CBSE Maths Sample Paper Class 12 2026-27 with Solutions PDF: Answer Key & Chapter-wise Analysis
CBSE Maths Sample Paper Class 12 2026-27 with Solutions PDF: Answer Key & Chapter-wise Analysis
Answer key, chapter-wise analysis and free step-by-step solutions PDF for the CBSE Class 12 Mathematics (041) sample paper 2026-27.
Table of Contents
The CBSE Maths sample paper class 12 for 2026-27 shows exactly how your 2027 board paper will be built, and this page gives you its full solutions. Download the free PDF with step-by-step working for all 38 questions, check the answer key below, and see which chapters carry the most marks.
Maths solutions need clean notation, fractions and integrals, so the complete working is in the PDF where every step is formatted properly. On this page you get what helps you plan revision: the answer key, the chapter behind every question, and a chapter-wise marks table from the paper.
Calculus, vectors and 3D geometry together decide most of the paper. If you are also preparing for JEE, these are the same chapters that carry the most weight there.
Download Cbse maths sample paper class 12 Solutions PDF
CBSE Class 12 Mathematics Sample Paper 2026-27: Complete Solutions
- All 38 questions solved, including internal choices
- Step-wise marking scheme for every answer
- Every derivation, diagram and calculation laid out clearly
- Free, no login, works on mobile
Maths Sample Paper Class 12 2026-27 Pattern
CBSE has stated that there is no change in the question paper design for the 2026-27 session, so this is the structure of your board paper.
| Section | Questions | Question Types | Marks |
|---|---|---|---|
| A | 1 to 20 | 18 MCQ + 2 Assertion-Reason (1 mark each) | 20 |
| B | 21 to 25 | Very short answer (2 marks each) | 10 |
| C | 26 to 31 | Short answer (3 marks each) | 18 |
| D | 32 to 35 | Long answer (5 marks each) | 20 |
| E | 36 to 38 | Case study (4 marks each) | 12 |
| Total | 38 | Duration: 3 hours | 80 |
Official source: CBSE Academic, Sample Question Papers Class XII 2026-27.
Maths Sample Paper Class 12 Answer Key: Section A (Q1 to Q20)
Use this to check your objective answers in seconds. The reasoning for each option is in the solutions PDF.
| Q1 | Q2 | Q3 | Q4 | Q5 | Q6 | Q7 | Q8 | Q9 | Q10 |
|---|---|---|---|---|---|---|---|---|---|
| D | D | A | B | C | D | A | D | D | A |
| Q11 | Q12 | Q13 | Q14 | Q15 | Q16 | Q17 | Q18 | Q19 | Q20 |
|---|---|---|---|---|---|---|---|---|---|
| B | D | C | B | B | D | D | A | B | A |
Chapter-wise Marks in the Maths Sample Paper Class 12 2026-27
We mapped every question to its NCERT chapter and added up the marks. This is based on the main option of each question; internal choices may come from a different chapter.
| Chapter | Questions | Marks |
|---|---|---|
| Matrices | Q4, Q6, Q26, Q38 | 9 |
| Application of Derivatives | Q9, Q10, Q27, Q37 | 9 |
| Integrals | Q12, Q19, Q21, Q33 | 9 |
| Three Dimensional Geometry | Q23, Q30, Q36 | 9 |
| Probability | Q16, Q17, Q18, Q35 | 8 |
| Continuity and Differentiability | Q7, Q8, Q22, Q28 | 7 |
| Differential Equations | Q11, Q20, Q29 | 5 |
| Vector Algebra | Q13, Q14, Q15, Q24 | 5 |
| Linear Programming | Q25, Q31 | 5 |
| Relations and Functions | Q32 | 5 |
| Application of Integrals | Q34 | 5 |
| Inverse Trigonometric Functions | Q1, Q2, Q3 | 3 |
| Determinants | Q5 | 1 |
| Total | 38 questions | 80 |
Maths Sample Paper Solutions: Section A: MCQ and Assertion-Reason (Q1 to Q20)
| Q | Chapter | What the question asks | Marks | Answer |
|---|---|---|---|---|
| Q1 | Inverse Trigonometric Functions | Identify the function from its graph (y = π + cos⁻¹x) | 1 | (D) |
| Q2 | Inverse Trigonometric Functions | Domain of sec⁻¹(2x) | 1 | (D) |
| Q3 | Inverse Trigonometric Functions | Value of cot⁻¹(2cos(½ sin⁻¹(√3/2))) | 1 | (A) |
| Q4 | Matrices | Symmetric matrix: find √(6a + 2b + 3c) | 1 | (B) |
| Q5 | Determinants | Value of |adj(PQ)| given P and |Q| | 1 | (C) |
| Q6 | Matrices | Matrix product equal to the zero matrix: find x | 1 | (D) |
| Q7 | Continuity and Differentiability | Derivative of cos⁻¹(2x² − 1) with respect to cos⁻¹x | 1 | (A) |
| Q8 | Continuity and Differentiability | Differentiability of a piecewise function at x = 1 | 1 | (D) |
| Q9 | Application of Derivatives | Interval in which a given function is decreasing | 1 | (D) |
| Q10 | Application of Derivatives | Values of x where the derivative of x⁴ exceeds the function | 1 | (A) |
| Q11 | Differential Equations | Degree and order of a differential equation | 1 | (B) |
| Q12 | Integrals | Find P in ∫ dx/(1 + e²ˣ) = x − P log(1 + e²ˣ) + C | 1 | (D) |
| Q13 | Vector Algebra | Value of u when 2î − uĵ + 3k̂ has magnitude 4 | 1 | (C) |
| Q14 | Vector Algebra | Angle of î + ĵ + √2k̂ with the positive z-axis | 1 | (B) |
| Q15 | Vector Algebra | A vector perpendicular to î + ĵ and î − ĵ | 1 | (B) |
| Q16 | Probability | Probability of rain on independent days | 1 | (D) |
| Q17 | Probability | Conditional probability: at least one 4 when the sum is a multiple of 5 | 1 | (D) |
| Q18 | Probability | Independent events with exactly one occurring: find k | 1 | (A) |
| Q19 | Integrals | Assertion-Reason: ∫ from −1 to 1 of |x − 2| dx | 1 | (B) |
| Q20 | Differential Equations | Assertion-Reason: general solution as a family of curves | 1 | (A) |
Full step-by-step solutions with marking scheme for this section are in the Maths solutions PDF.
Maths Sample Paper Solutions: Section B: Very Short Answers (Q21 to Q25)
| Q | Chapter | What the question asks | Marks | Answer |
|---|---|---|---|---|
| Q21 | Integrals | Evaluate ∫ dx / (x√(x⁶ − 1)) (with internal choice) | 2 | See PDF |
| Q22 | Continuity and Differentiability | If y = sin(x + y) and dy/dx = 0, prove y = ±1 | 2 | See PDF |
| Q23 | Three Dimensional Geometry | Parallel lines through given points: find p and q (with choice) | 2 | See PDF |
| Q24 | Vector Algebra | Find |b| from a·b and b × a | 2 | See PDF |
| Q25 | Linear Programming | Write the constraints from a feasible region graph | 2 | See PDF |
Full step-by-step solutions with marking scheme for this section are in the Maths solutions PDF.
Maths Sample Paper Solutions: Section C: Short Answers (Q26 to Q31)
| Q | Chapter | What the question asks | Marks | Answer |
|---|---|---|---|---|
| Q26 | Matrices | Find PQ or QP and the inverse of the product | 3 | See PDF |
| Q27 | Application of Derivatives | Rate of change: area of vision of a surveillance drone | 3 | See PDF |
| Q28 | Continuity and Differentiability | Second derivative of parametric x = sin³t, y = cos³t (with choice) | 3 | See PDF |
| Q29 | Differential Equations | Particular solution of x dy + (y − x²eˣ) dx = 0 (with choice) | 3 | See PDF |
| Q30 | Three Dimensional Geometry | Foot of perpendicular from a point to a line (with choice) | 3 | See PDF |
| Q31 | Linear Programming | Maximise Z = 40x + 30y graphically | 3 | See PDF |
Full step-by-step solutions with marking scheme for this section are in the Maths solutions PDF.
Maths Sample Paper Solutions: Section D: Long Answers (Q32 to Q35)
| Q | Chapter | What the question asks | Marks | Answer |
|---|---|---|---|---|
| Q32 | Relations and Functions | Equivalence relation on N × N (with choice) | 5 | See PDF |
| Q33 | Integrals | Evaluate ∫ from −a to a of √((a − x)/(a + x)) dx | 5 | See PDF |
| Q34 | Application of Integrals | Area of a region using integration (with choice) | 5 | See PDF |
| Q35 | Probability | Bayes' theorem: fitness app and smartwatch users | 5 | See PDF |
Full step-by-step solutions with marking scheme for this section are in the Maths solutions PDF.
Maths Sample Paper Solutions: Section E: Case Studies (Q36 to Q38)
| Q | Chapter | What the question asks | Marks | Answer |
|---|---|---|---|---|
| Q36 | Three Dimensional Geometry | Case study: security laser beams, parallel lines and shortest distance | 4 | See PDF |
| Q37 | Application of Derivatives | Case study: Yamuna Action Plan water quality index (maxima and minima) | 4 | See PDF |
| Q38 | Matrices | Case study: fruit baskets for orphanages using matrix inverse | 4 | See PDF |
Full step-by-step solutions with marking scheme for this section are in the Maths solutions PDF.
Chapter-wise Revision Checklist From the Maths Sample Paper
Based on what this paper asks, here is what to revise in each high-weight chapter before the board exam.
Matrices and Determinants
Symmetric matrices, products, inverse and adjoint, and solving real-life systems through the matrix method as in the case study. Practise inverse calculations for speed.
Calculus (Continuity, Derivatives, Integrals)
Differentiability of piecewise functions, parametric second derivatives, rate of change, definite integrals using properties and area under curves. Calculus alone decides a large share of the paper.
Three Dimensional Geometry and Vectors
Parallel lines, foot of perpendicular, shortest distance, and vector magnitude, angle and cross product. Write vector equations carefully with correct direction ratios.
Probability
Independent events, conditional probability and Bayes' theorem. Draw a tree or table before writing the formula in Bayes' questions.
Differential Equations
Order and degree, linear differential equations and particular solutions. Identify the type first; the method then follows.
Linear Programming
Writing constraints from a graph and maximising Z graphically. The graph, feasible region and corner-point table all carry marks.
Section-wise Strategy for the Maths Paper
How to approach each section of this paper in the exam hall, based on the questions CBSE has set.
Section A (20 marks)
Eighteen MCQs and two Assertion-Reason questions across inverse trigonometry, matrices, calculus, vectors and probability. Aim to finish in 25 to 30 minutes; do quick rough work rather than mental shortcuts that lead to sign errors.
Sections B and C (28 marks)
Integration, differentiability, 3D lines, vectors, LPP constraints, rate of change and differential equations. Show every step; the marking scheme gives separate marks for the method and the final answer.
Section D (20 marks)
Four long answers: equivalence relations, a definite integral using properties, area by integration and Bayes' theorem. These reward a clean, well-organised layout as much as the right answer.
A simple 3-hour time plan
Give Section A about 25 minutes, Sections B and C about 65 minutes, Section D around 45 minutes and Section E about 30 minutes. Keep the last 15 minutes to recheck calculations and copied values. A fixed plan stops one difficult question from eating the time meant for five easy ones.
Section E (12 marks)
Three case studies: laser beams as 3D lines, a water quality index modelled by a quadratic, and fruit baskets solved with matrices. Each has 1 + 1 + 2 marks; read the data carefully before choosing a method.
How to Use the Maths Sample Paper for Boards and JEE
- Attempt it in 3 hours first. Take the question paper from our sample paper page and solve it with a timer before opening the solutions.
- Check Section A with the answer key. Every wrong objective answer points to a concept to revise.
- Check written answers with the PDF. Compare your steps with the marking scheme and give yourself marks honestly.
- Revise by chapter weight. Use the chapter-wise table above to decide what to revise first.
- Push the same chapters for JEE. The high-weight board chapters are also high-weight in JEE Main; solve harder problems from them after boards practice.
Where Students Lose Marks in Class 12 Maths
The marking scheme of the cbse maths sample paper class 12 shows clear patterns in where marks slip away.
- Write every step. Marks are given step by step in 3 and 5-mark questions.
- Draw the graph in LPP and area questions. The graph and the shaded region carry marks.
- State the theorem or formula used. It earns method marks even if you slip later.
- Check the domain in inverse trig. Principal value ranges are a common trap.
- Practise case studies with a timer. Each has 1 + 1 + 2 marks and needs quick reading.
Preparing for JEE Main Alongside Boards?
Board preparation and JEE preparation use the same NCERT chapters. Once the board basics are in place, these two Competishun programmes help you take the same chapters to JEE level, depending on how much of the syllabus you have finished.
C³
Covers the complete JEE Main syllabus in a short, structured plan for students who still have chapters left.
- Topic-wise tests after every unit
- Senior faculty with 18 to 25 years' experience
- Built to run alongside board preparation
CHAMP
Sharp revision and exam-level practice for students who have finished the syllabus.
- High-yield revision of every chapter
- Full-length JEE Main mocks
- Detailed performance analysis
CBSE Class 12 Sample Paper Solutions: Other Subjects
Every subject in the same format. Or see all of them on the CBSE Class 12 sample paper solutions page.
