4 →
$\int_a^bf(g(x))\,g'(x)\,dx=\int_{g(a)}^{g(b)}f(u)\,du$ ($g'$ continuous on $[a,b]$)
Properties P₀–P₃§ 7.4 →
$\int_a^bf(x)\,dx=\int_a^bf(t)\,dt$; $\int_a^bf=-\int_b^af$, $\int_a^af=0$; $\int_a^bf=\int_a^cf+\int_c^bf$; $\int_a^bf(x)\,dx=\int_a^bf(a+b-x)\,dx$
Properties P₄–P₆§ 7.4 →
$\int_0^af(x)\,dx=\int_0^af(a-x)\,dx$; $\int_0^{2a}f=2\int_0^af$ if $f(2a-x)=f(x)$, $0$ if $f(2a-x)=-f(x)$; $\int_{-a}^af=2\int_0^af$ ($f$ even), $0$ ($f$ odd)
JEE: limit of a sum§ 7.4 →
$\lim_{n\to\infty}\frac1n\sum_{r=1}^nf\!\left(\frac rn\right)=\int_0^1f(x)\,dx$ ($f$ continuous on $[0,1]$)
Chapter 8 · Application of Integrals
Area under a curve (vertical strips)§ 8.1 →
$A=\displaystyle\int_a^b y\,dx=\int_a^b f(x)\,dx$, for $f$ continuous and $f(x)\ge0$ on $[a,b]$
Area between a curve and the $y$-axis (horizontal strips)§ 8.1 →
$A=\displaystyle\int_c^d x\,dy=\int_c^d g(y)\,dy$, for $g$ continuous and $g(y)\ge0$ on $[c,d]$
Curve below the $x$-axis§ 8.1 →
$f(x)\le0$ on $[a,b]$: $\ A=\left|\displaystyle\int_a^b f(x)\,dx\right|$
Curve crossing the $x$-axis at $c$§ 8.1 →
$A=\left|\displaystyle\int_a^c f(x)\,dx\right|+\left|\displaystyle\int_c^b f(x)\,dx\right|=\displaystyle\int_a^b|f(x)|\,dx$ (split at every sign change)
Parabolic segment§ 8.2 →
$y^2=4ax$ and $x=c$ ($a,c\gt0$): $\ A=\tfrac83\sqrt a\,c^{3/2}$ $=\tfrac23$ of the bounding rectangle; with the latus rectum $x=a$: $\ A=\tfrac83a^2$
Area of a circle§ 8.2 →
$x^2+y^2=a^2$: $\ A=4\displaystyle\int_0^a\sqrt{a^2-x^2}\,dx=\pi a^2$
Area of an ellipse§ 8.2 →
$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a,b\gt0$): $\ A=\pi ab$
Key integral for circles and ellipses§ 8.2 →
$\displaystyle\int\sqrt{a^2-x^2}\,dx=\dfrac x2\sqrt{a^2-x^2}+\dfrac{a^2}2\sin^{-1}\dfrac xa+C$, $\ |x|\le a$
Area between two curves (JEE)§ 8.2 →
$f(x)\ge g(x)$ on $[a,b]$, both continuous: $\ A=\displaystyle\int_a^b\big[f(x)-g(x)\big]\,dx$
Chapter 9 · Differential Equations
Order and degree§ 9.1 →
Order = order of the highest derivative; degree = its power once the equation is a polynomial in $y',y'',\dots$ (not defined otherwise, e.g. $\sin y'$)
Clearing a fractional power§ 9.1 →
$\left[1+(y')^2\right]^{3/2}=y''\ \Rightarrow\ \left[1+(y')^2\right]^3=(y'')^2$: order $2$, degree $2$
General vs particular solution§ 9.1 →
Order $n$ $\Rightarrow$ general solution has $n$ independent arbitrary constants; $n$ initial conditions give a particular solution (no constants)
$y''+k^2y=0$§ 9.1 →
$y=A\cos kx+B\sin kx$ ($k\ne0$)
Euler step§ 9.1 →
$\dfrac{dy}{dx}=f(x,y)$: $\ y_{n+1}=y_n+h\,f(x_n,y_n)$, $x_{n+1}=x_n+h$
Variables separable§ 9.2 →
$\dfrac{dy}{dx}=g(x)h(y)$, $h(y)\ne0$: $\ \displaystyle\int\frac{dy}{h(y)}=\int g(x)\,dx+C$; also check constant solutions $y=c$ with $h(c)=0$
Natural growth and decay§ 9.2 →
$\dfrac{dN}{dt}=kN,\ N(0)=N_0\ \Rightarrow\ N=N_0e^{kt}$
Half-life, doubling time§ 9.2 →
$T_{1/2}=\dfrac{\log2}{\lambda}$ for $N_0e^{-\lambda t}$; $\ T_d=\dfrac{\log2}{k}$ for $N_0e^{kt}$
Newton's law of cooling§ 9.2 →
$\dfrac{dT}{dt}=-k(T-T_s)$, $k\gt0$: $\ T=T_s+(T_0-T_s)e^{-kt}$
Tidying the constant§ 9.2 →
$\log|y|=G(x)+C\ \Rightarrow\ y=Ae^{G(x)}$, $A=\pm e^C$ (and $A=0$ if $y=0$ is a solution)
Homogeneous function of degree $n$§ 9.3 →
$F(\lambda x,\lambda y)=\lambda^nF(x,y)$ ($\lambda\ne0$), i.e. $F=x^n\,g\!\left(\tfrac yx\right)$
Homogeneous equation§ 9.3 →
$\dfrac{dy}{dx}=F(x,y)$, $F$ of degree $0$: $\ \dfrac{dy}{dx}=g\!\left(\dfrac yx\right)$
Substitution $y=vx$§ 9.3 →
$\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$; $\ x\dfrac{dv}{dx}=g(v)-v$; $\ \displaystyle\int\frac{dv}{g(v)-v}=\log|x|+C$, then $v=\tfrac yx$
Substitution $x=vy$§ 9.3 →
$\dfrac{dx}{dy}=h\!\left(\dfrac xy\right)$: $\ \dfrac{dx}{dy}=v+y\dfrac{dv}{dy}$, $\ y\dfrac{dv}{dy}=h(v)-v$
Linear equation in $y$§ 9.4 →
$\dfrac{dy}{dx}+Py=Q$ ($P,Q$ continuous functions of $x$): $\ \text{I.F.}=e^{\int P\,dx}$, $\ y\cdot\text{I.F.}=\displaystyle\int Q\cdot\text{I.F.}\,dx+C$
Linear equation in $x$§ 9.4 →
$\dfrac{dx}{dy}+P_1x=Q_1$: $\ x\,e^{\int P_1dy}=\displaystyle\int Q_1e^{\int P_1dy}\,dy+C$
Useful integrating factors§ 9.4 →
$P=\tfrac nx$: $x^n$; $\ P=\tan x$: $\sec x$; $\ P=\cot x$: $\sin x$; $\ P=k$: $e^{kx}$
Constant input, proportional loss§ 9.4 →
$\dfrac{dA}{dt}+kA=r$ ($k\gt0$): $\ A=\dfrac rk+\left(A_0-\dfrac rk\right)e^{-kt}\to\dfrac rk$; time constant $\dfrac1k$ ($RC$ for a circuit)
Chapter 10 · Vector Algebra
Position vector and magnitude§ 10.1 →
$\overrightarrow{OP}=x\hat i+y\hat j+z\hat k$, $|\overrightarrow{OP}|=\sqrt{x^2+y^2+z^2}$
Direction cosines (Theorem 10.1)§ 10.1 →
For $\vec a=a_1\hat i+a_2\hat j+a_3\hat k\ne\vec 0$: $\ l=\cos\alpha=\tfrac{a_1}{|\vec a|}$, $m=\tfrac{a_2}{|\vec a|}$, $n=\tfrac{a_3}{|\vec a|}$; $l^2+m^2+n^2=1$
From direction ratios to direction cosines§ 10.1 →
$a,b,c$ (not all $0$) $\Rightarrow$ $l,m,n=\pm\dfrac{a,\ b,\ c}{\sqrt{a^2+b^2+c^2}}$ (same sign throughout)
Unit vector§ 10.1 →
$\hat a=\dfrac{\vec a}{|\vec a|}=l\hat i+m\hat j+n\hat k$ ($\vec a\ne\vec 0$)
Equal vectors (Theorem 10.2)§ 10.1 →
$\vec a=\vec b\iff a_1=b_1,\ a_2=b_2,\ a_3=b_3$
Direction angles§ 10.1 →
$\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$, $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2$
Triangle and parallelogram laws§ 10.2 →
$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$; $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=\vec 0$
Operations in components (Theorem 10.6)§ 10.2 →
$\vec a\pm\vec b=(a_1\pm b_1)\hat i+(a_2\pm b_2)\hat j+(a_3\pm b_3)\hat k$; $\lambda\vec a=\lambda a_1\hat i+\lambda a_2\hat j+\lambda a_3\hat k$
Collinearity test§ 10.2 →
$\vec a\ne\vec 0$: $\ \vec b\parallel\vec a\iff\vec b=\lambda\vec a\iff\dfrac{b_1}{a_1}=\dfrac{b_2}{a_2}=\dfrac{b_3}{a_3}$ (zero components must match)
Vector joining two points (Theorem 10.7)§ 10.2 →
$\overrightarrow{P_1P_2}=(x_2-x_1)\hat i+(y_2-y_1)\hat j+(z_2-z_1)\hat k=\vec b-\vec a$
Section formula (Theorem 10.8)§ 10.2 →
Internal $m:n$: $\ \vec r=\dfrac{m\vec b+n\vec a}{m+n}$; external ($m\ne n$): $\ \vec r=\dfrac{m\vec b-n\vec a}{m-n}$; midpoint $\dfrac{\vec a+\vec b}2$
Centroid of a triangle§ 10.2 →
$\vec g=\dfrac{\vec a+\vec b+\vec c}{3}$
Scalar (dot) product§ 10.3 →
$\vec a\cdot\vec b=|\vec a|\,|\vec b|\cos\theta=a_1b_1+a_2b_2+a_3b_3$
Angle between two vectors§ 10.3 →
$\cos\theta=\dfrac{\vec a\cdot\vec b}{|\vec a|\,|\vec b|}$, $0\le\theta\le\pi$ ($\vec a,\vec b\ne\vec 0$)
Perpendicularity and unit vectors§ 10.3 →
$\vec a\perp\vec b\iff\vec a\cdot\vec b=0$ (non-zero); $\hat i\cdot\hat i=1$, $\hat i\cdot\hat j=0$, …
Projection (Definition 10.10)§ 10.3 →
Projection of $\vec a$ on $\vec b$: $\ \dfrac{\vec a\cdot\vec b}{|\vec b|}$; projection vector $\dfrac{\vec a\cdot\vec b}{|\vec b|^2}\,\vec b$ ($\vec b\ne\vec 0$)
Identities (Theorem 10.11)§ 10.3 →
$|\vec a\pm\vec b|^2=|\vec a|^2\pm2\,\vec a\cdot\vec b+|\vec b|^2$; $(\vec a+\vec b)\cdot(\vec a-\vec b)=|\vec a|^2-|\vec b|^2$; $|\vec a\cdot\vec b|\le|\vec a|\,|\vec b|$
Work done by a constant force§ 10.3 →
$W=\vec F\cdot\vec d=|\vec F|\,|\vec d|\cos\theta$ J
Vector (cross) product§ 10.4 →
$\vec a\times\vec b=|\vec a|\,|\vec b|\sin\theta\,\hat n$ ($\hat n$ by the right-hand rule); $\ \vec 0$ if $\vec a\parallel\vec b$
Determinant formula (Theorem 10.13)§ 10.4 →
$\vec a\times\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}$
Unit-vector table§ 10.4 →
$\hat i\times\hat j=\hat k$, $\hat j\times\hat k=\hat i$, $\hat k\times\hat i=\hat j$; $\vec b\times\vec a=-\vec a\times\vec b$; $\vec a\times\vec a=\vec 0$
Areas (Theorem 10.14)§ 10.4 →
Parallelogram $|\vec a\times\vec b|$; from diagonals $\tfrac12|\vec d_1\times\vec d_2|$; triangle $\tfrac12|\overrightarrow{AB}\times\overrightarrow{AC}|$
Unit vector perpendicular to $\vec a,\vec b$§ 10.4 →
$\pm\dfrac{\vec a\times\vec b}{|\vec a\times\vec b|}$
Lagrange's identity§ 10.4 →
$|\vec a\times\vec b|^2+(\vec a\cdot\vec b)^2=|\vec a|^2|\vec b|^2$
Chapter 11 · Three Dimensional Geometry
Direction cosines of a line (Theorem 11.1)§ 11.1 →
$l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$ ($\alpha,\beta,\gamma\in[0,\pi]$); $l^2+m^2+n^2=1$; the reversed line has $-l,-m,-n$
Direction ratios → direction cosines (Theorem 11.2)§ 11.1 →
$a,b,c$ not all $0$: $\ l=\pm\dfrac{a}{\sqrt{a^2+b^2+c^2}}$, $m=\pm\dfrac{b}{\sqrt{a^2+b^2+c^2}}$, $n=\pm\dfrac{c}{\sqrt{a^2+b^2+c^2}}$ (same sign throughout)
Line through two points (Theorem 11.3)§ 11.1 →
Direction ratios $x_2-x_1,\ y_2-y_1,\ z_2-z_1$; from $P$ to $Q$: $\ l=\dfrac{x_2-x_1}{PQ}$, $m=\dfrac{y_2-y_1}{PQ}$, $n=\dfrac{z_2-z_1}{PQ}$
Collinear points (Theorem 11.4)§ 11.1 →
$A,B,C$ collinear $\iff$ direction ratios of $AB$ and $BC$ proportional
Vector equation of a line (Theorem 11.5)§ 11.2 →
$\vec r=\vec a+\lambda\vec b$, $\ \lambda\in\mathbb R$ (through $\vec a$, parallel to $\vec b\ne\vec 0$)
Cartesian equation of a line (Theorem 11.6)§ 11.2 →
$\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}\ (=\lambda)$; parametric: $x=x_1+\lambda a$, $y=y_1+\lambda b$, $z=z_1+\lambda c$; a zero ratio $a=0$ means $x=x_1$
Line through two points (Theorem 11.7)§ 11.2 →
$\vec r=\vec a+\lambda(\vec b-\vec a)$; $\ \dfrac{x-x_1}{x_2-x_1}=\dfrac{y-y_1}{y_2-y_1}=\dfrac{z-z_1}{z_2-z_1}$
Distance along a line§ 11.2 →
The point with parameter $\lambda$ is $|\lambda|\,|\vec b|$ from $A$
Foot of the perpendicular, image§ 11.2 →
$F=\vec a+\lambda\vec b$ with $(F-P)\cdot\vec b=0$; distance $PF$; image $P'=2F-P$
Angle between two lines (Theorem 11.8)§ 11.3 →
$\cos\theta=\dfrac{|\vec b_1\cdot\vec b_2|}{|\vec b_1||\vec b_2|}=\dfrac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$, $\ 0\le\theta\le\tfrac\pi2$
Perpendicular / parallel (Theorem 11.9)§ 11.3 →
$\perp\iff a_1a_2+b_1b_2+c_1c_2=0$; $\ \parallel\iff\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$
Shortest distance, non-parallel lines (Theorem 11.10)§ 11.3 →
$d=\left|\dfrac{(\vec b_1\times\vec b_2)\cdot(\vec a_2-\vec a_1)}{|\vec b_1\times\vec b_2|}\right|$ ($\vec b_1\times\vec b_2\ne\vec 0$); $d=0\iff$ the lines intersect
Shortest distance, Cartesian form§ 11.3 →
$d=\dfrac{\left|\begin{vmatrix}x_2-x_1&y_2-y_1&z_2-z_1\\a_1&b_1&c_1\\a_2&b_2&c_2\end{vmatrix}\right|}{\sqrt{(b_1c_2-b_2c_1)^2+(c_1a_2-c_2a_1)^2+(a_1b_2-a_2b_1)^2}}$
Distance between parallel lines (Theorem 11.11)§ 11.3 →
$d=\dfrac{|\vec b\times(\vec a_2-\vec a_1)|}{|\vec b|}$
Chapter 12 · Linear Programming
Half-plane of a linear inequality (Theorem 12.1)§ 12.1 →
$ax+by\le c$ ($a,b$ not both $0$): closed half-plane on one side of $ax+by=c$; test one point not on the line (the origin if $c\ne0$). Solid line for $\le,\ge$; dashed for $\lt,\gt$.
Linear programming problem§ 12.1 →
Maximise or minimise $Z=ax+by$ subject to linear constraints in $x,y$, and $x\ge0,\ y\ge0$.
Feasible region§ 12.1 →
Set of points satisfying all constraints (incl. $x,y\ge0$); its points are feasible solutions.
Words to inequalities§ 12.1 →
“at most”, “available” $\to\ \le$; “at least”, “required” $\to\ \ge$; “$y$ at most $k$ times $x$” $\to\ y\le kx$.
Iso-profit / iso-cost lines§ 12.2 →
$Z=ax+by=k$: parallel lines of slope $-\tfrac ab$; increasing $k$ moves them in the direction $(a,b)$.
Corner point theorems (Theorems 12.2, 12.3)§ 12.2 →
If $Z$ has an optimal value on the feasible region $R$, it occurs at a corner point. If $R$ is bounded (and non-empty), $Z$ has both a maximum and a minimum, each at a corner.
$Z$ along a segment§ 12.2 →
$X=(1-t)P+tQ$, $0\le t\le1$ $\Rightarrow$ $Z(X)=(1-t)Z(P)+tZ(Q)$
Ties (Theorem 12.4)§ 12.2 →
$Z(P)=Z(Q)=$ optimal value $\Rightarrow$ every point of segment $PQ$ is optimal (none, one, or infinitely many optima).
Corner point method§ 12.2 →
Draw $R$ → find all corners (solve pairs of boundary lines, keep feasible ones) → table of $Z$ → bounded: largest = max, smallest = min.
Open half-plane test (Theorem 12.5)§ 12.3 →
$R$ unbounded, $M$/$m$ = largest/smallest corner value of $Z=ax+by$: if $ax+by\gt M$ has no point in common with $R$, $\max Z=M$; otherwise no maximum. If $ax+by\lt m$ has no point in common with $R$, $\min Z=m$; otherwise no minimum.
Proving a minimum by combining constraints§ 12.3 →
If $ax+by=p(\text{LHS}_1)+q(\text{LHS}_2)$ with $p,q\ge0$, then on $R$: $ax+by\ge p\,c_1+q\,c_2$.
Empty region§ 12.3 →
No common point of the constraints $\Rightarrow$ infeasible: no feasible, no optimal solution.
Possible outcomes of an LPP§ 12.3 →
Empty: none · Bounded: max and min both at corners · Unbounded: each exists only if the open half-plane test is passed · Tie at two points: infinitely many optima.
Chapter 13 · Probability
Conditional probability (Definition 13.1)§ 13.1 →
$P(E\mid F)=\dfrac{P(E\cap F)}{P(F)}$, $P(F)\ne0$
Equally likely outcomes§ 13.1 →
$P(E\mid F)=\dfrac{n(E\cap F)}{n(F)}$, $n(F)\ne0$
Properties (Theorem 13.1), $P(F)\ne0$§ 13.1 →
$P(S\mid F)=P(F\mid F)=1$; $P(E'\mid F)=1-P(E\mid F)$; $P\big((A\cup B)\mid F\big)=P(A\mid F)+P(B\mid F)-P\big((A\cap B)\mid F\big)$
Multiplication theorem (Theorem 13.2)§ 13.2 →
$P(E\cap F)=P(E)\,P(F\mid E)$, $P(E)\ne0$
Three stages, $P(E\cap F)\ne0$§ 13.2 →
$P(E\cap F\cap G)=P(E)\,P(F\mid E)\,P(G\mid E\cap F)$
Independent events (Definition 13.2)§ 13.2 →
$P(E\cap F)=P(E)\,P(F)$; equivalently $P(F\mid E)=P(F)$ if $P(E)\ne0$
Mutually independent $A,B,C$ (Definition 13.3)§ 13.2 →
all three pairs satisfy the product rule and $P(A\cap B\cap C)=P(A)P(B)P(C)$
At least one (Theorem 13.3, independent events)§ 13.2 →
$P(E\cup F)=1-P(E')\,P(F')$; $P(A\cup B\cup C)=1-P(A')P(B')P(C')$ (mutually independent)
Partition of $S$ (Definition 13.4)§ 13.3 →
$E_i\cap E_j=\phi$ ($i\ne j$), $E_1\cup\cdots\cup E_n=S$, $P(E_i)\gt0$
Total probability (Theorem 13.4)§ 13.3 →
$P(A)=\displaystyle\sum_{j=1}^{n}P(E_j)\,P(A\mid E_j)$, $E_1,\dots,E_n$ a partition of $S$
Bayes' theorem (Theorem 13.5), $P(A)\ne0$§ 13.3 →
$P(E_i\mid A)=\dfrac{P(E_i)\,P(A\mid E_i)}{\sum_{j}P(E_j)\,P(A\mid E_j)}$
Two causes $E$, $E'$§ 13.3 →
$P(E\mid A)=\dfrac{P(E)P(A\mid E)}{P(E)P(A\mid E)+P(E')P(A\mid E')}$
Probability distribution (Definition 13.6)§ 13.4 →
$p_i=P(X=x_i)\gt0$, $\sum_{i=1}^n p_i=1$
Mean / expectation (Definition 13.7)§ 13.4 →
$E(X)=\mu=\sum_{i=1}^n x_ip_i$
Properties of the mean (Theorem 13.6)§ 13.4 →
$E(aX+b)=a\,E(X)+b$; $x_{\min}\le E(X)\le x_{\max}$
Variance (JEE extension)§ 13.4 →
$\operatorname{Var}(X)=\sum(x_i-\mu)^2p_i=E(X^2)-[E(X)]^2$, $\sigma=\sqrt{\operatorname{Var}(X)}$