Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
TARGET STUDY MATERIAL
Plus-1
Mathematics
VOL – I
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
TARGET EDUCATIONAL INSTITUTION
Target Educational institution is the one and only Entrance coaching and CBSE 10th coaching centre
at Mukkam with advanced technologies and facilities started by a group of highly qualified
teachers and professionals. Our main idea is to provide a good teaching atmosphere with our Hi-Fi
features and well qualified teachers. Along with coaching we are also providing soft skill
development classes including interactive sessions by professionals, which will definitely help the
student to explore their skill to achieve their goal.
OUR FEATURES

JEE (main),NEET,KEAM (Med/Engg entrance coaching)

Providing Entrance oriented booklets and study materials

High facility A/C Class rooms

Highly experienced and talented faculties

Projector based video classrooms

Elaborated library facility
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
THIS VOLUME INCLUDES
1.
Sets
2.
Relations and Functions
3.
Trigonometric Functions
4.
Principle of Mathematical Induction
5.
Complex Numbers
6.
Linear Inequalities
7.
Permutations and Combinations
8.
Binomial Theorem
9.
Sequences and Series
10.
Straight Lines
11.
Conic Sections
12.
Introduction to Three Dimensional Geometry
13.
Limits and Derivatives
14.
Mathematical Reasoning
15.
Statistics
16.
Probability
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
Sets
Basic Concepts :
 A set is a well-defined collection of objects. Sets are usually represented by capital
letters A, B, C, D, X, Y, Z, etc.
 The objects inside a set are called elements or members of a set. They are denoted by small
letters a, b, c, d, x, y, z, etc.
 If a is an element of a set A, then we say that “a belongs to A” and mathematically we write it
as “a ∈ A”; if b is not an element of A, then we write “b ∉ A”. E.g. If Z is a set of all integers, then
5 ∈ Z but 0.6 ∉ Z.
Representation of Sets :
There are two methods of representing a set.

Roster or tabular form: In the roster form, all the elements of a set are listed in
such a manner that different elements are separated by commas and enclosed within
braces { }.
A set of all integers greater than 5 and less than 9 will be represented in roster form
as {6, 7, 8}. However, it must be noted that in roster form, the order in which the
elements are listed is immaterial. Hence, the set {6, 7, 8} can also be written as {7, 6,
8}.

Set-builder form: In set-builder representation of a set,all the elements of the set
have a single common property that is exclusive to the elements of the set i.e., no
other element outside the set has that property.
We have learnt how to write a set of all integers greater than 5 and less than 9 in
roster form. Now let us understand how we write the same set in set-builder form. Let
us denote this set by L.
L = {x : x is an integer greater than 5 and less than 9}
Hence, in set-builder form, we describe an element of a set by a symbol x (though we
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
may use any other small letter), which is followed by a colon (:). After the colon, we
describe the characteristic property possessed by all the elements of that set.
Special Cases of Sets :
A set that does not contain any element is called an empty set or a null set or a void set.
o
E.g., A = {x : x > 2, where x is an even prime number}
Now, as you can see, no value of x will satisfy the given property as 2 is the
only even prime number and no even number greater than 2 will be a prime
number. Hence, A will be an empty set as it has no elements.
o

Note that an empty set is denoted by the symbol Φ or { }.
A set that contains only one element is called a singleton set. E.g., {3} is a singleton
set.

A set that is empty or consists of a definite number of elements is called finite. Else,
the set is called infinite.
o
A set B = {1, 3, 5} has 3 elements i.e., 1, 3 and 5. Hence, the number of
elements in the set is definite and B is a finite set. However, if we take a
set V = {all stars in universe}, then the number of elements in set V is not
defined and it is an infinite set.
o
You must note that in case we have to write an infinite set in roster form, we
cannot write all the elements in braces as the number of elements in the set is
not defined. Hence, we write the first few elements followed by three dots i.e.,
ellipses.
E.g., If X is a set of all natural numbers, then X = {1, 2, 3, 4…}
Subsets :

“A set A is said to be the subset of a set B if every element of A is also an element
of B.”
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
o
Let A = {1, 5, 6, 4, 8} and B = {5, 4, 8}. Since every element of B is also an
element of A, B is a subset of A.
o
If a is any element of A and it implies that a is also an element of B, then we
can say that A ⊂ B.
Mathematically, we can write it as A ⊂ B if a ∈ A ⇒ a ∈ B
If a set A is not a subset of set B, then it is written as A ⊄ B.
o
Note that if A is a subset of B, then it implies that all elements of A also
belong to B but it’s not necessary that all elements of B also belong to A.
o
When all the elements of a set A belong to set A, it means that set A is a
subset of itself, i.e., A ⊂ A.
o
Also, note that an empty set Φ has no elements. Hence, we say that Φ is a
subset of every set.

If A and B are two sets such that every element of A is also an element of B and
every element of B is also an element of A, then A and B are said to be equal sets.
o
E.g., Let A = {1, 2, 3} and B = {2, 3, 1}. Then, A and B are equal sets and we
can write A = B.

If two sets A and B are such that A ⊂ B and A ≠ B, then A is called a proper
subset of Band B is called the superset of A. If A and B are two equal sets,
then A and B are the improper subsets of each other.
o
Look at this example to understand concept well.
A = {1, 3, 4} B = {3, 4} C = {3, 4, 1}
As you can see, all the elements of B are also in sets A and C.
Also, B ≠ A and B ≠ C. This means that B is a proper subset of A as well
as C.
B ⊂ A and B ⊂ C
A and C are supersets of B
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
Here, A and C are equal sets. Hence, A and C are improper subsets of each
other.

The collection of all subsets of a set A is called the power set of A. It is denoted by
P(A). In P(A), every element is a set.
o
If the number of elements in set A is m, then the number of elements in the
power set of A is 2m.
i.e., nP(A) = 2m, where n(A) = m

A universal set is a set that contains all objects including itself.
o
It is denoted by “U”. A basic set of all rational numbers will be an example of a
universal set for all sets of integers because a set of integers will always be a
subset of the set of all rational numbers.
Union and Intersection of Sets

Let A and B be any two sets. The union of A and B is the set that consists of all the
elements of A, all the elements of B and the common elements taken only once.
o
The symbol ‘∪’ is used for denoting the union.
o
E.g., If A = {7, 9, 5} and B = {4, 5, 6}, then A ∪ B = {4, 5, 6, 7, 9}
o
The properties of the union of two sets are as follows:
1) A ∪ B = B ∪ A (Commutative law)

2) (A ∪ B) ∪ C = A ∪ (B ∪ C)
(Associative law)
3) A ∪ Φ = A
(Law of identity element, Φ)
4) A ∪ A = A
(Idempotent law)
5) U ∪ A = U
Law of universal set, U)
The intersection of sets A and B is the set of all elements that are common to
both A and B.
o
The symbol ‘∩’ is used for denoting the intersection.
o
E.g., If A = {4, 5, 9} and B = {2, 3, 4}, then A ∩ B = {4}
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
o
The properties of the intersection of two sets are as follows:
1) A ∩ B = B ∩ A
2) (A ∩ B) ∩ C = A ∩ (B ∩ C)
(Associative law)
3) Φ ∩ A = Φ
(Law of Φ)
4) A ∩ A = A
(Idempotent law)
5) U ∩ A = A
(Law of U)
6) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(Commutative law)
(Distributive law)
If two sets, A and B, are such that A ∩ B = Φ, i.e., they have no element in common,
then A and B are called disjoint sets.E.g., the sets {1, 2, 3} and {4, 5} are disjoint
sets.
Difference of Sets :

The difference of sets A and B in this order is the set of elements that belongs
to A but not to B.
Symbolically, we write it as A − B and read as A minus B
E.g., If A = {1, 2, 3} and B = {2, 7, 10}, then A − B = {1, 3} and B − A = {7, 10}.
o

Note that sets A − B, A ∩ B and B − A are mutually disjoint sets. This means that if
we find the intersection of any of these sets, then we will get a null set as our answer.
Application of Union and Intersection

For any two finite sets A and B, n (A ∪ B) = n (A) + n (B) − n (A ∩ B)
o
If A and B are disjoint sets, i.e., A ∩ B = Φ, then n (A ∪ B) = n (A) + n (B).
a) Suppose a set A has 55 elements and a set B has 42 elements. It is also given
that the set A ∪ B has 85 elements. Can you find the number of elements in A ∩ B?
Just look at the formula described above.
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
We know that n (A ∪ B), n (A), n (B) and we have to find n (A ∩ B).
Hence, we just substitute and solve.
85 = 55 + 42 − n (A ∩ B)
n (A ∩ B) = 12

For anythree finite sets A, B and C, we have
n (A ∪ B ∪ C) = n (A) + n (B) + n (C) − n (A ∩ B) − n (B ∩ C) − n (A ∩ C)
+ n (A ∩ B ∩ C)
Complement of a Set

Let U be the universal set and A be a subset of U. Accordingly,
the complement of A is the set of all elements of U that are not the elements of A.
o
Symbolically, we write A′ to denote the complement of A with respect to U.
o
A′ can be defined as A′ = {x : x ∈ U and x ∉ A}
o
A′ = U – A
The complement of the union of two sets is the intersection of their complements and the
complement of the intersection of two sets is the union of their complements i.e.,
(A ∪ B)′ = A′ ∩ B′
(A ∩ B)′ = A′ ∪ B′
These are also known as De Morgan’s law.
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]
Target Tower Abhilash Junction Mukkam +91495-2294073
E-mail: [email protected]