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

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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

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)|

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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

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

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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

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|

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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.

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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.

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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!