Test-2

B.Sc. (Hons.) Mathematics – III year Test 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.