### M.sc Maths First Year Important Questions 2023

In this post msc 1st year maths subject wise important questions are given for 2023.

You can score good marks in the exam by solving this msc mathematics question paper.

Solving math papers will be easy, if you remember math formulas.

Along with this, you should have a clear concept regarding the questions.

Sometimes it becomes difficult to solve the questions. In this situation, you can take the help of your teacher.

The pattern of this msc math question paper has been prepared on the basis of the previous year's question paper.

If the questions do not match your syllabus then consider this mathematical question paper as a practice set or model paper.

You can also find m.sc maths question paper pdf on the official website, which will further increase your area of questions.

*FAQ*

Question - **Is MSc Maths tough?**

Answer - This subject can be tough for you if the math concepts are not clear.

It can be made easy. For this, you have to practice the questions on the subject again and again. Formulas have to be memorized and used properly. If math questions are solved step by step then this subject can be interesting. The duration of the MSc Mathematics course is 2 years. Getting a master's degree in this subject is a challenging task. In this degree, you get deep knowledge through the topics of Dynamical Systems, Geometry, Calculus, Number Theory, Differential Equations, and Algebra.

#### MSc 1st Year Question Paper

**Advanced Abstract Algebra Question Paper**

- Prove that a polynomial of degree n over a field can have at most n roots in any extension field.
- State and prove the Remainder theorem.
- Prove that every finite extension of a field F is algebraic.
- Show that every element in a finite field can be written as the sum of two squares.
- Prove that every finitely generated module is a homomorphic image of a finitely generated free module.
- Prove that every homomorphic image of a Noetherian (Artinian) module is Noetherian (Artinian).
- State and prove Schur's theorem.
- State and prove the fundamental theorem of Galois's theory.
- State and prove the Jordan-Holder theorem on any group.
- What do you mean by extension of a field? Establish the transitivity property of finite extension of a field.
- State and prove Kronecker's theorem.
- Prove that in every principal ideal domain, each pair of elements has a greatest common divisor.
- Prove that the range of homomorphism of a module is a sub-module of the module.
- Define algebraic and simple extensions of a field and give an example of each one.
- Define a sub-module of a module M. Show that the arbitrary intersection of sub-modules of a module M is a sub-module of M.
- Define a subnormal series of a group. Hence or otherwise form a subnormal series of the additive group of integers.
- Construct all the composition series of Z 60.

**Real Analysis Important Questions**

- State and prove Implicit function theorem.
- Define a function of bounded variation clearly and prove that a bounded monotonic function is a function of bounded variation.
- State and prove Schwarz's theorem for a function of two variables.
- State and prove a necessary and sufficient condition for a function f to be R-integrable over [a, b].
- State and prove the Bolzano-Weierstrass theorem and give a suitable example of it.
- Deduce the Bolzano-Weierstrass theorem from the Heine-Borel theorem.
- Prove that a necessary and sufficient condition for a function f on [a, b] to be of bounded variation is that it can be written as the difference of two monotonically increasing functions on [a, b].
- State and prove Abel's theorem.
- State and prove the inverse function theorem.
- State and prove implicit function theorem.

**Measure Theory Exam Questions**

- Show that the measure of an enumerable set is Zero.
- Prove that a necessary and sufficient condition for a function f to be L-measurable is that it is the limit of a convergent sequence of simple functions.
- Prove that the class M of L-measurable functions is closed with respect to all arithmetical operations.
- State and prove the bounded convergence theorem.
- State and prove Lebesgue dominated convergence theorem.
- Prove that every absolutely continuous function is an indefinite integral of its own derivative. If f is a measurable function then show that f is also a measurable function.
- Show that the class of all measurable functions is closed with respect to all algebraic operations.
- Give the analytic description of Cantor's Ternary set and show that it is an uncountable set of measure Zero.
- State and prove Fatou's Lemma.
- State and prove Lebesque's monotone convergence theorem.
- State and prove dominated convergence theorem.

**M.sc Topology Question Paper**

- What do you mean by T 2 -space or a Hausdorff space. Prove that every discrete space is a Hausdorff space.
- Show that every metric space is a normal space.
- Define a Topological space, Indiscrete Topology, Discrete topology, co-finite topology, and co-countable topology.
- Prove that a topological space (X, T) is normal space if and only if each neighborhood of a closed set F contains the closure of some neighborhoods of F.
- Define a metrizable space and Equivalent metrics giving one suitable example for each.
- Define hereditary and topological properties and show that the property of a T1 -space is both hereditary and topological.
- Prove that the union of any family of connected sets having a non-empty intersection is connected.
- Prove that every second countable space is separable.
- What do you mean by a regular space? Prove that a compact Hausdorff space is regular.
- Prove that a topological space (X, T) is a T1 -space if every singleton subset {x} of X is a T-closed set.
- Prove that every compact subspace of the real line is closed and bounded.
- Prove that every convergent sequence in a Hausdorff space has a unique limit.
- Give an example of topological space which is a T1 -space but not a T2 -space.
- Prove that a finite sub-set of T1 -space has no cluster point.
- If X and Y are topological spaces, then prove that X × Y is connected iff X and Y are connected.
- Prove that every closed subspace of a normal space is normal.
- Prove that every compact subspace of a Hausdorff space is closed.
- Prove that every closed subset of a compact space is compact.
- Prove that an arbitrary intersection of topological spaces is a topological space.
- Define T 3 -space and T 4 -space and prove that every T 4 -space is a T 3 -space.
- Show that connectedness is not a hereditary property.

**M.sc Linear Algebra, Lattice Theory And Boolean Algebra Questions**

- Define a Lattice and dual of a statement in a Lattice. Give two examples to make it clear.
- Define a sub lattices with two examples.
- What do you mean by a complemented lattice? Prove that if L is a bounded distributive lattice then complements are unique if they exist.
- If L is a complemented lattice with unique complements then the join irreducible elements of L other than O are its atom prove it.
- Consider the Boolean algebra D 210 the divisors of 210. Find the number of sub-algebras of D 210.
- Prove that the row space and the column space of matrix A have the same dimension.
- Prove that all bases for a vector space V have the same number of vectors.
- Prove that a linear operator E is a projection on some subspace if it is idempotent.
- Prove that two real quadratic forms are equivalent if they have the same rank and index.
- If f is a linear functional on a vector space V(K) then show that (i) f(0) = 0 and f(–x) = –f(x).
- If R is a ring and L is a lattice of all ideals of R, then prove that L is modular.
- Prove that a Boolean Algebra B is a complemented distributive lattice.
- Prove that in Boolean Algebra, the complement of an element is unique.
- Define a linear transformation and its null space. If U(f) and V(f) are two vector spaces and T is a linear transformation from U into V, then show that the kernel T or null space of T is a subspace of U.

**Complex Analysis Important Questions**

- State and prove poison's integral formula.
- State and prove Laurent's theorem.
- State and prove the maximum modulus principle.
- State and prove Cauchy-Hadmard's theorem for power series.
- Describe different kinds of singularities.
- What is the pole of a function? Also, introduce the residue at simple pole and pole of order m.
- By introducing the Bilinear transformation, derive the existence of fixed points of a Bilinear transformation.
- State and prove Poisson's integral formula.
- State and prove Cauchy's theorem.

**Theory Of Differential Equations Important Questions**

- State and prove Cauchy-Peano's existence theorem.
- State and prove Picard's-Lindelof theorem.
- State and prove Ascolli's Lemma.
- Explain different types of critical points for a system and give the geometrical meaning of each critical point.
- Define a linear system and show that it satisfies the Lipschitz condition and the set of solutions form a vector space.
- Define fundamental matrix.
- Describe the orthogonal property of the Laguerre polynomial.
- Find Rodrigue's formula for the Legendre polynomial.
- Derive an expression for the generating function for Bessel's function.
- What is the meaning of generating function for Legendre polynomial? Hence find it.

**Set Theory, Graph Theory, Number Theory And Differential Geometry Questions**

- Prove that the set of all real numbers R is uncountable.
- If A and B are countable sets then show that A × B is also countable.
- Prove that Zorn's lemma implies a well-ordering theorem.
- What is the Axiom of choice? Show that the Axiom of choice is equivalent to Zermelo's postulates.
- Prove that every set can be well ordered.
- Find g.c.d. of 28 and 49 and express it as a linear combination of 28 and 49.
- State and prove the division algorithm in the theory of numbers.
- Factorize 493 by Euler's Factorization method.
- Define an Umbilic. Prove that in general three lines of curvature pass through an umbilic.
- If a tree has n vertices of degree 1, two vertices of degree 2, and four vertices of degree 4 then find the value of n.
- If a tree has n vertices of degree 4 then find the value of n.
- Prove that an undirected graph is a tree if there is a unique path between any two vertices.

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