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B.Sc. (Hons.) Mathematics – III year
Analysis – IV
 Time: 1Hr. Max. Marks: 25
 (This question paper consists of 6 questions. All questions carry 5 marks each.)
(Attempt any 5 questions.)

  1. Test for convergence of the integral [math]\int_0^1x^p(log\frac{1}{x})^q~dx[/math]
  2. Show that [[Image:]]exists iff n < m+1
  3. Discuss the convergence of [[Image:]]
  4. State Green's theorem and verify it for I = [[Image:]]; where C is the closed curve formed by y3 = x2 and y=x between (0,0) and (1,1).
  5. Change the order of integration in the inegral[[Image:]]and evaluate it.
  6. Evaluate[[Image:]]taken over the upper half of the circle x2 + y2 = 2ax.
  7. State and prove Abel's Test for the convergence of Improper Integrals.