# 21
| |||

| |||

Re: Msc maths model question papers of previous years
SIR I HAVE MODEL QUESTION PAPERS IN BHARATHIDASAN UNIVERSITY IN MSC MATHEMATICS BEACAUSE PREPARING THE EXAMINATIONS PLEASE SIR KINDLY THE QUESTION PAPERS SEND MY EMAIL ID IS== abiramisanthi2012@gmail.com I AM CDE STUDENT OF THIS UNIVERSITY |

# 29
| |||

| |||

Re: Msc maths model question papers of previous years
Choose the correct answer: 1.The number of conjugate class in S5 is a)5 b) 7 c)3 d)4 2.If O(G)=28 then G has a normal subgroup of order a) 6 b) 8 c) 7 d) 9 3.An element a in a Euclidean ring R is a unit if a) d(a)=1 b)d(a)=0 c) d(a)=d(1) d) d(a)=d(0) 4.The units in Z[i] are a) ± 1b) ± i c) ± 1, ± i d)1,i 5.If L,K ,F are the finite fields such that L⊂ K⊂F then [L:F] is a)[L:K][K:F] b)[L:F][F:K] c)[K:L][L:F] d)[F:K][K:L] 6.If F is the of rational numbers and if f(x)= x^3-2 then [f(2 ):f] is a)2 b) 3 c)1/3 d)1/2 7. If F is the field of real numbers and K is the field of complex numbers then O(G(K,F)) is a)2 b) 3 c)1 d)0 8.If H is the subgroup of G(K,F) and KH is the fixed field of H then [K:KH] is a)O(K)b)O(H)c)O(K/KH) d)O(KH) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 2 of 57 9.If T∈A(V)is such that (vT,v)=0∀ v∈V then T is a) I b) o c)T-1 d) none of these 10. If A transformation T is normal if a) TT* =TT* b) TT* =I c) TT* =0 d) T=T* Section B (5×5=25 marks ) 11. a)Prove that the relation conjuncy is an equivalence relation or b) If O(G) =55 then Prove that its 11-sylow subgroup is normal 12. a)Prove that every Euclidean ring has a unit element or b) State and prove Euclid’s lemma 13. a)Prove that the set of algebraic elements in K over F form a subfield of K or b)If f(x)=F[x] is of degree n≥ 1 then Prove that there is an extension E of F of degree atmost n in which f(x ) has n roots 14 a) If F is field of real numbers and K is the field of complex numbers then prove that K is an extension of F. or b)If F and K are the two finite fields ∋ F⊂K and if O(F)=q then Prove that O(K)=qn where n=[K:F] 15.a) If A,B ∈Fn then prove that tr(AB)=tr(BA) or . b) If T∈A(V) then prove that T* is also in A(V) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 3 of 57 Section C–(5× 8=40marks) 16.a) If G is the finite group then prove that O(C(a))=O(G)/O(N (a)) or b) State and prove Sylow’s second theorem 17 a) Let R be an Euclidean ring and A be an ideal of R .Then prove that A=(a0) for some ao ∈A or b) Prove that J[i] is an Euclidean ring. 18 a)If a∈ K is an algebraic over F of degree n then prove that [F(a):F]=n or b) Prove that a polynomial of n degree over then F can have atmost n roots in any extension field . 19 a) If K is a field and if σ 1, σ 2 ,……,σ n are distinct automorphisms of K then prove that it is impossible to find elements a1,a2, ….,an not all of them 0 such that a1σ 1(u)+ a2σ 2(u)+…….+ anσ n(u) =0 ∀u or b)If K is a finite extension of field F then prove that O(G(K,F)) ≤ [K:F] 20 a) If T∈A(V) has all its characteristic roots in F then prove that there is a basis of V in which the matrix of T is regular or b) If F is a field of characteristic 0 and if T∈A(V) if such that trTi =0∀ i ≥ 1then prove that T is nilpotent. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 4 of 57 MODEL QUESTION PAPER For candidates admitted from 2007-2008 and onwards M.Sc Degree Examination Third Semester MATHEMATICS Time:3Hours TOPOLOGY Max.Marks:75 Answer All Questions Section A –(10*1=10 marks) Choose the correct answer: 1. In a topology of a set X, both X and φ are a) only open b) only closed c) both open and closed d) neither open nor closed 2.Assume that Y is a subspace of X .If U is open in Y and Y is open in X then U a) closed in X b) open in X c) Y is both open and closed in X d) none of these 3. Let C, D be a separation of X,Y be a connected subset of X then a)Y lies in C b) Y lies in D c) Y lies in C or D d) none of these 4. If L is a linear continuum in the order of topology then L is a)disconnected b)connected c) empty d)the whole space X 5. The set A={x×1/x /0 a) compact b) closed and compact c) closed but not compact d) none of these 6. A space X is said to be separable if it has a) a countable topology b) a countable basis c) a countable sub basis d) none of these 7. A product of normal spaces Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 5 of 57 a) is normal b) need not be normal c)not regular d) none of these 8.Two subsets A and B of a space X are said to be separated by a continuous function f:X→[0,1] such that a)f(A)=f(B)=0 b) f(A)= 0 ,f(B)=1 c f(A)=f(B)=1 d) none of these× 9. The space SΩ ×SΩ is a) completely regular and normal b)normal c) not normal d) completely regular and not normal 10. The Stone –Check compactification β (X) is a)X b) unique c)not unique d) X⊆ β (X ) and β (X)is unique Section B (5×5=25 marks ) 11 a) Assume that β and β ’ are respectively bases for the topologies and ‘. Show that ‘ is finer than iff for each x ∈ X and basis element B ∈ β ’ containing x there is a basis element B’∈ β ’ such that x∈ B’ ∈ β ’ . or b)Show that A =A∪ A’ 12. a) Show that the union of a collection of connected sets that a have a point in common is connected or b) State and prove Intermediate value theorem. 13 a) Show that every compact Hausdorff space is normal or Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 6 of 57 b) Show that a subspace of a regular space is regular and product of regular spaces is regular 14 a)Let A⊂X and f:A→Z be a continuous map of A into Hausdorff space Z .Show that there is atmost one extension of f to a continuous function g: π →Z or b)If Y is complete under d ,Show that YJ is complete in the union metric ρ 15 a)If X is locally compact or if X satisfies the first axiom of countability , show that X is compactly generated . or b)If C1 ⊃ C2 ⊃ C3 ⊃ ….. is a nested sequence of nonempty closed sets in a complete metric space X, and diam Cn→0,show that ∩ Cn ≠φ . Section C–(5*8=40marks) 16 a) Let f:X→Y be a map between two spaces X and Y .Show that the following statements are equivalent i) f is continuous ii) f( A ) ⊂ f(A) ,for every subset A of X iii) f-1(B) is closed in X , when ever B is closed in Y or b) Show that the topologies on Rn , induced by d and are the same as the product topology on Rn . Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 7 of 57 17 a) Show that Cartesian product of connected spaces is connected . or b) Show that the product of finitely many compact spaces is compact. 18 a) Show that every regular space with a countable basis is normal. b) State and prove the Uryshon ‘s lemma 19 a)Show that there is an isomorphic imbedding of a metric space (X ,d) into a complete metric space . or b) State and prove the Ascoli’s theorem. 20 a) Let h : [0,1] → R be a continuous function .For any∈ >0,show that there is a function g:[0,1] →R with h(x) g(x) − < ∈ for all x∈X ,such that g is continuous and no where differentiable . or b) State and prove Tietz –extension theorem. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 8 of 57 MODEL QUESTION PAPER For candidates admitted from 2007-2008 and onwards M.Sc Degree Examination Second semester MATHEMATICS Time:3Hours PARTIAL DIFFERENTIAL EQUATIONS Max.Marks:75 Answer All Questions Section A –(10*1=10 marks) Choose the correct answer: 1. In utt = c2 uxx which describes the vibration of a stretched string , is a) P/T b) T/P c)PT d) none of these 2. Along the curve ξ = constant it is true that dy/dx = a) -ξx / ξ y b) ξx / ξ y c)ξy / ξ x d)ξy / ξ x 3. The two characteristics of uxx _ uyy =1 are a) straight lines b) parabolas c) ellipses d)rectangular hyberbolas 4. A sine wave traveling with speed C in the negative x-direction without changing its shape is given by a)sin(x-ct) b)sin(x - (t/c)) c)sin(x + ct) d)sin(x + (t/c)) 5. The inequality │-an (nπ/l)2 K e -(nπ/l) 2 k t sin (nπx/l)│≤C(nπ/l)2K e -(nπ/l) 2 k t 0 holds if a) t 6. The solution of X ” + α 2 X =0 satisfying X’(0)=X’(a)=0 is a) A sin( n πx/a) b ) A cos( n πx/a) c) A sin( nx/ πa) d ) A cos(nx/ πa) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 9 of 57 7. If u ( x,y ) is harmonic in a bounded domain D and continuous in D = D+ B , then u attains its minimum on a) D b) the boundary B of D c) any point in D d) none of these. 8. In the Neumann problem for a rectangular there is the compatibility condition to be satisfied a) False b) Always True c) Occasionally true d) None of these 9. If δ (x- ξ ) and (y- η) are one dimensional delta functions, then ∫∫ F( x,y) δ (x- ξ ) δ (y- η)dxdy is R (a) F( x , y ) b) F( x, η ) c) F( ξ ,y ) d) F( ξ ,η) 10 The equation ∇ 2 u + k2 u=0 a) Laplace b) Poisson c) Helmholtz d) D’ Alembert s Section B (5×5=25 marks ) 11 a) List any four assumptions made in the derivations of the equations of the vibrating membrane. Or b)Find the general solution of x2 uxx +2xy uxy +y 2 uyy = 0 12 a) Define the Cauchy data for the equation A uxx + Buxy +Cuyy = F(x, y, u, ux, uy ) Or b) Interprêt the D’ Alembert s formula when g(x) = 0 13 a) Obtain u (x , t) = X(x)T(t ) for utt +a 2 uxxxx = 0 Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 10 of 57 Or b)Obtain u (x , t) = X(x)T(t ) for u tt =k u xx, 0 < x< l satisfying u ( o ,t ) = u ( , t) , t ≥ 0 14 a)Prove the solution of the Direchlet problem , if it exists , is unique . Or b) Explain the method of solution of the Direchlet problem involving the Poisson equation 15 a) state the three properties to be satisfied by the Green’s function for the Dirchilet problem involving the Laplace operator Or b) Show that ∂ G/ ∂ n is discontinuous at ( ξ η, ) and and 0 C lim G / n ∈→ ∈ ∂ ∂ ∫ ds =1, C∈ : ( x- ξ ) 2 + ( y- η) 2 =∈ 2 SECTION – C (5×8 =40 marks) 16 a) Reduce x2 uxx +2xy uxy +y 2 uyy = 0 to the canonical form Or b) Reduce uxx +x 2 uyy = 0 to the canonical form 17 a) Solve uxx - uyy = 1 , u(x , 0)=sin x , uy(x,0)=0 Or b) Solve utt =c 2 uxx , 0 0 ≤ x ≤ l and u(0 ,t) = u(l,t)= 0, t≥ 0 18 a)Prove that there exist atmost one solution of the wave equation utt =c 2 uxx 0 and the boundary conditions u ( l , t ) = 0 ,t≥ 0 where u(x, t )is a twice continuously differentiable function with respect to both x and t Or b) solve ∆ 2 u = 0 , 0 19 a) Find the solution of the Direchlet problem ∆ 2 u = -2 ,r< a , 0< θ < 2 π , u( a ,0 )=0 Or b) prove that the solution of the Direchlet problem depends continuously on the boundary data 20 a) Apply the eigen function method to obtain Green’s function of the Direchlet problem in the rectangular domain Or b) Solve the Direchlet problem in the rectangular domain MODEL QUESTION PAPER For candidates admitted from 2007-2008 and onwards M.Sc Degree Examination First semester MATHEMATICS Time:3Hours NUMERICAL METHODS Max.Marks:75 Answer All Questions Section A –(10 ×1=10 marks) Choose the correct answer: 1) Newton’s method is also called a)Newton’s Raphson b) Picard’s rule c) Euler d) none of these 2) Bairstows method is used to find of a polynomial Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 12 of 57 a) complex roots b) real roots c ) repratred roots d) none of these 3 ) is a direct method to solve a system of equation a) Fixed rule b) Runge kutta c) Gauss Elimination d) none of these 4) iteration method to solve a system equation converges faster a) Gauss Jordan b) Gauss seidal c) Taylor’s d) none of these 5) Value of y at specifed values of x can be found from methods coming under I Category a) Adams Bashforth b) Euler c)Bairstow d) none of these 6) is a mulistep method a) Milne b) Adam Moulton c) Euler d) none of these 7)1-D heat equation is an example of equation a)parabolic b) Exponential c) Hyberbolic d) none of these 8) uxx + uyy =f( x , y) is called equation a)Parabola b)Eponential c)Hyberbolic d) none of these 9) method is used to find the largest eigen value a) Power b) Relaxation c) Poission d) none of these 10) Pictorial operator ∆ 2 uij= 1 0 -1 a) 1 -4 1 uij b) 0 1 0 uij c) -1 -4 -1 uij 1 0 -1 d) None of these SECTION B(5×5 =25) 11 a) Using Newton’s method find the root between 0 &1 of x3=6x-4 correct to 3 decimal places Or b) Evaluate 1 0 (1/1 x2)dx + ∫ using Romberg method to find the approximate value of πAnx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 13 of 57 12 a) By Gauss Elimination method solve x+2y+z=3; 2x+3y+3z=10 ; 3x-y+2z =13 Or 1 -2 2 b)Find the inverse of A= 4 1 2 2 3 -1 13 a) Using Taylor’s series method solve dy/dx = x+y given y( 1)= 0 , find y(1.1)andy(1.2) Or b) Using Modified Euler method solve dy/dx = x 2 +y 2 given y(0)= 1 , find y(0.1)andy(0.2) 14 a) Solve the boundary value problem d2 x/dt2 –( 1-(t/5))x= t x(1)=2,x(3)=-1 using Shooting method assuming ,x’(1)=-1.5 and x’(2)= -3 or 2 -1 0 b) find the largest eigen value and vector of A= -1 2 -1 0 -1 2 Or 15 a) Derive the Laplace equation to the pictorial form Or b) Solve the elliptic equation uxx + uyy = 0 for 1 2 1 4 2 5 4 5 u1 u2 u3 u4 Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 14 of 57 SECTION – C (5×8 =40 marks) 16 a) The population of a town is as follows .Estimate the population increase during 1946 to 1976 year x : 1941 1951 1961 1971 1981 1991 poulation lakhs y : 20 24 29 36 46 51 Or b) Using Bairstows method obtain the quadratic factor of x4 -5x3 +20x2 -40x+60 = (taking p0=-4 ,q0=8) 17 a) Using Gauss Sedial method solve x+6y+10z = -3; 4x-10y+3z=-3 ; 10x-5y-2z =3 Or b)By Relation method solve 10 x-2y-2z=6; -x+10y-2z=7 ;-x-y+10z =8 18 a) Using Runge gutta method solve dy/dx = x+y given y( 0)= 1 , find y(0.2) Or b ) Using Milne’s method find y(4.4) given 5xy’+y2 -2 = 0 ,y(4)=1, y(4.1)=1.0049 y(4.2)=1.0097, y(4.3)=1.0143 19 a) d2 y/dx2 =y, y’(1)=1.1752, y’(3)=10.0179 convert this to a difference equation normalize to [0,1]with h =0.25 Or b) Solve the Characteristic value problem d2 y/dx2 +k2 y=0; y(0)=0; y(1)=0 . Convert this to a difference equation .What can you say about k ? 20 a) Solve the Laplace equation for the square region in the fig with boundary values (up to 3 iterations only ) 11.1 17.0 19.2 18.6 u11 u12 u13 u21 u22 u23 Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 15 of 57 0 21.0 0 21.0 0 17.0 0 9.0 8.7 12.1 12.8 Or b) Solve the heat flow problem u(x,0)=sin πx/2 u(0, t)= 0 , u(2 ,t)=0 M.Sc. DEGREE MODEL QUESTION PAPER ( For the candidates admitted from 2007 onwards ) FIRST SEMESTER Mathematics Time : 3hours Ordinary Differential Equations Max. marks:75 SECTION A- (10 × 1=10 marks) Choose the best option 1) The power series solution for x’=exp(-t-2),x(0)=0 a)exists b) does not exist c) exist and is not unique d) none of these 2) The Legendre polynomials form an orthogonal set of functions with weight function a) unity on [0,1] b)unity on [0,1] c)t on [0,1] d) none of these u31 u32 u33 Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 16 of 57 3) For a differentiable matrix X, dX-1/dt is a)-X-2 dX/dt b) –dX/dt X-2 c) –XdX/dt X-1 d) none of these 4) The fundamental system of solutions for the system x-1= 1 0 0 2 a) et 0 b) 0 e2t c) et 0 0 , e2t e t , 0 e2t , e t d) none of these 5) The solution of matrix differential X’= - AX , X(0) = -E is a) e-tA b) –etA c) –e-tA d) –EetA 6) A linear system x-1 = A x admits a non-zero periodic solution of period w iff E-eAw is a) singular b)non-singular c) both d) none o f these 7) If A = 0 1 ,then eAt is 1 0 a) cosht sinht b) cost sint c) 1 0 d) none o f these sinht cosht sint cost 0 1 8) The first approximation x1(t) of the IVP x’ =x/(1+ x2 ), x(0)=1 is a) 1+t b) 1 c)1/(1+t2 ) d) t/2 +1 9) The equation x’’ + x=0 ,t ≥ 0 is a) oscillatory b) non-oscillatory c) none o f these 10) If x(t) is a solution of x’’+a(t) x=0 ,t ≥ 0 where a(t)<0 is a continuous function for t ≥ 0 then x(t) has a) atmost one zero b) no zero c)atleast one zero d) none of these SECTION B- (5 × 5=25 MARKS) 11 a) Prove that Pn(t)=1/2n n! dn / dtn ( t2 -1)n Or b) Prove that t1/2 J1/2(t) = 2 sint /┌( 1/2) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 17 of 57 12 a) Solve x’’-2x’ +x=0 , x(0)=0 , x’(0)=1 Or b) Prove that a solution matrix φ of X’= A(t)X, t∈I is a fundamental matrix of x -1=A(t) x on I iff det φ (t) ≠ 0 13 a) Find the fundamental matrix of x-1=Ax ,where A= 3 -2 Or -2 3 b) State and prove Floquet theorem 14 a) Compute the first three successive approximations for the solution the following equation: x′ = x 2 , x(0)=1. Or b) Find the constants L, k, h for the IVP x′ = x 2 + cos2 t, x(0 =1, R={(t, x): 0 ≤ t ≤ a, │x│≤ b, a ≥ ½, b>0. 15 a) State and prove Strum’s separation theorem. Or b) If a(t) in x′′ + a(t)x = 0 is continuous on (0, ∞) and if a* = t lim→∞ sup t f(t) < ¼ then prove that x′′ + a(t)x = 0 is non oscillatory. SECTION C- (5 x 8 = 40 MARKS) 16 a) Solve: x′′-2tx′+2x = 0 Or b) If a1,a2,…, be the positive zeros of the Bessel function Jp(t), then prove that 1 0 ∫ t Jp(amt) Jp(ant) dt = 2 p 1 n 0, m n 1/ 2 (a ),m n J + ≠ = 16 a) State and prove existence and uniqueness theorem on IVP. Or Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 18 of 57 b)Find the four approximations of a solution to x′′-2x′+x = 0, x(0)=0, x′(0)=1. 18 a) Find the general solution of 1 0 0 6 x 1 0 11 0 1 6 − = − x . Or b) Determine e tA and a fundamental matrix for the system 1 1 2 3 x 0 2 1 x 0 3 0 − − = − 19 a) State and prove Picard’s theorem. ( or) b) State and prove the theorem on non-local existence of solution of IVP x ′ = f ( t, x), x(t0) = xo. 20 a) Show that x′′+a(t)x′+b(t)x = 0, t≥0, where a′(t) exists and is continuous for t≥0 is oscillatory iff x′′+c(t)x = 0, c(t)=b(t)-1/4 a2 (t)-1/2a′(t) is oscillatory. Or b) State and prove Hille – Wintner comparison theorem. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 19 of 57 M.Sc. DEGREE MODEL QUESTION PAPER ( For the candidates admitted from 2007 onwards ) THIRD SEMESTER BRANCH I - Mathematics Time : 3hours ELECTIVE I - NUMBER THEORY Max. marks:75 Answer All questions Section A- (10 × 1=10 marks) Choose the best option 1. The g.c.d of a and a+3 for any integer a is a) a b) a or 3 c) 1 or 3 d) 1or a Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 20 of 57 2. If (a, m) = g, ax ≡ b (mod m) and g does not divide b then the number of solutions of this congruences is a) -1 b) 2 c)1 d)0 3. State Euclidean algorithm. 4. The number of solutions modulo 35 15x ≡ 25 (mod 35) is _____________. 5. (963, 657) is a) 9 b) 27 c) 3 d) 12 6. The value of Φ(1896) is a) 246 b) 426 c) 624 d) 524. 7. Reduced residue system modulo 30 is a) 1 b)6 c)3 d)none of these 8. The degree of congruence 6x+7 ≡ 0 (mod 3) is 2 a) 8 b) 2 c)1 d)0 9. The number of primitive roots of 29 is a) 28 b) 12 c)1 d)0 10. G = { 7, -2, 17, 30, 8, 3} is a group under addition modulo 6. The additive inverse of 8 in this group is a) 7 b)17 c) -2 d) 3 SECTION – B(5 X 5 = 25 marks) 11. a) If (a, m) = (b, m) = 1. Prove that (ab, m) = 1. Or b) State and prove Euclid’s theorem. 12. a) Let p be a prime, show that x2 ≡ (-1) (mod p) has a solution iff p= 2 or p ≡1(mod 4). Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 21 of 57 Or b) Solve the congruence 6x ≡3(mod 9). 13. a) If a∈exponent modulo h, prove that h│(j-k). Or b) Let m>1 be a positive integer. Prove that any reduced residue system modulo is a group under multiplication modulo m. 14. a) Evaluate 23 83 − Or b) If Q is add and φ >0, prove that 2 Q 1 Q 1 2 8 1 2 ( 1) and ( 1) Q Q − − − = − = − 15. a) State and prove Moebius inversion formula. Or b) Show that an arithmetic function has a multiplicative inverse iff f(1)≠0. If the inverse exists, is it unique? SECTION – C (5 X 8 = 40 marks) 16. a) If g is the g.c.d. of b and c, prove that there exists integers x0,and y0 such that g = (b, c) = bx0 + c y0. Or b) (i) If m>0, prove that [ma, mb] = m [a, b] and [a, b] = ab (ii) Show that every integer n>1 can be expressed as product of primes. 17. a) State and prove Wilson’s theorem. Or b) Solve 2 x x 7 0(mod189) + + ≡ 18. a) Prove that the set m z {0,1,..., m 1} = − for m>1 is a ring with respect to Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 22 of 57 addition and multiplication defined modulo m. prove also that m z is a field iff m is a prime. Or b) State and prove lemma of Gauss. 19. a) State and prove the Gaussian reciprocity law. Or b) Prove that a (ab)! a!(b!) is an integer. 20. a) Let f(n) be a multiplicative function and let d\n F(n) f (d) = ∑ . Prove that F(n) is mulplicative. Or b) If F 1,F 1,F F F n 2 0 1 n 1 n n 1, = = = + ≥ + − find an expression for Fn. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 23 of 57 MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS FOURTH SEMESTER MATHEMATICS FUNCTIONAL ANALYSIS TIME: 3HRS MAXMARKS:75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 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 Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 24 of 57 (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 similar : cos sin sin cos θ θ θ θ − and 0 0 i i e e θ − θ (or) (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. SECTION-C- (5X8=40 MARKS) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 25 of 57 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 . Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 26 of 57 MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS THIRD SEMESTER MATHEMATICS FLUID DYNAMICS TIME: 3HRS MAXMARKS:75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. The condition for incompressible flow is (a) ∇ x q = 0 (b)∇ . q =0 (c)∇ q = 0 . 2. The equation of motion of an inviscid fluid is (a) d q 1 F p dt p = − ∇ (b) dq p F dt p ∇ = − (c) dq 1 F p dt p = − − ∇Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 27 of 57 3. For steady motion the Bernoulli’s equation is (a) 2 2 dp q p + + Ω = ∫ constant (b) 2 2 dp q p − + Ω = ∫ constant (c) 2 2 dp q p − − Ω = ∫ constant 4. The vorticity vector is (a)∇ x a (b)∇ . a (c)∇ x a 5. The velocity potential of a source of strength m is (a) mlnr (b) –mlnr (c) m2 lnr 6. The image of a sink is (a) source (b) sink (c) doublet. 7. If the velocity vector q is =x i – y j , then the equation of stream line is (a) xy = k (b) x log y = k (c) y log x = k 8. The velocity profile for a Poiseuille flow is (a) circular (b) parabolic (c) cycloid. 9. The formula for boundary layer thickness is (a) 0 1 u dy U ∞ ∞ + ∫ (b) 1 u dy U ∞ −∞ ∞ − ∫ c) 0 1 u dy U ∞ ∞ − ∫ 10. Prandtl number is the ratio of (a) kinematic viscocity to thermal diffusivity (b) dynamic pressure to shearing stress (c) density to velocity. SECTION-B (5X5=25 MARKS) 11. (a) Derive the equation of continuity. (or) (b) Derive Laplace equation for a liquid in irrotational motion. 12. (a) Obtain the expression for the equation of motion for conservative forces. (or) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 28 of 57 (b) Derive the energy equation when the forces are conservative. 13. (a) Show that both Φ and ψ satisfy Laplace equations. (or) (b) Define a doublet and obtain complex velocity potential for it. 14. (a) Define and explain vorticity and circulation in a viscuous flow. (or) (b) Explain the significance of Reynold’s number. 15. (a) Explain displacement thickness. (or) (b) Explain momentum thickness. SECTION-C- (5X8=40 MARKS) 16. (a) Derive the expression for the rate of change of linear momentum. (or) (b) Prove that the pressure at any point in an inviscid fluid is independent of direction. 17. (a) Derive the general form of Bernoulli’s equation for a fluid in steady motion. (or) (b) State and prove Kelocin’s theorem. 18. (a) State and prove Blasiu’s theorem . (or) (b) Describe the flow of a uniform stream past a circular cylinder of radius having a circulation K per unit arc around it. 19. (a) Derive Navier-stoke’s equation. (or) (b) Discuss the steady flow between parallel planes. 20. (a) Derive the integral equations of the boundary layer. (or) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 29 of 57 (b)Obtain the boundary layer equations for a two dimensional flow along a plane wall. MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS SECOND SEMESTER MATHEMATICS COMPLEX ANALYSIS TIME: 3HRS MAXMARKS:75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 30 of 57 CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. The order of the rational function R(z) = P(z)/Q(z) with deg P = m and deg Q = n is (a) m + n (b) mn (c) max(m, n) (d) min(m, n) 2. The radius of convergence of a polynomial considered as a power series is (a) 0 (b) 1 (c) degree of the polynomial (d) ∞. 3. If γ(t) is a curve with parametric interval [a, b] then the parametric interval of -γ(t) is (a) [-b, -a] (b) [b, a] (c) [-a, -b] (d) [a, b] 4. The value of f dz γ ∫ is (a) f dz γ ∫ (b) f dz γ ∫ (c) fd z γ ∫ (d) fdz γ ∫ 5. The residue of 1/z3 at z = 0 is (a) 1 (b) 0 (c) ∞ (d) not defined. 6. The function log | z | is harmonic in the region (a) C (b) C \ {0} (c) C \ {1, 2} (d) C \ {1}. 7. The value of 0 log(1 ) limx z → z + is (a) 1 (b) 0 (c) not defined. (d) ∞. 8. The function sin z is bounded in (a) C (b) outside a disc (c) inside a disc (d) C/ {0}. 9. An analytic branch of z a − can be defined in a region Ω if (a) Ω simply connected (b) Ω simply connected and a Є Ω (c) Ω is the whole plane (d) Ω is any region and a Є Ω. 10. The Riemann mapping theorem is not valid if the region Ω is (a) an open disc (b) an open rectangle (c) a half plane (d) the whole plane. SECTION-B (5X5=25 MARKS) 11.(a) State and prove Luca’s theorem . Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 31 of 57 (or) (b) Show that every rational function has a partial fraction expansion. 12.(a)Define the index n(γ ,a) of a closed curve γ (a∉γ) and prove that it is an integer. (or) (b) Using local correspondence theorem show that a non-constant analytic function region is an open map. 13.(a) State and prove Residue theorem. (or) (b) State and prove Poisson’s formula for functions harmonic in a closed disc. 14.(a) Show that every series n n n A z ∞ =−∞ ∑ represents an analytic function in an annular region R1< z (b) Using Miffag-Leffler’s theorem prove 2 2 2 1 sin ( ) z z n π ∞ −∞ = − ∑ 15.(a) Define {zn} or z(t) tending to the boundary of Ω and prove that if f: Ω→Ω1 is topological then if {zn} or z(t) tends to ∂ Ω then {f(zn)}or f(z(t)) tends to ∂ Ω1. (or) (b) Show that the Riemann mapping function of a simply connected region Ω can be extended to a one sided free boundary arc analytically SECTION-C- (5X8=40 MARKS) 16.(a) Show that if f(z)= u(z) + iv z is analytic in a region Ω (u ,v, Real and imaginary parts gf ) then show that f is constant if either u or v or u2 + v2 or u 2 – v 2 or uv. (or) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 32 of 57 (b) Show that A⇒B if, (i) The image of the real axis under any linear fractional transformation is a circle or a straight line (ii) The cross ratio of four points (a, b, c, d) is real iff the four points a, b, c, d lie on a circle or a straight line. 17.(a) Show that zeros of non-constant analytic functions are isolated and deduce that f(z) and g(z) are analytic in a Ω and if f(z) = g(z) over a set of points A⊂ Ω with a limit point in Ω then f(z) ≡ g(z) for all z in Ω. (or) (b) Describe isolated singularities of f analytic in 0< z a − <δ at z=a using algebraic order and state and prove Weierstrass theorem on isolated essential singularities. 18.(a) Compute 0 logsin d π θ θ ∫ using residue calculus. (or) (b) Prove the following: (i) If u1 and u2 are harmonic in a region Ω then 1 2 2 1 u du u du * * 0 γ − = ∫ where γ ~ 0 in Ω. (ii) If u is a harmonic in an annulus R1⊂ z ⊂ Rz then 1 log 2 z r ud r θ α β π = = + ∫ for R1 ⊂ r⊂ R2 and α= θ if u is harmonic in a disc. 19. (a) Show that every analytic function in a annulus R1< z a − (or) (b) State and prove Mittag Leffler’s theorem. 20. (a) Construct a bijective bi-continuous map of z <1 onto the whole w-plane. Can this be analytic justify. (or) (b) In the content of Riemann mapping theorem if Ω is symmetric with respect to Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 33 of 57 real axis and if z is real then prove that Riemann mapping function f(z) satisfies f z f z ( ) ( ) = MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS THIRD SEMESTER MATHEMATICS MATHEMATICAL STATISTICS TIME: 3HRS MAXMARKS:75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. Two events A and B are mutually exclusive if (a)A∪ B = φ (b) A∩B = φ (c) A∪ B c = φ (d) A c∩B = φ 2. If the distribution function of (X,Y) , X and Y are independent if (a) F(x, y) = F1(x)/F2(y) (a) F(x, y) = F1(x) + F2(y) (a) F(x, y) = F1(x)F2(y) (d) none of these. 3. If φ(t) is the characteristic function of the random variable X then φ(t) = (a) E[etx] (b) E[xk ] (a) E[eitx] (d) none. 4. The standard deviation of the Binomial distribution with parameters n and p is (a) np (b) npq (c) np p (1 ) − (d) None of these. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 34 of 57 5. If X is a normal variate and if µ2k+1 is the (2k + 1)th central moment of X, then µ2k+1 (a) 1 (b) 0 (c) 2k+1 (d) 2k2 +1. 6. If X is the random variable with p.d.f 0 for x ≤ 0 f(x) = n x 1 x e n − − Γ for x > 0 then the characteristic function of X is (a) (1-t)n (b) (1-it)n (c) (1-t)-n (d) (i-it)-n 7. The sum of squares of n independent standard normal variate is a ______ variate (a) t (b) x2 (c) F (d) none of these. 8. For large value of degrees of freedom the t-distribution tends to a _____ distribution (a) normal (b) chi-square (c) F (d) none of these. 9. An estimator Un of the parameters Q is called consistent if ______ for every ∈>0 (a)lim ( ) n n P U Q ε →∞ − < = 0 (b)lim ( n n P U Q ε →∞ − > = 0 (c) 0 lim ( ) n n P U Q ε → − < =0 (d) None of these 10. If Un is an estimator of Q and if E[Un] = Q , then Un is called_________ (a) consistent (b) unbiased (c) efficient (d) none of these. SECTION-B (5X5=25 MARKS) 11. (a) State and prove Baye’s theorem on probability. (or) (b) Define distribution function throwing an unbiased die , find the distribution function. 12. (a) If the lth moment of a random variable X exists then prove that it is given by the lth derivative of the characteristic function φ(t) of X at t=0. (or) Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 35 of 57 (b) Find the mean and variance of Binomial distribution with parameters n and p. 13. (a) Define Gamma distribution and find its characteristic function. (or) (b) State and prove Bernoulli’s law of large number. 14. (a) Define chi-square distribution. Find its characteristic function. (or) (b) Define Student’s t- distribution. Find its mean and variance. 15. (a) Define a sufficient estimator .Give an example of sufficient estimator. (or) (b)Describe the method of maximum likelihood for construction of the estimator’s. SECTION-C- (5X8=40 MARKS) 16. (a) The content’s of urns I , II , III are as follows: 1 white balls , 2 black balls and 3 red balls 2 white balls , 1 black balls and 1 red balls 4 white balls , 5 black balls and 3 red balls (or) (b) The joint distribution function of X and Y is given by, 2 2 ( ) ( , ) 4 x y f x y xye− + = , x ≥ 0 y ≥ 0. Test whether X and Y are independent. 17. (a) State and prove Levy’s theorem on characteristic function. (or) (b) State and prove additive property of poisson variates. 18. (a) Find the characteristic function Cauchy distribution. State and prove addition theorem for Cauchy distribution. (or) (b) State and prove Lindeberg-Levy theorem. 19. (a) Derive the joint distribution of the statistic ( , ) X SAnx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 36 of 57 (or) (b) The heights of 6 randomly chosen sailors are in inches 63, 65, 68, 69, 71, 72. Those of 10 randomly chosen soldiers are 61, 62, 65, 66, 69, 69, 70, 71, 72,73 Discuss the height that these data throw on the suggestion that soldiers are on the average taller than soldiers. 20. (a) State and prove Rao- Cramers inequality. (or) (b) What is meant by confident interval? Find the 99% confidence interval for the unknown mean of a normal population when its S.D σ is unknown. MODEL QUESTION PAPER Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 37 of 57 FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS FIRST SEMESTER MATHEMATICS REAL ANALYSIS TIME: 3HRS MAXMARKS: 75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. If ∆fi = f(xi) + (xi-1) and ∆fi > 0 , then f is (a) monotonically increasing function (b) monotonically decreasing function (c) strictly increasing function (d) strictly decreasing function. 2. If P is a partition of [a, b] and cЄ [a, b] then the partition of the interval [c, b] is (a) P∩ [a, b] (b) P∩ [a, c] (c) P∩ [c, b] (d) P∪ [c, b] 3. If x is a real axis n = 0,1,2,….. then 2 2 0 (1 )n n x x ∞ = + ∑ is (a) (1 + x2 ) -1 (a) (1 + x2 ) (a) (1 +1/ x2 ) (d) (1 - x2 ) 4. Under what conditions, the limit function f of a sequence of continuous functions {fn} is also continuous? (a) fn ‘s are all uniformly continuous functions (b) fn ‘s are all uniformly monotonically increasing functions (c) fn converges to f monotonically (d) fn converges to f uniformly 5. Let the vector space X is spanned by r vectors and dimX = n, then (a) n ≤ r (b) n ≥ r (c) n = r (d) n and r are not comparable. 6. Let φ: (X, d) → (X, d) and c < 1. If d(φ(x), φ(x))cd(x, y) then Q is known as (a) continuous function (b) contraction mapping (c) open mapping (d) closed mapping. 7. If A= {1, 2, 5, 7}, then m*A =_______ (a) 6 (b) 4 (c) 0 (d) 3.5 Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 38 of 57 8. Which one of the followings is not true (a) m*B ≤ m*(A∪ B) (b) m*A ≥ m*(A∩B) (c) m*B = m*(A - B) + m*(A∩B) (d) none of these.10 9. The Lebesgue interval of f over E is defined by the equation , E ∫ f(x) dx =______ (a) inf ψ ( ) x dx ∫ (b)sup ψ ( ) x dx ∫ (c) inf ψ ( ) x dx ∫ (d) none of these ψ ≤ f ψ ≥ f ψ ≥ f 10. Fatou’s Lemma is applied only for the sequence {} of (a) measurable functions (b) non-negative measurable functions (c)increasing non-negative measurable functions (d)non-negative integrable functions. SECTION-B (5X5=25 MARKS) 11. (a) If f is monotonic on [a, b] and α is continuous on [a, b] , then prove that f ЄR(α) on [a, b]. (or) (b) State and prove the fundamental theorem of calculus 12. (a) Limit of an integral need not be equal to integral of the limit .Give an example. (or) (b) If f ЄB then show that f ЄB. 13. (a) A linear operator A on a finite dimensional vector space X is one-to-one iff the range of A is all of X. (or) (b) Show that the determinant of the matrix of a linear operator doesnot depend on the basis. 14 (a) Let {Ei}be a sequence of measurable sets. Then prove that m (∪ Ei) ≤ ∑ mEi . Suppose En are pairwise disjoint. Then prove that m(∪ Ei) = ∑ mEi . (or) (b) Show that the cf and f + g are measurable, when f and g are measurable, C is a compact Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 39 of 57 15. (a) State and prove the relationship between Riemann integral and Lebesgue integral. (or) (b) State and prove Fatou’s Lemma. SECTION-C- (5X8=40 MARKS) 16 (a) Let α increasing on [a, b] and α′ ЄR on [a, b] . Let f be a bounded real function on [a, b] . Then fЄR(α) on [a, b] iff fα′ ЄR, further b a ∫ fdα = b a ∫ f(x) α′ (x)dx (or) (b) Define the rectifiable curve. If γ ′(t) is continuous on [a, b] , then γ ′ is rectifiable and Λ(γ) = b a ∫ |γ′(t) |dt. 17 (a) State and prove the relationship between uniform convergence and differentiation. (or) (b) State and prove Weierstrass theorem 18.(a) State and prove implicit function. (or) (b) State and prove inverse function theorem. 19.(a) Prove that the outer measure of an interval is its length. (or) (b) (i) Show that (a, ∞) is measurable. (ii) If f is measurable and f =g a.e , then prove that g is also measurable. 20 (a) Let f be bounded on a measurable set E with mE<∞.Then inf ∫ ψ = sup ∫ φ iff f is measurable. f ≤ ψ f ≤ φ (or) (b) State and prove (i) Monotone convergence theorem (ii) Lebesgue convergence theorem. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 40 of 57 MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS SECOND SEMESTER MATHEMATICS MECHANICS TIME: 3HRS MAXMARKS: 75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. Constraint of the form f(t, r1, r2,….) = 0 is called _________ (a) Holonomic (b) Scleronomic (c) Non Holonomic (d) None 2. Number of coordinates minus the number of independent equation of constraints equals ___________ (a) units (b) Dimensions (c) degrees of freedom (d) none of these 3. Hamiltonian function is equal to total energy in (a) Conservative system (b) Scleronomic system (c) Scleronomic and natural systems (d) a holonomic conservative system. 4. State true (or) false For a non-conservative system the Lagranges equation remains the same. 5. The Hamiltonian equals total energy when Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 41 of 57 (a)Generalised Co-ordinates don’t depend on time (b)Forces are derivable from a conservative potential V. (c)Both (a) and (b) (d) None 6. When the Lagrangian is not an explicit function of time in steady motion, the cyclic Co-ordinate are (a) Linear function of time (b) Non-linear function of time (c) Not an explicit function of time (d) None of these 7. State true (or) false If Qi and Pi are to be canonical co-ordinates then modified Hamilton’s principle is δ 2 1 t t ∫ (Pi Qi -K(Q, P, t)) dt = 0 8. The poisson bracket of u, v with respect to (q, p) is (a) i i u v q p ∂ ∂ ∂ ∂ - i i u v p q ∂ ∂ ∂ ∂ (b) i i u v q q ∂ ∂ ∂ ∂ - i i u v p p ∂ ∂ ∂ ∂ (c) i i u v p p ∂ ∂ ∂ ∂ - i i u v q p ∂ ∂ ∂ ∂ (d) None 9. The ________ function plays the role of the Hamiltonian in the new co-ordinate set (P, Q) (a) Routhian (b) Ignorable co-ordinates (c) Poisson bracket (d) None 10. Solution of Hamilton- Jacobi equation is called ___________ (a) Gibb’s function (b) Quadratic function (c)Routhian function (d)None SECTION-B (5X5=25 MARKS) 11. (a) Write short notes on degrees of freedom holonomic and non-holonomic system (or) (b) Derive D-Alemberts Principle 12. (a) Find the shortest distance between two points in a plane . (or) (b) Write Short notes on cyclic. (or) ignorable co-o0rdinates , what can you say about corresponding generalized momentum . 13. (a) Prove that for a Conservative holonomic system the Hamilton H is a constant (or) (b) Discuss Routh’s procedure and oscillations about steady motion Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 42 of 57 14. (a) Show that the transformation P= 1 2 2 2 ( ) p q− + Q = 1 tan q p − is canonical . (or) (b) Prove that the Lagrange brackets are invariant under contact transformation 15. (a) Write short notes on the physical significance of Hamilton Principal function (or) (b) Derive Hamilton- Jacobi equation in the form 2 2 1 2 1 , ,..., ,...., n n F F q q q t q q ∂ ∂ ∂ ∂ + F2 t ∂ ∂ = 0 . SECTION-C- (5X8=40 MARKS) 16. (a) Explain the motion of one particle using plane polar Co-ordinates . (or) (b) Show that the rate of dissipation of energy by frivtion is equal to twice the Rayleigh’;s dissipation function 17. (a) Solve the Branchistochrone problem. (or) (b) Find the curve for which some given line integral has a stationary value 18. (a) Stater and prove the principle of least action (or) (b) Derive Hamilton’s equation from a variational principle . 19.(a) Show that the integral J= i s ∫∫∑ dqi dpi is invariant under canonical transformation (or) (b) Find the relation between Lagrange and poisson brackets. 20.(a) Discuss about the H-J equation for Hamilton’s characteristic function (or) (b) By an example solve H-J equation by separation of variables. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 43 of 57 MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 44 of 57 M.SC., DEGREE EXAMINATIONS FOURTH SEMESTER MATHEMATICS COMPUTER PROOGRAMMING -II TIME: 3HRS MAXMARKS: 75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. Which of the following are good reasons to use an object oriented languages (a) Define an own data type (b) Program statement are sympler than the procedural language (c) An OOP can be taught to connect its mown errors (d) its easier to computize an OOP 2. A normal C++ operator that acts in special ways on newly defined data types is set tobe (a) glorified (b) encapsulated (c) classified (d) overloaded 3. Operator overloading is (a) making C++ operators works with objects (b) Giving C++ operators more than they can handle (c) Giving new meanings to existing C++ operators (d) making new C++ operators 4. _________ is used to allocate memory in the constructors (a) new (b) set (c) assign (d) none 5. __________ class contains basic facilities that are used by all input output classes (a) is (b) ios (c) os (d) iostream 6. __________ is a sequence of bytes that serves are source or destination for an I/O data (a) statements (b) stream (c) Cin and Cout 7._______ is a special member function whose task is to initialize the objects of its class (a) prototype (b) static (c) Abstract (d) constructors 8. The write( ) function handle the data is Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 45 of 57 (a) ascii form (b) ansi form (c) binary form 9. __________ achieve the runtime polymorphism (a) Virtus (b) Friend functions (c) Abs Functions (d) universal 10. In C++ the class variables are called as _______ (a) objects (b) Functions (c) prototype (d) static SECTION-B (5X5=25 MARKS) 11. (a) What are the basic concepts of OOP? (or) (b) Explain software crisis 12. (a) Write note on “Operator Overloading” (or) (b) How do you declare variables in C++ with suitable examples? 13. (a) Discuss C++ stream classes (or) (b) What is the meaning of call by reference? 14. (a) Explain nesting of member functions (or) (b) Discuss multiple constructors in a class 15. (a) Write hierarchical inheritance in C++ (or) (b) Discuss the overloading unary and binary operators in C++ SECTION-C- (5X8=40 MARKS) 16. (a) (i) What are the benefits of OOP? (ii) Discuss OOP paradigm (or) (b) (i) Write the applications of OOP. (ii) Discuss object oriented languages. 17. (a) How do you declare operators in C and C++ , with examples ,that are used for memory management ? Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 46 of 57 (or) (b) Discuss in details identifiers and constants. 18. (a) (i) Write the term “Friend and Virtual Functions” (ii) Explain the math library function (or) (b) Discuss the formatted I/O operations. 19. (a) Explain memory allocation for objects (or) (b) How do you declare private member functions and static member functions with examples? 20. (a) Explain data conversions with example which is basic to class type and class to basic type (or) (b) Write short notes on : (i) Virtual base classes (ii) Abstract classes. MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS FOURTH SEMESTER Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 47 of 57 MATHEMATICS MATHEMATICAL METHODS TIME: 3HRS MAXMARKS: 75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. Fourier sine transform of f(t)sinwt is (a) 1 [ ( ) ( )] 2 F F c s ξ ω ξ ω + − − (b) 1 [ ( ) ( )] 2 F F c c ξ ω ξ ω − − + (c) 1 [ ( ) ( )] 2 F F c c ξ ω ξ ω − + + (d) 1 [ ( ) ( )] 2 F F s c ξ ω ξ ω + − − 2. 1 2 2 t e − is a self – reciprocal function under (a) Fourier sine transform (b) Fourier cosine transform (c) Fourier transform (d) both (b) and (c) 3. The Hankel inversion theorem is valid when (a) υ ≥ 1 2 − (b) υ > 1 2 − (c) υ > 0 (d) υ > -1 4. The Hankel transform of order υ is equal to its inverse transform , when (a) υ = 0 (b) υ = -1,0 (c)υ = -1,1 (d) υ = 0, 1 2 − 5. The eigen value of 1 0 ( ) ( ) s t g s e s g t dt = λ ∫ is _________ (a) zero (b) non-zero (c) both (a) and (b) (d) None of above 6. The Newmann series for 0 ( ) (1 ) ( ) ( ) s g s s s t g t dt = + + − ∫ is (a) s e − (b) s e (c) st e (d) s t e 7. The boundary value problem y ''(t) + y '(t) + y(t) = f(t), y(0) = y(1) = 0 leads to Integral equation with (a) asymmetric Kernal (b) parametric Kernal (c) continuous Kernal (d) non-parametric Kernal 8. The integral equation 0 ( ) ( ) ( ) s t g s f s e g t dt λ ∞ − − = + ∫ is of Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 48 of 57 (a) Fredholm type (b) Volterra type (c) singular type (d) None of the above 9. The variational problem [ ( )] b a v y x ydx xdy = + ∫ , 0 y a y ( ) = , 1 y b y ( ) = has (a) two solutions corresponding to (a, 0 y ) and (b, 1 y ) (b) unique solution (c) no solution (d) none of the above. 10. The solution of minimum surface problem is (a) Sphere (b) catenoid (c) ellipsoid (d) none of the above. SECTION-B (5X5=25 MARKS) 11. (a) Find the Fourier transform of ia t e , a > 0 (or) (b) If a is a real find the Fourier transform F[f(at);ξ] 12. (a) Prove 1 H H 0 0 − = -Hankel transform of order zero (or) (b) Obtain the parseval relation for Hankel transform 13. (a) Show that the equation ψ λ ψ ( ) ( ) ( , ) ( ) s f s k t s t dt = + ∫ has a unique (or) (b) Find the approximate solution of 1 0 ( ) ( 1) ( ) s st g s e s S e g t dt = − − − ∫ 14. (a) Show that boundary value problems in ordinary differential equations lead to Fredholm-type integral equations (or) (b) Solve 1/ 2 0 ( ) ( ) s g t dt S s t = − ∫ 15. (a) State and prove the fundamental lemma of the calculus of variations (or) (b) Give an example to show that there is no extremal that satisfies the boundary Conditions. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 49 of 57 SECTION-C- (5X8=40 MARKS) 16. (a) Deduce the graphs of ( ) n δ x and ( ) n ∆ x for various values of n (or) (b) Find the Lap lace’s equations in a half-plane 17. (a) Find the relations between Fourier and Hankel transform (or) (b) Solve the axisymmetric Dirchilet problem for a thick plate 18. (a) Solve 2 0 g s f s s t g t dt ( ) ( ) cos( ) ( ) π = + + λ ∫ 2 0 g s f s s t g t dt ( ) ( ) cos( ) ( ) π = + + λ ∫ (Or) (b) Solve 1 0 g s s t g t dt ( ) 1 ( ) ( ) = + + λ ∫ 19. (a) State and solve the transverse oscillations of a homogeneous elastic bar (Or) (b) Solve 2 2 ( ) ( ) ( ) s a g t dt f s s t α = − ∫ , 0 < α<1 ; a < s < b 20. (a) Derive Euler’s equation (Or) (b) Investigate the following functional for an extremum: [ ( , )] , , , , D z z V z x y F x y z dxdy x y ∂ ∂ = = ∂ ∂ ∫∫ ; the values of the function z(x, y) are the given on the boundary C of domain D, a spatial path Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 50 of 57 MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS SECOND S SEMESTER MATHEMATICS OPERATIONS RESEARCH TIME: 3HRS MAXMARKS: 75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. The behaviour of the optimum solution has been studied in (a) Problem definition (b) Sensitivity analysis (c) implementation of the solution (d) Validation of the model 2. If the type of the objective function is maximization then the sign of coefficient of an artificial variable in the objective function is (a) Negative (b) positive (c) M (d) zero 3. 1 2 3 10 20 30 10 2 3 (10) 4 5 6 = _______ (a) 4 5 6 (b) 40 5 6 (c) Either (a) or (b) (d) 10 20 30 40 50 60 4. VAM is an improved version of (a) North-west corner method (b) row-minima method (c) least cost method (d) Modi method 5. The algorithm used for the construction of paved roads that links several rural towns is (a) Minimal spanning tree algorithm (b) shortest route method algorithm (c) Critical path method algorithm (d) maximal flow algorithm 6. The predecessor(s) of the activity C in the following network is (are) A C M B E (a) A (b) A and B (c) A and D (d) A, B and D Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 51 of 57 7. If the line segment joining any two distinct points in the set, also falls in the set, then The set is known as a (a) Concave set (b) convex set (c) extreme point set (d) linear point set 8. The net evaluation is given by the equation Z C j j − = __________ (a) 1 P B C C j B j − − (b) 1 C P B C B j j − − (c) 1 C B P C B j j − − (d) 1 B P C C j B j − − 9. Acceptance-rejection method is applied for generating successive __________ (a) Probabilistic samples (b) Probabilistic tables (c) Random sample (d) convolution samples 10. If u0 =11, b = 9, c = 5 and m = 12, then by multiplicative congruence method the Value of u1 is (a) 0.4167 (b) 0.1667 (c) 0.7776 (d) 0.6667 SECTION-B (5X5=25 MARKS) 11. (a) Solve graphically Maximize Z= 5x1 + x2 Subject constraints: 6x1 + 4x2 ≤ 24; x2 ≤ 2 and x1 , x2 ≥ 0 (Or) (b) Write down the four steps to be adopted in solving a LPP. 12. (a) Explain how the dual problem is constructed from the primal. (Or) (b)Write down the mathematical formulation of the following transportation problem 1 2 supply A 80 215 1000 B 100 108 1500 C 102 68 1200 Demand 2300 1400 (or) 13. (a) Write down Dijkstra’s algorithm (or) (b) Calculate mean and variance of each of the following activities: Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 52 of 57 Activity: A B C D E Times : (3, 5, 7) (4, 6, 8) (1, 3, 5) (5, 8, 11) (1, 2, 3) 14. (a) Classify all the basic solutions of the following systems of equations x1 1 3 -1 x2 = 4 2 -2 -2 x3 2 (Or) (b) Describe revised algorithm 15. (a) Write a short note on inverse method (Or) (b) Explain multiplication congruential method with an example. SECTION-C- (5X8=40 MARKS) 16. (a) Ozark farm uses at least 800kg of special feed daily. It is a mixture of corn and Soyabean meal with the following compositions: __________________________________________________ _________ Kg. per Kg. of feedstuff Feed stuff Protein Fiber Cost (in Rs.PerKg) Corn 0.09 0.02 30 Soyabean Meal 0.60 0.06 90 The dietary requirements of the special of the feed are at least 30% protein and at most 5% fiber . Determine the daily minimum-cost feed mix. (Or) (b) Solve Minimize Z= 4x1 + x2 Subject to constraints: 3x1 + x2 = 3; 4x1 + 3x2 ≥ 6; x1 + 2x2 ≤ 4 and x1, x2 ≥ 0 17. (a) Apply dual simplex method to Solve , Minimize Z= 3x1 + 2x2 + x3 Subject to constraints: Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 53 of 57 3x1 + x2 + x3 ≥ 3 ; -3x1 + 3x2 + x3 ≥ 6 ; x1 +x2 +x3 ≤ 3 and x1, x2 , x3 ≥ 0 (Or) (b) Solve the following transportation Problem: 1 2 3 4 supply 1 10 2 20 11 15 2 12 7 9 20 25 3 4 14 16 18 10 Demand 5 15 15 15 18. (a) The following network gives the permissible routes and their lengths in Km between city1 and four other cities. Determine the shortest routes between city1 and each of the remaining four cities. 2 15 4 100 20 10 50 1 3 5 30 60 (b) Determine the critical path for the following Project Network: Activity: (1, 2) (1, 3) (2, 3) (2, 4) (3, 5) (3, 6) (4, 6) (5, 6) Duration: 5 6 3 8 2 11 1 12 19. (a) Consider the following LP, Maximize Z=x1 + 4x2 + 7x3 + 5x4 Subject to constraints: 2x1 + x2 + 2x3 + 4x4 = 10 , 3x1 – x2 – 2x3 + 6x4 = 5 and x1,x2,x3,x4, ≥0. Generates the simplex tables associated with the bases B = (P1, P2) and B = (P3, P4) (Or) (b) Solve the following LP by the revised simplex method: Maximize Z = 6x1 – 2x2 +3x3 Subject to constraints: 2x1-x2 + 2x3 ≤ 2, x1 + 4x3 ≤ 4 and x1, x2, x3 ≥ 0 20. (a) Use Monte- Carlo Sampling to estimate the area of the circle Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 54 of 57 2 2 ( 1) ( 2) 25 x y − + − = (Or) (b) Describe acceptance-rejection method .Illustrate it, wrong the beta distribution f(x) = 6x (1- x) for 0 ≤ x ≤ 1. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 55 of 57 MODEL QUESTION PAPER FOR CANDIDATES ADMITTED FROM 2007-2008 AND ONWARDS M.SC., DEGREE EXAMINATIONS FOURTH SEMESTER MATHEMATICS GRAPH THEORY TIME: 3HRS MAXMARKS: 75 ANSWER ALL QUESTIONS SECTION A-(10 X 1=10 MARKS) CHOOSE THE BEST ANSWER FROM THE FOUR ALTERNATIVES GIIVEN BELOW EACH OF THE FOLLOWING QUESTIONS 1. A simple graph (a) can have self loops and parallel edges (b) can have self loops but not parallel edges (c) can have only parallel edges (d) can have neither self loops nor parallel edges 2. The incidence degree of the vertex v of the following graph is (a) 2 (b) 4 (c) 3 (d)0. 3. A graph in which all the vertices have the same degree is called (a)planar graph (b)regular graph (c) walk (d) circuit 4. A graph G is said to be disconnected if (a)there is exactly one path between any two vertices (b)there is atleast one path between any two vertices ( c) there is no path between any two vertices (d)there are two vertices so that there is no path between them. 5. An Euler graph has (a) an odd number of vertices (b)odd number of vertices of even degree ( c )every vertex is of even degree (d)there is no vertex of even degree. 6.The number of edge disjoint Hamiltonian circuits in a complete graph of n vertices where n≥3 is (a)n/2 (b)n(n-1)/2( c)(n-1)/4 (d)n-1/2. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 56 of 57 7.the eccentricity of a vertex v in a graph G is defined as (a)degree of v. (b) degree of v-1 (c)min d(v,vi) vi ε d(v,vi ) (e) max d(v,vi) viεG 8.Every connected graph has (a)exactly one spanning tree (b)no spanning tree (c )every tree in the graph is a spanning tree (d)atleast one spanning 9.The vertex connectivity of any graph is (a) always 1 (b)always equal to edge connectivity ( c) the number of vertices in the graph (d)always less than or equal to edge connectivity. 10.A connected planar graph with n vertices and e edges has (a)(e-n+1) regions (b)e regions(c)(n-1)regions(d)e-n+2 regions. SECTION-B (5X5=25 MARKS) 11. (a) Show that in any graph , the number of vertices of odd degree is alwaya even (or) (b) Show that an edge e of a graph Gis a cut edge iff e is contained in no cycle of G. 12 (a) If G is a block with υ ≥ 3 , show that any two edges of G lie on a common cycle (or) (b) If G is a Hamiltonian show that for every nonempty proper subsets S of V, w G S S ( ) − ≤ 13. (a) If a matching M in G is a maximum matching , show that G contains no M-augumenting path (or) (b) Show that every 3-regular graph without cut edges has a perfect matching. 14. (a) With usual notations , show that α + β = υ. (or) (b) Show that in a critical graph no vertex cut is a clique 15. (a) Show that K5 is non-planar. (or) (b) Show that a loopless digraph D has an independent set S every vertex of D not in S reachable from a vertex in S by a directed path of length almost 2. 16. (a) (i) Define a component and give an example (ii) Prove that ( ) 2 v V d v e ∈ ∑ = . (or) (b) (i) Show that in a tree show that e = υ – 1. Anx.17 A - M Sc Maths (Colleges) Model Q P M.Q.P. Page 57 of 57 17. (a) Show that a graph G with υ ≥ 3 is 2-connected iff any two vetices of G are connected by atleast two internally disjoint paths. (or) (b) (i) If G is a simple graph with υ ≥ 3 and δ ≥ 3/2 , show that G is Hamiltonian (ii) A connected graph has an Euler trail iff it has almost two vertices of odd degree. Prove 18. (a) State and Prove Vizing’s theorem (or) (b) Show that G has a perfect matching iff o G S S ( ) − ≤ for all S V ⊂ with usual notations. 19. (a) (i) If δ > 0 , show that α' + β' = υ (ii) Prove Brook’s theorem (or) (b) State and Prove Erdo’s theorem 20. (a) (i) Derive Euler’s formula (ii) Show that all planar embeddings of a given connected planar graph have the same number of all faces (or) (b) Show that a digraph D contains a directed path of length X-1 |