BSc Sem VI 2014 2015 2015 Math 1 Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks for respective subquestions

1. Q7 then prove that: = f @) where = is the partial sum of the Fourier series of f and D,, is the Dirichlet’s kernel
   • (b) Attempt any three questions:
   • (ii) Is the series series of a function f C ? Justfy your answer (4) (ili) If f(z) = -m<a< f ~ + (a, + bp sinnz) , compute State the result used. (4)
   • (iv) f(z) = cos and f ~ (an +b, Then - find the Fourier coefficients and by Euclidean), K is closed and bounded. Show that K is sequen- : 4
   • (ii) Show that the following function (distance in R is usual) f : x R, f(z,y) y (distance in R? Euclidean ) is uniformly continuous 4
   • (iii) Let (X,d) be a metric space and K C X be a compact set. Show that a closed subset F of K is compact 4
   • (iv) Prove or disprove: A closed and bounded subset of a metric space is compact 4
2. Q3 (a) Attempt any one question:
   • (i) Show that a metric space (X, d) is connected if and only if every continuous function f : X —> {1,—1} is constant 8
   • (ii) Let (X,d) be a metric space and A be a connected subset of X. If AC BC then Show that B is connected. In particular, prove that A is connected. Give an example to show that if A,C are connected subsets of X and AC then B need not be connected 8
   • (b) Attempt any three questions:
   • (i) Show that the set A = {(z,y) R? y? = is path subset of R?
   • (ii) If (X,d) be a connected metric space and f : X —> Z (distance in Z being usual distance) is a continuous function then prove that constant function. (4) Prove or disprove: The subset : y (d being Euclidean distance ) is connected. (4)
   • (iv) Let (X,d) be a metric space. If A is a of X having more than one element, show that A is disconnected. (4)
3. Q4 Attempt any three questions:
   • (a) f(z) = Find the Fourier series of f. Assuming that the Fourier series of f converges to f(z) find the sum 5
   • (b) Let f and f Fourier sereis 4 cos nx + , Show that that {(z,y) 2? + y? = 1} is a compact subset of distance being 5
   • (d) be a compact metric space. If is a sequence of non-empty closed sets that C A, for each n N, then show that An # (5) or disprove; A = {(z,y) zy = 0} is a connected subset of (distance Prove that a path conn ected subset of (distance being is connected

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