FIRST SEMESTER M.Sc. MATHEMATICS ASSIGNMENTS
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Course – 1: Math 1.1Algebra
1. Let 2 3 2 3
4 D e a a a b ba ba ba = { , , , , , , , } be set of 8 elements. Define the product in D4
by the relation 4
a e = ,
2
b e = and 1
ab ba−
= . Show that D4
is a group. Find all subgroups
od D4
and find all left cosets of one of the subgroup of D4
.
2. Define a group homomorphism and Kernel of the homomorphism. State and prove
fundamental theorem of homomorphism of groups. Illustrate with example.
3. Define Conjugacy relation; prove that it an equivalence relation. Find the partition of D4
induced by the Conjugacy relation and hence find the class equation of D4
.
Course – 2: Math 1.2Real Analysis – I
1. Define limit point of a set.Distinguish the limit point of a set and limit of a sequence with
illustration. Prove that every infinite bounded set has a limit point.
2. State and prove Cauchy’s second theorem on limit of a real sequence. If 3
b converges and
find their limits.
3. State and prove the Cauchy’s condensation test and Kummer’s test.Illustrate the
significance of these testes in finding the convergence of the series.
Course –3: Math 1.3Complex analysis– I
1. Define an analytic function. State the necessary and sufficient condition for the functions to
be analytic. Deduce the both Cartesian and Polar form of Cauchy – Riemann equations.
2. State and prove Weierstrass M-test.
3. State and prove the following:
i) Cauchy’s theorem for a disk,
ii) Cauchy’s integral formula.
iii) Taylor’s theorem.
iv) Fundamental theorem of Algebra.
Course –4: Math 1.4 Discrete Mathematics
1. Define Principle Disjunctive Normal Form (PDNF) and Principle Conjunctive Normal
Form (PCNF). Describe a method to covert given compound proposition to PDNF and
PCNF. Illustrate with examples. What are the applications of PDNF and PCNF?
2. State and prove the Pigeonhole Principle and the Generalized Pigeonhole Principle. Every
sequence of n
2 3 ( ) n n a n = and ( 1)( 2) ( ) n n n b n n n n = + + + ⋯ , thenshow that the sequences { } 1 n n a and { } 1 n n
+ 1 distinct real numbers contains a subsequence of length n + 1 that is
either strictly increasing or strictly decreasing.
3. Define connectivity relation R
∞
on R.State and explain Warshall’s algorithm with
illustration.
Course –5: Math 1.5 Differential Equations
1. State and prove Liouville’s theorem and hence verify the same for y y y y ′′′ ′′ ′ − − + = 0 in
the interval[0, 1] .
2. Deduce Green’s formula and verify the same for y y y ′′ ′ + + = 0.
3. Discuss the series solution of Hermite differential equation. Find the recurrence relation for
Hermite polynomials and hence deduce orthogonality of Hermite polynomials.
Click here to download pdf.
Course – 1: Math 1.1Algebra
1. Let 2 3 2 3
4 D e a a a b ba ba ba = { , , , , , , , } be set of 8 elements. Define the product in D4
by the relation 4
a e = ,
2
b e = and 1
ab ba−
= . Show that D4
is a group. Find all subgroups
od D4
and find all left cosets of one of the subgroup of D4
.
2. Define a group homomorphism and Kernel of the homomorphism. State and prove
fundamental theorem of homomorphism of groups. Illustrate with example.
3. Define Conjugacy relation; prove that it an equivalence relation. Find the partition of D4
induced by the Conjugacy relation and hence find the class equation of D4
.
Course – 2: Math 1.2Real Analysis – I
1. Define limit point of a set.Distinguish the limit point of a set and limit of a sequence with
illustration. Prove that every infinite bounded set has a limit point.
2. State and prove Cauchy’s second theorem on limit of a real sequence. If 3
b converges and
find their limits.
3. State and prove the Cauchy’s condensation test and Kummer’s test.Illustrate the
significance of these testes in finding the convergence of the series.
Course –3: Math 1.3Complex analysis– I
1. Define an analytic function. State the necessary and sufficient condition for the functions to
be analytic. Deduce the both Cartesian and Polar form of Cauchy – Riemann equations.
2. State and prove Weierstrass M-test.
3. State and prove the following:
i) Cauchy’s theorem for a disk,
ii) Cauchy’s integral formula.
iii) Taylor’s theorem.
iv) Fundamental theorem of Algebra.
Course –4: Math 1.4 Discrete Mathematics
1. Define Principle Disjunctive Normal Form (PDNF) and Principle Conjunctive Normal
Form (PCNF). Describe a method to covert given compound proposition to PDNF and
PCNF. Illustrate with examples. What are the applications of PDNF and PCNF?
2. State and prove the Pigeonhole Principle and the Generalized Pigeonhole Principle. Every
sequence of n
2 3 ( ) n n a n = and ( 1)( 2) ( ) n n n b n n n n = + + + ⋯ , thenshow that the sequences { } 1 n n a and { } 1 n n
+ 1 distinct real numbers contains a subsequence of length n + 1 that is
either strictly increasing or strictly decreasing.
3. Define connectivity relation R
∞
on R.State and explain Warshall’s algorithm with
illustration.
Course –5: Math 1.5 Differential Equations
1. State and prove Liouville’s theorem and hence verify the same for y y y y ′′′ ′′ ′ − − + = 0 in
the interval[0, 1] .
2. Deduce Green’s formula and verify the same for y y y ′′ ′ + + = 0.
3. Discuss the series solution of Hermite differential equation. Find the recurrence relation for
Hermite polynomials and hence deduce orthogonality of Hermite polynomials.
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