Mathematics

---
name: mathematics
description: >-
  CBSE Grade 12 Mathematics — Full Syllabus Mastery (2026-27). 
  Covers the 35-mark Calculus giant, Vectors, 3D Geometry, and Probability. 
  Target: 80/80 Theory + 20/20 Internal.
---

# Mathematics — CBSE Grade 12 (PCMB/PCMC)

## Syllabus Blueprint (2026–27)

| Unit | Chapters | Marks |
|------|----------|-------|
| I | Relations and Functions | 08 |
| II | Algebra (Matrices & Determinants) | 10 |
| III | Calculus (Ch 5-9) | 35 |
| IV | Vectors and 3D Geometry | 14 |
| V | Linear Programming | 05 |
| VI | Probability | 08 |
| **Total** | **13 Chapters** | **80** |

---

## Unit I: Relations & Algebra (18 Marks)

### Key Concepts
- **Relations:** Reflexive, Symmetric, Transitive, Equivalence relations.
- **Functions:** One-one (injective), Onto (surjective), Bijective functions.
- **Matrices:** Types, Transpose, Symmetric/Skew-symmetric, Inversion.
- **Determinants:** Area of triangle, Adjoint, System of equations (Cramer’s/Matrix method).

### Key Formulas
```latex
Matrix Inverse: A^{-1} = \frac{1}{|A|} adj(A)
System of Eq: AX = B \implies X = A^{-1}B
Area of \Delta: \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|
```

---

## Unit III: Calculus (35 Marks) - The Core

### 1. Continuity & Differentiability (Ch 5)
- **Continuity:** $LHL = RHL = f(a)$.
- **Derivatives:** Chain rule, Parametric differentiation, Logarithmic differentiation.
- **Higher Order:** $d^2y/dx^2$.

### 2. Applications of Derivatives (Ch 6)
- **Rate of Change:** $dy/dt$.
- **Increasing/Decreasing:** $f'(x) > 0$ or $f'(x) < 0$.
- **Maxima & Minima:** First and Second derivative tests. Absolute vs Local.

### 3. Integrals (Ch 7)
- **Indefinite:** Substitution, Partial Fractions, By Parts (UV Rule).
- **Definite:** Fundamental Theorem, Properties (P0 to P7).

### 4. Applications of Integrals & Differential Eq (Ch 8 & 9)
- **Area under Curves:** Simple curves (circles, parabolas, ellipses).
- **Differential Eq:** Order/Degree, Variable Separable, Homogeneous, Linear ($dy/dx + Py = Q$).

### Key Formulas
```latex
Integration by Parts: \int u v dx = u \int v dx - \int [u' \int v dx] dx
Property P4 (King): \int_0^a f(x) dx = \int_0^a f(a-x) dx
Linear Diff Eq: I.F. = e^{\int P dx} \implies y(I.F.) = \int Q(I.F.) dx + C
```

---

## Unit IV: Vectors & 3D Geometry (14 Marks)

### Key Concepts
- **Vectors:** Dot Product ($A·B = |A||B| \cos \theta$), Cross Product ($A \times B = |A||B| \sin \theta \hat{n}$).
- **3D Geometry:** Direction Cosines/Ratios, Equation of a line in space.
- **Shortest Distance:** Between skew lines and parallel lines.

### Key Formulas
```latex
Vector Line Eq: \vec{r} = \vec{a} + \lambda \vec{b}
Shortest Distance: d = \left| \frac{(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})}{|\vec{b_1} \times \vec{b_2}|} \right|
Angle between lines: \cos \theta = \left| \frac{b_1 \cdot b_2}{|b_1||b_2|} \right|
```

---

## Unit V & VI: LPP & Probability (13 Marks)

### Key Concepts
- **LPP:** Feasible region, Corner point method, Objective function maximization/minimization.
- **Probability:** Conditional Probability, Multiplication theorem, Bayes' Theorem, Random variables.

---

## High-Yield Strategy (Target 495+)
1. **The Calculus Sprint:** Focus on **Definite Integral Properties** and **Linear Differential Equations**. 5-mark LAs often come from Ch 6 (Max/Min) or Ch 8 (Area).
2. **Matrix Method:** One 5-mark question is almost guaranteed on solving systems of linear equations.
3. **3D SD:** Practice the **Shortest Distance** formula religiously. It's a high-frequency 3-5 mark question.
4. **Bayes' Theorem:** Understand the "Reverse Probability" logic. It's a standard Section D Case Study topic.

---

## Common Board Exam Traps
- **Indefinite Integrals:** Forgetting the constant $+C$ (Deducts 0.5 marks).
- **Matrix Inverse:** Calculation error in finding the determinant or cofactor matrix.
- **LPP Graph:** Not shading the feasible region correctly or missing the non-negativity ($x, y \ge 0$) constraint.
- **Probability:** Confusing $P(A|B)$ with $P(B|A)$.

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## Answer Writing Framework
1. **State the Property/Formula:** Write the property name (e.g., "Using property P4...").
2. **Steps:** Show the substitution. In Integration, show the transformation clearly.
3. **Final Result:** Must be clearly labeled (e.g., "$Area = [X] \text{ sq. units}$").

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## Internal Assessment (20 Marks)
- **Periodic Tests:** 10 Marks.
- **Mathematics Activity (Lab):** 10 Marks. Keep your record book updated with 10 activities!