📖 Important Theorems · All 13 Chapters

Ishu's Theorem Bible 📖

Every important theorem, with statement, conditions, and board exam tips — crafted with love for Ishu (My Wife). Learn once, never forget! 🎯

01

Relations & Functions

Key Theorems
⭐ Board Favourite
Equivalence Relation Theorem
A relation R on set A is an Equivalence Relation if and only if it is simultaneously Reflexive, Symmetric, and Transitive.
  • Reflexive: $(a,a) \in R$ for all $a \in A$
  • Symmetric: $(a,b)\in R \Rightarrow (b,a)\in R$
  • Transitive: $(a,b),(b,c)\in R \Rightarrow (a,c)\in R$
💡 Tip: Empty relation is NOT reflexive. Universal relation IS equivalence. Always verify all 3 conditions!
Bijective Function ↔ Inverse Exists
A function $f: A \to B$ has an inverse if and only if $f$ is bijective (both one-one and onto).
  • One-One: $f(a)=f(b) \Rightarrow a=b$
  • Onto: Range = Codomain
💡 Bijective = One-One + Onto. If either fails, inverse does NOT exist!
Composition of Functions
If $f:A\to B$ and $g:B\to C$, then $(g\circ f):A\to C$ defined as $(g\circ f)(x)=g(f(x))$ is also a function.
💡 $g\circ f \neq f\circ g$ in general. Apply the rightmost function FIRST.
02

Inverse Trigonometric Functions

Key Identities
⭐ 5 Marks
Complementary Angle Identity
$\sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2}, \quad x \in [-1,1]$
$\tan^{-1}x + \cot^{-1}x = \dfrac{\pi}{2}, \quad x \in \mathbb{R}$
💡 These are the most tested identities in CBSE boards! Memorise all three pairs.
Addition Formula
$\tan^{-1}x + \tan^{-1}y = \tan^{-1}\!\left(\dfrac{x+y}{1-xy}\right)$ if $xy < 1$
If $xy > 1$: add $\pi$ to the result. If $xy < -1$: subtract $\pi$.
💡 Classic board question: Prove $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$
Double Angle Identities
$2\tan^{-1}x = \sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right) = \tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)$
Valid for $|x| \leq 1$. Used to simplify complex inverse trig expressions.
03

Matrices

Properties & Results
⭐ Board Fav
Every Matrix = Symmetric + Skew-Symmetric
$A = \underbrace{\dfrac{A+A^T}{2}}_{\text{Symmetric}} + \underbrace{\dfrac{A-A^T}{2}}_{\text{Skew-Symmetric}}$
💡 This decomposition is unique. Directly asked in 3-mark board questions!
Transpose Properties
  • $(A^T)^T = A$
  • $(A+B)^T = A^T + B^T$
  • $(AB)^T = B^T A^T$ — order reverses!
  • $(kA)^T = k A^T$
Remember: $(AB)^T$ reverses order!
Invertibility Theorem
A square matrix $A$ is invertible if and only if $|A| \neq 0$ (non-singular).
$A^{-1} = \dfrac{1}{|A|} \cdot \text{adj}(A)$
  • $A \cdot A^{-1} = A^{-1} \cdot A = I$
  • $(AB)^{-1} = B^{-1}A^{-1}$
  • $(A^{-1})^T = (A^T)^{-1}$
04

Determinants

Key Theorems
Most Asked
Determinant Properties
  • $|AB| = |A| \cdot |B|$
  • $|A^T| = |A|$
  • $|kA| = k^n|A|$ for $n \times n$ matrix
  • Two identical rows → $|A| = 0$
  • Interchanging rows changes sign of det
  • Adding multiple of one row to another: det unchanged
⭐ Collinearity
Area & Collinearity Theorem
$\text{Area} = \dfrac{1}{2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}$
Points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are collinear if and only if this determinant equals zero.
Adjoint Theorem
$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| \cdot I$
  • $|\text{adj}(A)| = |A|^{n-1}$ for $n \times n$
  • $\text{adj}(AB) = \text{adj}(B)\cdot\text{adj}(A)$
💡 Adjoint = Transpose of Cofactor Matrix
Consistency of Linear System $AX=B$
  • $|A| \neq 0$ → Unique solution: $X = A^{-1}B$
  • $|A| = 0$ and $(\text{adj}\,A)B = 0$ → Infinitely many / no solution
  • $|A| = 0$ and $(\text{adj}\,A)B \neq 0$ → No solution (Inconsistent)
05

Continuity & Differentiability

Fundamental Theorems
⭐ Proof Asked in Boards
Differentiability ⟹ Continuity
If $f$ is differentiable at $c$, then $f$ is continuous at $c$. The converse is NOT true.
$\lim_{x\to c}(f(x)-f(c)) = \lim_{x\to c}\frac{f(x)-f(c)}{x-c}\cdot(x-c)=f'(c)\cdot 0=0$
💡 Counter-example: $f(x)=|x|$ is continuous but NOT differentiable at $x=0$.
Proof in Board
Rolle's Theorem
If $f$ is:
  • Continuous on $[a,b]$
  • Differentiable on $(a,b)$
  • $f(a) = f(b)$
Then $\exists$ at least one $c \in (a,b)$ such that $f'(c) = 0$.
💡 Geometric meaning: there's a point with horizontal tangent between $a$ and $b$.
Proof in Board
Lagrange's Mean Value Theorem (LMVT)
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then:
$\exists\; c \in (a,b) : f'(c) = \dfrac{f(b)-f(a)}{b-a}$
💡 LMVT is a generalisation of Rolle's Theorem (doesn't need $f(a)=f(b)$).
Chain Rule
If $y = f(u)$ and $u = g(x)$, then:
$\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$
💡 For parametric: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$
06

Application of Derivatives

Theorems & Criteria
⭐ 6-Mark Proof
Increasing / Decreasing Function Theorem
  • $f'(x) > 0$ on $(a,b)$ → Strictly Increasing
  • $f'(x) < 0$ on $(a,b)$ → Strictly Decreasing
  • $f'(x) = 0$ on $(a,b)$ → Constant
💡 Based on MVT. Solve $f'(x)=0$ first, then test sign in each interval.
Most Asked
First Derivative Test (Maxima/Minima)
If $f'(c) = 0$ and:
  • $f'$ changes $+$ to $-$ at $c$ → Local Maximum
  • $f'$ changes $-$ to $+$ at $c$ → Local Minimum
  • $f'$ doesn't change sign → Neither (Point of Inflection)
Second Derivative Test
If $f'(c) = 0$, then:
  • $f''(c) < 0$ → Local Maximum at $c$
  • $f''(c) > 0$ → Local Minimum at $c$
  • $f''(c) = 0$ → Test fails; use First Derivative Test
💡 Second Derivative Test is faster when $f''$ is easy to compute.
07

Integrals

Fundamental Theorems
⭐ Core Theorem
Fundamental Theorem of Calculus
If $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$:
$\int_a^b f(x)\,dx = F(b) - F(a) = \big[F(x)\big]_a^b$
💡 This theorem links differentiation and integration. Foundation of all definite integrals!
Must Know All
Properties of Definite Integrals
  • $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$
  • $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$
  • $\int_{-a}^{a} f\,dx = 2\int_0^a f\,dx$ (even); $0$ (odd)
  • $\int_0^{2a}f\,dx = 2\int_0^a f\,dx$ if $f(2a-x)=f(x)$; $0$ if $f(2a-x)=-f(x)$
Integration by Parts
$\int u\cdot v\,dx = u\int v\,dx - \int\left(\frac{du}{dx}\int v\,dx\right)dx$
ILATE Order for choosing $u$: Inverse Trig → Log → Algebra → Trig → Exponential
💡 Special: $\int e^x[f(x)+f'(x)]dx = e^x f(x) + C$
08

Application of Integrals

Area Theorems
⭐ 5 Marks
Area Between Two Curves
$A = \int_a^b \big[f(x) - g(x)\big]\,dx$
Where $f(x) \geq g(x)$ on $[a,b]$. Find intersection points first — those are $a$ and $b$!
💡 Always sketch the curve. If region crosses x-axis, split in two integrals and take absolute value.
Standard Areas
  • Circle $x^2+y^2=r^2$: Area $= \pi r^2$
  • Quarter circle by integration: $\int_0^r\sqrt{r^2-x^2}\,dx = \dfrac{\pi r^2}{4}$
  • Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$: Area $= \pi ab$
  • Parabola $y^2=4ax$ to $x=h$: Area $= \dfrac{8}{3}\sqrt{ah^3}$
09

Differential Equations

Solution Methods
⭐ 6 Marks
Linear DE — Integrating Factor Theorem
For $\dfrac{dy}{dx} + P(x)y = Q(x)$, the Integrating Factor is:
$\text{I.F.} = e^{\int P\,dx}$
Multiply both sides by I.F., LHS becomes $\dfrac{d}{dx}(y \cdot \text{I.F.})$
$y \cdot \text{I.F.} = \int Q \cdot \text{I.F.}\,dx + C$
Homogeneous DE Theorem
A DE $\dfrac{dy}{dx} = F(x,y)$ is homogeneous if $F(tx,ty) = F(x,y)$ for all $t$.
Substitution: $y = vx \Rightarrow \dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$, then separate variables.
Variable Separable
If $\dfrac{dy}{dx} = f(x)\cdot g(y)$, separate and integrate:
$\int \dfrac{dy}{g(y)} = \int f(x)\,dx + C$
💡 Most common type. Always write $+C$ on the right side only.
10

Vector Algebra

Vector Theorems
⭐ Triangle Law
Triangle Law of Vector Addition
If two sides of a triangle represent $\vec{a}$ and $\vec{b}$ in order, then the closing side represents $\vec{a}+\vec{b}$.
$|\vec{a}+\vec{b}| \leq |\vec{a}| + |\vec{b}|$ (Triangle Inequality)
Most Asked
Dot Product Theorem
$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$
  • $\vec{a} \perp \vec{b} \Leftrightarrow \vec{a}\cdot\vec{b}=0$
  • Projection of $\vec{a}$ on $\vec{b}$: $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$
  • $\vec{a}\cdot\vec{a} = |\vec{a}|^2$
Cross Product Theorem
$|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$
  • $\vec{a} \parallel \vec{b} \Leftrightarrow \vec{a}\times\vec{b}=\vec{0}$
  • Area of $\triangle = \dfrac{1}{2}|\vec{a}\times\vec{b}|$
  • Area of $\square = |\vec{a}\times\vec{b}|$
Scalar Triple Product
$[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c})$
  • Volume of parallelepiped $= |[\vec{a}\ \vec{b}\ \vec{c}]|$
  • Coplanar $\Leftrightarrow [\vec{a}\ \vec{b}\ \vec{c}]=0$
11

3D Geometry

Distance & Angle Theorems
⭐ 6 Marks
Shortest Distance (Skew Lines)
$d = \dfrac{|(\vec{a_2}-\vec{a_1})\cdot(\vec{b_1}\times\vec{b_2})|}{|\vec{b_1}\times\vec{b_2}|}$
💡 If $\vec{b_1}\times\vec{b_2}=\vec{0}$, lines are parallel — use a different formula.
Most Asked
Distance: Point from Plane
$d = \dfrac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$
Plane equation: $ax+by+cz+d=0$; Point: $(x_1,y_1,z_1)$. Normal vector: $(a,b,c)$.
Angle Between Lines / Planes
$\cos\theta = \dfrac{|\vec{b_1}\cdot\vec{b_2}|}{|\vec{b_1}||\vec{b_2}|}$ (lines)
$\cos\theta = \dfrac{|\vec{n_1}\cdot\vec{n_2}|}{|\vec{n_1}||\vec{n_2}|}$ (planes)
💡 Lines ⊥ when $\vec{b_1}\cdot\vec{b_2}=0$; Lines ∥ when $\vec{b_1}\times\vec{b_2}=\vec{0}$
12

Linear Programming

Fundamental Theorem
⭐ Core Theorem
Corner Point (Extreme Point) Theorem
If the feasible region of an LPP is a bounded convex polygon, the maximum and minimum values of the objective function $Z = ax + by$ occur at the corner (vertex) points of the feasible region.
💡 Evaluate $Z$ at ALL corner points. The largest value is max, smallest is min. Never skip endpoints!
Unbounded Feasible Region
If the feasible region is unbounded, the maximum or minimum of $Z$ may or may not exist. Check by drawing the line $ax+by=M$ and verifying no feasible point gives a better value.
13

Probability

Key Theorems · 8 Marks
⭐ 5 Mark Proof
Bayes' Theorem
If $E_1, E_2, \ldots, E_n$ are mutually exclusive and exhaustive events, and $A$ is any event with $P(A) > 0$:
$P(E_i|A) = \dfrac{P(E_i)\cdot P(A|E_i)}{\displaystyle\sum_{j=1}^n P(E_j)\cdot P(A|E_j)}$
💡 Always verify $\sum P(E_i)=1$ before applying Bayes'!
Must Know
Total Probability Theorem
If $E_1,\ldots,E_n$ are mutually exclusive and exhaustive:
$P(A) = \sum_{i=1}^n P(E_i)\cdot P(A|E_i)$
💡 This is the denominator in Bayes' theorem. Always compute this first!
Multiplication Theorem
$P(A\cap B) = P(A)\cdot P(B|A) = P(B)\cdot P(A|B)$
  • Independent events: $P(A\cap B)=P(A)\cdot P(B)$
  • $A,B$ independent $\Leftrightarrow P(B|A)=P(B)$
Binomial Distribution Theorem
$P(X=r) = \binom{n}{r}p^r q^{n-r}$
  • $n$ = no. of trials, $p$ = P(success), $q=1-p$
  • Mean $\mu = np$
  • Variance $\sigma^2 = npq$
  • Standard Deviation $= \sqrt{npq}$