ICSE Class 10 Mathematics Syllabus 2025-26

CLASS IX

There will be one paper of two and a half hours duration carrying 80 marks and Internal Assessment of 20 marks.
Certain questions may require the use of Mathematical tables (Logarithmic and Trigonometric tables).
The solution of a question may require the knowledge of more than one branch of the syllabus.

1. Pure Arithmetic
   • Rational and Irrational Numbers: Rational, irrational numbers as real numbers, their place in the number system. Surds and rationalisation of surds. Simplifying an expression by rationalizing the denominator. Representation of rational and irrational numbers on the number line. Proofs of irrationality.
2. Commercial Mathematics
   • Compound Interest
     • Compound interest as a repeated Simple Interest computation with a growing Principal. Use of this in computing Amount over a period of 2 or 3 years.
     • Use of formula n. Finding CI from the relation CI = A – P.
     • Interest compounded half-yearly included.
     • Using the formula to find one quantity given different combinations of A, P, r, n, CI and SI; difference between CI and SI type included. Rate of growth and depreciation.
3. Algebra
   1. Expansions
      • Recall of concepts learned in earlier classes.
      • (a ± b)², (a ± b)³, (x ± a)(x ± b), (a ± b ± c)²
   2. Factorisation
      • a² – b², a³ ± b³, ax² + bx + c (by splitting the middle term)
   3. Simultaneous Linear Equations in two variables (With numerical coefficients only)
      • Solving algebraically by:
        • Elimination
        • Substitution
        • Cross Multiplication method
      • Solving simple problems by framing appropriate equations.
   4. Indices/Exponents
      • Handling positive, fractional, negative and “zero” indices.
      • Simplification of expressions involving various exponents.
      • Use of laws of exponents.
   5. Logarithms
      • Logarithmic form vis-à-vis exponential form: interchanging.
      • Laws of Logarithms and their uses. Expansion of expression with the help of laws of logarithms.
        e.g. y = (a⁴ b²) / c³ → log y = 4 log a + 2 log b – 3 log c
4. Geometry
   1. Triangles
      1. Congruency: four cases: SSS, SAS, AAS, and RHS. Illustration through cutouts. Simple applications.
      2. Problems based on:
         • Angles opposite equal sides are equal and converse.
         • If two sides of a triangle are unequal, then the greater angle is opposite the greater side and converse.
         • Sum of any two sides of a triangle is greater than the third side.
         • Of all straight lines that can be drawn to a given line from a point outside it, the perpendicular is the shortest.
      3. Mid-Point Theorem and its converse, Equal intercept theorem
         1. Proof and simple applications of Mid-point theorem and its converse.
         2. Equal intercept theorem: proof and simple application.
      4. Pythagoras Theorem
         • Area based proof and simple applications of Pythagoras Theorem and its converse.
   2. Rectilinear Figures
      1. Proof and use of theorems on parallelogram
         • Both pairs of opposite sides equal (without proof).
         • Both pairs of opposite angles equal.
         • One pair of opposite sides equal and parallel (without proof).
         • Diagonals bisect each other and bisect the parallelogram.
         • Rhombus as a special parallelogram whose diagonals meet at right angles.
         • In a rectangle, diagonals are equal. In a square they are equal and meet at right angles.
      2. Constructions of Polygons: quadrilaterals (including parallelograms and rhombus) and regular hexagon using ruler and compasses only.
      3. Proof and use of Area theorems on parallelograms:
         • Parallelograms on the same base and between the same parallels are equal in area.
         • The area of a triangle is half that of a parallelogram on the same base and between the same parallels.
         • Triangles between the same base and between the same parallels are equal in area (without proof).
         • Triangles with equal areas on the same bases have equal corresponding altitudes.
   3. Circle
      1. Chord properties
         • A straight line drawn from the centre of a circle to bisect a chord which is not a diameter is at right angles to the chord.
         • The perpendicular to a chord from the centre bisects the chord (without proof).
         • Equal chords are equidistant from the centre.
         • Chords equidistant from the centre are equal (without proof).
         • There is one and only one circle that passes through three given points not in a straight line.
      2. Arc and chord properties
         • If two arcs subtend equal angles at the centre, they are equal, and its converse.
         • If two chords are equal, they cut off equal arcs, and its converse (without proof).
5. Statistics
   • Introduction, collection of data, presentation of data, Graphical representation of data, Mean, Median of ungrouped data.
   • Understanding and recognition of raw, arrayed and grouped data.
   • Tabulation of raw data using tally-marks.
   • Understanding and recognition of discrete and continuous variables.
   • Mean, median of ungrouped data.
   • Class intervals, class boundaries and limits, frequency, frequency table, class size for grouped data.
   • Grouped frequency distributions: the need to and how to convert discontinuous intervals to continuous intervals.
   • Drawing a frequency polygon.
6. Mensuration
   • Area and perimeter of a triangle and a quadrilateral. Area and circumference of circle. Surface area and volume of Cube and Cuboids.
   • Area and perimeter of triangle (including Heron’s formula), all types of Quadrilaterals.
   • Circle: Area and Circumference. Direct application problems including Inner and Outer area.
   • Surface area and volume of 3-D solids: cube and cuboid including problems of type involving:
     • Different internal and external dimensions of the solid.
     • Cost.
     • Concept of volume being equal to area of cross-section x height.
     • Open/closed cubes/cuboids.
7. Trigonometry
   • Trigonometric Ratios: sine, cosine, tangent of an angle and their reciprocals.
   • Trigonometric ratios of standard angles - 0, 30, 45, 60, 90 degrees. Evaluation of an expression involving these ratios.
   • Simple 2-D problems involving one right-angled triangle.
   • Concept of trigonometric ratios of complementary angles and their direct application:
     sin A = cos (90° - A), cos A = sin (90° – A)
     tan A = cot (90° – A), cot A = tan (90° - A)
     sec A = cosec (90° – A), cosec A = sec (90° – A)
8. Coordinate Geometry
   • Cartesian System, plotting of points in the plane for given coordinates, solving simultaneous linear equations in 2 variables graphically and finding the distance between two points using distance formula.
   • Dependent and independent variables.
   • Ordered pairs, coordinates of points and plotting them in the Cartesian plane.
   • Solution of Simultaneous Linear Equations graphically.
   • Distance formula.

CLASS X

There will be one paper of two and a half hours duration carrying 80 marks and Internal Assessment of 20 marks.
Certain questions may require the use of Mathematical tables (Logarithmic and Trigonometric tables).

1. Commercial Mathematics
   1. Goods and Services Tax (GST)
      • Computation of tax including problems involving discounts, list-price, profit, loss, basic/cost price including inverse cases.
      • 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 items will be provided. Problems based on corresponding inverse cases are also included.
   2. Banking
      • Recurring Deposit Accounts: computation of interest and maturity value using the formula:
        I = ^{P × n(n+1) × r} / _{2 × 12 × 100}
        MV = P × n + I
   3. Shares and Dividends
      • Face/Nominal Value, Market Value, Dividend, Rate of Dividend, Premium.
      • Formulae:
        • Income = number of shares × rate of dividend × FV.
        • Return = (Income/Investment) × 100.
2. Algebra
   1. 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.
   2. Quadratic Equations in one variable
      • 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
      • Solving Quadratic equations by Factorisation and Using Formula.
      • Solving simple quadratic equation problems.
   3. Ratio and Proportion
      • Proportion, Continued proportion, mean proportion
      • Componendo, dividendo, alternendo, invertendo properties and their combinations.
      • Direct simple applications on proportions only.
   4. Factorisation of polynomials
      • Factor Theorem.
      • Remainder Theorem.
      • Factorising a polynomial completely after obtaining one factor by factor theorem.
      • Note: f(x) not to exceed degree 3
   5. Matrices
      • Order of a matrix. Row and column matrices.
      • Compatibility for addition and multiplication.
      • Null and Identity matrices.
      • Addition and subtraction of 2×2 matrices.
      • Multiplication of a 2×2 matrix by
        • a non-zero rational number
        • a matrix
   6. Arithmetic and Geometric Progression
      • Finding their General term.
      • Finding Sum of their first ‘n’ terms.
      • Simple Applications.
   7. Co-ordinate Geometry
      • Reflection:
        • Reflection of a point in a line: x=0, y=0, x=a, y=a, the origin.
        • Reflection of a point in the origin.
        • Invariant points.
      • Section and Mid-point formula (Internal section only, co-ordinates of the centroid of a triangle included).
      • 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 / tanθ 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.
3. Geometry
   1. Similarity
      • Similarity, conditions of similar triangles.
      • As a size transformation.
      • Comparison with congruency, keyword being proportionality.
      • Three conditions: SSS, SAS, AA. Simple applications (proof not included).
      • Applications of Basic Proportionality Theorem.
      • Areas of similar triangles are proportional to the squares of corresponding sides.
      • Direct applications based on the above including applications to maps and models.
   2. Loci
      • Loci: Definition, meaning, Theorems and constructions based on Loci.
        • 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.
        • The locus of a point equidistant from two intersecting lines is the bisector of the angles between the lines.
        • The locus of a point equidistant from two given points is the perpendicular bisector of the line joining the points.
   3. Circles
      1. 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.
      2. 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).
      3. 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.
   4. Constructions
      • Construction of tangents to a circle from an external point.
      • Circumscribing and inscribing a circle on a triangle and a regular hexagon.
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.
5. Trigonometry
   • Using Identities to solve/prove simple algebraic trigonometric expressions:
   • sin²A + cos²A = 1
   • 1 + tan²A = sec²A
   • 1 + cot²A = cosec²A; 0 ≤ A ≤ 90°
   • Heights and distances: Solving 2-D problems involving angles of elevation and depression using trigonometric tables.
6. Statistics
   • Statistics – basic concepts, Mean, Median, Mode. Histograms and Ogive.
   • 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: Σf / Σfx
     Short-cut: A + (Σfd / Σf), where d = x - A
     Step-deviation: A + (Σft / Σf) × i, where t = (x-A)/i
   • 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.
7. Probability
   • Random experiments, Sample space, Events, definition of probability, Simple problems on single events.

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.

Award of Marks (20 Marks)

• Subject Teacher (Internal Examiner): 10 marks
• External Examiner: 10 marks

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 IN MATHEMATICS - GUIDELINES FOR MARKING WITH GRADES