ISC Class 11 Syllabus 2028 Applied Mathematics

ISC
INDIAN SCHOOL CERTIFICATE
EXAMINATION

YEAR 2028

APPLIED MATHEMATICS
(885)

Developed by:
Research, Development and Curriculum Division (RDCD)
CISCE

January 2026
____________________________________________________________________________________________

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APPLIED MATHEMATICS (885)

This subject may not be taken with Mathematics.
(Note: For candidates who wish to pursue a career in Humanities/ Commerce/ Economics/ Biosciences/ Social
Sciences and other related fields.)
Aims
1. To enable candidates to acquire knowledge and to develop an understanding of the terms, concepts, symbols,
definitions, principles, processes and formulae of Mathematics at the Senior Secondary stage.
2. To develop the ability to apply the knowledge and understanding of Mathematics to unfamiliar situations or
to new problems.
3. To enhance ability of analytical and rational thinking in young minds.
4. To develop mathematical thinking and ability to communicate mathematical ideas logically and precisely.
5. To develop skills of –
a. Computation.
b. Logical thinking.
c. Handling abstractions.
d. Generalising patterns.
e. Mathematical modeling to solve real-time problems.
f. Analysing the data and solving problems using multiple mathematical methods.
g. Reading and interpreting tables, charts, graphs, etc.
6. To enhance the ability to apply the mathematical skills in interdisciplinary subjects.
7. To develop an appreciation of the role of Mathematics in day-to-day life.
8. To develop a scientific attitude through the study of Mathematics.

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CLASS XI
There will be two papers in the subject:
Paper I: Theory (3 hours)…80 marks
Paper II: Project Work……20 marks

PAPER I (THEORY) : 80 MARKS

DISTRIBUTION OF MARKS FOR THE THEORY PAPER

S.No UNIT TOTAL WEIGHTAGE
1. Sets and Functions 12 Marks
2. Algebra 22 Marks
3. Coordinate Geometry 12 Marks
4. Calculus 6 Marks
5. Statistical methods & Probability 12 Marks
6. Mathematical Reasoning 4 Marks
7. Financial Mathematics 12 Marks
TOTAL 80 Marks

1. Sets and Functions
(i) Sets
Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set
of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams and
practical applications. Union and Intersection of sets. Difference of sets. Symmetric Difference.
Complement of a set. Properties of Complement of Sets. Algebra of sets.
- Idempotent laws. Identity laws. Boundedness Laws.
- Commutative laws. Associative laws. Distributive laws.
- Laws of complementation.
- De Morgan’s laws.
(ii) Relations & Functions
Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets.
Cartesian product of the set of reals with itself. Definition of relation, pictorial diagrams, domain,
co-domain and range of a relation. Function as a special type of relation. Function as a type of mapping,
domain, co-domain and range of a function. Real valued functions, domain and range of these functions,
constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer
functions. Sum, difference, product and quotient of functions.
• Sets: Self-explanatory.
• Basic concepts of Relations and Functions
- Ordered pairs, sets of ordered pairs.
- Cartesian Product (Cross) of two sets, cardinal number of a cross product.

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Relations as:
- an association between two sets.
- a subset of a Cross Product.
- Domain, Range and Co-domain of a Relation.
Functions:
- As special relations, concept of writing “y is a function of x” as y = f(x).
- Domain and range of a function
- Reading, sketching and understanding the graphs of all standard real valued functions.
(iii) Trigonometry
Positive and negative angles. Measuring angles in radians and in degrees and conversion from one
measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the
identity sin2𝑥𝑥 + cos2𝑥𝑥 =1, for all 𝑥𝑥. Signs of trigonometric functions. Domain and range of
trigonometric functions and their graphs. Expressing sin (𝑥𝑥 ±y) and cos (𝑥𝑥 ±y) in terms of sin𝑥𝑥, siny, cos𝑥𝑥
& cosy and their simple applications. Deducing the identities like the following:
tan x ± tan y
tan (𝑥𝑥 ± y) = ,
1  tan x tan y
cot x cot y  1
cot (𝑥𝑥 ± y)=
coty ± cotx
1 1
sin α ± sin β =2sin ( α ± β ) cos (α  β )
2 2
1 1
cos α + cos β = 2 cos ( α + β ) cos (α - β )
2 2
1 1
cos α - cos β = - 2sin ( α + β ) sin (α - β )
2 2
Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x.
• Angles and Arc lengths
- Angles: Convention of sign of angles.
- Magnitude of an angle: Measures of Angles; Circular measure.
- The relation S = rθ where θ is in radians. Relation between radians and degree.
- Definition of trigonometric functions with the help of unit circle.
- Truth of the identity sin2x+cos2x=1
NOTE: Questions on the area of a sector of a circle are required to be covered.
• Trigonometric Functions
- Relationship between trigonometric functions.
- Proving simple identities.
- Signs of trigonometric functions.
- Domain and range of the trigonometric functions.
- Trigonometric functions of all angles.
- Periods of trigonometric functions.
- Graphs of simple trigonometric functions (only sketches).
NOTE: Graphs of sin x, cos x, tan x, sec x, cosec x and cot x are to be included.

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• Compound and multiple angles
- Addition and subtraction formula: sin(A ± B); cos(A ± B); tan(A ± B); tan(A + B + C) etc., Double
angle, triple angle, half angle and one third angle formula as special cases.
C+D C−D
- Sum and differences as products i.e. sin C + sin D= 2sin   cos   , etc.
 2   2 
- Product to sum or difference i.e. 2sinAcosB = sin (A + B) + sin (A – B) etc.
- Simple problems based on above concepts

2. Algebra
(i) Logarithm
Introduction and definition of logarithm and anti-logarithm.
Properties: Common & Natural logarithms
Problems based on logarithm and anti-logarithm.
(ii) Complex Numbers
Introduction of complex numbers and their representation, Algebraic properties of complex numbers.
Argand plane and polar representation of complex numbers.
• Conjugate, modulus and argument of complex numbers and their properties.
• Sum, difference, product and quotient of two complex numbers, additive and multiplicative inverse of
a complex number.
(iii) Quadratic Equations
Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients).
• Use of the formula:

− b ± b 2 − 4ac
x=
2a
In solving quadratic equations.
• Equations reducible to quadratic form.
• Nature of roots
− Product and sum of roots.
− Roots are rational, irrational, equal, reciprocal, one square of the other.
− Complex roots.
− Framing quadratic equations with given roots.
NOTE: Questions on equations having common roots are to be covered.
• Quadratic Functions.
Given α, β as roots then find the equation whose roots are of the form α 3 , β 3 , etc.
Real roots
Case I: a > 0 Complex roots
Equal roots

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Real roots
Case II: a < 0 Complex roots
Equal roots
Where ‘a’ is the coefficient of x2 in the equations of the form ax2 + bx + c = 0.
• Sign of quadratic
Sign when the roots are real and when they are complex.
• Graph of quadratic function. Maximum/minimum value of quadratic function and value of x for which
maximum/minimum occurs.
• Inequalities
- Linear Inequalities
Algebraic solutions of linear inequalities in one variable and their representation on the number
line.
Self-explanatory.
- Quadratic Inequalities
Using method of intervals for solving problems of the type:
x2 + x − 6 ≥ 0
+ - +
-3 2
A perfect square e.g. x 2 − 6 x + 9 ≥ 0 .
(iv) Permutations and Combinations
Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of
formulae for n Pr and n Cr and their connections, simple application.
• Factorial notation n! , n! =n (n-1)!
• Fundamental principle of counting.
• Permutations
- nP r .
- Restricted permutation.
- Certain things always occur together.
- Certain things never occur.
- Formation of numbers with digits.
- Word building - repeated letters - No letters repeated.
- Permutation of alike things.
- Permutation of Repeated things.
- Circular permutation – clockwise counterclockwise – Distinguishable / not distinguishable.
• Combinations
- nC r , nC n =1, nC 0 = 1, nC r = nC n–r , nC x = nC y , then x + y = n or x = y, n+1C r = nC r-1 + nC r .
- Total number of combinations of n dissimilar things taking any number of them at a time
 When all things are different.
 When all things are not different.
- Division, distribution into groups.

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- Mixed problems on permutation and combinations.
(v) Binomial Theorem
History, statement of the binomial theorem for positive integral indices.
Pascal's triangle, General and middle term(s) in binomial expansion, simple applications.
• Significance of Pascal’s triangle.
• Binomial theorem for positive integral powers,
i.e. (x + y )n = nC0 x n + nC1 x n-1 y + ...... + nCn y n .
• Binomial coefficients.
Questions based on the above.

(vi) Sequence and Series.
Sequence and Series. Arithmetic Progression (A.P.). Arithmetic Mean (A.M.) Geometric Progression
(G.P.), general term of a G.P., sum of first n terms of a G.P., infinite G.P. and its sum, geometric mean
(G.M.), relation between A.M. and G.M. Formulae for the following special sums ∑ n, ∑ n 2 , ∑ n 3 .
• Arithmetic Progression (A.P.)
- T n = a + (n - 1)d
n
- Sn = {2a + (n − 1)d }
2
- Arithmetic mean: 2b = a + c
- Inserting two or more arithmetic means between any two numbers.
- Three terms in A.P. : a - d, a, a + d
- Four terms in A.P.: a - 3d, a - d, a + d, a + 3d
• Geometric Progression (G.P.)
-T n = arn-1,
a (r n − 1) 𝑎𝑎(1−𝑟𝑟 𝑛𝑛 )
- Sn = , |r|>1, 𝑆𝑆𝑛𝑛 = , |𝑟𝑟| < 1
r −1 1−𝑟𝑟

a
-=S∞ ; r <1
1− r

- Geometric Mean, b = ac
- Inserting two or more Geometric Means between any two numbers.
- Three terms are in G.P. ar, a, ar-1
- Four terms are in GP ar3, ar, ar-1, ar-3
• Special sums ∑ n, ∑ n 2 , ∑ n 3
Using these summations to sum up other related expression.
Finding nth term of a sequence using Method of difference.

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3. Coordinate Geometry
(i) Straight Lines
Brief recall of two-dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle
between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slope-
intercept form, two-point form, intercept form and normal form. General equation of a line. Equation of
family of lines passing through the point of intersection of two lines. Distance of a point from a line.
• Brief recall of basic concepts of Points and their coordinates.
- Section formula (internally/externally)
- Coordinates of incentre, Area of triangle when vertices are given
- Condition for collinearity of three points
• The straight line
- Slope or gradient of a line.
- Angle between two lines.
- Condition of perpendicularity and parallelism.
- Various forms of equation of lines.
- Slope intercept form.
- Two-point slope form.
- Intercept form.
- Perpendicular /normal form.
- General equation of a line.
- Distance of a point from a line.
- Distance between parallel lines.
- Equation of lines bisecting the angle between two lines.
- Equation of family of lines
- Definition of a locus.
- Equation of a locus.
(ii) Circles
• Equations of a circle in:
- Standard form.
- Diameter form.
- General form.
- Parametric form.
• Given the equation of a circle, to find the centre and the radius.
• Finding the equation of a circle.
- Given three non collinear points.
- Given other sufficient data for example centre is (h, k) and it lies on a line and two points on the
circle are given, etc.
(iii) Parabola
Standard equations and simple properties of parabola.
• Conics as a section of a cone.
- Definition of Foci, Directrix, Latus Rectum.

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- PS = ePL where P is a point on the conics, S is the focus, PL is the perpendicular distance of the
point from the directrix.
• Parabola
- e =1, y2 = 4ax, x2 = 4ay, y2 = -4ax, x2 = -4ay.
- Rough sketch of the above.
- The latus rectum; quadrants they lie in; coordinates of focus and vertex; and equations of
directrix and the axis.
- Finding equation of Parabola when Foci and directrix are given, etc.
- Application questions based on the above.

4. Calculus
(i) Limits and Derivatives
Derivative introduced as rate of change both as that of distance function and geometrically. Intuitive idea
of limit. Limits of polynomials and rational functions trigonometric, exponential and logarithmic
functions. Definition of derivative, relate it to scope of tangent of the curve, Derivative of sum,
difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.
• Limits
- Notion and meaning of limits.
- Fundamental theorems on limits (statement only).
- Existence of lim f(x)
x→a
- Left hand limit, Right hand limit.
- Limits of algebraic, trigonometric, exponential, and logarithmic functions.
NOTE: Indeterminate forms are to be introduced while calculating limits.
• Differentiation
- Meaning and geometrical interpretation of derivative.
- Derivatives of simple algebraic and trigonometric functions and their formulae.
- Differentiation using first principles.
- Derivatives of sum/difference.
- Derivatives of product of functions.
- Derivatives of quotients of functions.

5. Statistical Methods and Probability
Statistical Methods
(i) Measures of dispersion
Measures of dispersion: range, mean deviation, variance and standard deviation of ungrouped/grouped
data.
• Range
• Quartiles, Interquartile range, Quartile deviation, Coefficient of Quartile deviation
• Mean deviation about mean and median, coefficient of mean deviation
• Mean deviation about median.
• Standard deviation - by direct method, short cut method and step deviation method, Variance
and coefficient of variance
• Combined mean and standard deviation

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• Mode of grouped and ungrouped data.
• Differentiate between range, quartile deviation, mean deviation and standard deviation.
• Choose appropriate measure of dispersion to calculate spread of data.
(ii) Skewness and Kurtosis:
• Define Skewness and Kurtosis using graphical representation of a data set.
• Interpret Skewness and Kurtosis of a frequency distribution by plotting the graph.
• Calculate coefficient of Skewness and interpret the results.
• Empirical Relation between Mean, Median and Mode
• Moments
• Measures of skewness – Absolute and Relative
• Relative measures –
− Karl Pearson’s Coefficient of Skewness.
− Bowley's Coefficient of Skewness.
− β and γ Coefficient of Skewness
• Karl Pearson’s Measures of Kurtosis

(iii) Correlation Analysis
• Definition and meaning of covariance.
• Coefficient of Correlation by Karl Pearson.

If x - x, y - y are small non - fractional
numbers, we use

∑ ( x - x )( y - y )
r=
∑ (x - x ) ∑(y - y)
2 2

If x and y are small numbers, we use
1
∑ xy − ∑ x∑ y
r= N
1
∑x −
2
(∑ x )2 ∑ y 2 − 1 (∑ y )2
N N
Otherwise, we use assumed means
A and B, where u = x-A, v = y-B
1
∑ uv -( ∑ u )( ∑ v )
r= N
2 1 2 2 1 2
∑ u − (∑ u) ∑ v − (∑ v)
N N
• Differentiate between causation and correlation.
• Rank Correlation by Spearman’s (Correction included)

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(iv) Linear Regression
• Lines of regression of x on y and y on x.
• Scatter diagrams.
• The method of least squares.
• Lines of best fit.
• Regression coefficient of x on y and y on x.
• b xy × b yx = r 2 , 0 ≤ b xy × b yx ≤ 1
• Identification of regression equations.
• Properties of regression lines.
• Estimation of the value of one variable using the value of other variable from appropriate line of
regression.
Self-explanatory

(v) Probability
Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, 'not',
'and' and 'or' events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability,
connections with other theories studied in earlier classes. Probability of an event, probability of 'not',
'and' and 'or' events.
• Random experiments and their outcomes.
• Events: sure events, impossible events, mutually exclusive and exhaustive events.
- Definition of probability of an event
- Laws of probability addition theorem.

6. Mathematical Reasoning
(i) Mathematically acceptable statements. Connecting words/ phrases - consolidating the understanding of
"if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there
exists" and their use through variety of examples related to the Mathematics and real life. Validating
the statements involving the connecting words (Truth tables), Difference between contradiction,
converse and contrapositive.
Creating natural data set using random experiment such as tossing a coin multiple times.
(ii) Logical problems involving odd man out, syllogism, blood (family) relation and coding decoding.

7. Financial Mathematics
(i) Interest and Interest Rates: Define the concept of Interest Rates (simple and compound). Compare the
difference between Nominal Interest Rate, Effective Rate and Real Interest Rate, Concept of equivalency.
Compounding Frequency.
- APR (Annual Percentage Rate)
- AER (Annual Equivalent Rate).
- Annualised Yield

(ii) Present value, net present value and future value: Interpret the concept of compounding and discounting
along with practical applications. Compute net present value. Apply net present value in capital budgeting
decisions

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(iii) Annuities, Calculating value of Regular Annuity: Immediate Annuity, Annuity due and Deferred Annuity.
General Annuity. Calculate the future value of regular annuity, annuity due. Apply the concept of Annuity
in real life situations.
(iv) Fundamentals of Taxation: Differentiate between Direct and indirect tax. Define, explain and calculate
GST. Explain rules under SGST, CGST, and UTGST.
(v) Bills, tariff rates, fixed charge, surcharge, service charge: Interpret and analyse electricity bills, water bills
and other supply bills.

PAPER II (PROJECT WORK) : 20 MARKS
Candidates will be expected to have completed two projects
Mark allocation for each Project [10 marks]:

Overall format 1 mark
Content 4 marks
Findings 2 marks
Viva-voce based on the Project 3 marks
Total 10 marks

List of suggested assignments for Project Work:

1. Explore different methods to prove the result “If a set has ‘n’ number of elements, then the total number of
subsets is 2n”.
2. Verify that for two sets A and B, n(A × B) = pq, where n(A) = p and n(B)= q, the total number of relations
from A to B is 2pq.
3. Using Venn diagram, verify the distributive law for three given non-empty sets A, B and C.
4. Identify distinction between a relation and a function with suitable examples and illustrate graphically.
5. Establish the relationship between the measure of an angle in degrees and in radians with suitable examples
by drawing a rough sketch.
6. Illustrate with the help of a model, the values of sine and cosine functions for different angles which are
multiples of π/2 and π.
7. Draw the graphs of sin x, sin 2x, 2 sin x, and sin x/2 on the same graph using same coordinate axes and
interpret the same.
8. Draw the graph of cos x, cos 2x, 2 cos x, and cos x/2 on the same graph using same coordinate axes and
interpret the same.
9. Using argand plane, interpret geometrically, the meaning of 𝑖𝑖 = √−1 and its integral powers.
10. Draw the graph of quadratic function 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐. From the graph find maximum/minimum value
of the function. Also determine the sign of the expression.
11. Construct a Pascal’s triangle to write a binomial expansion for a given positive integral exponent.
12. Obtain a formula for the sum of the squares/sum of cubes of ‘n’ natural numbers.
13. Obtain the equation of the straight line in the normal form, for 𝛼𝛼 (the angle between the perpendicular to the
line from the origin and the x-axis) for each of the following, on the same graph:
(i) α < 90°
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(ii) 90° < α < 180°
(iii) 180° < α < 270°
(iv) 270° < α < 360°
14. Identify the variability and consistency of two sets of statistical data using the concept of coefficient of
variation.
15. Construct the tree structure of the outcomes of a random experiment, when elementary events are not equally
likely. Also construct a sample space by taking a suitable example.
16. Let S and S 1 be two (non-concentric) circles with centres A , B and radii r 1 , r 2 and d be the distance between
their centres. Relation between r 1 , r 2 and d with respect to relative position of two circles.
17. Obtain truth values of compound statements of the type 𝑝𝑝 ∧ 𝑞𝑞 by using switch connection in series.
18. Obtain truth values of compound statements of the type 𝑝𝑝 ∨ 𝑞𝑞 by using switch connection in parallel.
19. Explain the statistical significance of percentile and draw inferences of percentile for a given data.
20. Find median from the point of intersection of cumulative frequency curves (less than and more than
cumulative frequency curves).
21. Describe the limitations of Spearman’s rank correlation coefficient and illustrate with suitable examples.
22. Correlation between the height of the student and the proficiency in long jump.
23. Correlation between monthly income and education qualification.
24. Correlation between sleeping disorder and the usage of smart phone.
25. Correlation between the particular disease (like varicose vein pain/ back ache /migraine) and the profession
of the patient.
26. Smart purchasing during sale season.
27. Prepare the best option plan to buy a product by comparing cost, shipping charges, tax (under GST), and
hidden cost, overhead cost etc.

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