Visva Bharati University MSc Maths 4th Sem Question Paper

Will you please provide the MSc Maths 4TH sem Question Paper of
Visva Bharati University?

[Kiran Chandar]

Here is the list of few questions of MSc Maths 4th sem exam of Visva Bharati University which you are looking for .

Graph Theory (answer any two) 2×10=20
1. Introduce the concept of connectivity parameters kand l of a graph with suitable
examples. Does there always exist a graph with S r = = l k , and t = d for any
three integers r,s,t such that ? t s r o £ £ < It so establish the result. 10
2. Discuss the concept of planar graphs: starting with Euler’s polyhedron formula and
giving kuratowski’s characterization of planar graphs. 10
3. Introduce the concept of colorability in Graphs. Discuss four color conjecture and
prove five coler theorem. 10

402. Numerical Analysis (answer any two) 2´10=20
1. Solve 10
10x – 7y + 3z +5u = 6
-6x + 8y – z -4u =5
3x +y +4y +11u = 2
5x -9y -2z +4u = 7
By Gauss elimination method.
2. Given the values 10
x : 5 7 11 13 17
F(x) 150 392 1452 2366 5202
Evaluate f(9) using (i) Lagrange’s formula
(ii) Newton's divided difference formula
3. Evaluate ∫
+
6
0
2 1 x
dx
by using 10
(i) Simpson’s 3/8 rule
(ii) Weddle’s rule and compare the results with its actual value.
403. (A) Functional Analysis (optional) (answer any two parts) 10+10=20
1. (a) Prove that every linear operator is bounded in a finite dimensional named linear
space. 5
(b) show that the functionals defined as C[a, b] by ] b , a [ c y , dt ) t ( y ) t ( x ) x ( f
b
a
0 0 Î = ∫ is
linear and bounded. 5
2. (a) Show that the dual space of 1 l is 2 l 6
(b) Show that Rn is a Hilbert space with inner product defined by 4
< x,y >=Î1h1+..... +În h2
where x =( Î1 ,......, În ) and y=( h1,.......h n)
3. State fundamental theorems on Banachspaas. Prove any one of them in two disterent
methods.
Reference: (1) Functional Analysis with Application by kreyszig.
(2) Functional Analysis by Megginson.

(B) Fluid Dynamics (optional) 5´4=20
1. Discuss the velocity distribution in the flow of a viscous incompressible fluid
between two parallel plates taking the fluid properties as constant in the following
cases.
(i) Plane Couette flow. 5
(ii) Generalized plane Couette Flow. 5
2. Establish the relation between wave velocity and group velocity with emphasize on
dynamical significance of group velocity. 5
3. Write an essay about Prandtl’s boundary layer theory with its importance in Fluid
Dynamics. 5

404. (A) Mathematical Statistics (optional) (answer any two) 2´10=20
1. State and prove Baye’s theorem. 10
Show that if A and B are independent then A¢ and B¢ are independent. Also A and B¢
are independent.
2. Write short notes on any two of the following 10
(a) Binomial Distribution
(b) Normal Distribution
(c) Regression and correlation
3. State the properties of t-distribution. How does it differ from a standard normal
distribution? Mention some applications of t-distribution. 10
(B) Dynamical System and Fractal Geometry (optional) 10+10=20
1. Construct the Sierpinski triangle and explain why it is self similar. 10
2. What is box-dimension. Find the box dimensions of Sierpinskiu triangle and Cantor
Set. 10

405. (A) Fuzzy Sets and Their Applications (optional) (answer any two) 2´10=20
1. What is the role of α-cuts and strong α-cuts in fuzzy set theory? What are the
differences between them? Describe these concepts in your own words. 10
2. Prove that the properties of symmetry, reflexivity and transitivity are preserved under
inversion for both crisp and fuzzy relations. 10
3. Give examples from daily life of the following types of fuzzy propositions and
express the propositions in its canonical form:
(i) Unconditional and qualified propositions. 5
(ii) Conditional and unqualified propositions. 5

(B) General Theory of Relativity and Cosmology (optional) 5´4=20
1) State Einstein’s principle of equivalence and discuss its application in brief to red
shift of spectral times and curvature of light in gravitational field. 5
2) State Einstein’s law of gravitation. Modify it for empty space and obtain
Schwargschild’s solution for an isolated particle continually at rest at the origin. 5
3) Derive Friedman-Robertson-Wlaker (FRW) model and discuss in detail its dynamical
consequences. 5
4) Discuss the physical properties of de-sitter universe and compare it with those of the
actual universe. 5

I want to get Visva Bharati University MSc Maths 4th Sem Question Paper for doing prparation so will you please provide me that ?

[Kiran Chandar]

As you want to get Visva Bharati University MSc Maths 4th Sem Question Paper for doing prparation so here I am giving you some questions of that paper:

1. In a Banach space xn → x ; yn→y implies that xn+ yn→
(a)x+y (b)x/y (c)x-y (d)xy
2. l p
n is
(a) linear space (b) Banach space (c) not Banach space
(d) none of these
3. In a Hilbert space | (x,y) |
(a) ≤ || x || || y || (b) ≤ || x || (c) ≤ || y || (d) = || x || / || y ||
4.. A closed convex subset C of a Hilbert space H contains a unique
vector
(a)of smallest norm (b) which is negative (c) which is negative
(d) none of these
5. An orthonormal set is a Hilbert space is
(a) dependent (b) linearly independent (c) generates H
(d) none of these
6. || TT*|| is equal to
(a) || T || (b) || T*|| (c) || T||2 (d) || T|| / || T*||
7. If A is a positive operator then I + A is
(a) singular (b) singular and onto (c) non singular (d) none of these
8. An operator U on H is unitary it
(a) UU* = U*U (b) UU* = U*U=I (c) UU* = U (d) UU* = U*
9. det ([δij]) =
(a) 0 (b) 1 (c) ½ (d) 2
10. For elements x and x0 in G the value of || x0
-1x - 1|| is
(a) < 0 (b) =0 (c) < ½ (d) < 1
SECTION-B (5X5=25 MARKS)
11. (a) Prove that addition and scalar multiplication are continuous in a Banach space
(or)
(b) Prove that the mapping x→ Fx is an isometric isomorphism of N into
N**
12. (a) State and prove the Schwartz inequality.
(or)
(b) If {ei}is an orthogonal sets in H , and if x in H, then prove that
x-Σ (x, ei ) ei ^ ej for j.
13. (a) Prove that || T*T|| = || T||2
(or)
(b) Prove that the self adjoint operators on H satisfy:
(i) A1≤A2→ A1 + A≤ A2 + A for every A:
(ii) A1≤A2 and α ≥ 0 ⇒ α A1≤ α A2
14. (a) For a fixed real number θ , prove that the using two matrices are
(b) For a self-adjoint operator A on H , prove that A = ∫ λ d Eλ .
15. (a) For a regular element x in a Banach algebra,prove that
∞
x-1 =1 + Σ (1-x)n.
n=1
(or)
(b) Prove that σ(x) is nonempty.
16. (a) If M is a closed linear subspace of a Banach space N,prove that N/M
is a Banach space.
(or)
(b) State and prove the Hahn-Banach theorem.
17. (a) Prove that the mapping T→T* is an isometric isomorphism of B(N) into B(N*).
(or)
(b) If M is a proper closed linear subspace of H, prove that there exists a
non-zero Z0 in H such that Z0 ^M.
18. (a) For an arbitrary functional f in H* , prove that there exists a unique
vector y in H such that f (x) = (x, y) for every x in H.
(or)
(b) State and prove the conditions under which sum of projections is also
a projection.
19. (a) Prove that two matrices in An are similar and only if they are the
matrices of a single operator on H relative to different bases.
(or)
(b) For an arbitrary operator on H, prove that the eigen values of T
constitute a non-empty finite subset of the complex plane.
20. (a) Prove that the boundary of S is a subset of Z .
(or)
(b) Prove that r(x) = lim ||xn||1/n.

I would like get a chance to attend my examination if f it is there for the BA to go for further studies