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ISC Class 12 Maths Syllabus 2027: Units, Marks and Paper Pattern
The ISC Class 12 Maths syllabus for 2027 (Mathematics, subject code 860) has seven units, all compulsory, examined in an 80-mark, 3-hour theory paper, plus 20 marks for two projects. Calculus carries 35 of the 80 marks, followed by Relations and Functions (10), Algebra (10), Probability (9), Three-Dimensional Geometry (6), Vector Algebra (5) and Linear Programming (5). The biggest change from 2026 is that the old Section B or Section C choice has gone: every student now studies vectors, 3-D geometry and linear programming, and the commerce topics of the old Section C are no longer in Mathematics (860).
This guide sets out the official unit marks, explains what happened to Sections A, B and C, goes unit by unit through the scope and limits, and lists every change from the 2026 syllabus. It covers project work and the 2027 specimen paper format, and ends with a "what to master" table and three worked examples. It is part of our series for families looking for ISC home tutors in Gurgaon for Class 11 and 12.
ISC Class 12 Maths 2027 at a glance
Mathematics is marked out of 100 in ISC Class 12: 80 marks for the theory paper and 20 marks for project work.
| Item | ISC Class 12 Mathematics, examination year 2027 |
|---|---|
| Subject code | Mathematics (860). It may not be taken with Applied Mathematics (885). |
| Intended for | Candidates who wish to pursue Mathematics, Physics, Chemistry, Engineering, Architecture and related fields (CISCE's note in the syllabus) |
| Paper I: Theory | 80 marks, 3 hours, plus 15 minutes of reading time (2027 specimen paper) |
| Paper II: Project Work | 20 marks: two projects of 10 marks each |
| Units | 7, all compulsory: Relations and Functions; Algebra; Calculus; Vector Algebra; Three-Dimensional Geometry; Linear Programming; Probability |
| Section choice | None. The 2026 option of Section B or Section C has been removed. |
| Paper format (2027 specimen paper) | 20 compulsory questions in Sections A to D (20 + 14 + 21 + 25 marks), with internal choice in three questions each in Sections B, C and D. Mathematical tables and graph paper are provided. |
Sources: the CISCE ISC 2027 Mathematics (860) syllabus, linked from CISCE's "Regulations and Syllabus ISC 2027" page; the ISC Specimen Question Paper 2027 for Mathematics, linked from CISCE's specimen papers page for ISC Class XII 2027; and CISCE's revised Class XII Mathematics syllabus for the 2026 examination. Checked on 29 September 2026.
A caution for parents: several documents called "ISC Mathematics" circulate online, including an older version of the syllabus with Sections A, B and C and a 100-mark paper. The 2027 document says "ISC Examination Year 2027" at the top of its syllabus pages. If the copy your child is using does not, it is the wrong one.
Unit-wise marks for the theory paper
CISCE gives marks per unit. Calculus is split into five parts in the syllabus, but its 35 marks are not divided between them.
| Unit | Unit name | Marks (out of 80) | Share of theory paper (rounded) |
|---|---|---|---|
| 1 | Relations and Functions (including Inverse Trigonometric Functions) | 10 | 13% |
| 2 | Algebra (Matrices and Determinants) | 10 | 13% |
| 3 | Calculus (continuity and differentiation; applications of derivatives; integrals; application of integrals; differential equations) | 35 | 44% |
| 4 | Vector Algebra | 5 | 6% |
| 5 | Three-Dimensional Geometry | 6 | 8% |
| 6 | Linear Programming | 5 | 6% |
| 7 | Probability | 9 | 11% |
| Total | 80 |
The message is clear: Calculus is 35 of 80 marks, almost 44% of the paper, and that one unit includes differential equations and area problems as well as differentiation and integration. A student who is secure in differentiation and integration by October is in a strong position. The other six units together carry 45 marks and reward steady, topic-by-topic practice.
Sections A, B and C: what happened in 2027
Many parents and older guidebooks still describe ISC Class 12 Maths in terms of Sections A, B and C. That structure applied up to the 2026 examination. It does not apply to the 2027 syllabus.
The old structure (2026 examination)
In the revised syllabus for 2026, Section A (65 marks) was compulsory: Relations and Functions 10, Algebra 10, Calculus 32 and Probability 13. Candidates then attempted either Section B or Section C, each worth 15 marks:
- Section B: Vectors 5, Three-Dimensional Geometry 6, Application of Integrals 4.
- Section C: Application of Calculus (in commerce and economics) 5, Linear Regression 6, Linear Programming 4.
The 2026 specimen paper followed this: 14 compulsory questions in Section A, then four questions in Section B or four in Section C.
The new structure (2027 examination)
The 2027 syllabus has a single list of seven units with no sections and no choice. Vectors, 3-D geometry and linear programming are now compulsory for every Mathematics (860) candidate. Application of Integrals has moved inside Calculus. Application of Calculus in commerce and economics, and Linear Regression, are no longer in the Mathematics (860) syllabus; commerce applications of calculus appear in the separate Applied Mathematics (885) syllabus instead.
The 2027 specimen paper still uses the letters A, B, C and D, but they now mean something different: they group questions by length (1, 2, 3 and 5 marks), not by topic, and every section is compulsory. A student who says "I will skip Section B" is working from the old pattern.
From our tutors: the students most affected by this change are those whose Class 11 teacher, or an older sibling's notes, prepared them for Section C. They may have skipped 3-D geometry and vectors thinking they would take the commerce option. In our first sessions with an ISC Class 12 student, we check this directly. Vectors and 3-D geometry together are 11 compulsory marks now, and they build on Class 11 coordinate geometry, so any gap needs to be closed early in the year rather than in January.
Unit-by-unit syllabus and scope
The ISC syllabus lists both the broad topics and a detailed "scope" list under each. Here is a summary of the 2027 Class XII content.
Unit 1: Relations and Functions (10 marks)
- Relations: relation on a set; identity, empty and universal relations; reflexive, symmetric, transitive and equivalence relations.
- Functions: one-one, many-one, into and onto; real-valued functions; domain and range; conditions for invertibility; composite and invertible functions (algebraic functions only); sketching the graph of a function and its inverse.
- Inverse trigonometric functions: definition, domain, range and principal value branch; graphs; elementary properties such as sin−1x + cos−1x = π/2; the sum and difference formulae for sin−1, cos−1 and tan−1; and formulae for 2sin−1x, 2cos−1x, 2tan−1x and 3tan−1x, with applications.
Unit 2: Algebra, Matrices and Determinants (10 marks)
- Matrices: types (order up to 3), including diagonal, scalar, identity and triangular matrices; symmetric and skew-symmetric matrices and their properties; addition, scalar multiplication and multiplication, including non-commutativity and non-zero matrices whose product is zero; singular and non-singular matrices; uniqueness of the inverse.
- Determinants: up to 3 × 3; minors, cofactors and expansion; properties of determinants and problems based on them; area of a triangle and collinearity.
- Adjoint and inverse: properties of the adjoint and of the inverse; A−1 = adj A / |A| for 2 × 2 and 3 × 3 matrices.
- Linear equations: solving systems in two or three variables using the inverse matrix (Martin's rule, X = A−1B); the conditions for consistency in two and three variables are to be covered.
Unit 3: Calculus (35 marks)
- Continuity and differentiability: continuity at a point and on an interval; removable discontinuity and its types; continuity and differentiability of functions such as |x| and [x]; derivatives of trigonometric, exponential, logarithmic and inverse trigonometric functions; differentiation by substitution; implicit and parametric functions; differentiating one function with respect to another; logarithmic differentiation (including y = xx-type functions); second-order derivatives.
- Applications of derivatives: rate of change; increasing and decreasing functions; equations of the tangent and normal; the angle between two curves; critical, stationary and extreme points; local and absolute maxima and minima by the first- and second-derivative tests; applied maxima and minima problems. Our guide to application of derivatives for ISC Class 12 covers this part in depth.
- Integrals: integration as the inverse of differentiation; standard integrals; substitution; integration by parts; partial fractions (including the case where the numerator's degree is not less than the denominator's); the standard forms with x2 ± a2, a2 − x2 and ax2 + bx + c; trigonometric forms such as 1/(a + b cos x); the Fundamental Theorem of Calculus (without proof); and the listed properties of definite integrals, including the even/odd and f(a − x) properties.
- Application of integrals: area bounded by simple curves and the coordinate axes, and area between two curves. The curves listed are lines, circles, parabolas, ellipses, polynomial functions, the modulus function, and now exponential and logarithmic functions.
- Differential equations: order and degree; general and particular solutions; formation by eliminating arbitrary constants; variable separable (including equations reducible to it), homogeneous equations, and linear equations dy/dx + Py = Q (and dx/dy + Px = Q) where P and Q are functions of x (or y) or constants. Second-order differential equations are excluded.
Unit 4: Vector Algebra (5 marks)
- Types of vectors, position vectors, components, direction cosines and ratios, the section formula; vectors in two and three dimensions with i, j and k.
- The scalar (dot) product and its geometric meaning; scalar and vector projection; the vector (cross) product, its geometric meaning and properties, including the areas of a triangle and a parallelogram and collinearity. Proofs of geometrical theorems using vectors are excluded.
Unit 5: Three-Dimensional Geometry (6 marks)
- Lines: direction cosines and ratios; Cartesian and vector equations of a line through one or two points; the angle between two lines; conditions for perpendicular and parallel lines; coplanar and skew lines; conditions for two lines to intersect; distance of a point from a line; shortest distance between two lines.
- Planes: Cartesian and vector equations; normal, one-point, normal-form and intercept forms; distance of a point from a plane; intersection of a line and a plane; the angle between two planes and between a line and a plane.
Unit 6: Linear Programming (5 marks)
- Terminology (constraints, objective function, optimisation), advantages, limitations and application areas; formulation of LP problems; the graphical method for problems in two variables with up to three non-trivial constraints; feasible (bounded or unbounded) and infeasible regions; optimal feasible solutions, which may or may not exist.
Unit 7: Probability (9 marks)
- Independent and dependent events; the addition and multiplication theorems; conditional probability; the theorem of total probability; Bayes' theorem; random variables, probability distributions and the mean of a random variable.
Limits and exclusions to know
These limits come directly from the 2027 syllabus. They tell a student where not to spend time, and where the paper can and cannot go.
| Topic | Limit stated in the 2027 syllabus |
|---|---|
| Composite and invertible functions | Algebraic functions only |
| Matrices | Order up to 3 (m, n ≤ 3); determinants up to 3 × 3 |
| Solving equations with matrices | Systems in two or three variables with a unique solution, by the inverse matrix; consistency conditions are covered |
| Successive differentiation | Up to the second order |
| Fundamental Theorem of Calculus | Without proof |
| Differential equations | First order and first degree; second-order equations excluded |
| Area under curves | Lines, circles, parabolas, ellipses, polynomial, modulus, exponential and logarithmic functions |
| Vectors | Proofs of geometrical theorems by vectors excluded |
| Linear programming | Two variables, graphical method, up to three non-trivial constraints |
Some topics that older ISC books include are no longer listed for Class 12 Mathematics (860) in 2027, such as Rolle's and Lagrange's mean value theorems, L'Hospital's rule, the scalar triple product, linear regression, and cost, revenue and profit functions. If your child's book has these chapters, check them against the 2027 syllabus before spending time on them.
What changed from the 2026 syllabus
We compared the 2027 Class XII Mathematics syllabus word by word with CISCE's revised Class XII syllabus for the 2026 examination. Unlike Physics, where Class 12 is unchanged, Maths has real changes.
| Area | 2026 examination | 2027 examination |
|---|---|---|
| Structure | Section A (65) compulsory, plus Section B or Section C (15) | Seven compulsory units; no sections or choice |
| Calculus | 32 marks (Section A) | 35 marks; now includes Application of Integrals |
| Probability | 13 marks | 9 marks |
| Vectors; 3-D Geometry | 5; 6 (Section B option) | 5; 6 (compulsory) |
| Application of Integrals | 4 marks (Section B option) | Part of Calculus; exponential and logarithmic curves added |
| Linear Programming | 4 marks (Section C option) | 5 marks (compulsory); bounded and unbounded regions and "may or may not exist" added |
| Application of Calculus (commerce and economics) | 5 marks (Section C option) | Removed from Mathematics (860) |
| Linear Regression | 6 marks (Section C option) | Removed from Mathematics (860) |
| Relations and Functions | As listed | Added: composite functions, sketching a function and its inverse, graphs of inverse trigonometric functions |
| Matrices and Determinants | As listed | Added: scalar and triangular matrices, properties of symmetric and skew-symmetric matrices, properties of the adjoint and of the inverse |
| Differentiation and its applications | As listed | Added: types of removable discontinuity, angle between two curves, critical and extreme points |
| Vectors | As listed | Added: scalar and vector projection; geometric meaning of the cross product |
| Linear differential equations | P and Q functions of x only | P and Q functions of x or constants |
| Project work | Two projects: one from Section A, one from Section B or C | Two projects from a single list of 35 suggestions |
| Paper format (specimen paper) | 22 questions; Section A plus B or C | 20 compulsory questions in Sections A to D by mark value |
Two practical consequences. First, the paper now has more calculus and less probability than in 2026, so time spent on integration and applications of derivatives pays off even more. Second, past ISC papers from 2026 and earlier are still useful for practice, but students should skip the Section C questions and should not use those papers to judge the balance of topics.
Question paper format in the 2027 specimen paper
CISCE's 2027 specimen paper for Mathematics shows the format of the 80-mark paper. All twenty questions are compulsory; internal choice is given in three questions each in Sections B, C and D.
| Section | Questions | Type | Marks |
|---|---|---|---|
| A | Question 1, with 20 sub-parts | Very short answer, 1 mark each (sub-parts (i) to (xvii) are multiple choice) | 20 |
| B | Questions 2–8 | Short answer, 2 marks each | 14 |
| C | Questions 9–15 | Moderately long answer, 3 marks each | 21 |
| D | Questions 16–20 | Long answer, 5 marks each | 25 |
| 20 questions | 80 |
Each question is tagged with the thinking it tests, such as Recall, Understand, Apply, Analyse or Evaluate, and several use real-life contexts: a roller-coaster profile for maxima and minima, a draining conical tank for rates of change, and a spam-filter problem for Bayes' theorem. CISCE's note on the paper says the board paper "may not necessarily replicate" the specimen paper, but the weightage given in the syllabus will be strictly followed. Our guide to the ISC specimen papers 2027 explains how to use it well.
Project work: the 20 marks
Each candidate completes two projects of 10 marks each. The project work is assessed by the subject teacher and a Visiting Examiner appointed locally and approved by the Council. CISCE does not set a question paper for project work.
| Component (per project) | Marks | Who awards it |
|---|---|---|
| Overall format | 1 | Teacher and Visiting Examiner each award these 7 marks separately; the average is taken |
| Content | 4 | |
| Findings | 2 | |
| Viva-voce on the project | 3 | Visiting Examiner only |
| Total per project | 10 | Two projects = 20 marks |
The syllabus gives 35 suggested assignments. They include graphing a one-one but not onto function, exploring inverse trigonometric graphs as mirror images about y = x, checking the consistency of linear systems graphically, illustrating local and absolute maxima with graphs, explaining Bayes' theorem with examples, solving a diet or transportation problem by linear programming, applying differential equations to population growth, and finding areas bounded by pairs of curves. Because 3 of the 10 marks per project come from the viva, the student must understand every line of the project, not just present it neatly.
Want a clear plan for ISC Class 12 Maths this year? Book a free demo class. Our tutor will check your child's calculus, vectors and 3-D geometry and map the 2027 syllabus to the school's test calendar.
Book a Free Maths Demo +91 92204 75088Unit by unit: what to master
This table turns the syllabus into a checklist. "Must master" is what the syllabus and specimen paper clearly test; "common trap" is where we most often see students lose marks.
| Unit (marks) | Must master | Common trap |
|---|---|---|
| Relations and Functions (10) | Testing reflexive, symmetric and transitive properties; one-one and onto proofs; finding an inverse and a composite; principal values; inverse trigonometric identities | Giving a principal value outside the principal branch (for example, cos−1 answers must lie in [0, π]) |
| Algebra (10) | Properties of determinants; adjoint and inverse of a 3 × 3 matrix; solving three equations by X = A−1B; consistency; area and collinearity | Sign errors in cofactors; writing X = BA−1 instead of A−1B |
| Calculus: differentiation (part of 35) | Continuity and differentiability at a point (left and right limits); chain rule; implicit, parametric and logarithmic differentiation; second derivatives | Claiming differentiability from continuity alone; forgetting to differentiate the inner function |
| Calculus: applications of derivatives | Rates of change; increasing and decreasing intervals; tangents, normals and the angle between curves; maxima and minima with a clear test | Not justifying a maximum or minimum with the first- or second-derivative test |
| Calculus: integrals and areas | Substitution, parts, partial fractions and the standard forms; definite-integral properties; areas between curves including exponential and log curves | Missing the "+ C"; taking area below the x-axis as negative; wrong limits from an unsketched region |
| Calculus: differential equations | Order and degree; variable separable; homogeneous (y = vx); linear with integrating factor e∫P dx | Using the integrating factor without first writing the equation in standard form |
| Vector Algebra (5) | Dot and cross products; projections; area of triangle and parallelogram; unit vector perpendicular to two vectors | Confusing the scalar projection with the vector projection |
| 3-D Geometry (6) | Equations of lines and planes in both forms; angle between lines and planes; shortest distance between skew lines; distance of a point from a plane | Using direction ratios of a line where the normal of a plane is needed |
| Linear Programming (5) | Formulating constraints from a word problem; plotting the feasible region; corner-point method; recognising unbounded or infeasible cases | Missing a corner point or a non-negativity constraint |
| Probability (9) | Conditional probability; total probability; Bayes' theorem; probability distributions and the mean of a random variable | Mixing up P(A | B) and P(B | A) in Bayes' problems |
For all the formulas in one place, see our ISC Class 12 Maths formula sheet.
Worked examples in board style
These three questions cover calculus (the largest unit), linear programming (now compulsory) and probability. Each answer has been checked by calculation.
Example 1 (Application of integrals, logarithmic curve, 3 marks). Find the area of the region bounded by the curve y = loge x, the x-axis and the line x = e.
Answer: 1 square unit. The curve meets the x-axis at x = 1, and loge x ≥ 0 for 1 ≤ x ≤ e, so the region lies above the x-axis between x = 1 and x = e. Area = ∫1e log x dx. Integrate by parts with u = log x and dv = dx: ∫ log x dx = x log x − ∫ x · (1/x) dx = x log x − x + C. So the area = [x log x − x]1e = (e · 1 − e) − (1 · 0 − 1) = 0 + 1 = 1 square unit. Logarithmic and exponential curves are new to the area list in 2027, so expect this type of question. Always sketch the region first and state the limits.
Example 2 (Linear programming, 5 marks). Maximise Z = 3x + 5y subject to x + y ≤ 8, x + 3y ≤ 12, x ≥ 0, y ≥ 0.
Answer: maximum Z = 28 at x = 6, y = 2. Draw the lines x + y = 8 (through (8, 0) and (0, 8)) and x + 3y = 12 (through (12, 0) and (0, 4)). The feasible region is the bounded region on the origin side of both lines in the first quadrant. Its corner points are O(0, 0), A(8, 0), B (where the two lines meet) and C(0, 4). Solving x + y = 8 and x + 3y = 12: subtracting gives 2y = 4, so y = 2 and x = 6; B = (6, 2). Evaluate Z at each corner: O gives 0, A gives 24, B gives 3(6) + 5(2) = 28, C gives 20. The maximum is 28 at (6, 2). In the board answer, show the graph, the shaded feasible region and a corner-point table; each usually carries marks.
Example 3 (Bayes' theorem, 5 marks). A factory has three machines, A, B and C, producing 50%, 30% and 20% of its output. Of their output, 2%, 3% and 4% respectively is defective. An item chosen at random is found to be defective. Find the probability that it was made by machine A.
Answer: 10/27, about 0.370. Let E1, E2, E3 be the events that the item is made by A, B, C, and D the event that it is defective. P(E1) = 0.5, P(E2) = 0.3, P(E3) = 0.2; P(D | E1) = 0.02, P(D | E2) = 0.03, P(D | E3) = 0.04. By the theorem of total probability, P(D) = 0.5(0.02) + 0.3(0.03) + 0.2(0.04) = 0.010 + 0.009 + 0.008 = 0.027. By Bayes' theorem, P(E1 | D) = 0.010/0.027 = 10/27 ≈ 0.370. Defining the events in words at the start is part of the expected method, and it prevents the common mistake of reversing the conditional probabilities.
Using the syllabus through the year
The syllabus is a planning tool, not just a list. This is the approach our Home Tutors Team uses with ISC Class 12 Maths students; adjust it to your school's order and test dates.
| Period | Focus |
|---|---|
| April–July | Relations and Functions, Inverse Trigonometric Functions, Matrices and Determinants, then Continuity and Differentiation. Choose both project topics. |
| August–October | Applications of Derivatives, Integrals and Application of Integrals, Differential Equations. Timed practice of 3- and 5-mark calculus questions. Complete the first project. |
| November | Vectors, 3-D Geometry, Linear Programming and Probability. Complete the second project and prepare for the viva. Attempt the full 2027 specimen paper. |
| December–January | Pre-boards (in many schools). Full papers under time; an error log by unit; revise limits and standard integrals. |
| February onwards | Board examination. Past papers (skipping old Section C questions) and the specimen paper, checked against its answer key. |
For textbook and practice-book choices, see our guide to the best books for ISC Class 12 Maths, and for a full revision calendar, our ISC board exam 2027 preparation plan.
From our tutors: the pattern we see most in ISC Class 12 Maths is strong differentiation and weak integration. Students enjoy the rules of differentiation, but integration needs pattern recognition built from volume of practice, and the school term moves on before that happens. We give every student a short daily set of mixed integrals from August, about ten minutes a day, rather than one long session a week. With Calculus at 35 marks, including areas and differential equations that depend on integration, this small habit protects a large share of the paper.
Frequently asked questions
Is there a Section B or Section C choice in ISC Class 12 Maths 2027?
No. The 2027 syllabus has seven compulsory units and no section choice. The 2027 specimen paper uses Sections A to D only to group questions by mark value (1, 2, 3 and 5 marks), and all of them are compulsory.
Which unit carries the most marks in ISC Class 12 Maths 2027?
Calculus, with 35 of the 80 theory marks. It covers continuity and differentiation, applications of derivatives, integrals, application of integrals and differential equations.
Has linear regression been removed from ISC Class 12 Maths?
Yes, for Mathematics (860). Linear Regression and Application of Calculus in commerce and economics were Section C options in 2026 and are not in the 2027 Mathematics syllabus. Commerce applications of calculus appear in the separate Applied Mathematics (885) syllabus.
Is linear programming compulsory in ISC Class 12 Maths 2027?
Yes. It is a compulsory 5-mark unit in 2027, taught by the graphical method for two variables with up to three non-trivial constraints.
Can a student take both Mathematics and Applied Mathematics in ISC?
No. The 2027 Mathematics (860) syllabus states that it may not be taken with Applied Mathematics.
How is ISC Class 12 Maths project work marked?
Two projects of 10 marks each. For each, the teacher and the Visiting Examiner separately award up to 7 marks (format 1, content 4, findings 2), which are averaged, and the Visiting Examiner conducts a 3-mark viva-voce.
Are the mean value theorems and L'Hospital's rule in the ISC Class 12 Maths syllabus 2027?
They are not listed in the 2027 Class XII Mathematics syllabus. Some older ISC books still include them, so check each chapter against the current syllabus.
Can we still use old ISC Maths papers for practice?
Yes, for the topics that remain, but skip the old Section C questions (commerce applications, regression) and remember that the balance of topics has changed: more calculus, less probability, and compulsory vectors, 3-D geometry and linear programming.
Looking for an ISC Maths home tutor in Gurgaon or Gurugram? Ajay Vatsyayan Classes has 12+ years of home tuition in Gurgaon and a network of 500+ verified tutors, male and female, teaching at home or online. See our Class 12 tuition in Gurgaon page or our JEE Maths tutors in Gurgaon, or book a free demo class. Saraswati kunj II, Wazirabad, Sector 52, Gurugram, Haryana 122003.
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