Dear students and teachers,
This material is prepared by a team of higher secondary
Mathematics teachers in our District for Malappuram District
Panchayath Vijayabheri Programme, based on the focus point in
Mathematics formulated by SCERT for the academic year 2020-21.
This is useful for students of all levels.
This book includes short notes, important results, solved problems
and self-practice problems of every chapter in plus two Mathematics.
We hope that this humble attempt will build confidence among
students and with the help of teachers they are able to bring off atleast
pass mark in Mathematics.
With regards,
Team Mathematics
Malappuram
Prepared by:
• MOHAMMED ARIF P [GHSS Anamangad 9745866628]
• ANANDA KUMAR MK [SHM Govt VHSS Edavanna 9446543161]
• ASHFAQ SHANAVAS T.P [ASM HSS Velliyanchery 9846635762]
• ABOOBACKER A [GBHSS Malappuram 9745894348]
• ANEESH C [PKMM HSS Edarikode 9946007201]
• SAJITH K [MSM HSS Kallingaparamba 9567689927]
• MUHAMMED ILIYAS P [DUHSS Panakkad 9447927713]
• SUMA P [MSP HSS Malappuram [9446022444]
• Dr. SREEJA M [GHSS PANDIKKAD 9061318212]
വിജയഭേരി , മലപ്പുറം ജിലലാ പഞ്ചായത്ത്
1
, CONTENTS
Page
No. : Chapter
No.
1 RELATIONS AND FUNCTIONS 3
2 INVERSE TRIGONOMETRIC FUNCTIONS 8
3 MATRICES 11
4 DETERMINANTS 19
5 CONTINUITY AND DIFFERENTIABILITY 24
6 APPLICATION OF DERIVATIVES 30
7 INTEGRALS 33
8 APPLICATION OF INTEGRALS 38
9 DIFFERENTIAL EQUATIONS 42
10 VECTOR ALGEBRA 46
11 THREE-DIMENSIONAL GEOMETRY 51
12 LINEAR PROGRAMMING PROBLEMS 56
13 PROBABILITY 59
വിജയഭേരി , മലപ്പുറം ജിലലാ പഞ്ചായത്ത്
2
, 1
RELATIONS AND FUNCTIONS
KEY NOTES
❖ Functions (Mappings) : A relation R from a set A to another set B is called a
function or mapping if it satisfies the following conditions
(i) Every element in A should have an image in B
(ii) For any element in A, there should not be more than one image in B
Eg : Consider the sets A={1,2,3} and B={4,5,6,7} . Let R be a relation from A to
B such that R={(1,4), (2,5), (3,6)}. The arrow diagram of the above relation is
given by
A B
4
1
5
2
6
3 7
The relation R is a function since every element in A has image and no
element has more than one image.
Here, domain = {1,2,3} and Range = {4,5,6} , co-domain is {4,5,6,7}
Note :
If the elements of a set A of a given function are 𝑥1 , 𝑥2 , ⋯ ⋯ 𝑥𝑛 , the
images are represented by 𝑓(𝑥1 ), 𝑓(𝑥2 ), ⋯ ⋯ 𝑓(𝑥𝑛 )
❖ Types of functions
➢ One-one function
A function 𝑓 ∶ 𝐴 → 𝐵 is said to be one-one if different elements have different
images
i.e., if 𝑥1 and 𝑥2 are different then 𝑓(𝑥1 ) and 𝑓(𝑥2 ) are different.
In other words, if 𝑓(𝑥1 ) and 𝑓(𝑥2 ) are same, then 𝑥1 and 𝑥2 must be same.
or
𝒇(𝒙𝟏 ) = 𝒇(𝒙𝟐 ) ⟹ 𝒙𝟏 = 𝒙𝟐
➢ Many-one function
A function which is not one-one is called many one function.
വിജയഭേരി , മലപ്പുറം ജിലലാ പഞ്ചായത്ത്
3
, ➢ On-to function
If every elements in B have a preimage in A, then the mapping is said to be on-to.
In an on-to mapping; Range of 𝒇 = co-domain
➢ In-to function
A function which is not on-to is called into function.
In an in-to mapping; Range of 𝑓 ⊂ co-domain
➢ Bijective function
A function which is both one one and onto is called bijective function
• To find out whether a function is one-one or not, we use the following method
(i) Draw the graph of the function
(ii) Draw lines parallel to x-axis
(iii) If any of the above line intersects the function at more than one point, it is not
a one-one function.
(iv) If all the lines intersect the curve in at least one point, the function is on-to
❖ Inverse of a function
If 𝑓: A → B is a bijective function, then 𝑓 −1 ∶ 𝐵 → 𝐴 is the inverse of the 𝑓
defined by 𝑓 −1 (𝑦) = 𝑥 if and only if 𝑓(𝑥) = 𝑦
• Functions having inverse are called invertible functions
• A function is invertible if and only if it is bijective
വിജയഭേരി , മലപ്പുറം ജിലലാ പഞ്ചായത്ത്
4
This material is prepared by a team of higher secondary
Mathematics teachers in our District for Malappuram District
Panchayath Vijayabheri Programme, based on the focus point in
Mathematics formulated by SCERT for the academic year 2020-21.
This is useful for students of all levels.
This book includes short notes, important results, solved problems
and self-practice problems of every chapter in plus two Mathematics.
We hope that this humble attempt will build confidence among
students and with the help of teachers they are able to bring off atleast
pass mark in Mathematics.
With regards,
Team Mathematics
Malappuram
Prepared by:
• MOHAMMED ARIF P [GHSS Anamangad 9745866628]
• ANANDA KUMAR MK [SHM Govt VHSS Edavanna 9446543161]
• ASHFAQ SHANAVAS T.P [ASM HSS Velliyanchery 9846635762]
• ABOOBACKER A [GBHSS Malappuram 9745894348]
• ANEESH C [PKMM HSS Edarikode 9946007201]
• SAJITH K [MSM HSS Kallingaparamba 9567689927]
• MUHAMMED ILIYAS P [DUHSS Panakkad 9447927713]
• SUMA P [MSP HSS Malappuram [9446022444]
• Dr. SREEJA M [GHSS PANDIKKAD 9061318212]
വിജയഭേരി , മലപ്പുറം ജിലലാ പഞ്ചായത്ത്
1
, CONTENTS
Page
No. : Chapter
No.
1 RELATIONS AND FUNCTIONS 3
2 INVERSE TRIGONOMETRIC FUNCTIONS 8
3 MATRICES 11
4 DETERMINANTS 19
5 CONTINUITY AND DIFFERENTIABILITY 24
6 APPLICATION OF DERIVATIVES 30
7 INTEGRALS 33
8 APPLICATION OF INTEGRALS 38
9 DIFFERENTIAL EQUATIONS 42
10 VECTOR ALGEBRA 46
11 THREE-DIMENSIONAL GEOMETRY 51
12 LINEAR PROGRAMMING PROBLEMS 56
13 PROBABILITY 59
വിജയഭേരി , മലപ്പുറം ജിലലാ പഞ്ചായത്ത്
2
, 1
RELATIONS AND FUNCTIONS
KEY NOTES
❖ Functions (Mappings) : A relation R from a set A to another set B is called a
function or mapping if it satisfies the following conditions
(i) Every element in A should have an image in B
(ii) For any element in A, there should not be more than one image in B
Eg : Consider the sets A={1,2,3} and B={4,5,6,7} . Let R be a relation from A to
B such that R={(1,4), (2,5), (3,6)}. The arrow diagram of the above relation is
given by
A B
4
1
5
2
6
3 7
The relation R is a function since every element in A has image and no
element has more than one image.
Here, domain = {1,2,3} and Range = {4,5,6} , co-domain is {4,5,6,7}
Note :
If the elements of a set A of a given function are 𝑥1 , 𝑥2 , ⋯ ⋯ 𝑥𝑛 , the
images are represented by 𝑓(𝑥1 ), 𝑓(𝑥2 ), ⋯ ⋯ 𝑓(𝑥𝑛 )
❖ Types of functions
➢ One-one function
A function 𝑓 ∶ 𝐴 → 𝐵 is said to be one-one if different elements have different
images
i.e., if 𝑥1 and 𝑥2 are different then 𝑓(𝑥1 ) and 𝑓(𝑥2 ) are different.
In other words, if 𝑓(𝑥1 ) and 𝑓(𝑥2 ) are same, then 𝑥1 and 𝑥2 must be same.
or
𝒇(𝒙𝟏 ) = 𝒇(𝒙𝟐 ) ⟹ 𝒙𝟏 = 𝒙𝟐
➢ Many-one function
A function which is not one-one is called many one function.
വിജയഭേരി , മലപ്പുറം ജിലലാ പഞ്ചായത്ത്
3
, ➢ On-to function
If every elements in B have a preimage in A, then the mapping is said to be on-to.
In an on-to mapping; Range of 𝒇 = co-domain
➢ In-to function
A function which is not on-to is called into function.
In an in-to mapping; Range of 𝑓 ⊂ co-domain
➢ Bijective function
A function which is both one one and onto is called bijective function
• To find out whether a function is one-one or not, we use the following method
(i) Draw the graph of the function
(ii) Draw lines parallel to x-axis
(iii) If any of the above line intersects the function at more than one point, it is not
a one-one function.
(iv) If all the lines intersect the curve in at least one point, the function is on-to
❖ Inverse of a function
If 𝑓: A → B is a bijective function, then 𝑓 −1 ∶ 𝐵 → 𝐴 is the inverse of the 𝑓
defined by 𝑓 −1 (𝑦) = 𝑥 if and only if 𝑓(𝑥) = 𝑦
• Functions having inverse are called invertible functions
• A function is invertible if and only if it is bijective
വിജയഭേരി , മലപ്പുറം ജിലലാ പഞ്ചായത്ത്
4
