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Pokhara University
Faculty of Science and
Technology
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Full Marks: 100 |
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Course title: Algebra and
Geometry (3-2-0) |
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Pass Marks: 45 |
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Nature of the course: Theory |
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Total Lectures: 45 hours |
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Level: Bachelor |
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Program: BE |
1. Course Description
The course
covers matrices and determinants, system of linear equations, rank, vector
spaces, eigenvalues and eigenvectors. It also covers infinite series and
Fourier series. Moreover, it covers geometry of two and three dimensions.
2. General Objectives
The general
objective of the course is to provide the sound knowledge of Algebra and
Geometry of two and three dimensions.
3.
Methods
of Instruction
Lecture, Discussion, Readings and Class Work
4. Contents in Detail
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Specific
Objectives |
Contents |
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Solve
system of linear equations and find rank of a matrix |
Unit
I: Matrix and Determinants (7 hrs) 1.1 Type
of matrices, matrix multiplication, determinant, properties of determinant,
orthogonal matrix, inverse of a matrix. 1.2 System
of linear equations and Gauss elimination method 1.3 Linear
dependence and independence, row rank, column rank, rank of matrix,
consistency condition, Caley-Hamilton theorem and its use in finding inverse
of a matrix |
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•
Solve
the problems related to vector space and identify eigenvalues |
Unit
II: Vector Space
(6 hrs) 2.1 Vector
space, subspace, linear combination of vectors, Basis 2.2 Linear
transformation 2.3 Eigenvalues
and eigenvectors, some applications of eigenvalues problems |
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Test the
convergence and divergence of the series |
Unit
III: Infinite Series (8
hrs) 3.1 Introduction,
necessary condition for an infinite series to be convergent, properties,
p-series 3.2 Comparison
test, ratio test, root test, integral test 3.3 Alternating
series, Leibnitz test, absolute convergence, power series, interval of
convergence |
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•
Solve
the problems related to Fourier series |
Unit
IV: Fourier Series
(6 hrs) 4.1 Periodic function, even and
odd functions, Fourier coefficients 4.2 Fourier
series of a function with period 2Ï€ and arbitrary period 2L 4.3 Half-range
expansions |
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Solve
the problems related parabola, ellipse, and hyperbola |
Unit
V: Two-dimensional Geometry
(7 hrs) 5.1
Parabola Introduction, general equation
of a conic, equation of a parabola, length of the latus rectum, different
forms of parabola, condition for tangency 5.2
Ellipse Standard equation of an
ellipse, focal distance, auxiliary circle, condition for tangency, director
circle of an ellipse 5.3
Hyperbola Standard equation of a
hyperbola, rectangular hyperbola, conjugate hyperbola |
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Solve
the problems related straight lines and sphere |
Unit
VI: Three-dimensional Geometry (11 hrs) 6.1 Review
of plane 6.2 Straight
lines Introduction, line in symmetrical form, line passing through
two given points, transformation of a line in general form into symmetrical
form, angle between a plane and a line, condition for a line to lie on a
plane, length of a perpendicular from a given point on a line, coplanar
lines, condition for coplanarity of two lines, shortest distance 6.3
Sphere Equation of a sphere, condition for a general equation of
second degree to represent a sphere, sphere on the line joining two points as
diameter, plane section of a sphere, intersection of a plane and a sphere,
intersection of two spheres, equation of a tangent plane, condition of
tangency |
5.
List
of Tutorials
The
following tutorial activities of 30 hours per group of maximum 24 students
shall be conducted to cover all the required contents of this course. This will
enable the students to complete the math problems under the supervision of the
subject teacher.
a) Problems
on matrices and determinants (1 hr)
b) To
test the consistency and then solve the linear equations (2 hrs)
c) Determining
rank of a matrix (1 hr)
d) Problems
on vector space, linear transformation, eigenvalues, and eigenvectors (3 hrs)
MATLAB work
e)
Matrix
creation, addition, subtraction, multiplication, inverse, rank, solution of
linear equations, Eigen values and Eigen vector, characteristic equation using
command window (2 hrs)
f) To
test for convergence of a series (2 hrs)
g) Finding
radius of convergence and interval of convergence (2 hrs)
MATLAB work
h) Creation and summation
of series (1 hr)
i) Problems
on Fourier series (4 hrs)
j) To
obtain the standard equation of ellipse and hyperbola (1 hr)
k) Problems
on parabola, ellipse and hyperbola (3 hrs)
MATLAB work
l) Plotting
of 2D (1 hr)
m) Problems
on straight lines (4 hrs)
n) Problems
on sphere (2 hrs)
o) Plotting
of 3D (1 hr)
6.
Evaluation
System and Students’ Responsibilities
Evaluation System
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Internal
Evaluation |
Weight |
Marks |
External
Evaluation |
Marks |
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Theory |
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50 |
Semester-End examination |
50 |
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Attendance
& Class Participation |
10% |
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Assignments |
20% |
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Presentations/Quizzes |
10% |
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Internal
Assessment |
60% |
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Total
Internal |
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50 |
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Full Marks: 50 + 50 = 100 |
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Students’ Responsibilities
7.
Prescribed
Books and References
Text Books
1. Kreyszig, E. Advanced
Engineering Mathematics. New Delhi: John Wiley and Sons Inc.
2. Thomas, G. & Finney, R. Calculus
and Analytical Geometry. New Delhi: Narosa Publishing House.
3. Vittal, P. R. Analytical
Geometry 2D and 3D, Delhi: Pearson India.
References
1. E.W.
Swokoswski, Calculus with Analytic Geometry, Prindle, Weber and Schmidi
2. Shanti
Narayan, Analytical Solid Geometry, S. Chand and company
3. Chandrika
Prasad, Algebra and Theory of Equations, Pothishala Pvt. Ltd.
4. Ward
Cheney and David Kincaid, Linear Algebra: Theory and applications, Jones and
Bartlett Publisher