·

**SETS**
·

**RELATIONS**
·

**FUNCTIONS**
·

**MEASUREMENT OF ANGLES**
·

**TRIGONOMETRIC FUNCTIONS**
·

**GRAPHS OF TRIGONOMETRIC FUNCTIONS**
·

**TRIGONOMETRIC RATIOS OF COMPOUND ANGLES**
·

**TRANSFORMATION****FORMULA E**
·

**TRIGONOMETRIC RATIOS OF MULTIPLE AND SUB MULTIPLE ANGLES**
·

**SINE AND COSINE FORMULA E AND THEIR APPLICATION**
·

**TRIGONOMETRIC EQUATION**
·

**MATHEMATICAL INDUCTION**
·

**COMPLEX NUMBERS**
·

**QUADRATIC EQUATIONS**
·

**LINEAR****IN EQUATIONS**
·

**PERMUTATIONS**
·

**COMBINATIONS**
·

**BINOMIAL THEOREM**
·

**ARITHMETIC PROGRESSIONS**
·

**GEOMETRIC PROGRESSIONS**
·

**SOME SPECIAL SERIES**
·

**BRIEF REVIEW OF CARTESIAN SYSTEM OF RECTANGULAR CO-ORDINATES**
·

**THE STRAIGHT LINES**
·

**THE CIRCLES**
·

**PARABOLA**
·

**ELLIPSE**
·

**HYPERBOLA**
·

**INTRODUCTION TO 3-D COORDINATE GEOMETRY**
·

**LIMITS**
·

**DERIVATIVES**
·

**MATHEMATICAL REASONING**
·

**STATISTICS**
·

**PROBABILITY****Mathematics for class XI**

Mathematics is one of the most important subjects which not
only decides the careers of many a young students but also enhances their
ability of analytically and rational thinking.

**1. SETS**

It is
well known fact that any attempt to define a set has always led mathematicians
to unsurmountable difficulties .For example, suppose one defines the term set
as “a well defined collection of objects”. One may then ask what is meant by a
collection . if one answers that a collection is an aggregate is a class of
things. What is then an aggregate? Perhaps then one may define that an
aggregate is a class of things . what is then a class? Now, one may define a
class as a collection. In this manner question after question, since our
language is finite, we find that after some time we will have to use some words
which have already been questioned. The definition thus becomes circular and
worthless. Thus, mathematicians realized that there must be some undefined (or
primitive ) terms. In this chapter, we start with two undefined (or primitive)
terms – “element” and “set”. We assume that the words “set” is synonymous with
the words “collection”, “aggregate”, “class” and is comprised of elements. The
words “element”, “object”, “member” are synonymous.]

Throughout
this chapter we shall denote sets by capital alphabets e.g. A,B,C,X,Y,Z etc. and
the elements by small alphabets e.g.
a,b,c,x,y,z etc.

The following
are some illustrations of sets:

**ILLUSTRATION 1**The collection of vowels in English alphabets. This set contains five elements, namely, a,e,i,o,u.

**ILLUSTRATION 2**The collection of first five prime natural numbers is a set containing the elements 2,3,5,7,11.

**ILLUSTRATION 3**The collection of all states in the Indian Union is a set.

**ILLUSTRATION 4**The collection of past presidents of the Indian union is a set.

**ILLUSTRATION 5**The collection of cricketers in the world who were out for 99 runs in a test match is a set

**ILLUSTRATION 6**The collection of a good cricket players of India is not a set, since the term “good player is vague and it is not well defined”.

Sets are
listed below:

Z : for the set integers .

Z+ :
for the set of all positive integers.

Q : for the
set of all rational numbers.

Q+ : for
the set of all positive rational numbers.

R : for the set of all real numbers.

R+: for
the set of all positive real numbers.

C : for the set of all complex numbers.

In this chapter, we shall study about one of the
most important concepts in mathematics known as function. Functions from one of
most important building blocks of mathematics. The word “Functin” is derived from a Latin word
meaning operation and the words mapping and map are synonyms to it. Function play
a very important role in differential and integral calculus which will be
studied in XII class. In this chapter, we shall introduce the concept of
afuntion as a correspondence between two sets. We shall also study function as
a relation from one set to the other set.

**2. RELATIONS**

We have
discussed various operations on sets to create more sets out of given sets. In
this chapter, we shall study one more operation which is known as the Cartesian
product of sets. This will finally enable us to introduce the concept of
relation.

**3. FUNCTIONS**

**4. MEASUREMENT OF ANGLES**

The word ‘trigonometry’ is
derived from two Greek words: (1) trigonon and, (2) metron. The word trigonon
means a triangle and the word metron means a measure. Hence, trigonometry means
the science of measuring triangles. In broader sense it is that branch of
mathematics which deals with the measurement of the sides and the angles of a
triangle and the problems allied with angles.

**5. TRIGONOMETRIC FUNCTIONS**

In the present chapter, we
will first introduce trigonometric ratios which are also known as trigonometric
functions

and then the identities involving them.

**6. GRAPHS OF TRIGONOMETRIC FUNCTIONS**

We have already studied in
previous chapter that all trigonometric functions are periodic. For example, sine
and cosine functions are periodic with period 2, while
tangent and cotangent functions are periodic with period. We know
that if

*f*(x) is a periodic function with period T and a > 0, then*f*(ax + b) is periodic with period T / a. therefore, sin(ax + b ) and cos ax are periodic functions with period 2 /a. if the graph of a periodic function with period T is to be drawn in a given interval, then it is sufficient to draw its graph only in an interval of length T. because, once it is drawn in one such interval, it can be easily drawn completely by reputing it over the intervals of lengths T. The amplitude of a function is defined as the greatest numerical value which it can attain.
Using the knowledge acquired in
the above discussion let us now draw the graphs of various trigonometric
functions.

**7. TRIGONOMETRIC RATIOS OF COMPOUND ANGLES**

The algebraic sums of two or
more angles are generally called compound angles and the angles are known as
the constituent angles.

In this chapter, we shall
derive formulae which will express the trigonometric ratios of compound angles
in terms of trigonometric ratios of constituent angles.

In this chapter, we will establish two sets of transformation formulae: one to transform the products of two sines or two consines or one sine and one consine into the sum or difference of two sines or two consines and the other two convert the sum or difference of two sines or two consines in the product of two sines or two consines or one sine and one consine. These two sets of formulae are of fundamental importance and one should have thorough acquaintance with these formulae.

**8. TRANSFORMATION FORMULAE**

In this chapter, we will establish two sets of transformation formulae: one to transform the products of two sines or two consines or one sine and one consine into the sum or difference of two sines or two consines and the other two convert the sum or difference of two sines or two consines in the product of two sines or two consines or one sine and one consine. These two sets of formulae are of fundamental importance and one should have thorough acquaintance with these formulae.

**FORMULAE TO TRANSFORM THE PRODUCT INTO SUM OR DIFFERENCE**

In the previous chapter we have derived the following
formulae:

sin A cos B + cos A sin B = sin (A+B) .… (1)

sin A cos B - cos A sin B = sin (A-B) ......(2)

cos A cos B - sin A sin B = cos (A+B) ….(3)

cos A cos B + sin A sin B= cos (A-B) ….(4)

**9. TRIGONOMETRIC RATIOS OF MULTIPLE AND SUBMULTIPLE ANGLES**

In this chapter, we intend to
express the trigonometric ratios of multiple angles 2A, 3A,4A,… etc. in terms
of trigonometric ratios of angle A and the trigonometric ratios of angle A in
terms of the trigonometric ratios of sub-multiple angles A / 2,A /3,A /4…etc. These
results will be used to find the trigonometric ratio of some important angles
viz. 18

^{o }, 36^{ o }, 54^{o }^{ }etc.**10. SINE AND COSINE FORMULAE AND THEIR APPLICTIONS**

In any triangle the three sides
and the three angles are generally called the elements of the triangle. A triangle
which does not contain a right angle is called an

*oblique triangle*.
In addition to these relations,
the elements of triangle are connected by some trigonometric relations. We intend
to discuss those relations in the sections to follow of the chapter.

**11. TRIGONOMETRIC EQUATIONS**

Trigonometric equations The
equations containing trigonometric of unknown angles are known as

*trigonometric equations*.**12. MATHEMATICAL INDUCTION**

**STATEMENTS**

A sentence or description which
can be judged to be true is called a statement.

**MATHEMATICAL STATEMENTS**

Statements involving mathematical
relations are known as the

*mathematical statements.*

**13. COMPLEX NUMBERS**

If a, b are natural numbers such that
a > b, then the equation x + a = b is not solvable in N, the set of natural
numbers i.e. there is no natural number satisfying the equation x + a = b. So,
the set of natural numbers is extended to form the set

*I*of integers in which every equation of the form x + a = b; a, b N is solvable.**14. QUADRATIC EQUATIONS**

In earlier classes, we have studied about quadratic
equations with real coefficients and real roots only. In this chapter, we shall
study about quadratic equations with real coefficients and complex roots. We shall
also discuss quadratic equations with complex coefficients and their solutions
in the complex number system.

**15. LINEAR INEQUATIONS**

In this chapter, we will study linear in equations in one
and two variables. The knowledge of linear in equations is very helpful in
solving problems in Science, Mathematics, Engineering, Linear programming etc.

**INEQUATIONS**a statement involving (s) and the sign of inequality viz, > ,< ,or is called an inequation or an inequality.**16. PERMUTATIONS**

In these sections, we shall introduce the term and
notation of factorial which will be often used in this chapter and the next
three chapters.

**FACTORIAL**The continued product of first n natural numbers is called the “ n factorial” and is denoted by n!.

**17. COMBINATIONS**

In the previous chapter, we have studied arrangements of
a certain numbers of objects by taking some of them or all at a time. Most of
the times we are not interested in arranging the objects. In other words, we do
not want to specify the ordering of selected objects.

**Each of the different selections made by taking some or all of a number of objects, irrespective of their arrangements is called a combination.**

*COMBINATIONS***18. BINOMIAL THEOREM**

An algebraic expression containing two terms is called a
binomial expression.

Similarly, an algebraic expression containing three terms
is called a

*trinomial*. In general, expressions containing more than two terms two terms are known as multinomial expressions.

**19. ARITHMETIC PROGRESSIONS (A.P)**

A
sequence is called an arithmetic progression if the difference of a term and
the previous term is always same.

**20. GEOMETRIC PROGRESSIONS**

A
sequence of non-zero numbers is called a geometric progression (abbreviated as
G.P.) if the ratio of a term and the term preceding to it is always a constant
quantity.

The
constant ratio is called the common ratio of the G.P.

**21. SOME SPECIAL SERIES**

In
this chapter, we intend to discuss the sum to

*n*terms of some other special other special series viz. series of natural numbers, series of square of natural numbers, series of cubes of natural numbers etc.**22. BRIEF REVIEW OF CARTESIAN SYSTEM OF RECTANGULAR CO-ORDINATES**

**23. THE STRAIGHT LINES**

A
straight line is a curve such that every point on the line segment joining any
two points on it lies on it.

The fixed point is called the center of the circle and the constant distance is called the radius of the circle.

**24. THE CIRCLE****A circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that place is always constant.**

The fixed point is called the center of the circle and the constant distance is called the radius of the circle.

**25. PARABOLA****A conic section, as the name implies, is a section cut-off from a circular (not necessary a right circular) cone by a plane in various ways. The shape of the section depends upon the position of the cutting plane.**

**26. ELLIPSE****An**

*ellipse*is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point*(called focus)*in the same plane to its distance from a fixed straight line*(called directrix)*is always constant which is always less than unity.**27. HYPERBOLA**

A hyperbola is the locus of
a point in a plane which moves in the plane in such a way that the ratio of its
distance from a fixed point

*(called focus)*in the same plane to its distance from a fixed line*(called directrix)*is always constant which is always greater than unity.**28. INTRODUCTION TO THREE DIMENSIONAL COORDINATE GEOMETRY**

**29. LIMITS****30. DERIVATIVES**

**31. MATHEMATICAL REASONING**

In this chapter, we shall
learn about some basics of mathematical reasoning. As all of us know that the
main asset that makes humans far more superior than the other species is the
ability to reasoning. The ability of reasoning varies from person to person. Also,
it is the ability of reasoning which makes one person superior than the other. In
this chapter, we shall discuss the process of reasoning especially in the
context of mathematics. In mathematical language, there are two kinds of reasoning.

**32. STATISTICS**

**33. PROBABILITY**