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 · Sets
Two ways to write a set§ 1.1 →
Roster: $\{2,4,6,8\}$ · Set-builder: $\{x : x\in\mathbb{N},\ x \text{ even},\ x\lt10\}$; $\ a\in A$, $a\notin A$
Standard number sets§ 1.1 →
$\mathbb{N}=\{1,2,3,\dots\}$ ($0\notin\mathbb{N}$), $\ \mathbb{Z}=\{\dots,-1,0,1,\dots\}$, $\ \mathbb{Q}=\left\{\tfrac pq : p,q\in\mathbb{Z},\ q\ne0\right\}$, $\ \mathbb{R}$ = all real numbers
Empty set and $n(A)$§ 1.1 →
$\phi=\{\ \}$, $\ n(\phi)=0$; $\ n(A)$ counts distinct elements of a finite set; $\ \{0\}\ne\phi$, $\ \{\phi\}\ne\phi$
Equal sets§ 1.1 →
$A=B$ when every element of $A$ is in $B$ and every element of $B$ is in $A$ (order and repetition ignored)
Subset§ 1.2 →
$A\subset B\iff(a\in A\Rightarrow a\in B)$; proper subset: $A\subset B$ and $A\ne B$
Properties of subsets§ 1.2 →
$A\subset A$; $\ \phi\subset A$; $\ A=B\iff A\subset B$ and $B\subset A$; $\ A\subset B,\ B\subset C\Rightarrow A\subset C$
Number systems§ 1.2 →
$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$; irrationals $T=\{x : x\in\mathbb{R},\ x\notin\mathbb{Q}\}$
Intervals ($a\lt b$)§ 1.2 →
$(a,b)=\{x : a\lt x\lt b\}$, $\ [a,b]=\{x : a\le x\le b\}$, $\ [a,b)=\{x : a\le x\lt b\}$, $\ (a,b]=\{x : a\lt x\le b\}$; length $=b-a$
Unbounded intervals§ 1.2 →
$[a,\infty)=\{x : x\ge a\}$, $\ (-\infty,b)=\{x : x\lt b\}$, $\ (-\infty,\infty)=\mathbb{R}$; $\infty$ always gets a round bracket
Number of subsets (JEE)§ 1.2 →
If $n(A)=n$: $n\big(P(A)\big)=2^n$ subsets, $2^n-1$ proper subsets
Union and intersection§ 1.3 →
$A\cup B=\{x : x\in A \text{ or } x\in B\}$, $\ A\cap B=\{x : x\in A \text{ and } x\in B\}$; disjoint: $A\cap B=\phi$
Difference and complement§ 1.3 →
$A-B=\{x : x\in A,\ x\notin B\}=A\cap B'$; $\ A'=U-A$
Laws of union and intersection§ 1.3 →
$A\cup B=B\cup A$, $\ A\cap B=B\cap A$; $\ A\cup\phi=A$, $\ A\cap U=A$; $\ A\cup A=A\cap A=A$
Distributive laws§ 1.3 →
$A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$, $\ A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$
Complement laws and De Morgan's laws§ 1.3 →
$A\cup A'=U$, $\ A\cap A'=\phi$, $\ (A')'=A$; $\ (A\cup B)'=A'\cap B'$, $\ (A\cap B)'=A'\cup B'$
Counting a union (JEE)§ 1.3 →
For finite sets: $n(A\cup B)=n(A)+n(B)-n(A\cap B)$
Chapter 2 · Relations and Functions
Equal ordered pairs§ 2.1 →
$(a,b)=(c,d)\iff a=c$ and $b=d$
Cartesian product§ 2.1 →
$A\times B=\{(a,b) : a\in A,\ b\in B\}$; $\ A\times\phi=\phi$; $\ A\times B\ne B\times A$ in general
Counting a product§ 2.1 →
$n(A\times B)=n(A)\cdot n(B)$; $\ n(A\times B\times C)=n(A)\cdot n(B)\cdot n(C)$
Products and set operations§ 2.1 →
$A\times(B\cup C)=(A\times B)\cup(A\times C)$, $\ A\times(B\cap C)=(A\times B)\cap(A\times C)$, $\ A\times(B-C)=(A\times B)-(A\times C)$
The plane and space§ 2.1 →
$\mathbb{R}\times\mathbb{R}=\{(x,y) : x,y\in\mathbb{R}\}$; $\ \mathbb{R}\times\mathbb{R}\times\mathbb{R}$ = points of space
Relation§ 2.2 →
A relation from $A$ to $B$: any $R\subset A\times B$; a relation on $A$: $R\subset A\times A$
Domain, range, codomain§ 2.2 →
domain $=\{a : (a,b)\in R\}\subset A$; $\ $ range $=\{b : (a,b)\in R\}\subset B$; $\ $ codomain $=B$
Number of relations§ 2.2 →
$n(A)=p,\ n(B)=q$: $\ 2^{pq}$ relations from $A$ to $B$; $\ 2^{n^2}$ relations on a set of $n$ elements
Relations with domain $A$§ 2.2 →
$n(A)=p,\ n(B)=q$: $\ (2^q-1)^p$ relations from $A$ to $B$ have domain $A$
Function§ 2.3 →
$f : A\to B$: every $x\in A$ has exactly one image $f(x)\in B$; range $=\{f(x) : x\in A\}\subset B$
Natural domain§ 2.3 →
Remove from $\mathbb{R}$ the $x$ with a zero denominator or a negative number under $\sqrt{\ }$
Vertical line test§ 2.3 →
A curve is the graph of a function of $x$ $\iff$ no vertical line meets it more than once
Modulus and signum§ 2.3 →
$|x|=x$ ($x\ge0$), $-x$ ($x\lt0$); $\ \operatorname{sgn}x=\tfrac{|x|}{x}$ ($x\ne0$), $\operatorname{sgn}0=0$
Greatest integer function§ 2.3 →
$[x]=n\iff n\le x\lt n+1$ ($n\in\mathbb{Z}$); $\ [x+1]=[x]+1$
Domains and ranges of standard functions§ 2.3 →
$x$: $\mathbb{R},\mathbb{R}$; $\ c$: $\mathbb{R},\{c\}$; $\ x^2$: $\mathbb{R},[0,\infty)$; $\ \tfrac1x$: $\mathbb{R}-\{0\}$ both; $\ |x|$: $\mathbb{R},[0,\infty)$; $\ \operatorname{sgn}x$: $\mathbb{R},\{-1,0,1\}$; $\ [x]$: $\mathbb{R},\mathbb{Z}$
Sum and difference§ 2.4 →
$(f\pm g)(x)=f(x)\pm g(x)$, $\ x\in D_f\cap D_g$
Scalar multiple§ 2.4 →
$(\alpha f)(x)=\alpha\,f(x)$, $\ x\in D_f$
Product§ 2.4 →
$(fg)(x)=f(x)\,g(x)$, $\ x\in D_f\cap D_g$
Quotient§ 2.4 →
$\left(\dfrac fg\right)(x)=\dfrac{f(x)}{g(x)}$, $\ x\in(D_f\cap D_g)-\{x : g(x)=0\}$ — find the domain before simplifying
Profit and average cost§ 2.4 →
$P=R-C$; break-even where $P(x)=0$; average cost $\dfrac{C(x)}{x}$, $x\gt0$
Chapter 3 · Trigonometric Functions
Degrees, minutes, seconds§ 3.1 →
$1$ turn $=360^\circ$; $\ 1^\circ=60'$; $\ 1'=60''$
Radians ↔ degrees§ 3.1 →
$\pi\ \text{rad}=180^\circ$; $\ 1\ \text{rad}=\left(\tfrac{180}{\pi}\right)^\circ\approx57.3^\circ$; $\ 1^\circ=\tfrac{\pi}{180}\ \text{rad}$
Arc length§ 3.1 →
$s=r\theta$, $\ \theta=\dfrac{s}{r}$ — $\theta$ in radians only
Coterminal angles§ 3.1 →
$\theta$ and $\theta+n\cdot360^\circ$ (i.e. $\theta+2n\pi$), $n\in\mathbb{Z}$, have the same terminal side
Clock hands§ 3.1 →
Minute hand $6^\circ$ per minute; hour hand $30^\circ$ per hour $=0.5^\circ$ per minute
Unit-circle definition§ 3.2 →
$P=(\cos x,\sin x)$ on $a^2+b^2=1$; $\ \tan x=\tfrac{\sin x}{\cos x}$, $\ \sec x=\tfrac1{\cos x}$, $\ \operatorname{cosec}x=\tfrac1{\sin x}$, $\ \cot x=\tfrac{\cos x}{\sin x}$
Pythagorean identities§ 3.2 →
$\sin^2x+\cos^2x=1$; $\ 1+\tan^2x=\sec^2x$; $\ 1+\cot^2x=\operatorname{cosec}^2x$ (where defined)
Negative angles and full turns§ 3.2 →
$\sin(-x)=-\sin x$, $\ \cos(-x)=\cos x$, $\ \tan(-x)=-\tan x$; $\ \sin(x+2n\pi)=\sin x$, $\ \cos(x+2n\pi)=\cos x$
Signs by quadrant (ASTC)§ 3.2 →
I: all $+$; $\ $ II: $\sin,\operatorname{cosec}+$; $\ $ III: $\tan,\cot+$; $\ $ IV: $\cos,\sec+$
Zeros of sine and cosine§ 3.2 →
$\sin x=0\iff x=n\pi$; $\ \cos x=0\iff x=(2n+1)\tfrac{\pi}{2}$, $\ n\in\mathbb{Z}$
Domains, ranges, periods§ 3.2 →
$\sin,\cos$: $\mathbb{R}$, $[-1,1]$, $2\pi$; $\ \tan$: $x\ne(2n+1)\tfrac{\pi}{2}$, $\mathbb{R}$, $\pi$; $\ \cot$: $x\ne n\pi$, $\mathbb{R}$, $\pi$; $\ \sec$: $x\ne(2n+1)\tfrac{\pi}{2}$, $|y|\ge1$, $2\pi$; $\ \operatorname{cosec}$: $x\ne n\pi$, $|y|\ge1$, $2\pi$
Cosine of a sum and difference§ 3.3 →
$\cos(x+y)=\cos x\cos y-\sin x\sin y$; $\ \cos(x-y)=\cos x\cos y+\sin x\sin y$
Sine of a sum and difference§ 3.3 →
$\sin(x+y)=\sin x\cos y+\cos x\sin y$; $\ \sin(x-y)=\sin x\cos y-\cos x\sin y$
Tangent and cotangent of a sum§ 3.3 →
$\tan(x\pm y)=\dfrac{\tan x\pm\tan y}{1\mp\tan x\tan y}$ (no odd multiple of $\tfrac{\pi}{2}$ among $x,y,x\pm y$); $\ \cot(x\pm y)=\dfrac{\cot x\cot y\mp1}{\cot y\pm\cot x}$ (no multiple of $\pi$)
$\tan\left(\tfrac{\pi}{4}\pm x\right)$§ 3.3 →
$\tan\left(\tfrac{\pi}{4}+x\right)=\dfrac{1+\tan x}{1-\tan x}$, $\ \tan\left(\tfrac{\pi}{4}-x\right)=\dfrac{1-\tan x}{1+\tan x}$
Allied angles§ 3.3 →
$\sin(\pi\mp x)=\pm\sin x$, $\ \cos(\pi\mp x)=-\cos x$; $\ \sin\left(\tfrac{\pi}{2}\mp x\right)=\cos x$, $\ \cos\left(\tfrac{\pi}{2}\mp x\right)=\pm\sin x$; $\ \sin(2\pi-x)=-\sin x$, $\ \cos(2\pi-x)=\cos x$
Exact values at $15^\circ$ and $75^\circ$§ 3.3 →
$\sin15^\circ=\cos75^\circ=\tfrac{\sqrt3-1}{2\sqrt2}$; $\ \cos15^\circ=\sin75^\circ=\tfrac{\sqrt3+1}{2\sqrt2}$; $\ \tan15^\circ=2-\sqrt3$
Double angle§ 3.4 →
$\sin2x=2\sin x\cos x=\dfrac{2\tan x}{1+\tan^2x}$; $\ \cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x=\dfrac{1-\tan^2x}{1+\tan^2x}$; $\ \tan2x=\dfrac{2\tan x}{1-\tan^2x}$
Half angle / power reduction§ 3.4 →
$\sin^2x=\tfrac{1-\cos2x}{2}$, $\ \cos^2x=\tfrac{1+\cos2x}{2}$; sign of $\sin\tfrac x2$, $\cos\tfrac x2$ from the quadrant of $\tfrac x2$
Triple angle§ 3.4 →
$\sin3x=3\sin x-4\sin^3x$; $\ \cos3x=4\cos^3x-3\cos x$; $\ \tan3x=\dfrac{3\tan x-\tan^3x}{1-3\tan^2x}$ ($x,3x$ not odd multiples of $\tfrac{\pi}{2}$)
Products as sums§ 3.4 →
$2\sin x\cos y=\sin(x+y)+\sin(x-y)$; $\ 2\cos x\sin y=\sin(x+y)-\sin(x-y)$; $\ 2\cos x\cos y=\cos(x+y)+\cos(x-y)$; $\ 2\sin x\sin y=\cos(x-y)-\cos(x+y)$
Sums as products§ 3.4 →
$\sin x\pm\sin y=2\sin\tfrac{x\pm y}{2}\cos\tfrac{x\mp y}{2}$; $\ \cos x+\cos y=2\cos\tfrac{x+y}{2}\cos\tfrac{x-y}{2}$; $\ \cos x-\cos y=-2\sin\tfrac{x+y}{2}\sin\tfrac{x-y}{2}$
$\sin18^\circ$ and $\cos36^\circ$§ 3.4 →
$\sin18^\circ=\dfrac{\sqrt5-1}{4}$, $\ \cos36^\circ=\dfrac{\sqrt5+1}{4}$
Chapter 4 · Complex Numbers and Quadratic Equations
The imaginary unit§ 4.1 →
$i^2=-1$; $\ \sqrt{-a}=i\sqrt a$ for $a\gt0$; $\ \sqrt{-a}\,\sqrt{-b}=-\sqrt{ab}$ ($a,b\gt0$)
Equality of complex numbers§ 4.1 →
$a+ib=c+id\iff a=c$ and $b=d$ ($a,b,c,d\in\mathbb{R}$)
Sum and difference§ 4.1 →
$(a+ib)\pm(c+id)=(a\pm c)+i(b\pm d)$
Product§ 4.1 →
$(a+ib)(c+id)=(ac-bd)+i(ad+bc)$
Powers of $i$§ 4.1 →
$i^{4k}=1,\ i^{4k+1}=i,\ i^{4k+2}=-1,\ i^{4k+3}=-i$ ($k\in\mathbb{Z}$); $\ i^{-1}=-i$; $\ i^n+i^{n+1}+i^{n+2}+i^{n+3}=0$
Identities in $\mathbb{C}$§ 4.1 →
$(z_1\pm z_2)^2=z_1^2\pm2z_1z_2+z_2^2$; $\ z_1^2-z_2^2=(z_1+z_2)(z_1-z_2)$; $\ (z_1\pm z_2)^3=z_1^3\pm3z_1^2z_2+3z_1z_2^2\pm z_2^3$; $\ (1\pm i)^2=\pm2i$
Conjugate§ 4.2 →
$\overline{a+ib}=a-ib$; $\ z+\bar z=2\operatorname{Re}z$, $\ z-\bar z=2i\operatorname{Im}z$; $\ \overline{z_1z_2}=\bar z_1\bar z_2$, $\ \overline{z_1\pm z_2}=\bar z_1\pm\bar z_2$
Modulus§ 4.2 →
$|a+ib|=\sqrt{a^2+b^2}$ ($a,b$ real); $\ z\bar z=|z|^2$; $\ |\bar z|=|z|=|-z|$
Modulus of products and quotients§ 4.2 →
$|z_1z_2|=|z_1|\,|z_2|$; $\ \left|\tfrac{z_1}{z_2}\right|=\tfrac{|z_1|}{|z_2|}$ ($z_2\ne0$); in general $|z_1+z_2|\ne|z_1|+|z_2|$
Multiplicative inverse§ 4.2 →
$z^{-1}=\dfrac{\bar z}{|z|^2}=\dfrac{a-ib}{a^2+b^2}$ for $z=a+ib\ne0$; $\ i^{-1}=-i$
Division§ 4.2 →
$\dfrac{a+ib}{c+id}=\dfrac{(a+ib)(c-id)}{c^2+d^2}=\dfrac{(ac+bd)+i(bc-ad)}{c^2+d^2}$ ($c+id\ne0$)
Argand plane§ 4.3 →
$z=a+ib\leftrightarrow P(a,b)$; $\ |z|=OP=\sqrt{a^2+b^2}$
Reflections§ 4.3 →
$\bar z\to(a,-b)$: mirror in the real axis; $\ -z\to(-a,-b)$: half turn about $O$; $\ -\bar z\to(-a,b)$: mirror in the imaginary axis
Parallelogram law§ 4.3 →
$z_1+z_2$ is the fourth vertex of the parallelogram on $O$, $z_1$, $z_2$; midpoint of $z_1$, $z_2$ is $\tfrac{z_1+z_2}{2}$
Distance§ 4.3 →
$P_1P_2=|z_1-z_2|$; $\ |z-z_0|=r$ is the circle with centre $z_0$ and radius $r$
Multiplication by $i$§ 4.3 →
$i(a+ib)=-b+ia$: rotation through $90^\circ$ anticlockwise about $O$; $\ -i$: $90^\circ$ clockwise; $\ k\gt0$ real: stretch by $k$
Chapter 5 · Linear Inequalities
Rules for inequalities (for $a\lt b$)§ 5.1 →
$a\pm c\lt b\pm c$ for every real $c$; $\quad ac\lt bc$ and $\tfrac ac\lt\tfrac bc$ if $c\gt0$; $\quad ac\gt bc$ and $\tfrac ac\gt\tfrac bc$ if $c\lt0$ (sign reversed)
Strict and slack§ 5.1 →
$a\le b\iff a\lt b\ \text{or}\ a=b$; $\quad a\lt b\iff b-a\gt0$
Combining inequalities§ 5.1 →
$a\lt b,\ b\lt c\Rightarrow a\lt c$; $\quad a\lt b,\ c\lt d\Rightarrow a+c\lt b+d$ (never subtract)
Reciprocals and squares§ 5.1 →
$0\lt a\lt b\Rightarrow\dfrac1a\gt\dfrac1b$; $\quad 0\le a\lt b\Rightarrow a^2\lt b^2$
Linear inequality in one variable§ 5.1 →
$ax+b\lt0,\ \gt0,\ \le0,\ \ge0$ with $a\ne0$
Solving $ax+b\lt0$ ($a\ne0$)§ 5.2 →
$a\gt0$: $\ x\lt-\tfrac ba$, i.e. $x\in\left(-\infty,-\tfrac ba\right)$; $\quad a\lt0$: $\ x\gt-\tfrac ba$, i.e. $x\in\left(-\tfrac ba,\infty\right)$
When $x$ cancels§ 5.2 →
True numerical statement $\Rightarrow$ solution set $\mathbb{R}$; false $\Rightarrow$ $\phi$
Double inequality, $k\lt0$§ 5.2 →
$p\lt E\lt q\Rightarrow kq\lt kE\lt kp$ (both signs reverse)
System of inequalities§ 5.2 →
Solution set $=$ intersection of the solution sets, e.g. $[-4,\infty)\cap(-\infty,4)=[-4,4)$
Translating words§ 5.2 →
at least $\to\ \ge$; $\ $ at most, not more than $\to\ \le$; $\ $ more than $\to\ \gt$; $\ $ less than $\to\ \lt$
Modulus inequalities (JEE, $r\gt0$)§ 5.2 →
$|x-a|\lt r\iff a-r\lt x\lt a+r$; $\quad |x-a|\gt r\iff x\lt a-r\ \text{or}\ x\gt a+r$
Chapter 6 · Permutations and Combinations
Fundamental principle of counting§ 6.1 →
$k$ successive stages with $n_1,n_2,\dots,n_k$ ways (each count independent of earlier choices): $\ n_1\times n_2\times\cdots\times n_k$ ways
Addition principle§ 6.1 →
Disjoint cases with $m_1,\dots,m_k$ ways: $\ m_1+m_2+\cdots+m_k$ ways
Arrangements with repetition§ 6.1 →
Length $r$ from $n$ symbols, repetition allowed: $\ n^r$
"At least one" by subtraction§ 6.1 →
$\#(\text{at least one})=\#(\text{all})-\#(\text{none})$
Number of divisors§ 6.1 →
$N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ (distinct primes): $\ (a_1+1)(a_2+1)\cdots(a_k+1)$ divisors
Factorial§ 6.2 →
$n!=1\cdot2\cdots n$; $\ 0!=1$; $\ n!=n\,(n-1)!$ ($n\ge1$)
Permutations of different objects§ 6.2 →
${}^nP_r=n(n-1)\cdots(n-r+1)=\dfrac{n!}{(n-r)!}$ ($0\le r\le n$, no repetition); $\ {}^nP_n=n!$, $\ {}^nP_0=1$
Permutations with repetition§ 6.2 →
$r$ places from $n$ different objects, repetition allowed: $\ n^r$
Objects not all different§ 6.2 →
$n$ objects with $p_1,p_2,\dots,p_k$ alike of each kind: $\ \dfrac{n!}{p_1!\,p_2!\cdots p_k!}$ arrangements
Together / never together§ 6.2 →
Block: (arrangements of units) × (arrangements inside block); never together = all − together
Circular arrangements (JEE)§ 6.2 →
$n$ different objects round a circle: $(n-1)!$; if clockwise = anticlockwise ($n\ge3$): $\tfrac{(n-1)!}{2}$
Combinations§ 6.3 →
${}^nC_r=\dbinom{n}{r}=\dfrac{{}^nP_r}{r!}=\dfrac{n!}{r!\,(n-r)!}$ ($0\le r\le n$); $\ {}^nC_0={}^nC_n=1$, $\ {}^nC_1=n$
Select, then arrange§ 6.3 →
${}^nP_r={}^nC_r\times r!$
Symmetry§ 6.3 →
${}^nC_r={}^nC_{n-r}$; $\ {}^nC_a={}^nC_b\Rightarrow a=b$ or $a+b=n$
Addition rule§ 6.3 →
${}^nC_r+{}^nC_{r-1}={}^{n+1}C_r$ ($1\le r\le n$)
Geometry counts§ 6.3 →
$n$ points, no three collinear: ${}^nC_2$ lines, ${}^nC_3$ triangles; $n$-gon: $\tfrac{n(n-3)}{2}$ diagonals
Distributions (JEE)§ 6.3 →
$n$ identical objects into $r$ different boxes: ${}^{n+r-1}C_{r-1}$; $\ $ each box non-empty: ${}^{n-1}C_{r-1}$; $\ $ $2n$ different objects into two unlabelled groups of $n$: $\tfrac{(2n)!}{2!\,(n!)^2}$
Chapter 7 · Binomial Theorem
Binomial theorem§ 7.1 →
$(a+b)^n=\sum_{r=0}^{n}{}^nC_r\,a^{n-r}b^r={}^nC_0a^n+{}^nC_1a^{n-1}b+\cdots+{}^nC_nb^n$ ($n$ a positive integer); $n+1$ terms, indices add up to $n$
Pascal's triangle§ 7.1 →
Entry $r$ of row $n$ is ${}^nC_r$ (count from $r=0$); $\ {}^nC_{r-1}+{}^nC_r={}^{n+1}C_r$; $\ {}^nC_r={}^nC_{n-r}$
Expansion of $(a-b)^n$§ 7.1 →
$(a-b)^n=\sum_{r=0}^{n}(-1)^r\,{}^nC_r\,a^{n-r}b^r$ — signs alternate, starting with $+$
$(1+x)^n$ and $(1-x)^n$§ 7.1 →
$(1+x)^n=1+nx+\tfrac{n(n-1)}{2}x^2+\cdots+x^n$; $\ (1-x)^n=1-nx+\tfrac{n(n-1)}{2}x^2-\cdots+(-1)^nx^n$
Row sums§ 7.1 →
${}^nC_0+{}^nC_1+\cdots+{}^nC_n=2^n$; $\ {}^nC_0-{}^nC_1+\cdots+(-1)^n\,{}^nC_n=0$ ($n\ge1$)
Evaluating a power§ 7.2 →
Split the base: $99^5=(100-1)^5$, $\ (10.2)^4=(10+0.2)^4$; keep every term for an exact answer
Approximation§ 7.2 →
$(1+x)^n\approx1+nx+\tfrac{n(n-1)}{2}x^2$ — accurate only when $nx$ is small
Lower bound§ 7.2 →
$(1+x)^n\ge1+nx$ for $x\ge0$ and $n$ a positive integer (also $\ge$ any sum of first terms)
Divisibility§ 7.2 →
For integers $x$ and $n\ge1$: $x^2$ divides $(1+x)^n-1-nx$; $\ (1+x)^n$ leaves remainder $1$ on division by $x\ge2$
Chapter 8 · Sequences and Series
Fibonacci sequence§ 8.1 →
$F_1=F_2=1$, $\ F_n=F_{n-1}+F_{n-2}$ ($n\ge3$): $\ 1,1,2,3,5,8,13,\dots$
Rules for $\Sigma$§ 8.1 →
$\sum(a_k+b_k)=\sum a_k+\sum b_k$; $\ \sum c\,a_k=c\sum a_k$; $\ \sum_{k=1}^{n}c=nc$; $\ \sum_{k=m}^{n}$ has $n-m+1$ terms
Terms from partial sums§ 8.1 →
$a_1=S_1$; $\ a_n=S_n-S_{n-1}$ for $n\ge2$ only
Arithmetic progression (recall)§ 8.1 →
$a_n=a+(n-1)d$; $\ S_n=\tfrac n2[2a+(n-1)d]=\tfrac n2(a+l)$; $\ \sum_{k=1}^{n}k=\tfrac{n(n+1)}{2}$
Sums of squares and cubes (JEE)§ 8.1 →
$\sum_{k=1}^{n}k^2=\tfrac{n(n+1)(2n+1)}{6}$; $\ \sum_{k=1}^{n}k^3=\left[\tfrac{n(n+1)}{2}\right]^2$
$n$th term of a GP§ 8.2 →
$a_n=ar^{n-1}$ ($a\ne0$, $r\ne0$); $\ r=\dfrac{a_{n+1}}{a_n}$
Sum of $n$ terms of a GP§ 8.2 →
$S_n=\dfrac{a(r^n-1)}{r-1}=\dfrac{a(1-r^n)}{1-r}=\dfrac{lr-a}{r-1}$ ($r\ne1$); $\ S_n=na$ ($r=1$)
Sum to infinity§ 8.2 →
$S_\infty=\dfrac{a}{1-r}$, only when $|r|\lt1$; no sum if $|r|\ge1$
Three numbers in GP§ 8.2 →
$a,b,c$ in GP $\iff b^2=ac$ (non-zero); take $\tfrac ar,\ a,\ ar$ when the product is given
Growth by a fixed percentage§ 8.2 →
$A=P\left(1\pm\tfrac{R}{100}\right)^n$ (growth $+$, depreciation $-$)
Arithmetico-geometric sum (JEE)§ 8.2 →
$1+2x+3x^2+\cdots=\dfrac{1}{(1-x)^2}$ for $|x|\lt1$; method: subtract $x\cdot S$
A.M. and G.M.§ 8.3 →
$A=\dfrac{a+b}{2}$; $\ G=\sqrt{ab}$ ($a,b\gt0$)
Inserting $n$ arithmetic means§ 8.3 →
$d=\dfrac{b-a}{n+1}$, $\ A_k=a+kd$; $\ A_1+\cdots+A_n=n\cdot\dfrac{a+b}{2}$
Inserting $n$ geometric means§ 8.3 →
$r=\left(\dfrac ba\right)^{\frac1{n+1}}$, $\ G_k=ar^k$; $\ G_1G_2\cdots G_n=\left(\sqrt{ab}\right)^n$ ($a,b\gt0$)
A.M. $\ge$ G.M.§ 8.3 →
$a,b\gt0$: $\ \dfrac{a+b}{2}\ge\sqrt{ab}$, equality iff $a=b$; $\ A-G=\tfrac12(\sqrt a-\sqrt b)^2$
Numbers from their means§ 8.3 →
Roots of $x^2-2Ax+G^2=0$: $\ A\pm\sqrt{(A+G)(A-G)}$
Chapter 9 · Straight Lines
Distance, section and mid-point§ 9.1 →
$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$; internal division $p:q$: $\left(\tfrac{px_2+qx_1}{p+q},\tfrac{py_2+qy_1}{p+q}\right)$; mid-point $\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2\right)$
Slope of a line§ 9.1 →
$m=\tan\theta$ ($0^\circ\le\theta\lt180^\circ$, $\theta\ne90^\circ$); through $(x_1,y_1)$, $(x_2,y_2)$ with $x_1\ne x_2$: $\ m=\dfrac{y_2-y_1}{x_2-x_1}$; vertical lines have no slope
Parallel and perpendicular§ 9.1 →
Non-vertical lines: parallel $\iff m_1=m_2$; perpendicular $\iff m_1m_2=-1$
Angle between two lines§ 9.1 →
Acute angle: $\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1m_2}\right|$ (non-vertical lines, $1+m_1m_2\ne0$); obtuse angle $180^\circ-\theta$
Lines parallel to the axes§ 9.2 →
Through $(x_1,y_1)$: horizontal $y=y_1$; vertical $x=x_1$; $x$-axis $y=0$, $y$-axis $x=0$
Point-slope form§ 9.2 →
Through $(x_1,y_1)$ with slope $m$: $\ y-y_1=m(x-x_1)$
Two-point form§ 9.2 →
Through $(x_1,y_1)$, $(x_2,y_2)$, $x_1\ne x_2$: $\ y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)$ (if $x_1=x_2$: $x=x_1$)
Slope-intercept form§ 9.2 →
Slope $m$, $y$-intercept $c$: $\ y=mx+c$; slope $m$, $x$-intercept $d$: $\ y=m(x-d)$
Intercept form§ 9.2 →
Intercepts $a$, $b$ (both $\ne0$): $\ \dfrac xa+\dfrac yb=1$; triangle with the axes has area $\tfrac12|ab|$
Distance of a point from a line§ 9.3 →
$(x_1,y_1)$ from $Ax+By+C=0$ ($A,B$ not both $0$): $\ d=\dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$; from $y=mx+c$: $\ d=\dfrac{|y_1-mx_1-c|}{\sqrt{1+m^2}}$
Distance between parallel lines§ 9.3 →
$Ax+By+C_1=0$, $Ax+By+C_2=0$ (same $A$, $B$): $\ d=\dfrac{|C_1-C_2|}{\sqrt{A^2+B^2}}$; $\ y=mx+c_1$, $y=mx+c_2$: $\ d=\dfrac{|c_1-c_2|}{\sqrt{1+m^2}}$
Foot of the perpendicular and image§ 9.3 →
Foot $N$: intersect $L$ with the perpendicular through $P$ (slope $-\tfrac1m$); image $P'=2N-P$
Chapter 10 · Conic Sections
Sections of a double cone (semi-vertical angle $\alpha$, plane at angle $\beta$ to the axis)§ 10.1 →
Not through the vertex: $\beta=90^\circ$ circle; $\alpha\lt\beta\lt90^\circ$ ellipse; $\beta=\alpha$ parabola; $\beta\lt\alpha$ hyperbola. Through the vertex: point, line, pair of lines.
Circle, centre $(h,k)$, radius $r$§ 10.1 →
$(x-h)^2+(y-k)^2=r^2$; centre at the origin: $x^2+y^2=r^2$
General equation of a circle§ 10.1 →
$x^2+y^2+2gx+2fy+c=0$: centre $(-g,-f)$, $r=\sqrt{g^2+f^2-c}$ if $g^2+f^2-c\gt0$ (point circle if $=0$, no points if $\lt0$)
Circle on a diameter $AB$§ 10.1 →
Centre = mid-point of $AB$, $r=\tfrac12AB$; or $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$
Tangency to $x^2+y^2=a^2$ (JEE)§ 10.1 →
$y=mx+c$ touches $x^2+y^2=a^2\iff c^2=a^2(1+m^2)$
Parabola $y^2=4ax$ ($a\gt0$)§ 10.2 →
Focus $(a,0)$, directrix $x=-a$, axis $y=0$, vertex $(0,0)$, opens right
The other standard parabolas ($a\gt0$)§ 10.2 →
$y^2=-4ax$: focus $(-a,0)$, directrix $x=a$; $\ x^2=4ay$: focus $(0,a)$, directrix $y=-a$; $\ x^2=-4ay$: focus $(0,-a)$, directrix $y=a$
Latus rectum of a parabola§ 10.2 →
Length $4a$; for $y^2=4ax$ its ends are $(a,\pm2a)$
Definition of a parabola§ 10.2 →
$PS=PM$ (distance to focus = distance to directrix); on $y^2=4ax$, $PS=x+a$
Parametric point (JEE)§ 10.2 →
$(at^2,\,2at)$ lies on $y^2=4ax$; focal distance $a(1+t^2)$
Ellipse, major axis along the $x$-axis ($a\gt b\gt0$)§ 10.3 →
$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$: vertices $(\pm a,0)$, foci $(\pm c,0)$, $c^2=a^2-b^2$
Ellipse, major axis along the $y$-axis ($a\gt b\gt0$)§ 10.3 →
$\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1$: vertices $(0,\pm a)$, foci $(0,\pm c)$; the larger denominator marks the major axis
Defining property§ 10.3 →
$PF_1+PF_2=2a$ for every point $P$ of the ellipse
Eccentricity and latus rectum§ 10.3 →
$e=\dfrac ca=\sqrt{1-\dfrac{b^2}{a^2}}$, $0\lt e\lt1$; latus rectum $\dfrac{2b^2}{a}$
Orbits (focus at the centre of attraction)§ 10.3 →
$r_{\min}=a-c$, $r_{\max}=a+c$, $b^2=r_{\min}r_{\max}$
Focal distances (JEE)§ 10.3 →
$PF_1=a+ex$, $PF_2=a-ex$; directrices $x=\pm\dfrac ae$
Hyperbola, transverse axis along the $x$-axis§ 10.4 →
$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$: vertices $(\pm a,0)$, foci $(\pm c,0)$, $c^2=a^2+b^2$
Hyperbola, transverse axis along the $y$-axis§ 10.4 →
$\dfrac{y^2}{a^2}-\dfrac{x^2}{b^2}=1$: vertices $(0,\pm a)$, foci $(0,\pm c)$; $a^2$ is under the positive term
Defining property§ 10.4 →
$|PF_1-PF_2|=2a$ for every point $P$ of the hyperbola
Eccentricity and latus rectum§ 10.4 →
$e=\dfrac ca=\sqrt{1+\dfrac{b^2}{a^2}}\gt1$; latus rectum $\dfrac{2b^2}{a}$; rectangular ($a=b$): $e=\sqrt2$
Which conic?§ 10.4 →
One variable squared: parabola; both, same sign, equal coefficients: circle; same sign, unequal: ellipse ($e\lt1$); opposite signs: hyperbola ($e\gt1$)
Tangency to a hyperbola (JEE)§ 10.4 →
$y=mx+c$ touches $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\iff c^2=a^2m^2-b^2$
Chapter 11 · Introduction to Three Dimensional Geometry
Coordinate planes§ 11.1 →
$XY$-plane: $z=0$, $YZ$-plane: $x=0$, $ZX$-plane: $y=0$
Points on the axes§ 11.1 →
$x$-axis $(x,0,0)$, $y$-axis $(0,y,0)$, $z$-axis $(0,0,z)$, $O(0,0,0)$
Feet of perpendiculars from $P(x,y,z)$ and distances from the planes§ 11.1 →
on $XY$: $(x,y,0)$, distance $|z|$; on $YZ$: $(0,y,z)$, distance $|x|$; on $ZX$: $(x,0,z)$, distance $|y|$
Signs of $(x,y,z)$ in the octants§ 11.1 →
I $(+,+,+)$, II $(-,+,+)$, III $(-,-,+)$, IV $(+,-,+)$; V $(+,+,-)$, VI $(-,+,-)$, VII $(-,-,-)$, VIII $(+,-,-)$. A point with a zero coordinate is in no octant.
Image in a coordinate plane§ 11.1 →
In the $XY$-plane: $(x,y,z)\mapsto(x,y,-z)$ — only the coordinate measured from the mirror changes sign
Distance between two points in space§ 11.2 →
$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
Distance from the origin§ 11.2 →
$OP=\sqrt{x^2+y^2+z^2}$
Distance of $P(x,y,z)$ from the axes§ 11.2 →
from the $x$-axis $\sqrt{y^2+z^2}$, the $y$-axis $\sqrt{z^2+x^2}$, the $z$-axis $\sqrt{x^2+y^2}$
Collinearity test§ 11.2 →
$A,B,C$ are collinear (with $B$ between) exactly when $AB+BC=AC$, $AC$ being the longest of the three distances
Sphere with centre $(a,b,c)$, radius $r\gt0$§ 11.2 →
$(x-a)^2+(y-b)^2+(z-c)^2=r^2$
JEE: section formula and centroid in space§ 11.2 →
Ratio $m:n$ internally: $\left(\tfrac{mx_2+nx_1}{m+n},\tfrac{my_2+ny_1}{m+n},\tfrac{mz_2+nz_1}{m+n}\right)$; centroid $\left(\tfrac{\sum x_i}{3},\tfrac{\sum y_i}{3},\tfrac{\sum z_i}{3}\right)$ (not in the Board syllabus)
Chapter 12 · Limits and Derivatives
When a limit exists§ 12.1 →
$\displaystyle\lim_{x\to a}f(x)=l\iff\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=l$; the value $f(a)$ does not matter
Algebra of limits§ 12.1 →
If $\displaystyle\lim_{x\to a}f(x)=l$ and $\displaystyle\lim_{x\to a}g(x)=m$ exist: $\lim(f\pm g)=l\pm m$, $\lim\lambda f=\lambda l$, $\lim fg=lm$, $\lim\dfrac fg=\dfrac lm$ (if $m\ne0$)
Basic limits§ 12.1 →
$\displaystyle\lim_{x\to a}c=c,\ \ \lim_{x\to a}x=a,\ \ \lim_{x\to a}x^n=a^n$ ($n\in\mathbb{N}$)
Average and instantaneous speed§ 12.1 →
average over $[t,t+h]$: $\dfrac{s(t+h)-s(t)}{h}$; instantaneous: $\displaystyle\lim_{h\to0}\dfrac{s(t+h)-s(t)}{h}$
Polynomials and rational functions§ 12.2 →
$\displaystyle\lim_{x\to a}p(x)=p(a)$; $\ \displaystyle\lim_{x\to a}\dfrac{p(x)}{q(x)}=\dfrac{p(a)}{q(a)}$ if $q(a)\ne0$; for $\tfrac00$, cancel $x-a$ or rationalise
The standard limit for powers§ 12.2 →
$\displaystyle\lim_{x\to a}\dfrac{x^n-a^n}{x-a}=na^{n-1}$ ($n\in\mathbb{N}$; also rational $n$ with $a\gt0$)
Trigonometric limits ($x$ in radians)§ 12.2 →
$\displaystyle\lim_{x\to0}\dfrac{\sin x}{x}=1$, $\ \displaystyle\lim_{x\to0}\dfrac{1-\cos x}{x}=0$, $\ \displaystyle\lim_{x\to0}\dfrac{\tan x}{x}=1$, $\ \displaystyle\lim_{x\to0}\dfrac{\sin kx}{x}=k$, $\ \displaystyle\lim_{x\to0}\dfrac{1-\cos x}{x^2}=\dfrac12$
Sandwich theorem§ 12.2 →
$f\le g\le h$ near $a$ and $f,h\to l$ $\Rightarrow$ $g\to l$; e.g. $\cos x\lt\dfrac{\sin x}{x}\lt1$ for $0\lt|x|\lt\tfrac\pi2$
Exponential and log limits (JEE)§ 12.2 →
$\displaystyle\lim_{x\to0}\dfrac{e^x-1}{x}=1$, $\ \displaystyle\lim_{x\to0}\dfrac{\log(1+x)}{x}=1$ (natural log)
Derivative (first principles)§ 12.3 →
$f'(x)=\displaystyle\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$ (when the limit exists); at a point: $f'(a)=\displaystyle\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$
Tangent line at x = a§ 12.3 →
$y-f(a)=f'(a)\,(x-a)$
First-principles results§ 12.3 →
$(c)'=0$, $(x^2)'=2x$, $(x^3)'=3x^2$, $\left(\tfrac1x\right)'=-\tfrac1{x^2}$, $(\sqrt x)'=\tfrac{1}{2\sqrt x}$ ($x\gt0$)
Derivatives of sin, cos, tan§ 12.3 →
$\dfrac{d}{dx}\sin x=\cos x$, $\ \dfrac{d}{dx}\cos x=-\sin x$, $\ \dfrac{d}{dx}\tan x=\sec^2x$ ($x$ in radians)
Power rule§ 12.4 →
$\dfrac{d}{dx}x^n=nx^{n-1}$ ($n\in\mathbb{Z}$; $x\ne0$ if $n\le0$); $\ \dfrac{d}{dx}c=0$
Sum and constant-multiple rules§ 12.4 →
$(u\pm v)'=u'\pm v'$, $\ (cu)'=cu'$ ($u$, $v$ differentiable)
Product and quotient rules§ 12.4 →
$(uv)'=u'v+uv'$; $\ \left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}$ ($v\ne0$)
Derivatives of the six trigonometric functions§ 12.4 →
$(\sin x)'=\cos x$, $(\cos x)'=-\sin x$, $(\tan x)'=\sec^2x$, $(\cot x)'=-\operatorname{cosec}^2x$, $(\sec x)'=\sec x\tan x$, $(\operatorname{cosec}x)'=-\operatorname{cosec}x\cot x$ ($x$ in radians)
Chapter 13 · Statistics
Range§ 13.1 →
Range $=$ largest observation $-$ smallest observation (same unit as the data)
Deviations from the mean§ 13.1 →
$\displaystyle\sum_{i=1}^{n}(x_i-\bar x)=0$; $\ \displaystyle\sum_{i=1}^{n}(x_i-a)=n(\bar x-a)$
Mean deviation (ungrouped data)§ 13.1 →
$\text{M.D.}(a)=\dfrac1n\displaystyle\sum_{i=1}^{n}|x_i-a|$; $\ a=\bar x$ or $a=M$ (median of the ordered data)
Least mean deviation§ 13.1 →
$\text{M.D.}(M)\le\text{M.D.}(a)$ for every $a$; for even $n$ any $a$ between the two middle values gives the least value
Shifting and scaling§ 13.1 →
Data $kx_i+c$: mean deviation $=|k|\times$ old mean deviation (adding $c$ changes nothing)
Mean deviation (frequency data)§ 13.2 →
$\text{M.D.}(a)=\dfrac1N\displaystyle\sum f_i|x_i-a|$, $\ N=\sum f_i$; $\ x_i=$ class marks for class intervals
Mean of a frequency distribution§ 13.2 →
$\bar x=\dfrac{\sum f_ix_i}{N}$
Step-deviation method§ 13.2 →
$u_i=\dfrac{x_i-A}{h}$, $\ \bar x=A+h\cdot\dfrac{\sum f_iu_i}{N}$ (equal class widths $h$)
Median of grouped data§ 13.2 →
$M=l+\dfrac{\frac N2-\text{c.f.}}{f}\times h$ ($l$, $f$, $h$: median class; c.f.: class before it)
Median of a discrete distribution§ 13.2 →
$N$ odd: the $\tfrac{N+1}{2}$th value; $N$ even: mean of the $\tfrac N2$th and $(\tfrac N2+1)$th values (use c.f.)
Inclusive classes§ 13.2 →
$10$–$19$, $20$–$29$, … $\to$ $9.5$–$19.5$, $19.5$–$29.5$, … (half the gap each side)
Variance and standard deviation§ 13.3 →
$\sigma^2=\dfrac1n\displaystyle\sum(x_i-\bar x)^2$, $\ \sigma=\sqrt{\sigma^2}\ge0$ (divide by $n$)
Computing formula§ 13.3 →
$\sigma^2=\dfrac1n\displaystyle\sum x_i^2-\bar x^{\,2}$; $\ \sigma=\dfrac1n\sqrt{n\sum x_i^2-\left(\sum x_i\right)^2}$
Sum of squares from $\bar x$ and $\sigma$§ 13.3 →
$\sum x_i=n\bar x$, $\ \sum x_i^2=n\left(\sigma^2+\bar x^{\,2}\right)$
Variance of a frequency distribution§ 13.3 →
$\sigma^2=\dfrac1N\displaystyle\sum f_i(x_i-\bar x)^2=\dfrac1N\sum f_ix_i^2-\bar x^{\,2}$ ($x_i$: class marks)
Shortcut method§ 13.3 →
$u_i=\dfrac{x_i-A}{h}$: $\ \sigma_x=\dfrac hN\sqrt{N\sum f_iu_i^2-\left(\sum f_iu_i\right)^2}$
Coefficient of variation (JEE)§ 13.3 →
$\text{C.V.}=\dfrac{\sigma}{\bar x}\times100\%$ ($\bar x\gt0$); smaller C.V. $=$ more consistent
Chapter 14 · Probability
Sample space and events§ 14.1 →
Sample space $S$ = all outcomes (each trial gives exactly one); an event is any $E\subset S$. Impossible event $\phi$, sure event $S$, simple event $n(E)=1$, compound event $n(E)\gt1$
Algebra of events§ 14.1 →
not $A$: $A'=S-A$; $\ A$ or $B$: $A\cup B$; $\ A$ and $B$: $A\cap B$; $\ A$ but not $B$: $A-B=A\cap B'$; $\ $neither: $(A\cup B)'=A'\cap B'$
De Morgan's laws for events§ 14.1 →
$(A\cup B)'=A'\cap B'$, $\ (A\cap B)'=A'\cup B'$
Mutually exclusive and exhaustive§ 14.1 →
Mutually exclusive: $E_i\cap E_j=\phi$ for all $i\ne j$; exhaustive: $E_1\cup\cdots\cup E_n=S$; both $\Rightarrow$ exactly one $E_i$ occurs on each trial
Axioms of probability§ 14.2 →
$P(E)\ge0$; $\ P(S)=1$; $\ P(E\cup F)=P(E)+P(F)$ if $E\cap F=\phi$. On finite $S$: weights $p_i\ge0$ with $\sum p_i=1$, and $P(E)=\sum_{\omega_i\in E}p_i$
Basic rules§ 14.2 →
$P(\phi)=0$, $\ 0\le P(A)\le1$, $\ P(A')=1-P(A)$; $\ A\subset B\Rightarrow P(A)\le P(B)$
Equally likely outcomes§ 14.2 →
$P(E)=\dfrac{n(E)}{n(S)}$ (only when all sample points are equally likely)
Addition rule and relatives§ 14.2 →
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$; $\ P(A-B)=P(A)-P(A\cap B)$; $\ $neither: $1-P(A\cup B)$; $\ $exactly one: $P(A)+P(B)-2P(A\cap B)$
“At least one”§ 14.2 →
$P(\text{at least one})=1-P(\text{none})$