All-Chapters AI Mathematics Master Handbook

Curated by: Ajay Yadav (Math King of Katargam | 15+ Yrs Experience | 2nd Rank TAT High School Exam)
Covering: Class 9, Class 10, Class 11, Class 12 (CBSE & GSEB) & JEE Main/Advanced
Scope: Complete Chapter Summaries, Formula Cheatsheets & Theorem Proofs

📘 PART 1: CLASS 12 & JEE ADVANCED

1. Integrals & Calculus

Definite Integral Property (King's Property): \[ \int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx \] Integration by Parts: \[ \int u \cdot v \, dx = u \int v dx - \int \left( u' \int v dx ight) dx \]

2. Differential Equations

Order is highest derivative present; Degree is exponent of highest derivative.

Linear DE: \[ rac{dy}{dx} + P(x)y = Q(x) \implies ext{IF} = e^{\int P dx} \implies y \cdot ext{IF} = \int (Q \cdot ext{IF}) dx + C \]

3. Matrices & Determinants

Inverse: \[ A^{-1} = rac{1}{|A|} ext{adj}(A) \] Determinant Property: \[ | ext{adj}(A)| = |A|^{n-1}, \quad |k A| = k^n |A| \]

4. Vectors & 3D Geometry

Angle between Vectors: \[ \cos heta = rac{ec{a} \cdot ec{b}}{|ec{a}||ec{b}|} \] Shortest Distance (Skew Lines): \[ d = \left| rac{(ec{a}_2 - ec{a}_1) \cdot (ec{b}_1 imes ec{b}_2)}{|ec{b}_1 imes ec{b}_2|} ight| \]

5. Probability & Bayes Theorem

Conditional: \[ P(A|B) = rac{P(A \cap B)}{P(B)} \] Bayes Theorem: \[ P(E_i|A) = rac{P(E_i)P(A|E_i)}{\sum P(E_k)P(A|E_k)} \]

📗 PART 2: CLASS 11 & JEE MAIN

6. Trigonometric Functions

Sum & Difference Formulas: \[ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \] \[ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \] \[ an(A + B) = rac{ an A + an B}{1 - an A an B} \]

7. Coordinate Geometry (Conic Sections)

Conic Curve	Standard Equation	Focus / Directrix
Parabola	\(y^2 = 4ax\)	Focus: \((a,0)\), Directrix: \(x = -a\)
Ellipse	\(rac{x^2}{a^2} + rac{y^2}{b^2} = 1\)	Eccentricity \(e = \sqrt{1 - rac{b^2}{a^2}}\)
Hyperbola	\(rac{x^2}{a^2} - rac{y^2}{b^2} = 1\)	Eccentricity \(e = \sqrt{1 + rac{b^2}{a^2}}\)

8. Limits, Continuity & Derivatives

Standard Limits: \[ \lim_{x o 0} rac{\sin x}{x} = 1, \quad \lim_{x o 0} rac{e^x - 1}{x} = 1, \quad \lim_{x o 0} rac{\ln(1+x)}{x} = 1 \]

🏆 PART 3: CLASS 10 BOARD EXAM

9. Real Numbers & Polynomials

Fundamental Theorem of Arithmetic: Every composite number can be expressed as a unique product of primes.

10. Quadratic Equations & AP

Quadratic Formula: \[ x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] AP nth Term: \[ a_n = a + (n-1)d \] AP Sum: \[ S_n = rac{n}{2} [2a + (n-1)d] \]

11. Circles & Triangles

BPT (Thales Theorem): If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.

12. Surface Areas & Volumes

Shape	Curved Surface Area	Total Surface Area	Volume
Cylinder	\(2\pi r h\)	\(2\pi r(r+h)\)	\(\pi r^2 h\)
Cone	\(\pi r l\) (\(l=\sqrt{r^2+h^2}\))	\(\pi r(r+l)\)	\(rac{1}{3}\pi r^2 h\)
Sphere	\(4\pi r^2\)	\(4\pi r^2\)	\(rac{4}{3}\pi r^3\)

📙 PART 4: CLASS 9 FOUNDATIONAL

13. Number Systems & Exponents

Laws of Exponents: \[ a^m \cdot a^n = a^{m+n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = rac{1}{a^n} \]

14. Heron's Formula

Semi-perimeter: \[ s = rac{a+b+c}{2} \] Area of Triangle: \[ ext{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]

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