# Mathematics Class 11 ISC (Commerce) CISCE Topics and Syllabus

## Syllabus

100 Sets and Functions

101 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. Union and Intersection of sets.
• Practical problems on union and intersection of two and three sets. Difference of sets. Complement of a set. Properties of Complement of Sets
102 Relations and 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 (upto R x R x R).

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, types of functions (one to one, many to one, onto, into) 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, with their graphs. 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.
• 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).
• Introduction of Types: one to one, many to one, into, onto.
• Domain and range of a function.
• Sketches of graphs of exponential function, logarithmic function, modulus function, step function and rational function.
103 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 sin2 x+cos2 x=1, for all x. Signs of trigonometric functions. Domain and range of trignometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications. Deducing the identities like the following:

tan(x+-y)=(tanx+-tany)/(1+-tanxtany)

cot(x+-y)=(cotxcoty+-1)/(coty+-cotx)

sinalpha+-sinbeta=2sin

cosalpha+cosbeta=2cos

cosalpha-cosbeta=-2sin

Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type siny = sina, cosy = cosa and tany = tana.

Properties of triangles (proof and simple applications of sine rule cosine rule and area of triangle)

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

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)

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.
• Sum and differences as products
• sinC + sinD = 2sin((C+D)/2)cos((C-D)/2) , etc
• Product to sum or difference i.e. 2sinAcosB = sin(A + B) + sin(A – B) etc

Trigonometric Equations

• Solution of trigonometric equations (General solution and solution in the specified range).
• Equations expressible in terms of sinθ =0 etc.
• Equations expressible in terms i.e sinθ = sinalpha etc
• Equations expressible multiple and sub- multiple angles i.e. sin2θ = sin2alpha etc
• Linear equations of the form acosθ +bsinθ = c, where |c|<=sqrt(a^2+b^2)
• Properties of Δ
• Sine formula: a/sinA=b/sinB=c/sinC
• Cosine formula:cosA=(b^2+c^2-a^2)/(2bc), etc
• Area of triangle:Δ = 1/2bc A etc
• Simple applications of the above
200 Algebra

201 Principle of Mathematical Induction
• Process of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications
• Using induction to prove various summations, divisibility and inequalities of algebraic expressions only
202 Complex Numbers

Introduction of complex numbers and their representation, Algebraic properties of complex numbers. Argand plane and polar
representation of complex numbers. Square root of a complexnumber. Cube root of unity

• 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
• Locus questions on complex numbers.
• Triangle inequality.
• Square root of a complex number.
• Cube roots of unity and their properties

Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients).

Use of the formula:

x=(-b+-sqrt(b^2-4ac))/(2a)

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

Given alpha,beta as roots then find the equation whose roots are of the form alpha^3, beta^3 , etc

Case I:a>0 -> 1)Real roots, 2)Complex roots,3)Equal roots

Case II:a<0 -> 1)Real roots, 2)Complex roots,3)Equal roots

Where ‘a’ is the coefficient of x2 in the equations of the form ax2 + bx + c = 0.

Understanding the fact that a quadratic expression (when plotted on a graph) is a parabola.

Sign when the roots are real and when they are complex

Inequalities

Linear Inequalities

Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical representation of linear inequalities in two variables. Graphical method of finding a solution of system of linear inequalities in two variables.

Using method of intervals for solving problems of the type:

A perfect square e.g. x^2-6x+9>=0

Inequalities involving rational expression of type

f(x)/g(x)<=a  etc to be covered

204 Permutations and Combinations

Fundamental principle of counting. Factorial n (n!) Permutations and combinations, derivation of formulae for nPand nCr and their connections, simple application.

1. Factorial notation n! , n! =n (n-1)!
2. Fundamental principle of counting
3. Permutations
• nPr..
• 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

4. Combinations

• nCr , nCn =1, nC0 = 1, nCr = nCn–r, nCx = nCy, then x + y = n or x = y, n+1Cr = nCr-1 + nCr
• When all things are different
• When all things are not different.
• Mixed problems on permutation and combinations.
205 Binomial Theorem

History, statement and proof of the binomial theorem for positive integral indices. Pascal's triangle, General and middle term in binomial expansion, simple applications.

• Significance of Pascal’s triangle.
• Binomial theorem (proof using induction) for positive integral powers,

i.e. (x+y)^n=^nC_0x^n+^nC_1x^(n-1)y+.....+^nC_ny^n

206 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 sumn,sumn^2,sumn^3

Arithmetic Progression (A.P.)

• Tn = a + (n - 1)d
• S_n=n/2{2a+(n-1)d}
• 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.)

• Tn = arn-1, Sn = (a(r^n-1))/(r-1),
• S_oo = a/(1-r;)|r|<1
• 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

Arithmetico Geometric Series

• Identifying series as A.G.P. (when we substitute d = 0 in the series, we get a G.P. and when we substitute r =1 the
A.P).

Special sums sumn,sumn^2,sumn^3

• Using these summations to sum up other related expression.
300 Coordinate Geometry

301 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, pointslope 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.

1. Basic concepts of Points and their coordinates
2. 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
302 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.

Tangents:

• Condition for tangency
• Equation of a tangent to a circle
400 Calculus

401 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).
• Limits of algebraic and trigonometric functions.
• Limits involving 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
500 Statistics and Probability

501 Statistics - 1

Measures of dispersion: range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances

• Mean deviation about mean and median
• Standard deviation - by direct method, short cut method and step deviation method.

NOTE: Mean, Median and Mode of grouped and ungrouped data are required to be covered.

502 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.
600 Conic Section

Sections of a cone, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola.

Conics as a section of a cone.

Definition of Foci, Directrix, Latus Rectum.

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.

(i) Parabola

e =1, y2 = ±4ax, x2 = 4ay, y2 = -4ax

x2 = -4ay, (y - beta)2 =±4a (x - alpha),

(x - alpha)2 = ±4a (y - beta).

• 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.

(ii) Ellipse

• x^2/a^2+y^2/b^2=1, e<1,b^2=a^2(1-e^2)
•  (x-alpha)^2/a^2+(y-beta)^2/b^2=1
• Cases when a > b and a < b.
• Rough sketch of the above.
• Major axis, minor axis; latus rectum; coordinates of vertices, focus and centre; and equations of directrices and the axes.
• Finding equation of ellipse when focus and directrix are given.
• Simple and direct questions based on the above.
• Focal property i.e. SP + SP' = 2a.

(iii)Hyperbola

• x^2/a^2-y^2/b^2=1,e>1,b^2=a^2(e^2-1)
• (x-alpha)^2/a^2-(y-beta)^2/b^2=1
• Cases when coefficient y2 is negative and coefficient of x2 is negative
• Rough sketch of the above
• Focal property i.e. SP - S’P = 2a
• Transverse and Conjugate axes; Latus rectum; coordinates of vertices, foci and centre; and equations of the directrices and the axes.

General second degree equation

ax2 + 2hxy + by2 + 2gx + 2 fy + c = 0

• Case 1: pair of straight line if abc+2fgh-af2-bg2-ch2=0
• Case 2: abc+2fgh-af2-bg2-ch2≠0 then represents a parabola if h2= ab, ellipse if h2 < ab, and hyperbola if h2 > ab.

Condition that y = mx + c is a tangent to the conics, general equation of tangents, point of contact and locus problems.

700 Introduction to Three-dimensional Geometry

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula

• As an extension of 2-D
• Distance formula.
• Section and midpoint form
800 Mathematical Reasoning
• 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, Difference between contradiction, converse and contrapositive
900 Statistics - 2
• Combined mean and standard deviation
• The Median, Quartiles, Deciles, Percentiles and Mode of grouped and ungrouped data
1000 Correlation Analysis

Definition and meaning of covariance

Coefficient of Correlation by Karl Pearson

• if x-barx,y-bary are small non fractional numbers we use
• r=(sum(x-barx)(y-bary))/(sqrt(sum(x-barx)^2)sqrt(sum(y-bary)^2)
• If x and y are small numbers, we use
•  r=(sumxy-1/Nsumxsumy)/(sqrt(sumx^2-1/N(sumx)^2)sqrt(sumy^2-1/N(sumy)^2))
•  Otherwise, we use assumed means
• A and B, where u = x-A, v = y-B
• r=(sumuv-1/N(sumu)(sumv))/(sqrt(sumu^2-1/N(sumu)^2)sqrt(sumv^2-1/N(sumv)^2))

Rank correlation by Spearman’s (Correction included)

1100 Index Numbers and Moving Averages

1101 Index Numbers
• Price index or price relative.
• Simple aggregate method.
• Weighted aggregate method.
• Simple average of price relatives.
• Weighted average of price relatives (cost of living index, consumer price index)
1102 Moving Averages
• Meaning and purpose of the moving averages.
• Calculation of moving averages with the given periodicity and plotting them on a graph.
• If the period is even, then the centered moving average is to be found out and plotted.