### M.sc Mathematics 2nd Year Questions 2023

In this post m.sc mathematics 2nd year questions have been given for session 2023.

This m.sc mathematics exam's important questions have been prepared with hard work.

These questions have been asked many times in previous exams.

These are the probable questions for the upcoming examination.

Therefore, read this m.sc mathematics part 2 question paper carefully and prepare answers for it.

If there is any doubt in knowing the answers to the questions then contact your teacher and clear your concepts.

The pattern of these questions has been set on the basis of msc mathematics previous question papers.

To prepare answers to these questions, write them in your notebook.

Now read the relevant chapter in the textbook recommended by your university.

Read the chapter again and again if the concept is not clear.

Through this process, you will know the answers to many more questions.

If you want the m.sc question paper in pdf then you can visit the website of the university.

From there m sc question paper can be saved.

__FAQ__

Question - **What is the pass mark for MSc Maths?**

Answer - It depends on the university. By the way, in some universities, candidates pass on 60%, and in some even less.

Let's solve m.sc maths 2nd year question papers.

#### Msc Maths Question Paper

**Numerical Analysis Questions For M.sc Mathematics**

- Find the formula for Quadrature for equally spaced arguments and hence derive Simpson's three-eighth rule.
- Compare Newton's method with the Regula-Falsi method. Apply Newton's Raphson method to find the square root of 12 to five places of decimals.
- Prove that the divided differences can be expressed as the product of multiple integrals.
- Compare Newton's method with the Regula-Falsi method. Apply Newton's Raphson method to find the square root of 12 to five places of decimals.
- Find the formula for Quadrature for equally spaced arguments and hence derive Simpson's three-eighth rule.

**Functional Analysis Important Questions For M.sc Maths**

- Let X and Y be two normed linear spaces where X is finite-dimensional. Then show that every linear map from X to Y is continuous.
- Let L be a linear space over F, then show that the sum of two inner products on L is also an inner product on L.
- State and prove F. Riesz's lemma.
- State and prove the closed graph theorem.
- Show that the inner product space is jointly continuous.
- State and prove the Hahn-Banach theorem.
- State and prove open mapping theorem.
- State polarization identity and explain it in an inner product space.
- Give an example of a Banach space that is not a Hilbert space.
- If H is a Hilbert space, then show that the conjugate space H* is also a Hilbert space.
- If a Hilbert space H is separable, then show that every orthonormal set of H is countable.

**Partial Differential Equations Important Questions**

- Explain Charpit's method for the solution of non-linear partial differential equation of the first order.
- Derive the Fourier equation of heat conduction.
- Explain Charpit's method for the solution of non-linear partial differential equation of the first order.

**Analytical Dynamics In Mathematics Questions**

- Prove that Lagrange's Bracket does not obey the commutative law of algebra.
- A particle moves in a plane under a central force depending on its distance from the origin. Then construct the Hamiltonian of the system and derive Hamilton's equation of motion.
- Give the physical significance of Hamilton's characteristic function.
- Explain the principle of least action and hence establish it in terms of the arc length of a particle path.
- Derive the Hamilton-Jacobi equation and then find Hamilton's characteristic function.
- Discuss the motion of a sphere when the small sphere rolls without slipping on the rough interior of a fixed vertical cylinder of greater radius.
- Derive the formula for the Kinetic energy in terms of generalized coordinates and express generalized components of momentum in terms of Kinetic energy.
- Find the equation of motion of a simple pendulum by applying Lagrange's equation of motion.
- State and prove the Jacobi-Poisson theorem.
- Describe the motion of particles about revolving axes.
- Determine the kinetic energy and the moment of momentum of a rigid body rotating about a fixed axis.
- Derive Euler's equation of motion for the motion of a rigid body about a fixed point.
- Derive the formula for kinetic energy in terms of generalized coordinates and express generalized components of momentum in terms of kinetic energy.
- Derive Lagrange's equation of motion from Hamilton's canonical form of equations.
- Derive Lagrange's equation of impulsive motion in a Holonomic dynamical system.
- A bead is sliding on a uniformly rotating wire in a force-free space. Derive its equation of motion.
- Discuss the motion of a spherical pendulum deducing from Hamilton's canonical equations of motion.
- Define the generating function of a transformation and give an example of a generating function of transformation.
- Explain the terms (i) degree of freedom, (ii) Constraints, and (iii) generalized coordinates and classify the dynamical systems based on different types of constraints.
- Explain the difference between possible displacement and virtual displacement. Give one example of each.
- Use the Routhian equation of motion to determine the motion of a uniform heavy rod turning about one end which is fixed.
- Construct the Routhian function and Routh's equation for the solution of a problem involving cyclic and non-cyclic coordinates.
- What do you mean by Hamilton's function? Find the differential equations for Hamilton's function.

**Fluid Mechanics Important Questions**

- Derive Euler's equation of motion in cylindrical polar coordinates.
- State and prove Kelvin's circulation theorem.
- Derive the equation of continuity in Cartesian form.
- Derive Cauchy-Riemann differential equation in polar form.
- Describe the motion of fluid between rotating co-axial circular cylinders.
- Derive Euler's equation of Fluid motion.
- Derive the equation of motion under impulsive force.
- Derive the rate of strain tensor of fluid in motion.
- What do you mean by Source, Sink, and Doublet? Describe them with suitable examples of each.
- Obtain the equation of continuity in spherical polar coordinates.
- Write notes on the following- (i) Velocity Potential (ii) Velocity Vector (iii) Boundary Surface.
- Show that the two-dimensional irrotational motion, stream function satisfies Laplace's equation.
- Derive Navier-Stokes equation of motion of the viscous fluid.
- What do you mean by source and sink? Find the complex potential due to a source of strength m placed at the origin.
- Derive the equation of energy for an incompressible fluid motion with constant fluid properties.
- Obtain the boundary layer equations in two-dimensional flow.

**Operations Research Important Questions**

- Show that every extreme point of the convex set of feasible solutions is a B.F.S. (Basic Feasible Solution).
- Explain in detail the dual simplex method elaborating on each step.
- Describe the method of constructing the solution to the 'Game Problem' where the game is without saddle point.
- Define hyperplane and hypersphere. Prove that every hyperplane in Rn is a convex set.
- Prove that the intersection of any finite number of convex sets is a convex set.

**Tensor Algebra, Integral Transforms, Linear Integral Equations, Operational Research Modeling Questions**

- State and prove the quotient theorem of tensors; give an example.
- Show that the covariant derivative of a co-variant vector is a mixed tensor of rank two.
- State and prove the Convolution theorem for the inverse Laplace transform.
- Determine a deterministic model with instantaneous production. Shortage allowed.
- Define Fredholm integral and Volterra integral equations.
- State and prove the convolution theorem of Fourier transforms.
- Explain the Fredholm integral equations of three kinds.
- Show that any linear combination of tensors of type (r, s) is a tensor of type (r, s).

**Programming in C Important Questions**

- What are the different types of statements written in C? Explain each type with the help of an example.
- What is an Operator? Describe different types of operators in C with examples.
- Explain different types of Scalar data types in C with examples.
- What is a control statement in C Language? Explain with the help of an example.
- What is meant by looping in C? Explain some of the looping statements with examples.
- Discuss integer constant, floating point constant, and character constant. What are the rules for constructing integer contents?
- What is a function? Explain the function using an example. Discuss the advantages and disadvantages of using functions. Are functions required when writing a C program?
- What are the different types of If and else statements used in C? Explain each of them with help of an example.
- Write a program in C to find the roots of a quadratic equation.
- Why Switch statements are used in C? How do they differ from other conditional statements?
- What are logical errors and how does it differ from syntax errors? Write a program in C to swap the value of two variables.
- Give an example of a Switch statement.
- What are structures? When and why are they used in C? Give an example to explain them.
- Differentiate between arrays and pointers. Give examples for each of them.
- Write short notes on the following- (a) Operators (b) GOTO statement (c) Break and continue statement (d) Global variables (e) Branching statements (f) Recursion
- What is debugging? When is the “void” keyword used in a function?
- Write a program in C to print the multiplication table of a given number.

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