Engineering Mathematics for B.Tech Students

Ranjit Tutorials


Textbook, 2014

82 Pages, Grade: 13


Free online reading

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ENGINEERING MATHEMATICS
B.Tech,Semester-II
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
By
Ranjit Singh
Alumnus of Panjab University,Chandigarh
Associate Professor
CGC Technical Campus,Jhanjeri(Mohali)
Department of Applied Sciences
Ex-Head
Department of Applied Sciences
GGS College of Modern Technology,Kharar(Mohali)

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ORDINARY DIFFERENTIAL EQUATIONS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
rt
ORDINARY
DIFFERENTIAL EQUATIONS
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
1. Exact differential equation
2. Equations reducible to exact form by integrating factors
01

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
3. Equations of first order & higher degree
4. Clairaut's Equation
5. Linear differential equation of first order
02

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
6. Linear differential equations of second and higher order
03

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
7. Method of variation of parameters to find P.I.
04

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
ORDINARY DIFFERENTIAL EQUATIONS
05
8. Legendre's linear equation & Cauchy's homogeneous
equation
WRITE YOUR OWN NOTES HERE
rt
...
...
...
...
...
...

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
o6

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
07
(
)
)
(

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
08

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
09

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
10

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
11

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
12

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
13
(
)
(
(
(
)
)
)
(
)
(
)

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
14

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
15

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
Expected Questions
16

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
17
(
(
)
)
(
)
(
)
(
)
(
)
(
)

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
18

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
19

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
20

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
Problems for meritorious students
21

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
22

APPLICATIONS OF ODE
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
rt
APPLICATIONS OF ODE
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
Mean Position
Extreme Position
Extreme Position
o
P
x
a
1. Simple Harmonic Motion (S.H.M)
2. Electric LCR Circuits
L
C
R
E=kcosnt
i
23

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
APPLICATIONS OF ODE
EXPECTED QUESTIONS
24

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
APPLICATIONS OF ODE
25

MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
rt
MATRICES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
2. Rank of a matrix
(3)
1. Elementary Transformations
3. Normal form of a matrix
26

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
4. Consistency and Solution of linear equations
5. Linearly dependent & Linearly independent vectors
27

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
7. Cayley Hamilton Theorem (CHT)
8. Quadratic Form
6. Eigen Values & Eigen Vectors
28

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
29
9. Complex matrices
10. Orthogonal matrix
vely
( )
( )

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
11. Unitary matrix
12. Similar matrices
13. Linear transformation
14. Orthogonal transformation
30

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
31

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
32

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
33

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
34

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
35

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
36

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
37

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Expected Questions
38

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
39

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
40

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
41

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Problems for meritorious students
42

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
MATRICES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
WRITE YOUR OWN NOTES HERE
rt
...
...
...
...
...
...
...
...
...
...
...
...
...
43

INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
rt
INFINITE SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
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1. Infinite Series
2. Convergence,Divergence & Oscillation of an Infinite Series
3. Auxiliary Series
44

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
INFINITE SERIES
ALTERNATING SERIES
POSITIVE TERM SERIES
CONVERGENT
CONVERGENT
DIVERGENT
OSCILLATORY
4. Tests of Convergence for positive term series
45

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
6
7
8
46

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
5. Leibnitz's test for convergence of alternating series
6. Absolute and conditional convergence of an alternating
series
7. Important Limits
47

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
48

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
49

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
50

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
51

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
52

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
53

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
54

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Expected Questions
55

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
56

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
57

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Problems for meritorious students
58

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
WRITE YOUR OWN NOTES HERE
rt
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
59

COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
rt
COMPLEX NUMBERS
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
60
1. De-Moivre's Theorem
2. Properties of cis
3. Some important cis forms
4. Values of (cos +isin ) , where p,q are +ve coprime integers
p/q

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
61
5. Euler's Formula
6. Circular Functions
7. Hyperbolic Functions
8. Relation between Hyperbolic & Circular Functions
9. Inverse Hyperbolic Functions
10. Logarithm of Complex Numbers
(
)
(
)

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
62
11. Some Standard Series

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
63

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
64

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
65

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
66

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
67

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
68

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
69

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
WRITE YOUR OWN NOTES HERE
rt
...
...
...
...
...
...
...
...
70

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Expected Questions
71

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
72

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
73
(
)
(
)

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
74

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
75

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Problems for meritorious students
76

IMPORTANT FORMULAE
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
rt
IMPORTANT BASIC
FORMULAE & RESULTS
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
1. Differentiation
77

RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
IMPORTANT FORMULAE
2. Integration
78
82 of 82 pages

Details

Title
Engineering Mathematics for B.Tech Students
Subtitle
Ranjit Tutorials
Course
B.Tech, Semester-II
Grade
13
Author
Year
2014
Pages
82
Catalog Number
V267852
ISBN (Book)
9783656593744
File size
57245 KB
Language
English
Notes
Best notes for Engineering Mathematics.
Tags
engineering, mathematics, tech, students, ranjit, tutorials
Quote paper
Ranjit Singh (Author), 2014, Engineering Mathematics for B.Tech Students, Munich, GRIN Verlag, https://www.grin.com/document/267852

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