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Page 1
Relations and Functions
? A relation R from a set A to a set B is a subset of the cartesian product A × B obtained
by describing a relationship between the first element x and the second element y of the
ordered pairs in A × B.
? Function : A function f from a set A to a set B is a specific type of relation for which every
element x of set A has one and only one image y in set B. We write f : A ? B, where
f (x) = y.
? A function f : X ? Y is one-one (or injective) if
f(x
1
) = f(x
2
) ? x
1
= x
2
? x
1
, x
2
? X.
? A function f : X ? Y is onto (or surjective) if given any
y ? Y, ? x ? X such that f(x) = y.
? Many-One Function :
A function f : A ? B is called many- one, if two or more different elements of A have the
same f-image in B.
? Into function :
A function f : A ? B is into if there exist at least one element in B which is not the f -
image of any element in A.
? Many One-Onto function :
A function f : A ? R is said to be many one- onto if f is onto but not one-one.
? Many One-Into function :
A function is said to be many one-into if it is neither one-one nor onto.
? A function f : X ? Y is invertible if and only if f is one-one and onto.
Trigonometric Functions and Equations
? General Solution of the equation sin? = 0:
when sin? = 0
? = n? : n ? I i.e. n = 0, ± 1, ±2...........
General solution of the equation cos? = 0 :
when cos? = 0
? = (2n + 1)?/2, n ? I. i.e. n = 0, ±1, +2.......
General solution of the equation tan? = 0:
General solution of tan? = 0 is ? = n?; n ? I
? General solution of the equation
(a) sin? = sin? : ? = n? + (?1)
n
? ; n ? I
(b) sin? = k, where ?1 ? k ? 1.
? = n? + (?1)
n
?, where n ? I and ? = sin
?1
k
(c) cos? = cos? : ? = 2n? ± ?, n ? I
(d) cos? = k, where ?1 < k < 1.
? = 2n? ± ?, where n ? I and ? = cos
?1
k
(e) tan? = tan? : ? = n? + ? ; n ? I
(f) tan? = k, ? = n? + ?, where n ? I and ? = tan
?1
k
(g) sin
2
? = sin
2
? : ? = n? ± ?; n ? I
(h) cos
2
? = cos
2
? : ? = n? ± ? ; n ? I
(i) tan
2
? = tan
2
? : ? = n? ± ? ; n ? I
JEE Mathematics Imporatant Formulas
Page 2
Relations and Functions
? A relation R from a set A to a set B is a subset of the cartesian product A × B obtained
by describing a relationship between the first element x and the second element y of the
ordered pairs in A × B.
? Function : A function f from a set A to a set B is a specific type of relation for which every
element x of set A has one and only one image y in set B. We write f : A ? B, where
f (x) = y.
? A function f : X ? Y is one-one (or injective) if
f(x
1
) = f(x
2
) ? x
1
= x
2
? x
1
, x
2
? X.
? A function f : X ? Y is onto (or surjective) if given any
y ? Y, ? x ? X such that f(x) = y.
? Many-One Function :
A function f : A ? B is called many- one, if two or more different elements of A have the
same f-image in B.
? Into function :
A function f : A ? B is into if there exist at least one element in B which is not the f -
image of any element in A.
? Many One-Onto function :
A function f : A ? R is said to be many one- onto if f is onto but not one-one.
? Many One-Into function :
A function is said to be many one-into if it is neither one-one nor onto.
? A function f : X ? Y is invertible if and only if f is one-one and onto.
Trigonometric Functions and Equations
? General Solution of the equation sin? = 0:
when sin? = 0
? = n? : n ? I i.e. n = 0, ± 1, ±2...........
General solution of the equation cos? = 0 :
when cos? = 0
? = (2n + 1)?/2, n ? I. i.e. n = 0, ±1, +2.......
General solution of the equation tan? = 0:
General solution of tan? = 0 is ? = n?; n ? I
? General solution of the equation
(a) sin? = sin? : ? = n? + (?1)
n
? ; n ? I
(b) sin? = k, where ?1 ? k ? 1.
? = n? + (?1)
n
?, where n ? I and ? = sin
?1
k
(c) cos? = cos? : ? = 2n? ± ?, n ? I
(d) cos? = k, where ?1 < k < 1.
? = 2n? ± ?, where n ? I and ? = cos
?1
k
(e) tan? = tan? : ? = n? + ? ; n ? I
(f) tan? = k, ? = n? + ?, where n ? I and ? = tan
?1
k
(g) sin
2
? = sin
2
? : ? = n? ± ?; n ? I
(h) cos
2
? = cos
2
? : ? = n? ± ? ; n ? I
(i) tan
2
? = tan
2
? : ? = n? ± ? ; n ? I
JEE Mathematics Imporatant Formulas
? sin? + sin (? + ?) + sin (? +2?) +........ to n terms
? ?
n 1 n
sin sin
2 2
; 2n
sin / 2
? ?? ? ? ? ? ? ? ?
?? ?
? ? ? ? ? ?? ?
? ? ? ? ? ?? ?
? ?? ?
?
? cos? + cos (? + ?) + cos (? +2?) +........ to n terms
n 1 n
cos sin
2 2
; 2n
sin
2
? ?? ? ? ? ? ? ? ?
?? ?
? ? ? ? ? ?? ?
? ? ? ? ? ?? ?
? ?? ?
? ? ?
? ?
? ?
? tan
B C b c A
cot
2 b c 2
? ? ? ? ? ? ? ?
?
? ? ? ? ? ?
?
? ? ? ? ? ?
? sin
A (s b)(s c)
2 bc
? ? ? ?
?
? ?
? ?
? tan
A (s b)(s c)
2 s(s a)
? ? ? ?
?
? ?
?
? ?
? R =
a b c
2sin A 2sinB 2sinC
? ?
? R =
abc
4?
? r = 4R sin
A B C
.sin .sin
2 2 2
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? a = c cos B + b cos C
? Maximum value of a sin ? + b cos ? =
2 2
a b ? and minimum value of a sin ? + b cos
? = ?
2 2
a b ?
Inverse Trigonometric Functions
? Properties of inverse trigonometric function
? tan
?1
x + tan
?1
y =
1
1
1
x y
tan , if xy 1
1 xy
if x 0,y 0 x y
tan ,
and xy 1 1 xy
if x 0, y 0 x y
tan ,
and xy 1 1 xy
?
?
?
? ? ? ?
?
? ? ?
?
? ?
?
?
? ? ? ? ? ?
??
? ? ?
? ?
? ?
?
?
? ? ? ? ?
?
???
? ?
? ? ?
? ? ?
Page 3
Relations and Functions
? A relation R from a set A to a set B is a subset of the cartesian product A × B obtained
by describing a relationship between the first element x and the second element y of the
ordered pairs in A × B.
? Function : A function f from a set A to a set B is a specific type of relation for which every
element x of set A has one and only one image y in set B. We write f : A ? B, where
f (x) = y.
? A function f : X ? Y is one-one (or injective) if
f(x
1
) = f(x
2
) ? x
1
= x
2
? x
1
, x
2
? X.
? A function f : X ? Y is onto (or surjective) if given any
y ? Y, ? x ? X such that f(x) = y.
? Many-One Function :
A function f : A ? B is called many- one, if two or more different elements of A have the
same f-image in B.
? Into function :
A function f : A ? B is into if there exist at least one element in B which is not the f -
image of any element in A.
? Many One-Onto function :
A function f : A ? R is said to be many one- onto if f is onto but not one-one.
? Many One-Into function :
A function is said to be many one-into if it is neither one-one nor onto.
? A function f : X ? Y is invertible if and only if f is one-one and onto.
Trigonometric Functions and Equations
? General Solution of the equation sin? = 0:
when sin? = 0
? = n? : n ? I i.e. n = 0, ± 1, ±2...........
General solution of the equation cos? = 0 :
when cos? = 0
? = (2n + 1)?/2, n ? I. i.e. n = 0, ±1, +2.......
General solution of the equation tan? = 0:
General solution of tan? = 0 is ? = n?; n ? I
? General solution of the equation
(a) sin? = sin? : ? = n? + (?1)
n
? ; n ? I
(b) sin? = k, where ?1 ? k ? 1.
? = n? + (?1)
n
?, where n ? I and ? = sin
?1
k
(c) cos? = cos? : ? = 2n? ± ?, n ? I
(d) cos? = k, where ?1 < k < 1.
? = 2n? ± ?, where n ? I and ? = cos
?1
k
(e) tan? = tan? : ? = n? + ? ; n ? I
(f) tan? = k, ? = n? + ?, where n ? I and ? = tan
?1
k
(g) sin
2
? = sin
2
? : ? = n? ± ?; n ? I
(h) cos
2
? = cos
2
? : ? = n? ± ? ; n ? I
(i) tan
2
? = tan
2
? : ? = n? ± ? ; n ? I
JEE Mathematics Imporatant Formulas
? sin? + sin (? + ?) + sin (? +2?) +........ to n terms
? ?
n 1 n
sin sin
2 2
; 2n
sin / 2
? ?? ? ? ? ? ? ? ?
?? ?
? ? ? ? ? ?? ?
? ? ? ? ? ?? ?
? ?? ?
?
? cos? + cos (? + ?) + cos (? +2?) +........ to n terms
n 1 n
cos sin
2 2
; 2n
sin
2
? ?? ? ? ? ? ? ? ?
?? ?
? ? ? ? ? ?? ?
? ? ? ? ? ?? ?
? ?? ?
? ? ?
? ?
? ?
? tan
B C b c A
cot
2 b c 2
? ? ? ? ? ? ? ?
?
? ? ? ? ? ?
?
? ? ? ? ? ?
? sin
A (s b)(s c)
2 bc
? ? ? ?
?
? ?
? ?
? tan
A (s b)(s c)
2 s(s a)
? ? ? ?
?
? ?
?
? ?
? R =
a b c
2sin A 2sinB 2sinC
? ?
? R =
abc
4?
? r = 4R sin
A B C
.sin .sin
2 2 2
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? a = c cos B + b cos C
? Maximum value of a sin ? + b cos ? =
2 2
a b ? and minimum value of a sin ? + b cos
? = ?
2 2
a b ?
Inverse Trigonometric Functions
? Properties of inverse trigonometric function
? tan
?1
x + tan
?1
y =
1
1
1
x y
tan , if xy 1
1 xy
if x 0,y 0 x y
tan ,
and xy 1 1 xy
if x 0, y 0 x y
tan ,
and xy 1 1 xy
?
?
?
? ? ? ?
?
? ? ?
?
? ?
?
?
? ? ? ? ? ?
??
? ? ?
? ?
? ?
?
?
? ? ? ? ?
?
???
? ?
? ? ?
? ? ?
? tan
?1
x ? tan
?1
y =
1
1
1
x y
tan , if xy 1
1 xy
x y
tan , if x 0, y 0 and xy 1
1 xy
x y
tan , if x 0, y 0 and xy 1
1 xy
?
?
?
? ? ? ?
??
? ? ?
?
? ?
?
?
? ? ? ?
?? ? ? ??
? ? ?
?
? ?
?
?
? ? ?
?
??? ? ? ??
? ?
? ?
? ? ?
? sin
?1
x + sin
?1
y =
2 2
1 2 2
2 2
1 2 2
2 2
1 2 2 2 2
if 1 x, y 1 and x y 1
sin {x 1 y y 1 x } ,
or if xy 0 and x y 1
if 0 x, y 1
sin {x 1 y y 1 x } ,
and x y 1
sin {x 1 y y 1 x } , if 1 x, y 0 and x y 1
?
?
?
? ? ? ? ? ?
? ? ?
?
? ? ?
?
?
? ?
?
?? ? ? ?
?
? ?
?
?
??? ? ? ? ? ? ? ? ?
?
?
?
? cos
?1
x + cos
?1
y =
1 2 2
1 2 2
cos {xy 1 x 1 y } , if 1 x, y 1 and x y 0
2 cos {xy 1 x 1 y }, if 1 x, y 1 and x y 0
?
?
?
? ? ? ? ? ? ? ?
?
?
?? ? ? ? ? ? ? ? ? ?
?
1 2
1 1 2
1 2
1 1
sin (2x 1 x ) , if x
2 2
1
2sin x sin (2x 1 x ) , if x 1
2
1
sin (2x 1 x ) , if 1 x
2
?
? ?
?
?
? ? ? ?
?
?
?
? ?? ? ? ?
?
?
?
??? ? ? ? ??
?
?
1
2
1 1
2
1
2
2x
tan , if 1 x 1
1 x
2x
2tan x tan , if x 1
1 x
2x
tan , if x 1
1 x
?
? ?
?
? ? ?
? ? ?
? ? ?
?
? ?
?
?
? ? ?
? ?? ?
?
? ?
?
? ?
?
?
? ?
??? ?? ?
? ?
? ? ? ? ?
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FAQs on Important Formulas: Functions With Examples - Mathematics (Maths) Class 12 - JEE
| 1. What are the different types of functions in commerce? |  |
Ans. In commerce, there are various types of functions, including:
1. Marketing Functions: These functions involve activities such as product development, pricing, promotion, and distribution to satisfy customer needs.
2. Financial Functions: These functions focus on managing and optimizing financial resources, including activities such as budgeting, financial planning, investment decisions, and capital structure management.
3. Human Resource Functions: These functions deal with managing the organization's workforce, including recruitment, selection, training, performance appraisal, and employee relations.
4. Production Functions: These functions involve activities related to the creation and delivery of goods or services, such as production planning, sourcing materials, quality control, and inventory management.
5. Accounting Functions: These functions encompass activities like bookkeeping, financial reporting, auditing, taxation, and cost control to ensure accurate financial records and compliance with regulations.
| 2. What is an example of a marketing function in commerce? | |
Ans. An example of a marketing function in commerce is product development. This function involves researching, designing, and creating new products or improving existing ones to meet customer demands and preferences. It includes activities such as market research, identifying consumer needs, conceptualizing product ideas, prototyping, and testing. The goal of product development is to introduce innovative and competitive products that can generate sales and contribute to the company's growth.
| 3. Can you provide an example of a financial function in commerce? | |
Ans. One example of a financial function in commerce is budgeting. Budgeting involves the process of planning and allocating financial resources for various activities within an organization. It includes estimating income and expenses, setting financial targets, and monitoring actual performance against the budget. Budgeting helps in financial planning, controlling costs, prioritizing investments, and ensuring the efficient utilization of resources. It is a crucial financial function that assists in achieving the organization's goals and objectives.
| 4. What is an example of a human resource function in commerce? | |
Ans. Recruitment is an example of a human resource function in commerce. This function involves attracting, selecting, and hiring suitable candidates to fill job vacancies within an organization. It includes activities such as job analysis, job posting, screening resumes, conducting interviews, and making job offers. Recruitment plays a vital role in ensuring that the organization has the right talent and skills to meet its objectives. It aims to find the best-fit candidates who align with the organization's culture and contribute to its success.
| 5. Can you provide an example of a production function in commerce? | |
Ans. An example of a production function in commerce is inventory management. This function involves monitoring and controlling the flow of goods or materials within an organization's supply chain. It includes activities such as forecasting demand, ordering and receiving inventory, storing, tracking, and optimizing inventory levels. Effective inventory management ensures that the organization has an adequate stock of products to meet customer demand while minimizing storage costs and the risk of stockouts. It plays a crucial role in maintaining smooth production and timely delivery of goods or services.
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