JEE Challenger

Mathematics

Master calculus, algebra, and geometry with comprehensive mathematical resources

Formula Reference

Access essential equations and laws in a clean, interactive format.

29 Chapters • 217 Formulas

Subset

A⊆B⇔∀x∈A,  x∈BA \subseteq B \Leftrightarrow \forall x \in A,\; x \in B

Every element of A is in B

Power Set Cardinality

∣P(A)∣=2n where n=∣A∣|P(A)| = 2^n \text{ where } n = |A|

Number of subsets of A

Union

A∪B={x:x∈A or x∈B}A \cup B = \{x : x \in A \text{ or } x \in B\}

All elements in A or B or both

Intersection

A∩B={x:x∈A and x∈B}A \cap B = \{x : x \in A \text{ and } x \in B\}

Elements common to both A and B

Complement

A′=U−A={x:x∈U,  x∉A}A' = U - A = \{x : x \in U,\; x \notin A\}

Elements in universal set but not in A

De Morgan's Laws

(A∪B)′=A′∩B′,(A∩B)′=A′∪B′(A \cup B)' = A' \cap B',\quad (A \cap B)' = A' \cup B'

Complement distributes over union/intersection

Inclusion-Exclusion (2 sets)

∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|

Avoids double counting

Inclusion-Exclusion (3 sets)

∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣B∩C∣−∣A∩C∣+∣A∩B∩C∣|A \cup B \cup C| = |A|+|B|+|C|-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|

Generalization for 3 sets

Cartesian Product

A×B={(a,b):a∈A,  b∈B},∣A×B∣=∣A∣⋅∣B∣A \times B = \{(a,b) : a \in A,\; b \in B\},\quad |A\times B| = |A|\cdot|B|

Set of all ordered pairs

Number of Relations

From A to B: 2∣A∣×∣B∣\text{From A to B: } 2^{|A|\times|B|}

Each ordered pair is either included or not

Number of Functions

From A to B: ∣B∣∣A∣\text{From A to B: } |B|^{|A|}

Each element of A maps to one of |B| elements

Number of One-One Functions

From A to B: ∣B∣P∣A∣=∣B∣!(∣B∣−∣A∣)!\text{From A to B: } ^{|B|}P_{|A|} = \frac{|B|!}{(|B|-|A|)!}

Only possible when |B| ≥ |A|

Composition of Functions

(g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))

Apply f first, then g

Inverse Function Condition

f−1 exists iff f is bijective (one-one and onto)f^{-1}\text{ exists iff }f\text{ is bijective (one-one and onto)}

Bijective = injective + surjective

Equivalence Relation

Reflexive  (aRa)+Symmetric  (aRb⇒bRa)+Transitive  (aRb,bRc⇒aRc)\text{Reflexive}\;(aRa) + \text{Symmetric}\;(aRb\Rightarrow bRa) + \text{Transitive}\;(aRb,bRc\Rightarrow aRc)

All three properties together

Pythagorean Identities

sin⁡2θ+cos⁡2θ=1,1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=csc⁡2θ\sin^2\theta + \cos^2\theta = 1,\quad 1+\tan^2\theta = \sec^2\theta,\quad 1+\cot^2\theta = \csc^2\theta

Fundamental identities derived from unit circle

Angle Sum (sin)

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B

Addition and subtraction formula

Angle Sum (cos)

cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

Note: sign flips for cos

Angle Sum (tan)

tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

Denominator sign flips

Double Angle Formulas

sin⁡2θ=2sin⁡θcos⁡θ,cos⁡2θ=cos⁡2θ−sin⁡2θ=1−2sin⁡2θ=2cos⁡2θ−1\sin 2\theta = 2\sin\theta\cos\theta,\quad \cos 2\theta = \cos^2\theta-\sin^2\theta = 1-2\sin^2\theta = 2\cos^2\theta-1

Special case of angle sum

tan Double Angle

tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}

Valid when θ ≠ 45° + n·90°

Product-to-Sum

2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A\cos B = \sin(A+B)+\sin(A-B)

Used to simplify products

Sum-to-Product (sin)

sin⁡C+sin⁡D=2sin⁡C+D2cos⁡C−D2\sin C + \sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2}

C = A+B, D = A−B substitution

Sum-to-Product (cos)

cos⁡C+cos⁡D=2cos⁡C+D2cos⁡C−D2\cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2}

cos C − cos D: sine version with negative sign

General Solution (sin)

sin⁡θ=sin⁡α⇒θ=nπ+(−1)nα,  n∈Z\sin\theta = \sin\alpha \Rightarrow \theta = n\pi + (-1)^n\alpha,\;n\in\mathbb{Z}

All solutions of sin equation

General Solution (cos)

cos⁡θ=cos⁡α⇒θ=2nπ±α,  n∈Z\cos\theta = \cos\alpha \Rightarrow \theta = 2n\pi \pm \alpha,\;n\in\mathbb{Z}

All solutions of cos equation

General Solution (tan)

tan⁡θ=tan⁡α⇒θ=nπ+α,  n∈Z\tan\theta = \tan\alpha \Rightarrow \theta = n\pi + \alpha,\;n\in\mathbb{Z}

All solutions of tan equation

Sum of first n natural numbers

∑k=1nk=n(n+1)2\sum_{k=1}^n k = \frac{n(n+1)}{2}

Proved by induction; Gauss's formula

Sum of squares

∑k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}

Standard PMI result

Sum of cubes

∑k=1nk3=[n(n+1)2]2\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2

Square of sum of first n naturals

Geometric Series Sum

∑k=0n−1ark=a⋅rn−1r−1,r≠1\sum_{k=0}^{n-1} ar^k = a\cdot\frac{r^n-1}{r-1},\quad r \neq 1

Proved by induction; base case then inductive step

PMI Steps

1. Base case: P(1) true2. Inductive step: P(k)⇒P(k+1)\text{1. Base case: }P(1)\text{ true}\quad\text{2. Inductive step: }P(k)\Rightarrow P(k+1)

Both steps necessary for complete proof

Imaginary Unit

i=−1,i2=−1,i3=−i,i4=1i = \sqrt{-1},\quad i^2=-1,\quad i^3=-i,\quad i^4=1

Powers of i repeat with period 4

Modulus

∣z∣=∣a+ib∣=a2+b2|z| = |a+ib| = \sqrt{a^2+b^2}

Distance from origin in Argand plane

Argument

arg⁡(z)=θ=tan⁡−1ba\arg(z) = \theta = \tan^{-1}\frac{b}{a}

Angle with positive real axis; note quadrant

Polar Form

z=r(cos⁡θ+isin⁡θ)=reiθz = r(\cos\theta + i\sin\theta) = re^{i\theta}

r = |z|, θ = arg(z); Euler's formula: e^{iθ} = cosθ + i sinθ

De Moivre's Theorem

(cos⁡θ+isin⁡θ)n=cos⁡nθ+isin⁡nθ(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

For integer n; used to find nth roots

Conjugate

zˉ=a−ib,zzˉ=∣z∣2=a2+b2\bar{z} = a-ib,\quad z\bar{z} = |z|^2 = a^2+b^2

Reflect across real axis

Cube Roots of Unity

ω=e2πi/3=−1+i32,1+ω+ω2=0,ω3=1\omega = e^{2\pi i/3} = \frac{-1+i\sqrt3}{2},\quad 1+\omega+\omega^2=0,\quad\omega^3=1

ω and ω² are the complex cube roots; ω² = conjugate of ω

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

Solutions of ax² + bx + c = 0

Discriminant

D=b2−4ac:D>0 (real distinct),  D=0 (equal),  D<0 (complex)D = b^2-4ac:\quad D>0\text{ (real distinct)},\;D=0\text{ (equal)},\;D<0\text{ (complex)}

Nature of roots of quadratic

Vieta's Formulas (Quadratic)

α+β=−ba,αβ=ca\alpha+\beta = -\frac{b}{a},\quad \alpha\beta = \frac{c}{a}

Sum and product of roots

Absolute Value Inequality

∣x∣<a⇒−a<x<a,∣x∣>a⇒x<−a or x>a|x| < a \Rightarrow -a < x < a,\quad |x| > a \Rightarrow x < -a \text{ or } x > a

a > 0; open intervals

Rules for Inequalities

a<b⇒a+c<b+c;a<b,  c<0⇒ac>bca < b \Rightarrow a+c < b+c;\quad a < b,\;c < 0 \Rightarrow ac > bc

Multiplying/dividing by negative flips inequality

Solution of Linear Inequality

ax+b>0,  a>0⇒x>−baax + b > 0,\;a>0 \Rightarrow x > -\frac{b}{a}

Solve same as linear equation; watch for sign flip

Compound Inequality

a<x<b⇔x>a AND x<ba < x < b \Leftrightarrow x > a \text{ AND } x < b

Intersection of two solution sets

Factorial

n!=n⋅(n−1)⋅(n−2)⋯2⋅1,0!=1n! = n\cdot(n-1)\cdot(n-2)\cdots2\cdot1,\quad 0! = 1

Grows rapidly: 10! = 3,628,800

Permutation

nPr=n!(n−r)!^nP_r = \frac{n!}{(n-r)!}

Ordered arrangements of r from n distinct objects

Combination

nCr=(nr)=n!r!(n−r)!^nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Unordered selections; ^nC_r = ^nC_{n-r}

Relation P and C

nPr=r!⋅nCr^nP_r = r! \cdot {}^nC_r

Permutation = r! × combination

Permutations with Repetition

nrn^r

r items from n with replacement (ordered)

Permutations with Identical Items

n!p! q! r!\frac{n!}{p!\,q!\,r!}

n items with p alike, q alike, r alike

Circular Permutations

(n−1)!(n-1)!

Arrangements of n distinct objects in a circle

Combinations with Repetition

n+r−1Cr^{n+r-1}C_r

Choosing r from n types with repetition allowed

Binomial Theorem

(a+b)n=∑r=0n(nr)an−rbr(a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r}b^r

Expansion of (a+b)^n for non-negative integer n

General Term

Tr+1=(nr)an−rbrT_{r+1} = \binom{n}{r}a^{n-r}b^r

(r+1)th term from beginning

Middle Term (n even)

Tn2+1=(nn/2)an/2bn/2T_{\frac{n}{2}+1} = \binom{n}{n/2}a^{n/2}b^{n/2}

Single middle term when n is even

Middle Terms (n odd)

Tn+12 and Tn+32T_{\frac{n+1}{2}}\text{ and }T_{\frac{n+3}{2}}

Two middle terms when n is odd

Number of Terms

n+1n+1

Expansion of (a+b)^n has n+1 terms

Sum of Binomial Coefficients

∑r=0n(nr)=2n\sum_{r=0}^n \binom{n}{r} = 2^n

Put a = b = 1

Sum of Alternating Coefficients

∑r=0n(−1)r(nr)=0\sum_{r=0}^n (-1)^r\binom{n}{r} = 0

Put a = 1, b = −1

Approximation (small x)

(1+x)n≈1+nx,∣x∣≪1(1+x)^n \approx 1 + nx,\quad |x| \ll 1

First-order binomial approximation

AP — nth Term

an=a+(n−1)da_n = a + (n-1)d

a = first term; d = common difference

AP — Sum of n Terms

Sn=n2[2a+(n−1)d]=n2(a+l)S_n = \frac{n}{2}[2a+(n-1)d] = \frac{n}{2}(a+l)

l = last term

GP — nth Term

an=arn−1a_n = ar^{n-1}

a = first term; r = common ratio

GP — Sum of n Terms

Sn=arn−1r−1  (r≠1),Sn=na  (r=1)S_n = a\frac{r^n-1}{r-1}\;(r\neq1),\quad S_n = na\;(r=1)

Finite geometric sum

GP — Sum to Infinity

S∞=a1−r,∣r∣<1S_\infty = \frac{a}{1-r},\quad |r| < 1

Convergent geometric series

Arithmetic Mean (AM)

AM=a+b2AM = \frac{a+b}{2}

AM between a and b

Geometric Mean (GM)

GM=abGM = \sqrt{ab}

GM between a and b (both positive)

Harmonic Mean (HM)

HM=2aba+bHM = \frac{2ab}{a+b}

HM between a and b

AM–GM–HM Inequality

AM≥GM≥HMAM \geq GM \geq HM

Equality holds iff a = b

Sum of AGP

S=a1−r+dr(1−r)2,∣r∣<1S = \frac{a}{1-r}+\frac{dr}{(1-r)^2},\quad|r|<1

Arithmetic-Geometric Progression infinite sum

Slope of Line

m=tan⁡θ=y2−y1x2−x1m = \tan\theta = \frac{y_2-y_1}{x_2-x_1}

θ = angle with positive x-axis

Slope-Intercept Form

y=mx+cy = mx + c

m = slope, c = y-intercept

Two-Point Form

y−y1x−x1=y2−y1x2−x1\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}

Line through two given points

Intercept Form

xa+yb=1\frac{x}{a}+\frac{y}{b} = 1

a = x-intercept, b = y-intercept

General Form

ax+by+c=0ax+by+c = 0

Slope = −a/b; y-intercept = −c/b

Distance from Point to Line

d=∣ax1+by1+c∣a2+b2d = \frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}

Perpendicular distance from (x₁, y₁) to ax+by+c=0

Angle between Two Lines

tan⁡θ=∣m1−m21+m1m2∣\tan\theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|

θ = acute angle between lines with slopes m₁ and m₂

Parallel Lines Condition

m1=m2m_1 = m_2

Equal slopes; distance = |c₁−c₂|/√(a²+b²)

Perpendicular Lines Condition

m1⋅m2=−1m_1\cdot m_2 = -1

Product of slopes = −1

Section Formula (Internal)

P=(mx2+nx1m+n,my2+ny1m+n)P = \left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)

Point dividing (x₁,y₁)–(x₂,y₂) in ratio m:n internally

Circle

(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2 = r^2

Center (h,k), radius r

General Circle Equation

x2+y2+2gx+2fy+c=0x^2+y^2+2gx+2fy+c = 0

Center (−g,−f); radius = √(g²+f²−c)

Parabola (standard)

y2=4axy^2 = 4ax

Focus (a,0); directrix x=−a; axis along x-axis

Parabola — Other Forms

y2=−4ax,  x2=4ay,  x2=−4ayy^2=-4ax,\;x^2=4ay,\;x^2=-4ay

Corresponding foci and directrices

Ellipse (standard)

x2a2+y2b2=1,a>b>0\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1,\quad a > b > 0

Foci: (±c,0); c²=a²−b²; e=c/a<1

Ellipse — Key Relations

b2=a2(1−e2),e=ca<1b^2 = a^2(1-e^2),\quad e = \frac{c}{a} < 1

e = eccentricity; b²+c²=a²

Hyperbola (standard)

x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1

Foci: (±c,0); c²=a²+b²; e=c/a>1

Hyperbola — Asymptotes

y=±baxy = \pm\frac{b}{a}x

Lines approached but never touched

Rectangular Hyperbola

xy=c2xy = c^2

Asymptotes are coordinate axes; eccentricity = √2

Focal Chord (Parabola)

1SP+1SQ=1a\frac{1}{SP}+\frac{1}{SQ} = \frac{1}{a}

P, Q are ends of focal chord; S is focus

Distance in 3D

PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2PQ = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

3D extension of distance formula

Section Formula (3D)

P=(mx2+nx1m+n,my2+ny1m+n,mz2+nz1m+n)P = \left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n}\right)

Internal division in ratio m:n

Centroid of Triangle (3D)

G=(x1+x2+x33,y1+y2+y33,z1+z2+z33)G = \left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3}\right)

Average of all three vertices

Direction Cosines

l2+m2+n2=1,l=cos⁡α,  m=cos⁡β,  n=cos⁡γl^2+m^2+n^2 = 1,\quad l=\cos\alpha,\;m=\cos\beta,\;n=\cos\gamma

α,β,γ = angles with x,y,z axes

Direction Ratios to Direction Cosines

l=aa2+b2+c2,  m=b...,  n=c...l = \frac{a}{\sqrt{a^2+b^2+c^2}},\;m=\frac{b}{\sqrt{...}},\;n=\frac{c}{\sqrt{...}}

(a,b,c) are direction ratios

Standard Limit

lim⁡x→0sin⁡xx=1\lim_{x\to0}\frac{\sin x}{x} = 1

x must be in radians; fundamental trigonometric limit

Standard Limit (exponential)

lim⁡x→0ex−1x=1,lim⁡x→0ln⁡(1+x)x=1\lim_{x\to0}\frac{e^x-1}{x} = 1,\quad\lim_{x\to0}\frac{\ln(1+x)}{x} = 1

Exponential and log limits

Standard Limit (algebraic)

lim⁡x→axn−anx−a=nan−1\lim_{x\to a}\frac{x^n-a^n}{x-a} = na^{n-1}

Works for all real n

L'Hôpital's Rule

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x),if 00 or ∞∞\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)},\quad\text{if }\frac{0}{0}\text{ or }\frac{\infty}{\infty}

Differentiate numerator and denominator separately

Derivative from First Principles

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}

Definition of derivative

Power Rule

ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}

For all real n

Product Rule

(uv)′=u′v+uv′(uv)' = u'v + uv'

Derivative of product of two functions

Quotient Rule

(uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Derivative of quotient; v ≠ 0

Negation

∼p  (not p)\sim p\;(\text{not }p)

True when p is false; false when p is true

Conjunction (AND)

p∧qp \land q

True only when both p and q are true

Disjunction (OR)

p∨qp \lor q

True when at least one of p, q is true

Implication (If-Then)

p⇒q≡∼p∨qp \Rightarrow q\equiv \sim p\lor q

False only when p is true and q is false

Biconditional (Iff)

p⇔q≡(p⇒q)∧(q⇒p)p \Leftrightarrow q\equiv (p\Rightarrow q)\land(q\Rightarrow p)

True when p and q have same truth value

Contrapositive

p⇒q≡∼q⇒∼pp\Rightarrow q\equiv\sim q\Rightarrow\sim p

Always logically equivalent to the original implication

Converse

Converse of (p⇒q) is (q⇒p)\text{Converse of }(p\Rightarrow q)\text{ is }(q\Rightarrow p)

Not equivalent to original implication

Mean

xˉ=∑xin=∑fixi∑fi\bar{x} = \frac{\sum x_i}{n} = \frac{\sum f_i x_i}{\sum f_i}

Arithmetic average; second form for frequency distribution

Median

M=L+N2−CFf×hM = L + \frac{\frac{N}{2}-CF}{f}\times h

L=lower class limit; CF=cumulative freq before class; f=freq; h=class width

Mode

Z=L+f1−f02f1−f0−f2×hZ = L + \frac{f_1-f_0}{2f_1-f_0-f_2}\times h

f₁=modal class freq; f₀=preceding freq; f₂=following freq

Variance

σ2=∑(xi−xˉ)2n=∑xi2n−xˉ2\sigma^2 = \frac{\sum(x_i-\bar{x})^2}{n} = \frac{\sum x_i^2}{n} - \bar{x}^2

Mean of squared deviations

Standard Deviation

σ=∑(xi−xˉ)2n\sigma = \sqrt{\frac{\sum(x_i-\bar{x})^2}{n}}

Square root of variance; same units as data

Coefficient of Variation

CV=σxˉ×100%CV = \frac{\sigma}{\bar{x}}\times100\%

Relative measure of dispersion; unitless

Empirical Relation

Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}

Approximate relation for moderately skewed distributions

Classical Probability

P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}

Favorable outcomes / total equally likely outcomes

Complementary Events

P(A′)=1−P(A)P(A') = 1 - P(A)

P(A) + P(A') = 1 always

Addition Rule

P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B) = P(A)+P(B)-P(A\cap B)

For mutually exclusive: P(A∩B) = 0

Conditional Probability

P(A∣B)=P(A∩B)P(B),P(B)≠0P(A|B) = \frac{P(A\cap B)}{P(B)},\quad P(B)\neq0

Probability of A given B has occurred

Multiplication Rule

P(A∩B)=P(A)⋅P(B∣A)=P(B)⋅P(A∣B)P(A\cap B) = P(A)\cdot P(B|A) = P(B)\cdot P(A|B)

Joint probability

Independent Events

P(A∩B)=P(A)⋅P(B)P(A\cap B) = P(A)\cdot P(B)

A and B are independent iff this holds

Types of Functions

Injective (1-1): f(x1)=f(x2)⇒x1=x2\text{Injective (1-1): }f(x_1)=f(x_2)\Rightarrow x_1=x_2

Each output comes from at most one input

Surjective (Onto)

∀y∈B,  ∃x∈A:  f(x)=y\forall y\in B,\;\exists x\in A:\;f(x)=y

Every element of codomain is in range

Bijective

One-one  (injective)+onto  (surjective)\text{One-one}\;(\text{injective}) + \text{onto}\;(\text{surjective})

Perfect pairing; invertible

Even and Odd Functions

f(−x)=f(x) (even);f(−x)=−f(x) (odd)f(-x)=f(x)\text{ (even)};\quad f(-x)=-f(x)\text{ (odd)}

Even: symmetric about y-axis; Odd: 180° rotational symmetry

Periodic Function

f(x+T)=f(x)  ∀x,T=periodf(x+T) = f(x)\;\forall x,\quad T = \text{period}

Smallest positive T is the fundamental period

Domain and Range

sin⁡−1x:[−1,1]→[−π2,π2],cos⁡−1x:[−1,1]→[0,π]\sin^{-1}x: [-1,1]\to[-\tfrac{\pi}{2},\tfrac{\pi}{2}],\quad\cos^{-1}x:[-1,1]\to[0,\pi]

Principal value branches

tan⁻¹ and cot⁻¹

tan⁡−1x:R→(−π2,π2),cot⁡−1x:R→(0,π)\tan^{-1}x: \mathbb{R}\to(-\tfrac{\pi}{2},\tfrac{\pi}{2}),\quad\cot^{-1}x:\mathbb{R}\to(0,\pi)

Open interval for tan⁻¹

Identity: sin⁻¹ + cos⁻¹

sin⁡−1x+cos⁡−1x=π2,x∈[−1,1]\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2},\quad x\in[-1,1]

Complementary angle identity

Identity: tan⁻¹ + cot⁻¹

tan⁡−1x+cot⁡−1x=π2,x∈R\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2},\quad x\in\mathbb{R}

Complementary angle identity

tan⁻¹ Addition

tan⁡−1x+tan⁡−1y=tan⁡−1x+y1−xy,xy<1\tan^{-1}x + \tan^{-1}y = \tan^{-1}\frac{x+y}{1-xy},\quad xy<1

xy > 1: add ±π to result (based on signs of x,y)

tan⁻¹ Subtraction

tan⁡−1x−tan⁡−1y=tan⁡−1x−y1+xy,xy>−1\tan^{-1}x - \tan^{-1}y = \tan^{-1}\frac{x-y}{1+xy},\quad xy>-1

Useful for simplification

2·tan⁻¹ Formula

2tan⁡−1x=sin⁡−12x1+x2=cos⁡−11−x21+x2=tan⁡−12x1−x22\tan^{-1}x = \sin^{-1}\frac{2x}{1+x^2} = \cos^{-1}\frac{1-x^2}{1+x^2} = \tan^{-1}\frac{2x}{1-x^2}

Connects inverse trig functions

Matrix Addition/Subtraction

(A±B)ij=Aij±Bij(A \pm B)_{ij} = A_{ij} \pm B_{ij}

Same order matrices; element-wise operation

Matrix Multiplication

(AB)ij=∑kAikBkj(AB)_{ij} = \sum_k A_{ik}B_{kj}

A is m×n, B is n×p → AB is m×p

Transpose

(AT)ij=Aji,(AB)T=BTAT(A^T)_{ij} = A_{ji},\quad (AB)^T = B^T A^T

Rows become columns

Symmetric Matrix

AT=AA^T = A

a_{ij} = a_{ji} for all i, j

Skew-Symmetric Matrix

AT=−A,aii=0A^T = -A,\quad a_{ii} = 0

Diagonal elements are always zero

Any Matrix Decomposition

A=A+AT2+A−AT2A = \frac{A+A^T}{2} + \frac{A-A^T}{2}

Sum of symmetric and skew-symmetric parts

Properties of Transpose

(A+B)T=AT+BT,(kA)T=kAT(A+B)^T = A^T+B^T,\quad (kA)^T = kA^T

Linearity of transpose

2×2 Determinant

det⁡(abcd)=ad−bc\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad-bc

Product of main diagonal minus anti-diagonal

3×3 Determinant (expansion)

det⁡(A)=a11C11+a12C12+a13C13\det(A) = a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}

Expand along any row or column; C_{ij} = cofactor

Cofactor

Cij=(−1)i+jMijC_{ij} = (-1)^{i+j}M_{ij}

M_{ij} = minor = det of matrix with row i, col j deleted

Adjugate (Adjoint)

adj(A)=(Cij)T\text{adj}(A) = (C_{ij})^T

Transpose of cofactor matrix

Inverse of Matrix

A−1=1∣A∣adj(A)A^{-1} = \frac{1}{|A|}\text{adj}(A)

Exists only if |A| ≠ 0 (non-singular)

Cramer's Rule

xi=DiDx_i = \frac{D_i}{D}

D = det(A); D_i = det with ith column replaced by constants

Properties of Determinant

det⁡(AB)=det⁡(A)det⁡(B),det⁡(AT)=det⁡(A)\det(AB) = \det(A)\det(B),\quad \det(A^T) = \det(A)

Key properties for calculations

Area of Triangle (Determinant)

Δ=12∣x1y11x2y21x3y31∣\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|

Vertices (x₁,y₁), (x₂,y₂), (x₃,y₃)

Continuity Condition

f continuous at x=a⇔lim⁡x→af(x)=f(a)f\text{ continuous at }x=a \Leftrightarrow \lim_{x\to a}f(x) = f(a)

Left limit = right limit = value at a

Chain Rule

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)

Derivative of composite function

Standard Derivatives

ddx(sin⁡x)=cos⁡x,  (cos⁡x)=−sin⁡x,  (tan⁡x)=sec⁡2x\frac{d}{dx}(\sin x)=\cos x,\;(\cos x)=-\sin x,\;(\tan x)=\sec^2x

Most used; memorise all six trig derivatives

Exponential & Log Derivatives

ddx(ex)=ex,  (ax)=axln⁡a,  (ln⁡x)=1x\frac{d}{dx}(e^x)=e^x,\;(a^x)=a^x\ln a,\;(\ln x)=\frac{1}{x}

e^x is its own derivative

Parametric Differentiation

dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

When x = f(t), y = g(t)

Implicit Differentiation

ddx[F(x,y)=0]⇒dydx=−∂F/∂x∂F/∂y\frac{d}{dx}[F(x,y)=0]\Rightarrow\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}

Differentiate entire equation w.r.t. x

Logarithmic Differentiation

y=uv⇒ln⁡y=vln⁡u⇒y′y=v′ln⁡u+vu′uy = u^v \Rightarrow \ln y = v\ln u \Rightarrow \frac{y'}{y} = v'\ln u + \frac{vu'}{u}

Used when base and exponent both contain x

Rolle's Theorem

f cont on [a,b],  f(a)=f(b)⇒∃c:  f′(c)=0f\text{ cont on }[a,b],\;f(a)=f(b)\Rightarrow\exists c:\;f'(c)=0

Guarantees horizontal tangent between equal function values

Mean Value Theorem (Lagrange)

f′(c)=f(b)−f(a)b−a for some c∈(a,b)f'(c) = \frac{f(b)-f(a)}{b-a}\text{ for some }c\in(a,b)

Generalisation of Rolle's theorem

Equation of Tangent

y−y1=f′(x1)(x−x1)y - y_1 = f'(x_1)(x - x_1)

Slope = f'(x₁) at point (x₁, y₁)

Equation of Normal

y−y1=−1f′(x1)(x−x1)y - y_1 = -\frac{1}{f'(x_1)}(x - x_1)

Perpendicular to tangent; slope = −1/f'(x₁)

Angle of Intersection of Curves

tan⁡θ=∣m1−m21+m1m2∣\tan\theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|

m₁, m₂ = slopes of tangents at intersection point

Increasing Function

f′(x)>0 in interval ⇒f is increasingf'(x) > 0 \text{ in interval }\Rightarrow f\text{ is increasing}

Derivative positive → function rising

Critical Points

f′(x)=0 or f′(x) does not existf'(x) = 0 \text{ or } f'(x)\text{ does not exist}

Candidates for maxima, minima or inflection points

Second Derivative Test

f′(c)=0:  f′′(c)<0⇒max;  f′′(c)>0⇒minf'(c)=0:\;f''(c)<0\Rightarrow\text{max};\;f''(c)>0\Rightarrow\text{min}

Conclusive when f''(c) ≠ 0

Approximation

Δy≈f′(x)Δx,f(x+h)≈f(x)+hf′(x)\Delta y \approx f'(x)\Delta x,\quad f(x+h)\approx f(x)+hf'(x)

Linear approximation using derivative

Absolute Max/Min on [a,b]

Compare values at critical points and endpoints\text{Compare values at critical points and endpoints}

Global extrema occur at one of these

Fundamental Theorem of Calculus

∫abf(x) dx=F(b)−F(a),F′(x)=f(x)\int_a^b f(x)\,dx = F(b)-F(a),\quad F'(x)=f(x)

Connects differentiation and integration

Power Rule (Integration)

∫xn dx=xn+1n+1+C,n≠−1\int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\quad n\neq-1

Add 1 to power, divide by new power

Standard Integrals

∫exdx=ex+C,  ∫1xdx=ln⁡∣x∣+C,  ∫sin⁡x dx=−cos⁡x+C\int e^x dx = e^x+C,\;\int\frac{1}{x}dx = \ln|x|+C,\;\int\sin x\,dx = -\cos x+C

Most frequently used

Integration by Parts

∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du

ILATE: Inverse trig > Log > Algebraic > Trig > Exponential (choose u)

Substitution

∫f(g(x))g′(x) dx=∫f(t) dt,t=g(x)\int f(g(x))g'(x)\,dx = \int f(t)\,dt,\quad t=g(x)

Reversal of chain rule

Partial Fractions

P(x)Q(x)=Ax−a+Bx−b+...\frac{P(x)}{Q(x)} = \frac{A}{x-a}+\frac{B}{x-b}+...

When degree of P < degree of Q and Q factorises

King's Property

∫abf(x) dx=∫abf(a+b−x) dx\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx

Substitute x → a+b−x; very useful for definite integrals

Even/Odd Function Property

∫−aaf(x) dx={2∫0af dxf even0f odd\int_{-a}^a f(x)\,dx = \begin{cases}2\int_0^a f\,dx & f\text{ even}\\0 & f\text{ odd}\end{cases}

Simplifies symmetric integrals

Area under Curve

A=∫ab∣f(x)∣ dxA = \int_a^b |f(x)|\,dx

Absolute value ensures positive area

Area between Two Curves

A=∫ab[f(x)−g(x)] dx,f(x)≥g(x) on [a,b]A = \int_a^b [f(x)-g(x)]\,dx,\quad f(x)\geq g(x)\text{ on }[a,b]

Upper curve minus lower curve

Area using y-integration

A=∫cd[f(y)−g(y)] dyA = \int_c^d [f(y)-g(y)]\,dy

Integrate horizontally; useful when curves are easier expressed as x=f(y)

Area of Ellipse

A=πabA = \pi ab

Semi-axes a and b; circle: a=b=r gives πr²

Area of Parabola (y²=4ax, x=h)

A=43×base×height=43⋅24ah⋅hA = \frac{4}{3}\times\text{base}\times\text{height}= \frac{4}{3}\cdot2\sqrt{4ah}\cdot h

Area enclosed by chord and parabola

Order and Degree

Order: highest derivative; Degree: power of highest derivative (after rationalising)\text{Order: highest derivative; Degree: power of highest derivative (after rationalising)}

Degree defined only for polynomial differential equations

Variable Separable

dydx=f(x)g(y)⇒∫dyg(y)=∫f(x) dx\frac{dy}{dx} = f(x)g(y)\Rightarrow\int\frac{dy}{g(y)} = \int f(x)\,dx

Separate variables, integrate both sides

Homogeneous DE

dydx=F ⁣(yx)→y=vxseparable in v,x\frac{dy}{dx} = F\!\left(\frac{y}{x}\right)\xrightarrow{y=vx}\text{separable in }v,x

Substitute y = vx to reduce to separable form

Linear DE (First Order)

dydx+P(x)y=Q(x)\frac{dy}{dx}+P(x)y = Q(x)

Standard form; solve using integrating factor

Integrating Factor

μ=e∫P(x) dx\mu = e^{\int P(x)\,dx}

Multiply both sides by μ to make left side exact

Solution of Linear DE

y⋅μ=∫Q(x)⋅μ dx+Cy\cdot\mu = \int Q(x)\cdot\mu\,dx + C

General solution after applying integrating factor

Bernoulli's DE

dydx+P(x)y=Q(x)yn→v=y1−nlinear in v\frac{dy}{dx}+P(x)y = Q(x)y^n\xrightarrow{v=y^{1-n}}\text{linear in }v

Reduce to linear DE by substitution

Magnitude of Vector

∣a⃗∣=ax2+ay2+az2|\vec{a}| = \sqrt{a_x^2+a_y^2+a_z^2}

Length of vector \vec{a} = a_x\hat{i}+a_y\hat{j}+a_z\hat{k}

Unit Vector

a^=a⃗∣a⃗∣\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Vector of magnitude 1 in direction of \vec{a}

Dot (Scalar) Product

a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ=axbx+ayby+azbz\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_xb_x+a_yb_y+a_zb_z

Scalar result; θ = angle between vectors

Cross (Vector) Product

∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣sin⁡θ|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta

Vector perpendicular to both; magnitude = area of parallelogram

Cross Product (Components)

a⃗×b⃗=∣i^j^k^axayazbxbybz∣\vec{a}\times\vec{b} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\a_x&a_y&a_z\\b_x&b_y&b_z\end{vmatrix}

Determinant formula

Scalar Triple Product

[a⃗  b⃗  c⃗]=a⃗⋅(b⃗×c⃗)[\vec{a}\;\vec{b}\;\vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c})

Volume of parallelepiped; zero if coplanar

Projection of \vec{a} on \vec{b}

projb⃗a⃗=a⃗⋅b⃗∣b⃗∣\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}

Scalar projection; vector projection = (scalar proj)×\hat{b}

Angle between Vectors

cos⁡θ=a⃗⋅b⃗∣a⃗∣∣b⃗∣\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}

0 ≤ θ ≤ π

Equation of Line (Vector Form)

r⃗=a⃗+λb⃗\vec{r} = \vec{a} + \lambda\vec{b}

\vec{a} = position vector of point on line; \vec{b} = direction vector

Equation of Line (Cartesian)

x−x1l=y−y1m=z−z1n\frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{n}

(l,m,n) = direction cosines or ratios

Angle between Two Lines

cos⁡θ=∣l1l2+m1m2+n1n2∣\cos\theta = |l_1l_2+m_1m_2+n_1n_2|

Acute angle between lines with direction cosines (l₁,m₁,n₁) and (l₂,m₂,n₂)

Equation of Plane

r⃗⋅n^=d,ax+by+cz=d\vec{r}\cdot\hat{n} = d,\quad ax+by+cz = d

\hat{n} = normal unit vector; d = perpendicular distance from origin

Plane through Three Points

∣x−x1y−y1z−z1x2−x1y2−y1z2−z1x3−x1y3−y1z3−z1∣=0\begin{vmatrix}x-x_1&y-y_1&z-z_1\\x_2-x_1&y_2-y_1&z_2-z_1\\x_3-x_1&y_3-y_1&z_3-z_1\end{vmatrix}=0

Determinant form

Distance — Point to Plane

d=∣ax1+by1+cz1−d∣a2+b2+c2d = \frac{|ax_1+by_1+cz_1-d|}{\sqrt{a^2+b^2+c^2}}

Perpendicular distance from (x₁,y₁,z₁) to ax+by+cz=d

Angle between Line and Plane

sin⁡θ=∣al+bm+cn∣a2+b2+c2l2+m2+n2\sin\theta = \frac{|al+bm+cn|}{\sqrt{a^2+b^2+c^2}\sqrt{l^2+m^2+n^2}}

θ = angle between line direction and plane normal complement

Angle between Two Planes

cos⁡θ=∣a1a2+b1b2+c1c2∣a12+b12+c12a22+b22+c22\cos\theta = \frac{|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}}

Angle between their normal vectors

Skew Lines — Shortest Distance

d=∣(a⃗2−a⃗1)⋅(b⃗1×b⃗2)∣∣b⃗1×b⃗2∣d = \frac{|(\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|}

Distance between two non-parallel, non-intersecting lines

Objective Function

Z=ax+by, maximise or minimiseZ = ax + by,\text{ maximise or minimise}

Linear function to optimise

Constraints

a1x+b1y≤c1,  a2x+b2y≤c2,  x≥0,  y≥0a_1x+b_1y \leq c_1,\;a_2x+b_2y \leq c_2,\;x\geq0,\;y\geq0

Inequality constraints; feasible region = intersection

Optimal Solution Location

Optimum occurs at a vertex (corner point) of feasible region\text{Optimum occurs at a vertex (corner point) of feasible region}

Corner point theorem; evaluate Z at all vertices

Unbounded Solution Condition

If feasible region is unbounded: check if optimum exists\text{If feasible region is unbounded: check if optimum exists}

Maximum may not exist; minimum may still exist

Bayes' Theorem

P(Ai∣B)=P(Ai)P(B∣Ai)∑jP(Aj)P(B∣Aj)P(A_i|B) = \frac{P(A_i)P(B|A_i)}{\sum_j P(A_j)P(B|A_j)}

Posterior probability; updates prior with evidence

Total Probability Theorem

P(B)=∑i=1nP(Ai)P(B∣Ai)P(B) = \sum_{i=1}^n P(A_i)P(B|A_i)

A₁, A₂,..., Aₙ form a partition of sample space

Binomial Distribution

P(X=r)=(nr)prqn−r,q=1−pP(X=r) = \binom{n}{r}p^r q^{n-r},\quad q=1-p

n = trials, p = success probability, r = successes

Binomial Mean and Variance

μ=np,σ2=npq\mu = np,\quad\sigma^2 = npq

q = 1−p

Poisson Distribution

P(X=r)=e−λλrr!P(X=r) = \frac{e^{-\lambda}\lambda^r}{r!}

λ = mean = np for large n, small p

Expected Value

E(X)=∑xiP(xi)E(X) = \sum x_i P(x_i)

Weighted average of outcomes by probability

Variance of Random Variable

Var(X)=E(X2)−[E(X)]2\text{Var}(X) = E(X^2) - [E(X)]^2

Second moment minus square of first moment

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