Download PTU B.Sc (Non-Medical) 2020 March 3rd Sem 76904 Analysis I Question Paper

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Roll No. [ ] [ ] [ ] [ ] Total No. of Pages : 02

Total No. of Questions : 09

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B.Sc. (Non Medical) (2018 Batch) (Sem.-3)

ANALYSIS-I

Subject Code : BSNM-305-18

M.Code : 76904

Time : 3 Hrs. Max. Marks : 50

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying ONE marks each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

SECTION-A

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1. Write briefly :
   1. Prove that S(1/n) is divergent.
   2. Define Absolute and Conditional Convergence.
   3. Define lower & upper Riemann integral.
   4. If f ? R [0, a], then prove that ?₀^af(x)dx = ?₀^af(a—x)dx, a>0
   5. Define improper integral of first kind.

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   6. Examine the convergence of ?₁⁸ e^-x2dx
   7. Define Gamma Function.
   8. Prove symmetry of Beta Function.
   9. State comparison test for series.
   10. Write the relation between Beta & Gamma function.

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SECTION-B

1. Test the following series for convergence.
2. State and prove Necessary & Sufficient condition for a bounded function to be R-integrable on [a, b].

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3. State and prove Abel’s Test.
4. Prove that B(m, n) = ?₀⁸ x^m-1/(1+x)^m+n dx = ?₀⁸ x^n-1/(x+1)^m+n dx; m,n > 0.
5. If a function f is R-integrable on [a, b] then f² is also R-integrable on [a, b]

SECTION-C

1. a) Show that the series S (n+2)/(2ⁿ+5) is absolutely convergent.
   

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   b) If a function f is integrable on [a,b] then m (b —a) = ?_a^b f(x)dx = M(b-a).
2. a) Show that ?₀¹ sinⁿ(x)/x^p dx, p>0 converges absolutely for p < 1.
   b) Prove that G(n+1) = nG(n)
3. a) If f₁ & f₂ ? R (a, b) & f₁(x) = f₂(x) ? x ? [a, b] then ?_a^b f₁(x)dx = ?_a^b f₂(x)dx.
   b) Prove that B (m, n)=B (m,n+ 1)+B (m + 1, n)

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NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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