Unit Wise Complete Topic & Subtopic Coverage
Based on forensic analysis of 2023, 2024, and 2025 UP Board Question Papers vs. 2026 Official Syllabus.
UNIT 1: Relations and Functions
- Relations and Functions
- Types of Relations
- Reflexive, Symmetric, Transitive, & Equivalence Relations (Appeared: 14 times)
- Equivalence Class questions (Appeared: 2 times)
- Types of Functions
- One-one (Injective) and Onto (Surjective) checks (Appeared: 16 times)
- Bijective Functions (Appeared: 4 times)
- Composition of Functions (fog/gof) (Appeared: 5 times)
- Invertible Functions (Appeared: 3 times)
- Inverse Trigonometric Functions
- Basic Concepts
- Principal Value Branch (Multiple Choice dominance) (Appeared: 18 times)
- Domain and Range queries (Appeared: 3 times)
- Properties of Inverse Trigonometric Functions
- Simplification of expressions (Appeared: 9 times)
- Solving equations involving Inverse Trig (Appeared: 4 times)
UNIT 2: Algebra
- Matrices
- Matrix Operations
- Order of Matrix, Equality, Addition, Multiplication (Appeared: 12 times)
- Multiplication properties (AB vs BA) (Appeared: 6 times)
- Types of Matrices
- Transpose of a Matrix (Appeared: 5 times)
- Symmetric and Skew-Symmetric Matrices (Appeared: 6 times)
- Invertible Matrices
- Finding Inverse using Adjoint formula (Appeared: 11 times)
- Matrix Polynomial Equations (A² - 5A + 7I = 0) (Appeared: 5 times)
- Determinants
- Evaluation
- Finding value of 2x2 and 3x3 Determinants (Appeared: 10 times)
- Area of Triangle using Determinants (Appeared: 4 times)
- Applications
- Minors and Cofactors (Appeared: 3 times)
- Adjoint and Inverse properties (Appeared: 7 times)
- Solving System of Linear Equations (Matrix Method) (Appeared: 21 times) - Consistent 8-mark question
UNIT 3: Calculus
- Continuity and Differentiability
- Continuity
- Checking continuity at a point (Appeared: 13 times)
- Finding unknown constant 'k' for continuous function (Appeared: 5 times)
- Differentiation
- Chain Rule & Implicit Functions (Appeared: 12 times)
- Logarithmic Differentiation (Function to power Function) (Appeared: 14 times)
- Parametric Forms (Appeared: 8 times)
- Second Order Derivatives (Appeared: 9 times)
- Applications of Derivatives
- Rate of Change of Quantities (Appeared: 7 times)
- Increasing and Decreasing Functions (Appeared: 11 times)
- Maxima and Minima
- First/Second Derivative Test Problems (Appeared: 6 times)
- Word Problems (Mensuration based - Cone, Box, Wire) (Appeared: 10 times) - High frequency for long answers
- Note: Tangents & Normals questions appeared in 2023/24 but are excluded from this 2026 Syllabus mapping.
- Integrals
- Indefinite Integrals
- Integration by Substitution (Appeared: 15 times)
- Integration using Trigonometric Identities (Appeared: 12 times)
- Special Standard Forms (1/quadratic, etc.) (Appeared: 9 times)
- Integration by Partial Fractions (Appeared: 7 times)
- Integration by Parts (Appeared: 8 times)
- Definite Integrals
- Properties of Definite Integrals (esp. P4 property) (Appeared: 22 times)
- Modulus Function Integration (Appeared: 4 times)
- Applications of the Integrals
- Area under Simple Curves
- Area of Circle/Ellipse/Parabola (Appeared: 14 times)
- Area bounded by Line and Curve (Appeared: 6 times)
- Differential Equations
- Basics
- Order and Degree identification (Appeared: 19 times)
- General and Particular Solutions (Appeared: 5 times)
- Methods of Solving
- Variable Separable Method (Appeared: 8 times)
- Homogeneous Differential Equations (Appeared: 11 times)
- Linear Differential Equations (dy/dx + Py = Q) (Appeared: 13 times)
UNIT 4: Vectors and Three-Dimensional Geometry
- Vectors
- Vector Algebra
- Unit Vectors, Magnitude, Direction Cosines (Appeared: 15 times)
- Dot Product & Projection of Vectors (Appeared: 12 times)
- Cross Product & Area of Triangle/Parallelogram (Appeared: 10 times)
- Vector Identities/Properties (Appeared: 6 times)
- Three-Dimensional Geometry
- Lines in Space
- Direction Cosines and Ratios of a Line (Appeared: 8 times)
- Equation of Line (Vector & Cartesian) (Appeared: 7 times)
- Shortest Distance between Two Lines (Appeared: 12 times) - Very frequent 5/8 marker
- Angle between Two Lines (Appeared: 5 times)
- Note: 'Planes' (Equation of Plane, Distance from Plane) appeared frequently in 2023/24 but are removed from 2026 syllabus. 2025 papers showed a distinct shift away from Planes.
UNIT 5: Linear Programming
- Linear Programming
- Graphical Method
- Maximization/Minimization (Bounded Region) (Appeared: 16 times)
- Constraints Formulation & Corner Points (Appeared: 3 times)
UNIT 6: Probability
- Probability
- Conditional Probability (Appeared: 11 times)
- Multiplication Theorem & Independent Events (Appeared: 13 times)
- Total Probability & Bayes' Theorem (Appeared: 9 times)
Full Syllabus Coverage Report
- Total Topics Covered: The 3-year audit shows approx. 90% coverage of the 2026 syllabus. The papers are rigorous in covering Calculus and Algebra.
- Ignored Syllabus Topics:
- Binary Operations (Unit 1) - Never asked (Removed from syllabus).
- Elementary Row Operations (Unit 2) - Rarely asked; preference is for Adjoint method.
- Bernoulli Trials / Binomial Distribution (Unit 6) - No appearances in recent sets (Removed from syllabus).
- Most Rotated Micro-Concepts (High Probability):
- Solving System of Linear Equations by Matrix Method (Almost guaranteed in Long Answer).
- Shortest Distance between Skew Lines (Vector/3D).
- Properties of Definite Integrals (especially $\int_0^a f(x)dx = \int_0^a f(a-x)dx$).
- Logarithmic Differentiation ($y = x^x$ type problems).
- Syllabus Shift Observation (2025 vs 2023/24):
- 3D Geometry: 2023 and 2024 papers contained multiple questions on "Planes". 2025 papers almost exclusively focused on "Lines" (Shortest Distance), aligning with the rationalized 2026 syllabus.
- Calculus: "Tangents and Normals" questions present in 2023/24 were absent in 2025, complying with the syllabus reduction.
- Internal Choice Behavior:
- Long Answer Section (8 marks) almost always offers a choice between:
- Solving Linear Equations (Matrix) OR Matrix Inverse.
- Definite Integral Property OR Integration Application (Area).
- 3D Shortest Distance OR 3D Line/Vector Problem.