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Total No. of Questions : 05
Total No. of Printed Pages : 04
RJ-196
B.Sc-B.Ed. (Secondary) Examination, 2024
(Second Semester)
MATHEMATICS
DC-III
Real analysis
Time : 3 Hours
Maximum Marks : 60
Note : All questions are compulsory. Attempt any two parts from each questions.
1.
(a) Define the Countable Set. Prove that every subset of Countable sets is countable.
6
(b) Define limit point. Can a set a possible where its limit point is not an element of this set S ? If yes give example If no, Justify your answer.
6
2.
(a) State and prove the Bolzano-Weierstrass Theorem.
6
(b) Show that the sequence {x_n}, where x_1 = 1 and x_n = sqrt(2 + x_{n-1}) for all n >= 2 is convergent and converges to 2.
6
(c) By definition show that :
lim (2n+1)/(n+3) = 2
6
(d) Using Cauchy first theorem on limits, show that
6
3.
(a) State and prove Rolle's theorem. Is it necessary or sufficient or both ? Justify.
6
4.
(a) If f(x) = x for x ∈ [0, 1] and p = {0, 1/3, 2/3, 1} is a partition of [0, 1]. Compute L(P, f) and U(P, f).
6
(b) Show that the following function, defined on [0, 1] is not R-integrable :
f(x) = { 1 when x is rational ; -1 when x is irrational
6
(c) Examine the convergence of the integral :
Integral from 0 to 1 of 1/(x * sqrt(1+x^2)) dx
6
5.
(a) State and prove Darboux's intermediate value theorem for derivatives.
6
(b) Show that function : f(x) = x^2 is not uniformly continuous on the interval (0, ∞).
6
(c) Show that the sequence of function (f_n) defined as f_n(x) = x^n / n on (-∞, ∞) is not uniformly convergent.
6
(d) Find radius of convergence of given power series :
Sum from n=1 to ∞ of (3x-2)^n / n
6
(e) Show that uniformly convergence implies pointwise convergence but the converse is not true.
6