ICSE Class 10 Mathematics Syllabus 2026-27

ICSE CLASS 10 — MATHEMATICS

Complete Study Guide 2026-27

Board: CISCE | Exam Year: 2028 | Theory: 80 Marks | Internal Assessment: 20 Marks

Exam Structure

Note: Certain questions may require the use of Mathematical tables (Logarithmic and Trigonometric tables).

THEORY SYLLABUS — 80 Marks

Unit 1 — Commercial Mathematics

(i) Goods and Services Tax (GST)

• Computation of tax including problems involving discounts, list-price, profit, loss, basic/cost price including inverse cases.

• Candidates are also expected to find price paid by the consumer after paying State Goods and Service Tax (SGST) and Central Goods and Service Tax (CGST) — the different rates as in vogue on different types of items will be provided.

• Problems based on corresponding inverse cases are also included.

  
  

(ii) Banking

Recurring Deposit Accounts: computation of interest and maturity value using the formula:

I = Pn(n+1) / (2 × 12) × r/100

MV = P × n + I

(iii) Shares and Dividends

(a) Face/Nominal Value, Market Value, Dividend, Rate of Dividend, Premium.

(b) Formulae:

• Income = number of shares × rate of dividend × FV

• Return = (Income / Investment) × 100

Note: Brokerage and fractional shares not included.

Unit 2 — Algebra

(i) Linear Inequations

Linear Inequations in one unknown for x ∈ N, W, Z, R.

Solving:

• Algebraically and writing the solution in set notation form

• Representation of solution on the number line

  
  

(ii) Quadratic Equations in One Variable

(a) Nature of roots:

• Two distinct real roots if b² – 4ac > 0

• Two equal real roots if b² – 4ac = 0

• No real roots if b² – 4ac < 0

  
  

(b) Solving Quadratic equations by:

• Factorisation

• Using Formula

  
  

(c) Solving simple quadratic equation problems.

(iii) Ratio and Proportion

(a) Proportion, Continued proportion, mean proportion.

(b) Componendo, dividendo, alternendo, invertendo properties and their combinations.

(c) Direct simple applications on proportions only.

(iv) Factorisation of Polynomials

(a) Factor Theorem.

(b) Remainder Theorem.

(c) Factorising a polynomial completely after obtaining one factor by factor theorem.

Note: f(x) not to exceed degree 3.

(v) Matrices

(a) Order of a matrix. Row and column matrices.

(b) Compatibility for addition and multiplication.

(c) Null and Identity matrices.

(d) Addition and subtraction of 2×2 matrices.

(e) Multiplication of a 2×2 matrix by:

• A non-zero rational number

• A matrix

  
  

(vi) Arithmetic and Geometric Progression

• Finding their General term.

• Finding Sum of their first 'n' terms.

• Simple Applications.

  
  

(vii) Co-ordinate Geometry

(a) Reflection:

(i) Reflection of a point in a line: x=0, y=0, x=a, y=a, the origin. (ii) Reflection of a point in the origin. (iii) Invariant points.

(b) Co-ordinates expressed as (x, y), Section formula, Midpoint formula, Concept of slope, Equation of a line, Various forms of straight lines.

(i) Section and Mid-point formula (Internal section only, co-ordinates of the centroid of a triangle included).

(ii) Equation of a line:

• Slope-intercept form: y = mx + c

• Two-point form: (y – y₁) = m(x – x₁)

• Geometric understanding of 'm' as slope/gradient/tan θ where θ is the angle the line makes with the positive direction of the x axis

• Geometric understanding of 'c' as the y-intercept/the ordinate of the point where the line intercepts the y axis/the point on the line where x = 0

• Conditions for two lines to be parallel or perpendicular

Simple applications of all the above.

Unit 3 — Geometry

(a) Similarity

Similarity, conditions of similar triangles.

(i) As a size transformation. (ii) Comparison with congruency, keyword being proportionality. (iii) Three conditions: SSS, SAS, AA. Simple applications (proof not included). (iv) Applications of Basic Proportionality Theorem. (v) Areas of similar triangles are proportional to the squares of corresponding sides. (vi) Direct applications based on the above including applications to maps and models.

(b) Loci

Loci: Definition, meaning, Theorems and constructions based on Loci.

(i) The locus of a point at a fixed distance from a fixed point is a circle with the fixed point as centre and fixed distance as radius. (ii) The locus of a point equidistant from two intersecting lines is the bisector of the angles between the lines. (iii) The locus of a point equidistant from two given points is the perpendicular bisector of the line joining the points.

Proofs not required.

(c) Circles

(i) Angle Properties:

• The angle that an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circle.

• Angles in the same segment of a circle are equal (without proof).

• Angle in a semi-circle is a right angle.

(ii) Cyclic Properties:

• Opposite angles of a cyclic quadrilateral are supplementary.

• The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle (without proof).

(iii) Tangent and Secant Properties:

• The tangent at any point of a circle and the radius through the point are perpendicular to each other.

• If two circles touch, the point of contact lies on the straight line joining their centres.

• From any point outside a circle, two tangents can be drawn, and they are equal in length.

• If two chords intersect internally or externally then the product of the lengths of the segments are equal.

• If a chord and a tangent intersect externally, then the product of the lengths of segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.

• If a line touches a circle and from the point of contact, a chord is drawn, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments.

Note: Proofs of the theorems given above are to be taught unless specified otherwise.

(iv) Constructions:

(a) Construction of tangents to a circle from an external point.

(b) Circumscribing and inscribing a circle on a triangle and a regular hexagon.

Unit 4 — Mensuration

Area and volume of solids — Cylinder, Cone and Sphere.

Three-dimensional solids — right circular cylinder, right circular cone and sphere: Area (total surface and curved surface) and Volume.

Direct application problems including cost, Inner and Outer volume and melting and recasting method to find the volume or surface area of a new solid. Combination of solids included.

Note: Problems on Frustum are not included.

Unit 5 — Trigonometry

(a) Using Identities to solve/prove simple algebraic trigonometric expressions:

• sin²A + cos²A = 1

• 1 + tan²A = sec²A

• 1 + cot²A = cosec²A; 0 ≤ A ≤ 90°

(b) Heights and distances: Solving 2-D problems involving angles of elevation and depression using trigonometric tables.

Note: Cases involving more than two right angled triangles excluded.

Unit 6 — Statistics

Statistics — basic concepts, Mean, Median, Mode. Histograms and Ogive.

(a) Computation of:

Measures of Central Tendency: Mean, median, mode for raw and arrayed data. Mean, median class and modal class for grouped data (both continuous and discontinuous).

Mean by all 3 methods included:

• Direct: Σfx / Σf

• Short-cut: A + Σfd / Σf

• Step-deviation: A + (Σft / Σf) × i, where t = (x – A) / i

(b) Graphical Representation: Histograms and Less than Ogive.

• Finding the mode from the histogram, the upper quartile, lower quartile and median etc. from the ogive.

• Calculation of inter-quartile range.

  
  

Unit 7 — Probability

Random experiments, Sample space, Events, definition of probability, Simple problems on single events.

(a) Units may be written in full or using the agreed symbols, but no other abbreviation may be used.

(b) The letter 's' is never added to symbols to indicate the plural form.

(c) A full stop is not written after symbols for units unless it occurs at the end of a sentence.

(d) When unit symbols are combined as a quotient, e.g., metre per second, it is recommended that it should be written as m/s, or as m s⁻¹.

(e) Three decimal signs are in common international use: the full point, the mid-point and the comma. Since the full point is sometimes used for multiplication and the comma for spacing digits in large numbers, it is recommended that the mid-point be used for decimals.

Names and Symbols

In general:

In set language:

In measures:

Internal Assessment — 20 Marks

Assignment Requirements

Minimum number of assignments: Two assignments as prescribed by the teacher.

Suggested Assignments

• Comparative newspaper coverage of different items.

• Survey of various types of Bank accounts, rates of interest offered.

• Planning a home budget.

• Conduct a survey in your locality to study the mode of conveyance / Price of various essential commodities / favourite sports. Represent the data using a bar graph/histogram and estimate the mode.

• To use a newspaper to study and report on shares and dividends.

• Set up a dropper with ink in it vertical at a height say 20 cm above a horizontally placed sheet of plain paper. Release one ink drop; observe the pattern, if any, on the paper. Vary the vertical distance and repeat. Discover any pattern of relationship between the vertical height and the ink drop observed.

• You are provided (or you construct a model as shown) — three vertical sticks (size of a pencil) stuck to a horizontal board. You should also have discs of varying sizes with holes (like a doughnut). Start with one disc; place it on (in) stick A. Transfer it to another stick (B or C); this is one move (m). Now try with two discs placed in A such that the large disc is below, and the smaller disc is above (number of discs = n = 2 now). Now transfer them one at a time in B or C to obtain similar situation (larger disc below). How many moves? Try with more discs (n = 1, 2, 3, etc.) and generalise.

• The board has some holes to hold marbles, red on one side and blue on the other. Start with one pair. Interchange the positions by making one move at a time. A marble can jump over another to fill the hole behind. The move (m) equals 3. Try with 2 (n = 2) and more. Find the relationship between n and m.

• Take a square sheet of paper of side 10 cm. Four small squares are to be cut from the corners of the square sheet and then the paper folded at the cuts to form an open box. What should be the size of the squares cut so that the volume of the open box is maximum?

• Take an open box, four sets of marbles (ensuring that marbles in each set are of the same size) and some water. By placing the marbles and water in the box, attempt to answer the question: do larger marbles or smaller marbles occupy more volume in a given space?

• An eccentric artist says that the best paintings have the same area as their perimeter (numerically). Let us not argue whether such sizes increase the viewer's appreciation, but only try and find what sides (in integers only) a rectangle must have if its area and perimeter are to be equal. Note: there are only two such rectangles.

• Find by construction the centre of a circle, using only a 60-30 set square and a pencil.

• Various types of "cryptarithm".

  
  

Evaluation

The assignments/project work are to be evaluated by the subject teacher and by an External Examiner. The External Examiner may be a teacher nominated by the Head of the school, who could be from the faculty, but not teaching the subject in the section/class. For example, a teacher of Mathematics of Class VIII may be deputed to be an External Examiner for Class X, Mathematics projects.

The Internal Examiner and the External Examiner will assess the assignments independently.

The total marks obtained out of 20 are to be sent to CISCE by the Head of the school. The Head of the school will be responsible for the online entry of marks on CISCE's CAREERS portal by the due date.

Internal Assessment Marking Criteria

Preparation Tips — Theory (80 Marks)

General:

• The paper is 3 hours for 80 marks. Time management is critical — do not spend too long on one question.

• Mathematical tables (Logarithmic and Trigonometric) are permitted for certain questions. Practise using them efficiently.

• Show all working clearly — method marks are awarded even if the final answer is incorrect.

• Use correct units in all answers involving measurement.

  
  

Unit 1 — Commercial Mathematics:

• GST: practise both forward (find GST amount) and inverse cases (find original price when final price given). Know the difference between SGST and CGST.

• Banking: memorise both formulas for Recurring Deposit — the Interest formula and the Maturity Value formula. Apply them carefully without mixing up n and r.

• Shares and Dividends: know the difference between Face Value and Market Value clearly. Practise Income and Return calculations. Remember brokerage and fractional shares are excluded.

  
  

Unit 2 — Algebra:

• Linear Inequations: when multiplying or dividing by a negative number, the inequality sign reverses. Always write solution in set notation and represent on number line.

• Quadratic Equations: know both methods (factorisation and formula). Memorise the discriminant conditions for nature of roots — frequently asked as a short question.

• Ratio and Proportion: know all four properties (componendo, dividendo, alternendo, invertendo) and be able to apply them in combination.

• Factor and Remainder Theorem: f(a) = 0 means (x – a) is a factor. f(a) = remainder when divided by (x – a). Maximum degree is 3 — after finding one factor by factor theorem, divide to get the quadratic, then factorise further.

• Matrices: check compatibility before operating. For multiplication, number of columns in first matrix must equal number of rows in second. Know Null matrix (all zeros) and Identity matrix (diagonal ones, rest zeros).

• AP and GP: know the general term formulas (aₙ = a + (n–1)d for AP; aₙ = arⁿ⁻¹ for GP) and sum formulas. Practise identifying AP/GP from a sequence.

• Co-ordinate Geometry: practise all reflection rules — across x-axis, y-axis, y = x, y = –x, origin. For Section formula, only internal division is required. Know both forms of equation of a line and conditions for parallel (equal slopes) and perpendicular (product of slopes = –1) lines.

  
  

Unit 3 — Geometry:

• Similarity: know all three conditions (SSS, SAS, AA). Note: proofs of similarity conditions are NOT required. Areas of similar triangles are proportional to squares of corresponding sides — this ratio is frequently used in problems.

• Loci: know all three loci theorems exactly as stated. Proofs are not required. Practise constructing loci.

• Circles: know which theorems require proof and which are "without proof." Angles in the same segment and exterior angle of cyclic quadrilateral — without proof. All tangent-secant properties — proofs are required unless specified otherwise.

• Constructions: practise tangent construction from external point and circumscribing/inscribing circles on a triangle and regular hexagon with accurate compass and ruler technique. Marks are awarded for accuracy.

  
  

Unit 4 — Mensuration:

• Know all six formulas: Cylinder (CSA = 2πrh, TSA = 2πr(r+h), V = πr²h), Cone (CSA = πrl, TSA = πr(r+l), V = ⅓πr²h), Sphere (SA = 4πr², V = ⁴⁄₃πr³).

• For combination of solids: add/subtract volumes and surfaces as appropriate — watch out for which surfaces are internal and not counted in TSA.

• Melting and recasting: volume is conserved — set volumes equal and solve.

• Remember: Frustum problems are NOT included.

  
  

Unit 5 — Trigonometry:

• Memorise all three identities and their derived forms (e.g., sin²A = 1 – cos²A, tan²A = sec²A – 1, etc.)

• Heights and distances: draw a clear diagram first. Label angles of elevation and depression correctly. Maximum two right-angled triangles in any problem.

• Use trigonometric tables accurately — practise reading both the main table and the mean difference columns.

  
  

Unit 6 — Statistics:

• Know all three methods for Mean (Direct, Short-cut, Step-deviation) — all three are included in the syllabus.

• Median class and modal class for grouped data — know how to identify each correctly.

• Ogive: always use "Less than Ogive." Practise reading median, quartiles from the ogive. Know how to calculate inter-quartile range (Q3 – Q1).

• Histogram: practise finding mode from the histogram using the modal bar method.

  
  

Unit 7 — Probability:

• Know the definition: Probability of an event = (number of favourable outcomes) / (total number of outcomes in sample space).

• Only single events are required — no compound events or conditional probability.

• Always write the sample space when solving probability problems.

  
  

How to Score Grade I in Internal Assessment

FAQs — Mathematics

Q1. How long is the theory paper and how many marks is it? The theory paper is 3 hours for 80 marks.

Q2. Are mathematical tables allowed in the exam? Yes. Certain questions may require the use of Mathematical tables — Logarithmic and Trigonometric tables.

Q3. Are Frustum problems included in Mensuration? No. The syllabus explicitly states: "Problems on Frustum are not included."

Q4. Are proofs required for similarity conditions? No. The syllabus states: "Simple applications (proof not included)" for the three conditions SSS, SAS and AA.

Q5. Are proofs required for circle theorems? For the angle properties and cyclic properties: angles in the same segment and the exterior angle of a cyclic quadrilateral are stated as "without proof." For tangent and secant properties, the note states: "Proofs of the theorems given above are to be taught unless specified otherwise" — meaning proofs are required unless the theorem is individually marked "without proof."

Q6. Are proofs required for Loci? No. The syllabus explicitly states: "Proofs not required" for all three loci theorems.

Q7. Is the discriminant formula included? Yes. The nature of roots using b² – 4ac is explicitly listed in the syllabus — greater than zero (two distinct real roots), equal to zero (two equal real roots), less than zero (no real roots).

Q8. Is external section included in Section formula? No. The syllabus specifies: "Internal section only." External division is not required.

Q9. What sets are included in Linear Inequations? x ∈ N (natural numbers), W (whole numbers), Z (integers), R (real numbers) — all four are included.

Q10. How many assignments are needed for internal assessment in Mathematics? A minimum of two assignments as prescribed by the teacher.

Q11. Is brokerage included in Shares and Dividends? No. The syllabus explicitly states: "Brokerage and fractional shares not included."

Q12. What methods of finding Mean are included for grouped data? All three methods: Direct method, Short-cut method, and Step-deviation method. All three are explicitly listed in the syllabus.

Q13. Is the Less than Ogive or More than Ogive required? Less than Ogive only. The syllabus specifies "Less than Ogive" in the graphical representation section.

Q14. What cases are excluded in Heights and Distances? The syllabus states: "Cases involving more than two right angled triangles excluded."

Q15. What is the maximum degree of polynomial allowed in Factorisation? The syllabus states: "f(x) not to exceed degree 3." Polynomials of degree greater than 3 are not included.

Q16. Is the concept of centroid included? Yes. The syllabus includes "co-ordinates of the centroid of a triangle" within the Section and Mid-point formula topic.

All content above is based directly on the official CISCE ICSE Mathematics Syllabus, Examination Year 2028. Verify with the latest document at cisce.org.