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Master Class: Derivations 📐

The Proof Guardian

Detailed step-by-step derivations of the most important Math proofs asked in CBSE Board Exams for Ishu (My Wife). Target 80/80.

Chapter 1: Relations & Functions

Theorem: Equivalence Relation Proof

ReflexivityFor any $a \in A$, if $(a, a) \in R$, relation is Reflexive.
SymmetryIf $(a, b) \in R \implies (b, a) \in R$.
TransitivityIf $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$.
Important: A relation is 'Equivalence' ONLY if it satisfies all three!
Chapter 2: Inverse Trigo

Proof: $\sin^{-1}x + \cos^{-1}x = \pi/2$

AssumptionLet $\sin^{-1}x = \theta \implies x = \sin\theta$.
Complementary Angle$\sin\theta = \cos(\pi/2 - \theta)$.
Final StepSubstitute $\theta = \sin^{-1}x$:
$\sin^{-1}x + \cos^{-1}x = \pi/2$
Q.E.D.
Chapter 3: Matrices

Proof: Inverse of 2x2 Matrix Shortcut

AlgorithmIf $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $|A| \neq 0$:
$A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$
Tested via $A \cdot A^{-1} = I$
Chapter 4: Determinants

Derivation: Area of Triangle using Determinants

Coordinate SetupLet vertices be $(x_1, y_1), (x_2, y_2), (x_3, y_3)$.
Determinant FormThe area $\Delta$ is given by:
$\Delta = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$
If $\Delta = 0$, the points are collinear!
Q.E.D.
Chapter 5: Continuity & Diff

Theorem: Differentiability $\implies$ Continuity

AssumptionIf $f(x)$ is differentiable at $c$:
$f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x-c}$ exists.
Limit CheckTaking limit $x \to c$ on $f(x) - f(c) = \frac{f(x) - f(c)}{x-c} \cdot (x-c)$:
$\lim_{x \to c} (f(x) - f(c)) = f'(c) \cdot 0 = 0$
$\lim_{x \to c} f(x) = f(c)$, hence it is continuous.
Chapter 6: Application of Derivatives

Criteria: Increasing/Decreasing Functions

MVT FoundationLet $x_1 < x_2 \in [a, b]$.
Tangent SlopeIf $f'(x) > 0$ for all $x$, then the slope of the tangent is always positive.
This proves $f(x_1) < f(x_2)$, strictly increasing!
Chapter 7: Integrals

Derivation: $\int \frac{dx}{x^2+a^2} = \frac{1}{a} \tan^{-1}(x/a) + C$

SubstitutionUse $x = a\tan\theta$, $dx = a\sec^2\theta d\theta$.
$\int \frac{a\sec^2\theta}{a^2\sec^2\theta} d\theta = \frac{1}{a} \theta$
Q.E.D.
Chapter 8: Application of Integrals

Proof: Area of Parabola $y^2 = 4ax$ up to $x = x_1$

Integral Setup$A = 2 \int_0^{x_1} \sqrt{4ax} \, dx = 4\sqrt{a} \int_0^{x_1} x^{1/2} \, dx$
$A = 4\sqrt{a} \left[ \frac{x^{3/2}}{3/2} \right]_0^{x_1} = \frac{8}{3} \sqrt{a} x_1^{3/2}$
Q.E.D.
Chapter 9: Differential Equations

Integrating Factor (I.F.) Derivation

Form$\frac{dy}{dx} + Py = Q$.
Result$M = e^{\int P dx}$ makes the LHS a perfect derivative $(My)'$.
I.F. Locked.
Chapter 10: Vector Algebra

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

ComponentProjection length $= |\vec{a}| \cos \theta$.
Component $= \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$
It is the scalar length of $\vec{a}$ along $\vec{b}$.
Chapter 11: 3D Geometry

Proof: Distance of Point from Plane

Distance $= \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}$
Verified via Vector Projection
Chapter 12: Linear Programming

Corner Point Principle

Feasible RegionOptimal value occurs at extreme points of the convex polygon.
Chapter 13: Probability

Proof: Bayes' Theorem Formula

$P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum P(E_j)P(A|E_j)}$
Q.E.D.