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1. Prove: Characteristic of a field F is 0 then F is infinite. (3)
2. Explain: Is every field F has an infinite extension of F. (3)
3. Find the value of the improper integral ∫₁^∞ (1/tᵖ) dt for p ∈ ℝ. (3)
4. What is the cardinality of the set of all complex functions f(z) = u(x, y) + iv(x, y), z = x + iy such that u is a harmonic conjugate of v and v is a harmonic conjugate of u? Justify. (3)
5. Find the remainder when 6ⁿ⁺² + 7²ⁿ⁺¹ is divided by 43. (3)
6. Prove: The center of the set of all n × n matrices over a field F is isomorphic to F. (4)
7. Find the number of similarity classes of idempotent matrices of order n over a field F.
Explain your answer. (4)
8. Prove: S₅ is not solvable. (4)
9. Check whether the sequence of functions gₙ(x) = 1/[n(1 + x²)] converges uniformly or diverges on ℝ. (4)
10. Find the points on ℝ² where the directional derivative of the function f: ℝ² → ℝ defined by f(x, y) = √(x² + y²) exists. Also find the directional derivative(s) if it exists. (4)
11. Let f(z) be an entire function and M is a constant such that for a positive real number R and for an integer n ≥ 1, |f(z)| ≤ M|z|ⁿ for |z| > R. Prove that f(z) is a polynomial of degree less than or equal to n. (4)
12. Let X be a normed linear space, S = {x ∈ X : ‖x‖ ≤ 1}, and f be a map from S into ℝ such that f(αx + βy) = αf(x) + βf(y) for all x, y ∈ S and αx + βy ∈ S, where α and β are scalars. Is there an extension of f to all of X? Justify. (4)
13. Let X, Y, Z be topological space. If f: X → Y and g: Y → Z are continuous then show that g∘f: X → Z is continuous. (4)
14. A metric space X is connected iff every continuous function f : X → (0, 1) is not onto. (4)
15. Prove or disprove if A and C are connected subsets of a metric space X and if A ⊆ B ⊆ C then B is connected. (4)
16. Explain: ℝ is (1) not algebraic, (2) not finite, (3) not simple extension of ℚ. (5)
17. The set of all algebraic numbers over ℚ, in ℂ is countable and algebraically closed (prove). (5)
18. Consider the function f: (a, b] → ℝ defined by f(x) = 1/(x − a). Check the continuity and uniform continuity of the function f on (a, b]. (5)
19. Consider the function f : ℝ² → ℝ defined by f(x, y) = x² + y² − 1. Find the solution of f(x, y) = 0. (5)
20. Evaluate the improper integral ∫₀^∞ cos(αx)/(x² + β²) dx, α, β > 0. (5)
21. Let T be a bounded linear operator on a Hilbert space H. Show that there are unique self adjoint
operators P and Q on H so that T = P + i Q (5)
22. Three salesmen A, B and C visited a city on different routine. If A, B and C visit city after every 10 days, 7 days and 3 days respectively and A had last been to city 8 days ago B had last been to city Yesterday and C has visited city today. When will all three salesmen meet again ? (5)
23. Show that there are infinitely many primes of the form 6n + 5. (5)
24. A particle moves along the x-axis according to d²x/dt² + 6 dx/dt + 25x = 0. If the particle is started at x = 0 with an initial velocity of 12 ft/sec to the left, find x in terms of t. (5)
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