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Au Degree 4th Sem Maths (2019) Question Paper

By Venkat | February 06, 2020
Andhra University Degree 4th Semester Mathematics Question Paper of the year 2019 is available below. Have a look at it.
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Year - 2019
[BS - S 2207 / BA-S 2207]
B.Sc. (CBCS) DEGREE EXAMINATION.
Fourth Semester
Part II — Mathematics
Paper IV — REAL ANALYSIS
(Common for B.A./B.Sc.)
(With Effective from 2015-2016 admitted batch)
Time: Three hours      Maximum: 75 marks
SECTION A — (5 x 5 = 25. marks)
Answer any FIVE of the following.
1. Prove that the Sequence {Sn} where Sn = 1/(n+1) + 1/(n+2) +.....+1/(n+n)  is convergent.
    Sn = 1/(n+1) + 1/(n+2) +.....+1/(n+n) తో నిర్వచింపబడిన {Sn} అనే అనుక్రమం అభిసరిస్తుంది అని రుజువు చేయండి.
2. If {Sn} is a Cauchy sequence then prove that {Sn} is convergent.
   {Sn} కోషి అనుక్రమం అయితే {Sn} అభిసరించే అనుక్రమం అని నిరూఎంచంది.
3. Test for the convergence of  n=1∑∞ [ ∛(n3+1) - n].
    n=1∑∞ [ ∛(n3+1) - n]. యొక్క అభిసరణతను పరిక్షించండి.
4. Test for the Convergence of n=1∑∞ (-1)n 1/n [ 1+ 1/2 + 1/3 +.......+ 1/n ]
    n=1∑∞ (-1)n 1/n [ 1+ 1/2 + 1/3 +.......+ 1/n ]  శ్రేణి స్వభావాన్ని పరిక్షించండి.
5. If  Limx→a f(x)=l, then show that Limx→a |f(x)|=| l |. Is the converse true? Justify your answer.
   Limx→a f(x)=l అయితే Limx→a |f(x)|=| l | అని చూపండి. దీని విపర్యయం  నిజమగునా? మీ సమాధానాన్ని సమర్ధించుకోండి.
6. Let f : R→R be such that f(x) = (e1/x - e-1/x )/(e1/x + e-1/x ) if  x≠0 and f(0)= 1. Disscus the continuity of f at x=0.
   f(x) = (e1/x - e-1/x )/(e1/x + e-1/x ), x≠0 మరియు f(0) =1 అయ్యేటట్లు నిర్వచింపబడిన f : R→R ప్రమేయానికి x=0 వద్ద అవిచ్చిన్నతను చర్చించండి.
7. Find 'c' of Cauchy's mean value theorem for f(x)=√x and g(x) =1/√x in [a,b] where 0<a<b.
   0<a<b అయిన [a,b] లో f(x)=√x , g(x) = 1/√x లకు సంబంధించిన కోస సిద్దాంతములొని c∈(a,b) కనుక్కోండి.
8. If f : [a,b]→R is continues on [a,b] then show that f is integrable on [a, b].
    f : [a,b]→R అనే ప్రమయము [a,b] మీద అవిచ్చిన్నమైతే [a, b] మీద f సమాకలనీయం అనిచూపండి.
SECTION B — (5 x 10 = 50 marks) 
Answer. ALL the following questions.
9. (a) When do we say that a sequence is bounded? Prove that every convergent sequence is bounded. Is the converse of this result true? Justify your claim.
ఒక అనుక్రమం ఎప్పుడూ పరిబద్దమవుతుందనీ, అయితే అభిసరణమైన ప్రతి అనుక్రమం పరిబద్దమవుతుందని నిరూపించండి. ఈ ఫలితం యొక్క విపర్యయం సత్యమగునా మీ సమాధానాన్ని సమర్ధించుకోండి.
              Or
(b) State and prove Cauchy's first theorem on limits.
అవధులపై కోషి మొదిటి సిద్దాంతాన్ని ప్రవచించి నిరూపించిండి.
10. (a) State and prove the p-test.
     p - శ్రేణి పరీక్షను ప్రవచించి నిరూపించండి.
              Or
(b) Test for the convergence of ∑ [(2n - 2)/(2n + 2)] xn (x>0).
    ∑ [(2n - 2)/(2n + 2)] xn  (x>0) శ్రేణి యొక్క స్వభావాన్ని చర్చించండి.
11 (a) If f : [a,b]→R is continuous on [a, b], then show that f is bounded on [a,b]. Is the converse of this result 1s true? Justify your claim.
   f : [a,b]→R ప్రమేయం  [a, b] లో అవిచ్చినము అయితే  [a, b] లో f పరిబద్దం అని చూపండి. ఈ ఫలితం యొక్క విపర్యయం సత్యమగునా? మీ సమాధానాన్ని సమర్ధించుకోండి?
              Or
(b) If  f : [a,b]→R is uniformly continuous, then prove that f is continuous on [a, b]. Show that the converse of this theorem need not be true.
      f : [a,b]→R ఏకరూప  అవిచ్చిన్నం అయితే, [a,b] లో f  అవిచ్చిన్నం అని నిరూపించండి. ఈ సిద్ధాంతం యొక్క విపర్యయం సత్యం కానక్కరలేదని చూపండి.
12. (a) Show that f(x)=|x-l| + |x-2| is continuous but not derivable at x =1, 2.
    f(x)=|x-l| + |x-2| అనే ప్రమేయం  x =1, 2 వద్ద అవిచ్చిన్నం అవుతుందనీ మరియు అవకలనం కాదనీ చూపండి.
              Or
(b) State and prove Lagrange's mean value theorem.
    లెగ్రాంజ్‌ మధ్యమ మూల్య సిద్ధాంతాన్ని ప్రవచించి నిరూపించండి.
13. (a) A bounded function f : [a,b]→R is Riemann integrable on [a, b] iff for each є>0, there exists a partition P of [a,b] such that 0≤U(P, f)-L(P, f) <є.
   f : [a,b]→R  పరిబద్ధ ప్రమేయం [a,b] మీద రీమాన్‌ సమాకలనీయం కావడానికి ఆవశ్యక పర్వ్యాప్త నియమం. ప్రతి є>0 కు అనుగుణంగా 0≤U(P, f)-L(P, f) <є  అయ్యేటట్టు P ∈ Q(a,b)  వ్యవభ్ధతం.
              Or
(b) State and prove the First mean value theorem of integral and hence show that 1/𝝅 ≤  0∫1 [ sin 𝝅x/ (1+x2)]dx ≤ 2/𝝅.
   సమాకలన మొదటి మధ్యమ మూల్య సిద్దాంతాన్ని ప్రవచించి నిరూపించండి మరియు దానిని ఉపయోగించి 1/𝝅 ≤  0∫1 [ sin 𝝅x/ (1+x2)]dx ≤ 2/𝝅 అని చూపండి.
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