1
ISTE -SRINIVASA RAMANUJAN
MATHEMATICAL COMPETITIONS- 2013
(SRMC-2013)
Instruction Manual
INDIAN SOCIETY FOR TECHNICAL EDUCATION
Shaheed Jeet Sing Marg, near Katwaria Sarai, Neew Delhi 110 016
Tel: 011-2651342, 26963431, Fax: 91-11-26852421
email:
[email protected]t website: www.isteonline.in
2
INDIAN SOCIETY FOR TECHNICAL EDUCATION
ISTE -SRINIVASA RAMANUJAN MATHEMATICAL COMPETITIONS- 2013
(SRMC-2013)
Indian Society for Technical Education has successfully completed the Srinivasa
Ramanujan Mathematical competitions – 2012 and paid tribute to the great son of
India from Tamilnadu ‘Srinivasa Ramanujan’ during his 125 birth anniversary with the
support of the National Board of Higher Mathematics (NBHM) and Institute of
Mathematical Sciences, Madras.
NBHM was kind enough to accord approval to conduct the competitions in the
same manner with more participation from students and teachers this year also. ISTE
will utilize this opportunity and conduct the same in an appropriate manner. We
solicit the support of all ISTE Chapters and institutions as well.
Details regarding the competitions may be obtained from ISTE Chapter
Chairman of the College. The application form and general guidelines are also made
available in our website
www.isteonline.in. For more details you may contact us at
istedhq@vsnl.net
TAKE PART IN THE COMPETETION AND WIN PRIZES
3
ISTE -Srinivasa Ramanujan Mathematical Competitions
General Guidelines
1
. General Information
•
Section Chairman will be the Coordinator for the particular Section for the Project
•
There will be one examination on the same day for Students of Polytechnic and
Engineering Colleges
•
There will be one examination on the same day for Teachers of Polytechnic and
Engineering Colleges
•
There will be four different Question Papers one each for different category for
Stage I (Chapter level)
•
The examination will be only Multiple Choice Questions (MCQ) type with duration
of two hours for the Chapter level
.
•
There will be a separate application format for Students and Teachers
•
Application forms can be downloaded from ISTE website www.isteonline.in
•
Filled in application will be received by the Chapter Coordinator
2
. Students Awards
A
. Chapter Level
•
Each students chapter will conduct exam at the chapter level and select a
maximum of FIVE and a minimum of ONE student based on the ranks to the
next level
.
B. Zonal Level
•
There will be a total of 38 zones for the Degree level and 23 zones for the
diploma level
.
•
Three students from each stream (Degree and Diploma) will be selected from
each zone
.
(3 Degree + 3 Diploma from one Section)
•
Section Chairman will conduct the exam in the respective zones within the
Section and select the students with the help of Zonal Coordinator
•
Three prizes in each stream (Diploma and Degree) for students will be given
for each Section at a special function organized at the Zonal level
.
C
. National Level
•
The selected candidates from the zonal level will be qualified to take the
national level examination.
4
•
National level examination will be conducted at common centers for degree
and diploma stream.
•
Five students from degree level and five students from diploma level will be
selected and awarded suitably
.
•
Five consolation prizes will be given to degree and diploma level students
apart from the first five prize winners
•
Certificates will be given to all the participants
.
3
. Teachers Awards
A
.
Chapter Level
•
Each faculty chapter will conduct exam at the chapter level and select a
maximum of FIVE and a minimum of ONE teacher based on the ranks to the
next level.
B
. Zonal Level
•
There will be a total of 38 zones for the Degree level and 23 zones for the
diploma level
.
•
Three teachers from each stream (Degree and Diploma) will be selected from
each zone (3 Degree + 3 Diploma from one Section)
•
Section Chairman will conduct the exam in the respective zones and select
the teachers with the help of Zonal Coordinator
•
Three prizes in each stream (Diploma and Degree) for teachers will be given
for each zone at a special function organized at the Zonal level.
C
. National Level
•
The selected candidates from the zonal level will be qualified to take the
national level examination.
•
Five teachers from degree level and five teachers from diploma level will be
selected and awarded suitably.
•
Five consolation prizes will be given to degree and diploma level teachers
apart from the first five prize winners
•
Certificates will be given to all the participants.
4
. Responsibilities of the Chapter Coordinator:
•
ISTE Chapter Chairman will be the Chapter Coordinator for this project
•
He will invite application from Students and Teachers
•
He will compile the list of eligible candidates and send soft and hard copies to
Section Chairman and ISTE New Delhi
5
•
Collect an amount of Rs. 100/- from each candidate (Students and Teachers)
and send a consolidated DD drawn in favour of “ISTE New Delhi’ payable at
New Delhi along with the list of registered candidates.
•
Send the Hall ticket informing the date, exam centre and time with
instructions to candidates
.
•
Conduct the exam on the stipulated dates
•
Question papers will be sent by ISTE New Delhi
•
Evaluate the answers based on the keys provided by ISTE-NBHM and send
the rank list to ISTE NEW DELHI
•
Send the certified expenses and statement of accounts to ISTE NEW DELHI
within 10 days after the completion of exams
5
. Responsibilities of the Section Coordinator:
•
Give wide publicity to the project in all the institution through available methods
•
Coordinate the activities with the Chapter Chairmen
•
Compile the list of eligible candidates sent by Chapter Chairman and send to
ISTE New Delhi.
•
Supervise the conduct of the exams on the stipulated dates by chapters
•
Coordinate with the Chapters to send the certified expenses and statement of
accounts to ISTE New Delhi within 10 days after the completion of exams
6
. Calendar of Events:
Announcement in the website for downloading: 01 July 2013
Last date of sending list of registered applicants to ISTE, ND
by ISTE Chapters: 26 th August 2013
Dispatch of CL-Question paper booklet to ISTE Chapters: 29 August 2013
Date of Chapter Level Examination: 30 August 2013
Last date of sending Rank list to ISTE, ND by Chapters: 16 September 2013
Publication of List of candidates appearing for Zonal level Exam: 24 September 2013
Dispatch of Question paper to Zonal Centers: 08 November 2013
Date of Zonal Level Examination: 09 November 2013
Late date of sending ZL-Answer Papers to Evaluation Center: 20 November 2013
Publication of Result of the Zonal level Exam: 09 December 2013
6
Dispatch of National Level Question paper to National Centers: 24 January 2014
Date of National Level Examination: 25 January 2014
Late date of sending NL-Answer Papers to Evaluation Center: 14 February 2014
Announcement of winners: 28 February 2014
Prize distribution function March 2014
7
. Prize Money
National Level
•
First prize in each category :
Rs
.
30,000
•
Second prize in each category:
Rs
.
25,000
•
Third prize in each category:
Rs
. 20,000
•
Fourth prize in each category:
Rs
.
15,000
•
Fifth prize in each category:
Rs
.
10,000
•
Consolation prize in each category:
Rs
. 5,000
•
Prize winners will be given certificate, medallion and cash prize.
Zonal Level
•
First prize in each category:
Rs
.
3,000
•
Second prize in each category:
Rs
.
2,000
•
Third prize in each category:
Rs
. 1,000
•
Prize winners will be given certificate and cash prize.
Chapter Level
•
Certificate of participation will be given to all the participants
8
.
Finance
The guidelines for expenditure like remuneration to the coordinator, supervisors,
assistants, travel allowances and other misc expense is attached at the annexure.
7
INDIAN SOCIETY FOR TECHNICAL EDUCATION
SRINIVASA RAMANUJAN MATHEMATICS
COMPETITION-2013 EXAMINATION
Examination General Instructions for Section Chairman
1
. Question papers in pdf format will be sent to you on or before 2 9 t h
A u g u s t 2 0 1 3 through e-mail.
2
. There will be FOUR sets of each category of question papers for Teachers and
Students namely Type A,B,C and D.
3
. There are two zipped files, each of them is protected by password
.
4
. The passwords will be sent to you separately on 30 th Augus t 2013 before 10
am to your email and in your mobile by sms
.
5
. The following instructions are to be given to the Chapter Chairman by you for the
smooth conduct of the examination.
·
To issue Hall tickets to the eligible candidates (Teachers and Students)
·
To take sufficient number of question papers and response sheets as per
·
the requirement
.
·
Question papers are to be distributed in manner such that candidates gets
·
in the order A,B,C, D
.
·
No two adjacent candidates should get the same Type and seating
·
arrangements should be done accordingly
.
·
To inform the candidate to write the relevant question paper Type in the
·
response sheet
.
·
To inform the candidate to return the question paper along with the
·
marked response sheet
.
6
. For Chapter level (level- I) evaluation to be done by the Chapter Chairman in his
centre and merit list to be sent to ISTE, New Delhi. (see instructions for the
chapter level evaluation)
8
INDIAN SOCIETY FOR TECHNICAL EDUCATION
SRINIVASA RAMANUJAN MATHEMATICS
COMPETITION-2013 EXAMINATION
General Instructions for Chapter Level Evaluation
1
. For evaluating the response sheets for the chapter level, answer keys will be
sent to the Section Chairman by email on 03 September 2013.
2
. Section Chairmen are requested to send the answer keys to the respective
Chapter Chairman
.
3
. The Chapter Chairman shall appoint examiners from the Department of
Mathematics of his college for the evaluation of the response sheets
.
4
. The remuneration for evaluating response sheets has been fixed as Rs. 15/- per
response sheet
5
. The Chapter Chairman should send the following to the ISTE HQ, New Delhi on or
before: 16 September 2013
·
Application forms registered (teachers and Students) with details
·
Attendance sheet
·
Used Question papers
·
Corrected Response sheets
·
Total registration fee collected from teachers and students in a single DD
drawn in favour of “ISTE New Delhi”
·
Merit list (1 to 5) for both teachers and students with contact details.
·
Statement of expenditure and remuneration claim
9
Indian Society for Technical Education
Srinivasa Ramanujan Mathematics
Competitions Examination - 2013
Teacher’s Application Form
Name of the Candidate :
Father’s Name :
Mother’s Name :
Date of Birth (attach proof) :
Sex
:
Male/Female
Place of Birth :
(write Taluk, District and State)
Mother tongue :
Nationality/Religion :
Category (General/BC/OBC/SC/ST) :
College in which you are studying :
(give full address)
Pin
Tel:
Qualification:
No. of year of experiences:
Postal Address (all correspondence will be sent to this address, the same should be entered in
the HT also)
Phone Nos. Landline (with code):
Mobile:
E-mail ID
10
Demand Draft details (Λ/C Payee only) DD No.:______________Date _____________
Drawn in favour of “ISTE” Drawn on Bank__________________________
Amount: Rs.100/-
Payable at: New Delhi
Any other remarks you wish to make
The above statement made by me is true to the best of my knowledge, information and belief.
(Signature of the applicant with date)
Send the filled in application form along with the demand draft to the ISTE
Chapter Chairman of your institution.
ISTE – SRINIVASA RAMANUJAM MATHEMATICS AWARD
COMPETITIONS EXAMINATION – 2013
HALL TICKET
Name of the Candidate:
Registration No. :
(To be allotted by Office)
Λddress of the Centre :
Place:
Date: Authorized Signatory
11
Indian Society for Technical Education
Srinivasa Ramanujan Mathematics
Competitions Examination - 2013
Student’s Application Form
Name of the Candidate :
Father’s Name :
Mother’s Name :
Date of Birth (attach proof) :
Sex
:
Male/Female
Place of Birth :
(write Taluk, District and State)
Mother tongue :
Nationality/Religion :
Category (General/BC/OBC/SC/ST) :
College in which you are studying :
(give full address)
Pin
Tel:
Course of Study: Branch :
Year of study 1/2/3/4
Postal Address (all correspondence will be sent to this address, the same should be entered in
the HT also)
Permanent Address :
Phone Nos: Landline (with code):
Mobile:
E-mail ID
12
Demand Draft details (Λ/C Payee only) DD No.:______________Date _____________
Drawn in favour of “ISTE” Drawn on Bank__________________________
Amount: Rs.100/-
Payable at: New Delhi
Any other remarks you wish to make
The above statement made by me is true to the best of my knowledge, information and belief.
(Signature of the applicant with date)
Send the filled in application form along with the demand draft to the ISTE
Chapter Chairman of your Institution.
ISTE – SRINIVASA RAMANUJAM MATHEMATICS
COMPETITIONS EXAMINATION – 2013
HALL TICKET
Name of the Candidate :
Registration No. :
(To be allotted by Office)
Address of the Centre :
Place:
Date: Authorized Signatory
13
INDIAN SOCIETY FOR TECHNICAL EDUCATION
SRINIVASA RAMANUJAN MATHEMATICS
COMPETITION-2013 EXAMINATION
Exam Center:
Date:
Attendance Sheet
S.No Hall Ticket No Name Mobile Signature
14
Indian Society for Technical Education, New Delhi
Address Format
Name of the Institution
Address
ISTE Student Chapter Number
ISTE Faculty Chapter Number
Institutional Membership Number
Name of the Principal
Telephone
Mobile
email
Fax
Name of the Chapter Chairman
Telephone
Mobile
email
Name of the Chapter Secretary
Telephone
Mobile
email
Place:
Date:
15
INDIAN SOCIETY FOR TECHNICAL EDUCATION
ISTE-SRMC 2013
Remuneration Details for an Exam Center
Co-ordinator (1): Principal/ISTE Chapter Chairman
Rs
.
500 per session
Supervisor (Invigilator)(1):
Rs
.
300 per session
Office Staff (1):
Rs
. 150 per session
Assistant/Waterman (1):
Rs
.
100 per day
Security:
Rs
. 100 per day
*(For 50 candidates or part thereof; more than 50, more supervisors be engaged)
16
INDIAN SOCIETY FOR TECHNICAL E DUCATION
SRINIVASA RAMANUJAN MATHEMATICS
C
CC
CO
OO
OMPETITI
MPETITIMPETITI
MPETITION
ONON
ON
EXAMINATION-2013
Name of the candidate:
Hall Ticket Number:
Date: Time:
Centre Code / Name:
Level of Examination :( Chapter /Zonal/National):
(For office use)
Signature:
Name of the Examiner: (CAPITAL LETTERS)
17
INDIAN SOCIETY FOR TECHNICAL EDUCATION
Srinivasa Ramanujan Mathematics Competition-2013 Examination
Maximum: 40 Marks
Response Sheet
30 Aug 2013
Call Letter No
.
Instructions
Time: 2.00 to 4.00 p.m.
No cell phones, or electronic devices will be allowed during the examination.
This
question book contains 40 questions
.
Each question carries 1 mark.
1.
Please read the instructions first; and
do not open
the question book until
you are instructed do so.
2.
You may take first 5 minutes to look for any defects (such as missing a page,
printing defects, etc.)
in the question book. Any such defects should be
brought to the attention of the invigilator within first five minutes.
3.
Please enter your Hall ticket number on your question book and response sheet
in the space provided
.
Please write the question paper type A or B or
C or D in the response sheet.
4.
Do not write anything on the question book.
Answers must be circled
only on the Response sheet.
5.
You may not leave the exam Hall within the first half hour of the examination.
6.
Every question has four possible responses
.
You have to select the most accu-
rate one and circle your answer on your response sheet.
7.
Make sure that your responses are clearly circled
. Avoid errors or
changes on the response sheet.
8.
At the end of the two hours you must leave your question book, the
response sheet and the blank sheets on your table with face down.
You should not take any part of this examination with you.
9.
Announcement of time will be made every 15 minutes and also 5 minutes
before the end of two hours.
10.
If you are found cheating on this examination, you will be disquali-
fied for the competition.
18
INDIAN SOCIETY FOR TECHNICAL EDUCATION
Srinivasa Ramanujan Mathematics Competition-2013 Examination
Maximum Marks: 40
.
Response Sheet
30 Aug 2013 : 2.00 to 4 p.m.
Call Letter No
.
Question Paper Type:
(1)
A
B
C
D
(21)
A
B
C
D
(2)
A
B
C
D
(22)
A
B
C
D
(3)
A
B
C
D
(23)
A
B
C
D
(4)
A
B
C
D
(24)
A
B
C
D
(5)
A
B
C
D
(25)
A
B
C
D
(6)
A
B
C
D
(26)
A
B
C
D
(7)
A
B
C
D
(27)
A
B
C
D
(8)
A
B
C
D
(28)
A
B
C
D
(9)
A
B
C
D
(29)
A
B
C
D
(10)
A
B
C
D
(30)
A
B
C
D
(11)
A
B
C
D
(31)
A
B
C
D
(12)
A
B
C
D
(32)
A
B
C
D
(13)
A
B
C
D
(33)
A
B
C
D
(14)
A
B
C
D
(34)
A
B
C
D
(15)
A
B
C
D
(35)
A
B
C
D
(16)
A
B
C
D
(36)
A
B
C
D
(17)
A
B
C
D
(37)
A
B
C
D
(18)
A
B
C
D
(38)
A
B
C
D
(19)
A
B
C
D
(39)
A
B
C
D
(20)
A
B
C
D
(40)
A
B
C
D
19
Indian Society for Technical Education
Srinivasa Ramanujan Mathematical Competitions- 2013
Syllabus
Polytechnic Colleges
Basics:
Sets and functions-injective and surjective functions (1-1 and onto function)-
composition of functions- simple counting and equivalence of sets- finite and
infinite sets-knowledge of polynomial functions- rational functions-exponential
function-logarithmic
function-trigonometric functions-DeMoivre’s
theorem-
inverse
trigonometric
functions-
Real
and
complex
numbers (basic
understanding) _ Simple Combinatoric problems-simple probability problems
Algebra
Algebra of complex numbers–Real and Imaginary parts- Polar form of
complex number – Modulus and amplitude form multiplication and
division of complex numbers in polar form- Argand plane–collinear points,
four points forming square, rectangle, rhombus. Demoivre's Theorem
related Problems - Finding the nth roots of unity
.
Quadratic equations- polynomials- relation between roots and coefficients-
Remainder theorem- factorization-real roots and complex roots
.
Determinants:
Definition and expansion of determinants of order 2 and 3 -
Properties of determinants .Cramer's rule to solve simultaneous equations
in 2 and 3 unknowns-. Problems involving properties of determinants
-
Matrices :Definition of matrix - Types of matrices - Algebra of matrices
such as equality, addition, subtraction, scalar multiplication and
multiplication of matrices- Transpose of a matrix-adjoint matrix and
inverse matrix-.
permutation and combination- Binomial theorem for
positive integral index (statement only), finding of general and middle
terms - Problems finding co-efficient of
, independent terms. . Binomial
Theorem for rational index, expansions- Simple Expansions - Partial
Fractions -
To
resolve proper fraction into partial fraction with
denominator containing non repeated linear factors, repeated linear
20
factors and irreducible non repeated quadratic factors-
.
Geometry
Length of perpendicular distance from a point to the line and
perpendicular distance between parallel lines- -
Angle between two
straight lines and condition for parallel and perpendicular lines-
Pair of
straight lines - angle between pair of straight lines. Condition for parallel
and perpendicular lines - Condition for general equation of the second
degree
+
+ 2ℎ + 2 + 2 + = 0
to represent pair of lines -
Angle between them - condition for parallel and perpendicular lines
.
CIRCLES
Equation of circle – given centre and radius. General Equation of circle –
finding center and radius- Equation of circle through three non collinear
points – concyclic points- Equation of circle on the line joining the points
,
and points
,
as diameter- Length of the tangent-Position of
a point with respect to a circle- Equation of tangent -Concentric circles –
contact of circles (internal and external circles) – orthogonal circles –
condition for orthogonal circles
Trigonometry
Trigonometrical ratio of allied angles-Expansion of Sin(A+B) and
cos(A+B)- problems using above expansion
-
Expansion of tan(A+B) -
Trigonometrical ratios of multiple angles and sub-multiple angles
-
Sum
and Product formulae-Definition of inverse trigonometric ratios- relation
between inverse trigonometric ratios.
Calculus:
Differential Calculus:
Limits-
continuous
functions-
intermediate
value
theorem -differentiable
functions-
derivatives
of
sum, difference, product and
quotient
of two
differentiable functions- derivatives of elementary functions – second derivative
and higher
order
derivative-
successive
differentiation-
derivatives of
standard functions
.
Derivative as a rate measure- Velocity and Acceleration- Tangents and Normals-
Increasing function- Decreasing function and turning points -Maxima and
Minima (for single variable only). Partial Differentiation- Partial differentiation
of two variable up to second order only-
Definition of Homogeneous functions-
Euler’s theorem.
21
Integral Calculus:
Standard integrals- integration of rational functions- integration by parts-
definite integrals and their simple properties-
Area –volume – volume of cone
and sphere.
Ordinary Differential Equations:
Definition of order and degree of differential equation – solution of first
order variable separable type differential equation
-
Solution of first order
linear differential equation - Solution of second order differential
equations with constant
coefficients
VECTOR ALGEBRA
Vectors – types, addition and subtraction of vectors- Properties, of
addition and subtraction, position vector-Resolution of vector in two and
three dimensions- Direction cosines, direction ratios- Definition of scalar
product of two vectors – Properties – Angle between two vectors -
Geometrical meaning of scalar Product. Work done by Force. Definition of
vector product of two vectors-Geometrical meaning- properties-angle
between two vectors–unit vector perpendicular to two vectors-
Definition
of moment of a force - scalar triple product - geometrical meaning –
coplanar vectors - Vector Triple Product - Scalar and Vector product of
four vectors
.
PROBABLITY DISTRIBTION
Definition of Random Variable – Type –Probability Mass Function –
Probability density function -Mathematical expectation of discrete
variable, mean and variance –
Binomial distribution
-
Expressions for mean
and variance.
Poisson distribution-
Expression for mean and Variance -
Normal distribution -
standard normal distribution- Constants of normal
distribution (results only) – Properties of normal distribution
Curve Fitting
Fitting of a straight line using least square method
.
22
Indian Society for Technical Education
Srinivasa Ramanujan Mathematical Competitions- 2013
Syllabus
Engineering Colleges
Basics:
Sets and functions-injective and surjective functions (1-1 and onto function)-
composition of functions- simple counting and equivalence of sets- finite and
infinite sets-knowledge of polynomial functions- rational functions-exponential
function-logarithmic
function-trigonometric functions-DeMoivre’s
theorem-
inverse trigonometric functions- hyperbolic functions and inverse hyperbolic
functions
.
Real
and
complex numbers
(basic
understanding) _ Simple
Combinatoric problems-simple probability problems
Algebra:
Quadratic equations- polynomials- relation between roots and coefficients-
Remainder theorem- factorization-real roots and complex roots
.
Matrices- algebra of matrices- determinants- inverse of a matrix- row vectors
and column vectors-linear dependence and linear independence- span of
vectors- basis –rank and consistency of linear systems-characteristic equation-
eigen values-eigen vectors- Cayley Hamilton theorem- symmetric matrices-
orthogonal matrices-Hermitian matrices- diagonal matrices and diagonalization
.
Geometry
Triangles and their basic properties-Polygons-circles- simple mensuration
formulae-conics and their shapes
.
Coordinate Geometry
Two dimensional coordinate geometry:
Rectangular coordinates-points and distance between points-point dividing a
line segment in a given ratio- equations of curves- straight lines- slope of a
straight line- various standard equations of straight lines- circles- tangent and
normal- standard equations of conics-eccentricity of a conic- tangent and normal
of a conic-asymptotes
.
23
Three dimensional coordinate geometry:
Straight lines-skew lines-sphere-plane section of a sphere-tangent plane-
equation of a cone-right circular cone-cylinder – right circular cylinder
.
Convergence:
Sequences – Convergence of series – General properties – Series of positive
terms – Tests of convergence (Comparison test, Integral test, Comparison of
ratios and D'Alembert's ratio test) – Alternating series – Series of positive and
negative terms – Absolute and conditional convergence – Power Series –
Convergence of exponential, logarithmic and Binomial Series
.
Calculus:
Differential Calculus:
Limits-
continuous
functions-
intermediate
value
theorem -differentiable
functions-
derivatives
of
sum, difference, product and
quotient
of two
differentiable functions- derivatives of elementary functions – second derivative
and higher
order
derivative-
successive
differentiation- derivatives of
standard functions
.
Derivative as a rate of change- simple applications-
Roll’s Theorem- Mean Value Theorem
Maxima and minima using calculus
Taylor series- Maclaurin series- binomial series-exponential series-logarithmic
series
.
Tangent and
normal
of differentiable curves- increasing and
decreasing
functions- curvature in Cartesian coordinates- center and radius of curvature-
evolutes and involutes- envelopes
.
Functions
of
several variables-partial
derivatives-Euler’s
theorem
on
homogeneous
functions-total derivatives-jacobians-maxima and
minima
of
functions two/three variables- Lagrange multipliers
.
Integral Calculus:
Standard integrals- integration of rational functions- integration by parts-
definite integrals and their simple properties-area length, surface area and
volume of revolution – improper integrals- beta gamma functions
Double and triple integration – Cartesian and polar coordinates- change of order
of integration- change of variables Cartesian to polar coordinates- Area as a
double integral and volume as a triple integral
.
Ordinary Differential Equations:
Linear ordinary differential equations of first order, second order and higher
order equations with constant coefficients – variation of parameters- System of
first order linear equations with constant coefficients
24
Laplace Transforms:
Laplace transform – Conditions for existence – Transform of elementary
functions –Basic properties – Transform of derivatives and integrals – Transform
of unit step function and impulse functions – Transform of periodic functions
.
Definition of Inverse Laplace transform as contour integral – Convolution
theorem (excluding proof) – Initial and Final value theorems – Solution of linear
ODE of second order with constant coefficients using Laplace transformation
techniques
.
Complex Variables:
Functions of a complex variable – Analytic functions – Necessary conditions,
Cauchy –Riemann equation and Sufficient conditions (excluding proofs) –
Harmonic and orthogonal properties of analytic function – Harmonic conjugate –
Construction of analytic functions – Conformal mapping : w= z+c, cz, 1/z, and
bilinear transformation
.
Complex integration – Statement and applications of Cauchy’s integral theorem
and Cauchy’s integral formula – Taylor and Laurent expansions – Singular points
– Residues– Residue theorem – Application of residue theorem to evaluate real
integrals
.
Vector Calculus :
Vector algebra – vector equations of straight lines and planes-Gradient
Divergence and Curl – Directional derivative – Irrotational and solenoidal vector
fields – Vector integration – Green’s theorem in a plane, Gauss divergence
theorem and stokes’ theorem (excluding proofs) – Simple applications involving
cubes and rectangular parallelepipeds
.
Fourier Series:
Dirichlet’s conditions – General Fourier series – Odd and even functions – Half
range sine series –Half range cosine series – Complex form of Fourier series –
Parseval’s identity
.
Fourier Transforms:
Fourier integral theorem (without proof) – Fourier transform pair – Sine and
Cosine transforms –Properties – Transforms of simple functions – Convolution
theorem – Parseval’s identity
.
Partial Differential Equations:
Formation of partial differential equations – Lagrange’s linear equation –
Solutions of standard types of first order partial differential equations - Linear
partial differential equations of second and higher order with constant
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coefficients. Solutions of one dimensional wave equation – One dimensional
equation of heat conduction –Steady state solution of two-dimensional equation
of heat conduction (Insulated edges excluded) –Fourier series solutions in
Cartesian coordinates
.
Transforms and Difference equations:
Z-transforms - Elementary properties – Inverse Z-transform – Convolution
theorem - Formation of
difference equations – Solution of difference equations using Z-transform
.
Probability, Statistics and Numerical Methods
:
Discrete and continuous random variables – Moments - Moment generating
functions and their properties
.
Binomial, Poisson ,Geometric, Uniform,
Exponential, Gamma and normal distributions – Function of Random Variable.
Sampling distributions - Tests for single mean, Proportion, Difference of means
(large and small samples) – Tests for single variance and equality of variances –
chi-square test for goodness of fit – Independence of attributes
.
Completely randomized design – Randomized block design – Latin square design
-factorial design
.
Newton-Raphson method- Gauss Elimination method – Pivoting - Gauss-Jordan
methods – Iterative methods of Gauss-Jacobi and Gauss-Seidel - Matrix Inversion
by Gauss-Jordan method - Eigenvalues of a matrix by Power method
.
Lagrange’s and Newton’s divided difference interpolation –Newton’s forward
and backward difference interpolation - Approximation of derivatives using
interpolation polynomials - Numerical integration using Trapezoidal and
Simpson’s 1/3 rules
.
Taylor’s series method - Euler’s method - Modified Euler’s method - Fourth order
Runge-Kutta method for solving first and second order equations - Milne’s
predictor-corrector methods for solving first order equations - Finite difference
methods for solving second order equation
.
