BMTC-133 · June 2025 · English
IGNOU BMTC-133 June 2025 Previous Year Question Paper
REAL ANALYSIS
Structured previous year question paper for BMTC-133, June 2025 session.
Max marks: 100 · Questions: 11
Verified: 22 Aug 2026
BACHELOR OF SCIENCE
(GENERAL)/BACHELOR OF ARTS
(GENERAL) (BSCG/BAG)
Term-End Examination
June, 2025
BMTC-133 : REAL ANALYSIS
Time : 3 Hours Maximum Marks : 100
Note:
(i) Question No. 1 is compulsory.
Do any six questions from Q. Nos. 2
to 8.
(ii) Use of calculator is not allowed.
Q1.Which of the following statements are True or False ? Give reason for your answer in the form of a short proof or a counter- example, whichever is appropriate for each statement : 5x2=10
(a) -2 is not a limit point of the empty set >.
(b) If (@,)nen is a convergent sequence, then >a, is a convergent series. n=1
(c) Every continuous function is differentiable.
(d) Every integrable function is monotonic.
(e) The set s-|Epinen| is bounded above.
Q2.(a) For any subset S of R, show that S isa closed set. 5
(b) Find the radius of convergence of a series La,x", where a, =—. 6 © 7 h nol? f c) Test the series > or convergence.
Q3.(a) Show that f(x) =x° is Riemann integrable on [0, 1]. Deduce that 0 4
(b) Show that the series sin(v'x) is x pole uniformly convergent on [0, [. 5
(c) Using the definition of limit, prove that [= 2n+cos3n) ] 2 n neN
Q4.(a) Find infimum and supremum of the set S={——:neN,, ifthe same exist. 5
(b) Check whether the series xt + *) . 3" is convergent or not. 5
(c) Let a function f be defined as : 5 1 > x=2 Is f continuous at x = 2 ? If not, find the nature of discontinuity and redefine f to make it continuous, if possible.
Q5.(a) Prove the following : 5 @ lim &a1e1
(i) lim X+2sinx _ 1 x0 x + 2cosx
(b) Apply Cauchy integral test tofind: 5 n>0 Vn? Vn? 1 1 Vn? -2 _ 92 oa? : nr? —(n—1)2
(c) Prove or disprove the following : 5 “The set of rational numbers is countable.”
Q6.(a) Show that the series La,, where ntn+2 4. a, =, diverges. 4 n> -n+2
(b) Show that the equation 2x3 -18x +7 =0, does not have two distinct real roots in the interval
(c) Prove or disprove the following statements :
Q7.The sequence (हा) nr /neN converges.
(ii) Every sequence, which is _ not monotone, diverges.
Q8.(a) Prove that 10 is not a_ rational number. 5
(b) Evaluate : 5 3 |r
(c) Let f : [0, 1] ~ R be the Dirichlet function, defined by :
Q9.ifx €[0,lJisrational
Q10.ifx e[0,lJisirrational Check whether the function f is Riemann integrable ?
Q11.(a) Prove that 1-x>e™, ifx>0. 5
(b) Find Maclaurin’s series for sin x, VxeR.
(c) For the sequence (f,)nen, Where cos 2nx f,:(01J>R, f,(*)= = check whether or not, it is uniformly convergent on [0, 1]. 5 nal 5/2 5 3. n neN fey=} yg? 7? 1 > x=2 eat wen f oes 81 ee HQ HS x0 e* 41 Gi) im 2*28mx _y x0 x + 2cosx 1 1 1 (@) lim Ve Ven jee bin| me Vn 2?