# JEE Main and JEE Advanced Syllabus of Algebra

Algebra is one of the building blocks of Mathematics in __IIT JEE
examination__. In fact, in the preparation of JEE this is the starting
point. Algebra is a very scoring and an easy portion in the Mathematics syllabus
of JEE. Though Algebra begins with Sets and Relations but we seldom get any direct
question from this portion. Functions can be said to be a prerequisite to Calculus
and hence it is critical in __IIT JEE preparation__. Sequence
and series is another section which is mixed with other concepts and then asked
in the examination. Quadratic equations fetch direct questions too and are also
easy to grasp. Binomial Theorem is also a marks fetching topic as the questions
on this topic are quite easy. Permutations and Combinations along with Probability
is the most important section in Algebra. IIT JEE exam fetches a lot of questions
on them. Those who get good IIT JEE rank always do well in this section. Complex
Numbers are also important as this fetches question in the IIT JEE exam almost every
year. Matrices and Determinant mostly give direct question and there are no twist
and turns in the questions based on them.

We cover some of these contents here in brief as they have been discussed in detail
in the coming sections:

**Sets**

A set is a well-defined collection
of distinct objects. The different members of a set are called the elements of the
set. Moreover all the elements are unique.

Example: {1, 2, 3, 6} represents a set of numbers less than 10. Note that there
is no repetition of elements in a set.

**Relations and Functions**

A relation is a set of ordered pairs. A function is a relation for
which each value from the set the first components of the ordered pairs is associated
with exactly one value from the set of second components of the ordered pair.

The first elements in the ordered pairs i.e. the x values form the domain while
the second elements i.e. the y-values form the range. But, only the elements used
by the relation are counted in the range.

The figure given below describes a mapping which shows a relation from set A to
set B. As is clear from the figure, the relation consists of ordered pairs (1, 2),
(3, 2), (5, 7), and (9, 8). The domain is the set {1, 3, 5, 9} and the range which
is the dependent variable is the set {2, 7, 8}. Note that 3, 5, 6 are not a part
of the range as they are not associated to any member of the first set.

**Example:** f(x) = x/2 is a function because for every value
of x we get another value x/2, so f(2) = 1

f (3) = 3/2

f (-6) = -3

**Quadratic Equations: **An equation of the form ax^{2}
+ bx +c = 0

where x represents the variable and a,b,c are constants with a not equal to zero.

If a = 0, the equation becomes linear and is no more a quadratic equation. An
equation must have a second degree term in order to be a quadratic equation.

We give the formula of solving the quadratic equation:

The general form of quadratic equation is ax^{2}+bx+ c

Dividing by a, we obtain

x^{2} + b x/a = -c /a

(x+ b/2a)^{2} = -c/a + b^{2} / 4a^{2} = (b^{2}-4ac)
/ 4a^{2}

x+ b/2a = ±√b^{2}-4ac / 2a

Solving this for x we get

x = (-b±√b^{2}-4ac) / 2a

The above equation is called the quadratic formula.

**Binomial Theorem: **The binomial theorem is used for expanding
binomial expressions (a+b) raised to any given power without direct multiplication.
Mathematically a binomial theorem can be defined as the theorem that gives the expansion
of any binomial raised to a positive integral power say n. Such an expansion contains
(n+1) terms.

The general expression for it is

(* x* + *a*)* ^{n}*
=

*x*+

^{n}*nx*

^{n-1}

*a*+ [

*n*(

*n*-1)/2]

*x*

^{n}^{-2}

*a*

^{2}+…+ (

*)*

^{n}_{k}*x*

^{n}^{-}

*+ … +*

^{k}a^{k}*a*, where (

^{n}*) =*

^{n}_{k}*n*!/(

*n-k*)!

*k*!, the number of combinations of

*k*items selected from

*n.*

**Permutation is an ordered arrangement of the numbers, terms, etc., of a set into specified groups. The number of permutations of n objects taken r at a time is given by n! / (n-r)! The permutations of a, b, and c, taken two at a time, are ab, ba, ac, ca, bc, cb.**

__Permutation__
** Combinations: **A combination is also a
way selecting certain things out of a larger group. But here order does not matter
unlike permutation.

For example: If we need to form combination of two out f given three balls as in the previous case, only three cases are possible. As in the previous figure the first two constitute the same case in combination. Similarly the middle two and the last two also represent the same case.

The number of combinations of n objects taken r at a time is given by n! / r! (n-r)!

**Complex Numbers:**A complex number is a combination of a real and a imaginary number. They are written as a+ib, where a and b are real and I is an imaginary number with value √-1. Even 0 is also a complex number as 0 = 0 + 0i. Examples: 1+i, 2-3i, 6i, 3.