ISC Class 11 Syllabus 2027 Applied Mathematics

ISC YEAR 2027

INDIAN SCHOOL CERTIFICATE
EXAMINATION

APPLIED
MATHEMATICS
(885)

February 2025
____________________________________________________________________________________________

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Council for the Indian School Certificate Examinations (CISCE)

MISSION STATEMENT

The Council for the Indian School Certificate
Examinations is committed to serving the nation's
children, through high quality educational
endeavours, empowering them to contribute towards
a humane, just and pluralistic society, promoting
introspective living, by creating exciting learning
opportunities, with a commitment to excellence.

ETHOS OF CISCE

Trust and fair play.
Minimum monitoring.
Allowing schools to evolve their own niche.
Catering to the needs of the children.
Giving freedom to experiment with new ideas
and practices.
Diversity and plurality - the basic strength for
evolution of ideas.
Schools to motivate pupils towards the
cultivation of:
Excellence - The Indian and Global
experience.
Values - Spiritual and cultural - to be the bedrock
of the educational experience.
Schools to have an 'Indian Ethos', strong roots in
the national psyche and be sensitive to national
aspirations.

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 b. Logical thinking.
c. Handling abstractions.
1. To enable candidates to acquire knowledge and to
d. Generalizing patterns.
develop an understanding of the terms, concepts,
e. Mathematical modeling to solve real-time
symbols, definitions, principles, processes and
problems.
formulae of Mathematics at the Senior Secondary
f. Analyzing the data and solving problems
stage.
using multiple mathematical methods.
2. To develop the ability to apply the knowledge and
g. Reading and interpreting tables, charts,
understanding of Mathematics to unfamiliar
graphs, etc.
situations or to new problems.
6. To enhance the ability to apply the
3. To enhance ability of analytical and rational
mathematical skills in interdisciplinary
thinking in young minds.
subjects.
4. To develop mathematical thinking and ability to
7. To develop an appreciation of the role of
communicate mathematical ideas logically and
Mathematics in day-to-day life.
precisely.
8. To develop a scientific attitude through the
5. To develop skills of –
study of Mathematics.
a. Computation.

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

1. Sets and Functions (iii) Trigonometry
(i) Sets Positive and negative angles. Measuring
angles in radians and in degrees and
Sets and their representations. Empty set.
conversion from one measure to another.
Finite and Infinite sets. Equal sets. Subsets.
Definition of trigonometric functions with
Subsets of a set of real numbers especially
the help of unit circle. Truth of the identity
intervals (with notations). Power set.
Universal set. Venn diagrams and practical sin2x + cos2x=1, for all x. Signs of
applications. Union and Intersection of sets. trigonometric functions. Domain and range
Difference of sets. Complement of a set. of trignometric functions and their graphs.
Properties of Complement of Sets. Expressing sin (x±y) and cos (x±y) in terms of
sinx, siny, cosx & cosy and their simple
(ii) Relations & Functions applications. Deducing the identities like the
Ordered pairs, Cartesian product of sets. following:
Number of elements in the cartesian product
of two finite sets. Cartesian product of the set tan x ± tan y
tan (x ± y) = ,
of reals with itself Definition of relation, 1  tan x tan y
pictorial diagrams, domain, co-domain and
range of a relation. Function as a special type cot x cot y  1
cot(x ± y)=
of relation. Function as a type of mapping, coty ± cotx
domain, co-domain and range of a function.
Real valued functions, domain and range of 1 1
sin α ± sin β =2sin ( α ± β )cos (
these functions, constant, identity, 2 2
polynomial, rational, modulus, signum, α β)
exponential, logarithmic and greatest integer
functions. Sum, difference, product and 1 1
quotient of functions. cos α + cos β = 2 cos ( α + β ) cos
2 2
• Sets: Self-explanatory. (α - β )
• Basic concepts of Relations and 1 1
Functions cos α - cos β = - 2sin ( α + β ) sin
2 2
- Ordered pairs, sets of ordered pairs. (α - β )
- Cartesian Product (Cross) of two
sets, cardinal number of a cross Identities related to sin 2x, cos2x, tan 2x,
product. sin3x, cos3x and tan3x.
Relations as:
- an association between two sets. • Angles and Arc lengths
- a subset of a Cross Product. - Angles: Convention of sign of angles.
- Domain, Range and Co-domain of a - Magnitude of an angle: Measures of
Relation. Angles; Circular measure.
- Functions: - The relation S = rθ where θ is in
- As special relations, concept of radians. Relation between radians
writing “y is a function of x” as y = and degree.
f(x). - Definition of trigonometric functions
- Domain and range of a function with the help of unit circle.
- Reading, sketching and - Truth of the identity sin2x+cos2x=1
understanding the graphs of all NOTE: Questions on the area of a sector
standard real valued functions. of a circle are required to be covered.

2

• Trigonometric Functions multiplicative inverse of a complex
- Relationship between trigonometric number.
functions. (iii) Quadratic Equations
- Proving simple identities. Statement of Fundamental Theorem of
- Signs of trigonometric functions. Algebra, solution of quadratic equations (with
- Domain and range of the real coefficients).
trigonometric functions.
- Trigonometric functions of all angles. • Use of the formula:
- Periods of trigonometric functions.
- Graphs of simple trigonometric − b ± b 2 − 4ac
x=
functions (only sketches). 2a
NOTE: Graphs of sin x, cos x, tan x, sec x, In solving quadratic equations.
cosec x and cot x are to be included.
• Equations reducible to quadratic form.
• Compound and multiple angles • Nature of roots
- Addition and subtraction formula: − Product and sum of roots.
sin(A ± B); cos(A ± B); tan(A ± B); − Roots are rational, irrational, equal,
tan(A + B + C) etc., Double angle, reciprocal, one square of the other.
triple angle, half angle and one − Complex roots.
third angle formula as special cases. − Framing quadratic equations with
- Sum and differences as products sin given roots.
C + sin D= NOTE: Questions on equations having
C+D C−D common roots are to be covered.
2sin   cos   , etc.
 2   2  • Quadratic Functions.
- Product to sum or difference Given α, β as roots then find the equation
i.e. 2sinAcosB = sin (A + B) + sin (A whose roots are of the form α 3 , β 3 , etc.
– B) etc.
Real roots
- Simple problems based on above
concepts Case I: a > 0 Complex roots
Equal roots
2. Algebra
Case II: a < 0 Real roots
(i) Logarithm
Introduction and definition of logarithm and Complex roots,
anti-logarithm. Equal roots
Properties: Common & Natural logarithms Where ‘a’ is the coefficient of x2 in the
Problems based logarithm and anti- equations of the form ax2 + bx + c = 0.
logarithm. • Sign of quadratic
(ii) Complex Numbers Sign when the roots are real and when
Introduction of complex numbers and their they are complex.
representation, Algebraic properties of • Graph of quadratic function.
complex numbers. Argand plane and polar Maximum/minimum value of quadratic
representation of complex numbers. function and value of x for which
• Conjugate, modulus and argument of maximum/minimum occurs.
complex numbers and their properties.
• Inequalities
• Sum, difference, product and quotient of
two complex numbers additive and - Linear Inequalities

3

Algebraic solutions of linear Pascal's triangle, General and middle term(s)
inequalities in one variable and their in binomial expansion, simple applications.
representation on the number line.
• Significance of Pascal’s triangle.
Self-explanatory. • Binomial theorem for positive integral
- Quadratic Inequalities powers,
Using method of intervals for solving i.e. (x + y )n = nC0 x n + nC1 x n-1 y + ...... + nCn y n .
problems of the type:
• Binomial coefficients.
x2 + x − 6 ≥ 0 Questions based on the above.
+ - + (vi) Sequence and Series
-3 2 Sequence and Series. Arithmetic
Progression (A.P.). Arithmetic Mean (A.M.)
A perfect square e.g. x 2 − 6 x + 9 ≥ 0 . Geometric Progression (G.P.), general term of
a G.P., sum of first n terms of a G.P., infinite
(iv) Permutations and Combinations G.P. and its sum, geometric mean (G.M.),
Fundamental principle of counting. Factorial relation between A.M. and G.M. Formulae for
n. (n!) Permutations and combinations, the following special sums ∑ n, ∑ n 2 , ∑ n 3 .
derivation of formulae for n Pr and n Cr and
their connections, simple application. • Arithmetic Progression (A.P.)
• Factorial notation n! , n! =n (n-1)! - T n = a + (n - 1)d
• Fundamental principle of counting.
• Permutations n
- Sn = {2a + (n − 1)d }
- nP r . 2
- Restricted permutation. - Arithmetic mean: 2b = a + c
- Certain things always occur together. - Inserting two or more arithmetic
- Certain things never occur. means between any two numbers.
- Formation of numbers with digits. - Three terms in A.P. : a - d, a, a + d
- Word building - repeated letters - No - Four terms in A.P.: a - 3d, a - d, a
letters repeated. + d, a + 3d
- Permutation of alike things.
• Geometric Progression (G.P.)
- Permutation of Repeated things.
-T n = arn-1,
- Circular permutation – clockwise
counterclockwise – Distinguishable / a (r n − 1)
not distinguishable.
- Sn = , |r|>1, 𝑺𝑺𝒏𝒏 =
r −1
𝒂𝒂(𝟏𝟏−𝒓𝒓𝒏𝒏 )
• Combinations , |𝒓𝒓| < 𝟏𝟏
𝟏𝟏−𝒓𝒓
- nC r , nC n =1, nC 0 = 1, nC r = nC n–r ,
n
C x = nC y , then x + y = n or x = y, a
n+1
-=S∞ ; r <1
C r = nC r-1 + nC r . 1− r
- When all things are different.
- Geometric Mean, b = ac
- When all things are not different.
- Inserting two or more Geometric
- Mixed problems on permutation and
Means between any two numbers.
combinations.
- Three terms are in G.P. ar, a, ar-1
(v) Binomial Theorem
- Four terms are in GP ar3, ar, ar-1, ar-
History, statement and proof of the binomial 3
theorem for positive integral indices.

4

-
• Special sums ∑ n, ∑ n 2 , ∑ n 3 (ii) Circles
Using these summations to sum up other • Equations of a circle in:
related expression. - Standard form.
Finding nth. term of a sequence using - Diameter form.
Method of difference. - General form.
3. Coordinate Geometry - Parametric form.
(i) Straight Lines • Given the equation of a circle, to find the
centre and the radius.
Brief recall of two-dimensional geometry from
earlier classes. Shifting of origin. Slope of a • Finding the equation of a circle.
line and angle between two lines. Various - Given three non collinear points.
forms of equations of a line: parallel to
- Given other sufficient data for
axis, point-slope form, slope- intercept form,
example centre is (h, k) and it lies on
two-point form, intercept form and normal
a line and two points on the circle are
form. General equation of a line. Equation of
given, etc.
family of lines passing through the point of
intersection of two lines. Distance of a point (iii) Parabola
from a line. Standard equations and simple properties of
• Brief recall of basic concepts of Points parabola.
and their coordinates. • Conics as a section of a cone.
- Section formula (internally/externally) - Definition of Foci, Directrix, Latus
- Coordinates of incentre, Area of Rectum.
triangle when vertices are given
- PS = ePL where P is a point on the
- Condition for collinearity of three
conics, S is the focus, PL is the
points
perpendicular distance of the point
• The straight line from the directrix.
- Slope or gradient of a line. • Parabola
- Angle between two lines.
- e =1, y2 = ±4ax, x2 = 4ay, y2 = -4ax,
- Condition of perpendicularity and
x2 = -4ay.
parallelism.
- Rough sketch of the above.
- Various forms of equation of lines.
- Slope intercept form. - The latus rectum; quadrants they lie
in; coordinates of focus and vertex;
- Two-point slope form. and equations of directrix and the
- Intercept form. axis.
- Perpendicular /normal form. - Finding equation of Parabola when
- General equation of a line. Foci and directrix are given, etc.
- Distance of a point from a line. - Application questions based on the
- Distance between parallel lines. above.
- Equation of lines bisecting the angle
4. Calculus
between two lines.
- Equation of family of lines (i) Limits and Derivatives
- Definition of a locus. Derivative introduced as rate of change both
as that of distance function and geometrically.
- Equation of a locus.

5

Intuitive idea of limit. Limits of polynomials • Differentiate between range, quartile
and rational functions trigonometric, deviation, mean deviation and standard
exponential and logarithmic functions. deviation.
Definition of derivative relate it to scope of • Calculate range, quartile deviation for
tangent of the curve, Derivative of sum, ungrouped and grouped data set.
difference, product and quotient of functions.
• Choose appropriate measure of
Derivatives of polynomial and trigonometric
functions. dispersion to calculate spread of data.
• Limits (ii) Skewness and Kurtosis:
- Notion and meaning of limits. • Define Skewness and Kurtosis using
- Fundamental theorems on limits graphical representation of a data set.
(statement only). • Interpret Skewness and Kurtosis of a
- Existence of lim f(x) frequency distribution by plotting the
graph.
x→a • Calculate coefficient of Skewness and
- Left hand limit , Right hand limit interpret the results.
- Limits of algebraic, trigonometric (iii) Correlation Analysis
exponential and logarithmic
functions. • Definition and meaning of covariance.
NOTE: Indeterminate forms are to be • Coefficient of Correlation by Karl
introduced while calculating limits. Pearson.
• Differentiation If x - x, y - y are small non - fractional
- Meaning and geometrical numbers, we use
interpretation of derivative.
- Derivatives of simple algebraic and ∑ ( x - x )( y - y )
r=
trigonometric functions and their
∑ (x - x ) ∑(y - y)
2 2
formulae.
- Differentiation using first principles. If x and y are small numbers, we use
- Derivatives of sum/difference.
1
- Derivatives of product of functions. ∑ xy − ∑ x∑ y
r= N
Derivatives of quotients of functions.
1
∑x −
2
(∑ x )2 ∑ y 2 − 1 (∑ y )2
5. Statistical Methods and Probability N N
Statistical Methods Otherwise, we use assumed means
(i) Measures of dispersion A and B, where u = x-A, v = y-B
Measures of dispersion: range, mean
1
deviation, variance and standard deviation of ∑ uv -( ∑ u )( ∑ v )
ungrouped/grouped data. r= N
• Mean deviation about mean. 2 1 2 2 1 2
∑ u − (∑ u) ∑ v − (∑ v)
• Standard deviation - by direct method, N N
short cut method and step deviation
method. • Differentiate between causation and
• Combined mean and standard deviation correlation.
• The Median, Quartiles and Mode of • Rank Correlation by Spearman’s
grouped and ungrouped data.
• Mean deviation around median. (Correction included)

6

(iv) Linear Regression (ii) Logical problems involving odd man out,
• Lines of regression of x on y and y on x. syllogism, blood relation and coding
decoding.
• Scatter diagrams
• The method of least squares. 7. Financial Mathematics
• Lines of best fit. (i) Interest and Interest Rates: Define the concept
• Regression coefficient of x on y and y on of Interest Rates (simple and compound).
x. Compare the difference between Nominal
Interest Rate, Effective Rate and Real Interest
• b xy × b yx = r 2 , 0 ≤ b xy × b yx ≤ 1 Rate.
(ii) Present value, net present value and future
• Identification of regression equations value: Interpret the concept of compounding
• Properties of regression lines. and discounting along with practical
applications. Compute net present value.
• Estimation of the value of one variable
Apply net present value in capital budgeting
using the value of other variable from
decisions
appropriate line of regression.
(iii) Annuities, Calculating value of Regular
Self-explanatory Annuity: Immediate Annuity, Annuity due
and Deferred Annuity. General Annuity.
(v) Probability Calculate the future value of regular annuity,
Random experiments; outcomes, sample annuity due. Apply the concept of Annuity in
spaces (set representation). Events; real life situations
occurrence of events, 'not', 'and' and 'or' (iv) Fundamentals of Taxation: Differentiate
events, exhaustive events, mutually exclusive between Direct and indirect tax. Define,
events, Axiomatic (set theoretic) probability, explain and calculate GST. Explain rules
connections with other theories studied in under SGST, CGST, and UTGST.
earlier classes. Probability of an event,
probability of 'not', 'and' and 'or' events. (v) Bills, tariff rates, fixed charge, surcharge,
• Random experiments and their outcomes. service charge: Interpret and analyze
• Events: sure events, impossible events, electricity bills, water bills and other supply
mutually exclusive and exhaustive events. bills.
- Definition of probability of an event
- Laws of probability addition theorem.
PAPER II – PROJECT WORK – 20 Marks
Candidates will be expected to have completed two
6. Mathematical Reasoning projects, one from Section A and one from either
(i) Mathematically acceptable statements. Section B or Section C.
Connecting words/ phrases - consolidating the Mark allocation for each Project [10 marks]:
understanding of "if and only if (necessary
and sufficient) condition", "implies", Overall format 1 mark
"and/or", "implied by", "and", "or", "there Content 4 marks
exists" and their use through variety of
examples related to the Mathematics and real Findings 2 marks
life. Validating the statements involving the Viva-voce based on the Project 3 marks
connecting words, Difference between
contradiction, converse and contrapositive. Total 10 marks
Creating natural data set using random
experiment such as tossing a coin multiple
times.

7

List of suggested assignments for Project Work: (iii) 180° < α < 270°
1. Explore different methods to prove the result “If (iv) 270° < α < 360°
a set has ‘n’ number of elements, then the total
number of subsets is 2n”. 14. Identify the variability and consistency of two
sets of statistical data using the concept of
2. Verify that for two sets A and B, n(A × B) = coefficient of variation.
pq, where n(A) = p and n(B)= q, the total
15. Construct the tree structure of the outcomes of a
number of relations from A to B is 2pq.
random experiment, when elementary events
3. Using Venn diagram, verify the distributive law are not equally likely. Also construct a sample
for three given non-empty sets A, B and C. space by taking a suitable example.
4. Identify distinction between a relation and a 16. Let S and S1 be two (non-concentric) circles
function with suitable examples and illustrate with centres A , B and radii r1, r2 and d be the
graphically. distance between their centres. Relation
5. Establish the relationship between the measure between r1, r2 and d with respect to relative
of an angle in degrees and in radians with position of two circles.
suitable examples by drawing a rough sketch. 17. Obtain truth values of compound statements of
6. Illustrate with the help of a model, the values of the type 𝑝𝑝 ∧ 𝑞𝑞 by using switch connection in
sine and cosine functions for different angles series.
which are multiples of π/2 and π. 18. Obtain truth values of compound statements of
7. Draw the graphs of sin x, sin 2x, 2 sin x, and sin the type 𝑝𝑝 ∨ 𝑞𝑞 by using switch connection in
x/2 on the same graph using same coordinate parallel.
axes and interpret the same. 19. Explain the statistical significance of percentile
8. Draw the graph of cos x, cos 2x, 2 cos x, and cos and draw inferences of percentile for a given
x/2 on the same graph using same coordinate data.
axes and interpret the same. 20. Find median from the point of intersection of
9. Using argand plane, interpret geometrically, the cumulative frequency curves (less than and
more than cumulative frequency curves).
meaning of 𝑖𝑖 = √−1 and its integral powers.
21. Describe the limitations of Spearman’s rank
10. Draw the graph of quadratic function 𝑓𝑓(𝑥𝑥) =
correlation coefficient and illustrate with
𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐. From the graph find suitable examples.
maximum/minimum value of the function. Also
determine the sign of the expression. 22. Correlation between the height of the student
11. Construct a Pascal’s triangle to write a binomial and the proficiency in long jump.
expansion for a given positive integral 23. Correlation between monthly income and
exponent. education qualification.
12. Obtain a formula for the sum of the squares/sum 24. Correlation between sleeping disorder and the
of cubes of ’n’ natural numbers. usage of smart phone.
25. Correlation between the particular disease (like
13. Obtain the equation of the straight line in the
normal form, for 𝛼𝛼 (the angle between the varicose vein pain/ back ache /migraine) and the
perpendicular to the line from the origin and the profession of the patient.
x-axis) for each of the following, on the same 26. Smart purchasing during sale season.
graph: 27. Prepare the best option plan to buy a product by
(i) α < 90° comparing cost, shipping charges, tax (under
GST), and hidden cost, overhead cost etc.
(ii) 90° < α < 180°

8