# Mathematics (M 1108) - Grade 11

Unit | Area Covered | Marks | |
---|---|---|---|

Unit 1 |
SETS | 05 | Read more |

Unit 2 |
Relations and Functions | 10 | Read more |

Unit 3 |
Trigonometric Functions | 10 | Read more |

Unit 4 |
Complex Numbers and Quadratic Equations | 10 | Read more |

Unit 5 |
Linear and Quadratic Inequalities | 10 | Read more |

Unit 6 |
Permutation and Combination | 10 | Read more |

Unit 7 |
Binomial Theorem | 05 | Read more |

Unit 8 |
Sequences and Series | 05 | Read more |

Unit 9 |
Straight Lines | 05 | Read more |

Unit 10 |
Conic Section | 05 | Read more |

Unit 11 |
Introduction to Three dimensional Geometry | 10 | Read more |

Unit 12 |
Limits and Continuity | 10 | Read more |

Unit 13 |
Probability | 05 | Read more |

Total Marks |
100 |

# Unit 1

**Sets and their representations**

- identify sets as well defined collections.
- represent sets in roster and set builder form.
- identify the symbols and and understand the difference between the two.
- Conversion from set builder form to roster form and vice versa.

**Empty Set**

- • Identify empty sets (null sets).

**Singleton Set**

- Identify singleton set and frame examples

**Finite and infinite Sets**

- identify finite and infinite sets; and their respective representations.

**Equivalent and Equal Sets**

- understand meaning of equal and equivalent sets.
- differentiate between equal and equivalent sets.
- determine whether the given pair of sets is equal or not.

**Subsets**

- identify the subsets of a given set and its symbol ( )
- understand that every set has two trivial subsets - null set and the set itself.
- understand the difference between a subset and proper subset.

**Power Set**

- identify power set as set of subsets.

**Universal Set**

- identify universal set and its symbol ( ).

**Complement of a Set**

- find the complement of a subset of a given set, within a given universe.

**Intervals as Subsets of R**

- closed interval, open interval, right half open interval, left half open interval.

**Venn Diagrams**

- represent sets using venn diagrams.

**Union and Intersection of Sets**

- find the intersection of sets and union of sets.
- show the intersection and union of sets using Venn diagrams.
- identify disjoint sets and its representation using venn diagram.

**Difference of Sets**

- find the difference of sets and their representation using venn diagram.

**Laws of Operations on Sets**

- apply the following laws of algebra on sets:
- Laws of union of sets (commutative law, associative law, idempotent law, identity law)
- laws of intersection of sets
- distributive laws

- De Morgan's law

**Properties of Complement Sets**

- apply properties of complement sets

**Practical Problems on union and Intersection of Sets**

- solve practical problems on union and intersection of sets.
- apply results and solve problems on number of elements of sets using properties like.

# Unit 2

**Ordered Pairs**

- identify the equality of two ordered pairs.
- identify an ordered pair.

**Cartesian Product of Sets**

- identify a cartesian product of two non empty sets.
- identify the two sets given their cartesian product.
- find the union and intersection on cartesian products.
- find ordered triplets (R R R).
- identify the number of elements in the cartesian product of two finite sets.
- identify Cartesian product of set of all real numbers with itself.

**Definition of Relation**

- understand relation of two sets as a subset of their cartesian product.

**Arrow Diagram**

- pictorial representation of a relation between two sets.

**Domain, Codomain and Range of a Relation**

- identify domain, co-domain and range of a relation.

**Function as a Special Kind of Relation from one Set to another**

- identify function as a special kind of relation from one set to another.
- determine when a relation is a function.
- describe and write functional relationships for given problem situations.
- understand that f c R c A X A.

**Pictorial representation of a Function**

- represent functions using graphs.
- to understand that every graph does not represent a function.

**Domain, Codomain and Range of a Function**

- identify domain, co-domain and range of a function.
- finding domain and range of a given function.
- identify even and odd functions.
- find specific function values
- find the algebra of functions covering:

(f g)(x) = f(x) g(x) = g(x) f(x).

**Real valued functions and their graphs**

- recognise the following real valued functions
- constant function
- identity function
- linear function
- quadratic function
- polynomial function
- rational function
- modulus function
- signum function
- greatest integer function

# Unit 3

**Positive and negative angles**

- identify positive and negative angles.

**Measuring angles in radians and in degrees and conversion from one measure to another**

- measure angles in both degrees and in radians, and convert between these measures.

**Definition of trigonometric functions with the help of unit circle**

- define trigonometric functions with the help of unit circle.

**Sign of trigonometric functions**

- identify the change of signs of trigonometric functions in different quadrants.
- develop and apply the value of trigonometric functions at 0, /6,/4, /3, /2 radians and their multiples*.
- use the reciprocal and co-function relationships to find the values of the secant, cosecant and cotangent 0, /6, /4, /3, /2 radians value of trigonometric functions at n , where n is a positive integer

**Domain and range of trigonometric functions**

- identify the domain and range of trigonometric functions.

**Trigonometric functions as periodic functions, their amplitude, argument, period and graph **

- identify trigonometric functions as periodic functions with sine and cosine functions having a period of 2 , tangent and cotangent functions having a period of , secant and cosecant functions having a period of 2 .
- construct the graphs of trigonometric functions and describe their behaviour, including periodicity, amplitude, zeros and symmetry.

**Trigonometric functions of sum and difference of two angles **

- express sin (x ± y) and cos (x ± y) in terms of sin x, sin y, cos x and cos y.

**Express sum and difference of T-Functions as the product of T-ratios**

**Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x**

- deduce identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x, and apply them to simplify trigonometric equations.

**General solution of trigonometric equations of the type sin = sin , cos = cos and tan = tan**

- find the general solution of the trigonometric equations of the type sin = sin , cos = cos and tan = tan .

**Proof and simple applications of sine and cosine rules**

- prove the law of sines and law of cosine.
- solve for an unknown side or angle, using the law of sines or the law of cosine.
- apply law of sines and law of cosine in various problems.
- determine the area of a triangle or parallelogram, given the measure of two sides and the included angle.

# Unit 4

**Need for complex number,especially iota to be motivated by inability to solve some of quadratic equation**

- understand the need of Imaginary Quantities.
- understand the concept of iota and its application.

**Standard form of complex number**

- define a complex number (z = a+ib) and identify its real and imaginary parts
- concept of purely real and purely imaginary complex number
- get familiar with equality of complex numbers
- understand the addition and subtraction of complex numbers and its properties.

**Modulus and conjugate of complex number**

- identify the conjugate of a complex number and familiarized with its properties
- identify the modulus of a complex number and familiarized with its properties.

**Multiplication and division of complex numbers**

- understand the multiplication of complex numbers and its properties.
- understand the division of complex numbers and its properties.
- identify the multiplicative inverse or reciprocal of a complex number.

**Polar representation Of complex number Polar Representation of Complex number**

- understand the polar or trigonometrical form of a complex number.
- find the modulus of a complex number.
- find the argument of a complex number.

**Argand Plane**

- geometrical representation of a complex number.
- understand different properties of complex numbers and its representation on argand plane.
- solve different mathematical problems using argand plane.

**Statement of Fundamental theorem of algebra**

- get familiar with fundamental theorem of algebra.

**Square root of a complex number**

- find the square root of a complex number.

**Solution of quadratic equations in the complex number system **

- solve the quadratic equations in the complex number system.

**Cube root of unity and its properties**

- familiar with cube roots of unity and their properties.

**De Moivre's theorem**

- prove and apply De-Moivre's theorem.

# Unit 5

**Linear Inequations**

- understand linear inequalities.

**Algebraic solutions of linear inequations in one variable**

- find algebraic solutions of linear inequalities in one variable.
- represent the solution of linear inequalities in one variable on a number line.
- simultaneous solution of two linear inequalities algebraically as well as on number line.

**Algebraic solutions of linear in equations in two variables**

- find algebraic solutions of linear inequalities in two variables.

**Graphical solution of linear inequations in two variables**

- Solution of linear inequality in two variables and the graph of its solution set.
- Solution of system of linear inequalities in two variables and the graph of its solution set.

**Inequations solving modulus functions**

- inequalities involving modulus function.
- understand wavy curve method for 2nd degree and higher degree polynomials expressed in the form (x+a)(x+b) ...... (the number of such terms corresponding to the degree of the polynomial).

# Unit 6

**Fundamental principles of counting**

- know the fundamental addition principles of counting and apply it to find out number of ways particular event can occur.
- know the fundamental multiplication principles of counting and apply it to find out number of ways particular event can occur.

**Factorial n (n!)**

- know the meaning of factorial and its symbol.
- know how to compute factorial.
- know how to represent product of consecutive numbers in factorial.
- know how to represent product of consecutive numbers in factorial.

**Permutation**

**Combinations**

**Derivation of properties of combination**

**Types of permutations**

- linear permutations.
- circular permutation.
- restricted permutation.
- permutations when particular thing is to be included everytime.
- permutations when particular thing is never to be included.
- permutation of objects are not all different Permutation with repetition.

**Simple applications**

- solve the simple practical problems on permutation.
- solve the simple practical problems on combination.

# Unit 7

**Pascal's Triangle**

- get familiar with the Pascal's triangle.
- observe different patterns of numbers followed in pascals triangle.

**History, statement and proof of the binomial theorems for positive integral indices**

- know the binomial theorems for positive integral indices and their proof.
- expand an expression using binomial theorem.

**General and middle term in binomial expansion**

- get familiar with the general term in binomial expansion.
- get familiar with middle term in binomial expansion when number of terms are even/odd.
- get familiar with pth term from the end.

**Application of binomial theorem**

- compute simple application problems using binomial theorems.

# Unit 8

**Arithmetic Progression, Geometric Progression**

- identify an arithmetic or geometric sequence.
- find the formula for the nth term of an arithmetic sequence.
- find the formula for the nth term of a geometric sequence.
- prove a given sequence from an arithmetic progression or a geometric progression.
- determine a specified term of an arithmetic sequence.
- determine a specified term of a geometric sequence.
- generate or construct sequences from given recursive relationships.

**Arithmetic Mean**

- find the arithmetic mean.
- insert 'n' arithmetic means between 2 given numbers.

**Geometric Mean**

- find the geometric mean.
- insert 'n' geometric means between 2 given numbers.

**Sum to n terms of an A.P.**

- find the sum of finite terms of an arithmetic progression.

**Sum to n terms of a G.P.**

- find the sum of finite terms of a geometric progression.

**Infinite G.P. and its sum**

- find the sum of an infinite geometric progression.

**Relation between A.M. and G.M.**

- identify and apply the relation between arithmetic mean and geometric mean.

# Unit 9

**Brief recall of two dimensional geometry from earlier classes**

- distance between two points.
- area of triangle whose vertices are given.
- co-ordinates of a point divides the join of two given coordinates in the particular ratio.
- co-ordinates of midpoint of a line segment joining two coordinates.
- co-ordinates of centroid and incenter of a triangle.

**Shifting of origin**

- comprehend the change in equation on shifting the point of origin.

**Slope of a line**

- find the slope of a line when angle of inclination is given.
- identify the slope of a line in terms of co-ordinates of any two points on it.
- familiar with condition of parallel lines and perpendicular lines in terms of slope.
- use slopes of lines to investigate geometric relationships, including parallel lines, perpendicular lines.

**Angle between two lines**

- Angle between two lines.

**Various forms of equation of a line: parallel to axis, point slop form, slopintercept form, two point form, intercept form, and normal form**

- equation of lines parallel to the co-ordinate axis.
- form the equation of line when co-ordinates of point through which line passes and slope is given (point-slope form).
- form the equation of line when co-ordinates of two points through which line passes are given (two point form).
- familiar with intercepts of a line on the axes.
- form the equation of line making slope m and making an intercept c on y/x axis (slope intercept form).
- form the equation of line when a line cuts off intercepts a & b respectively on x and y axis (intercept form).
- form the equation of line when the length of the perpendicular on it and angle of that perpendicular is given (normal form of line).
- use different forms of a line to find out missing parameters of a line in symmetric form.

**General equation of a line**

- identify general equation and transform it in different standard forms.

**Equation of family of lines passing through the point of intersection of two lines**

- find the point of intersection of two lines.
- understand the concept of family of lines passing through the intersection of lines ll and l2 in terms of l1 + k l2 = 0.
- give the equation of lines passing through the point of intersection of two lines under given conditions.

**Distance of a point from a line**

- compute the distance of a point from a line.

**Distance between parallel lines**

- compute the distance between parallel lines.

# Unit 10

**Introduction to section of a cone**

- identify the circle, parabola, ellipse and hyperbola as cross sections of a double napped cone by a plane.

**Circle (Standard form)**

- identify the equation of a circle in standard form having the Centre (h, k) and radius r.
- equation of a circle having centre at origin and radius r.
- equation of a circle when the end points of a diameter are given.

**Circle (general form)**

**Parabola (standard form)**

- identify the standard parabola (right handed, left handed, upward and downward parabola).
- find the axis, vertex, focus, directrix and the latus rectum of the standard parabola.

**Parabola (general form)**

- identify the general equation of a parabola.
- reduction of general form of parabola to the standard form.
- find the axis, vertex, focus, directory and the latus rectum from the general equation of the parabola.
- find the equation of parabola under given condition.

**Ellipse (standard form) horizontal & vertical ellipse**

- identify the vertical and horizontal ellipse.
- find the vertices, major and minor axis, foci, directrix, centre, eccentricity and latus rectum of the vertical and horizontal ellipse.

**Ellipse (general form)**

- identify the general form of an ellipse (vertical & horizontal).
- reduction of general form of ellipse to the standard form.
- find the vertices, major and minor axis, foci, directrix, centre, eccentricity and latus rectum from the general from of ellipse.
- find the equation of an ellipse under given conditions.

**Hyperbole (standard form)**

- identify the hyperbola in standard form (also conjugate hyperbola).
- find the centre, vertices, foci, directrix, transverse and conjugate axes, eccentricity and length of latus rectum.

**Hyperbole (general form)**

- identify the general form of hyperbole.
- reduction of general form of hyperbola to standard form.
- find the centre, vertices, foci directrix, transverse and conjugate axes, eccentricity & latus rectum from the general equation of hyperbola.
- find the equation of hyperbole under given condition.

**Application of conic section**

- apply the concepts of parabola, ellipse and hyperbola in the given problems.

# Unit 11

**Co-ordinate axes and co-ordinate planes in three dimensions**

- identify co-ordinate axes in three dimensions.
- identify co-ordinate planes in three dimensions.
- find co-ordinates of a point in space.

**Distance between two points and section formula**

- find distance between two points.
- apply section formula.

**Some results on line in space**

- direction cosines of a line.
- direction ratios of a line.
- angle between two lines.

# Unit 12

**Limits and Continuity**

- Limit of function
- Fundamental theorem on limits
- Standard results on limits and their application
- Trigonometric limits
- Infinite limits
- One sided limit
- Continuity

# Unit 13

**Random experiment: outcomes, sample spaces (set representation)**

- learn the concept of random experiment, outcomes of random experiment and sample spaces.
- list the sample spaces of a random experiment.

**Events: occurrence of events, 'or', 'and', & 'not' events**

- understand the term event as a subset of sample space.
- write events/sample space for a given experiment.
- recognise 'or', 'and' & 'not' events.

**Exhaustive events, mutually exclusive events Axiomatic (set theoretic) probability**

- identify impossible events and sure events.
- Identify simple and compound events.
- identify mutually exclusive events.
- identify exhaustive events.
- get familiar with independent events, equally likely events, and complementary events*.

**Probability of an event**

- find the probability of occurrence of an event.

**Odds of an event**

- Odds in favour of an event.
- Odds against an event.

**Probability of occurrence of a complementary events**

- Find the probability of complement of an event using the relation P(E) = 1 – P( E).