Question 61. The number of 5-Sylow subgroup(s) in a group of order 45 is

Question 62. Let X = {(x, y)∈ℝ2: x2 + y2 = 1} ∪ ([-1, 1] * {0}) ∪ ({0} * [-1, 1]). Let n0 = max{k: k < ∞, there are k distinct points p1,..., pk ∈ X such that X∖{p1,..., pk}is connected}Let q1,¦,qn0+1 be n0+1 distinct points and Y = X\{q1,¦,qn0+1}. Let m be the number of connected components of Y. The maximum possible value of m is ______

Question 63. Let X = {(x, y)∈ℝ2: x2 + y2 = 1} ∪ ([-1, 1] * {0}) ∪ ({0} * [-1, 1]). Let n0 = max{k: k < ∞, there are k distinct points p1,..., pk ∈ X such that X∖{p1,..., pk}is connected}The value of n0 is ______

Question 64. Suppose the random variable U has uniform distribution on [0,1] and X =−2logU. The density of X is

Question 65. Let f be an entire function on ℂ such that |f(z)| ≤ 100 log|z| for each z with |z| ≥ 2. If F(i) = 2i then f(1)

Question 66. The number of group homomorphisms from ℤ3 to ℤ9 is ______

Question 67. Let X be a convex region in the plane bounded by straight lines. Let X have 7 vertices. Suppose f(x,y) = ax + by + c has maximum value M and minimum value N on X and N < M. Let S = {P ∶ P is a vertex of X and N < f(P) < M}. If S has n elements, then which of the following statements is TRUE?

Question 68. Which of the following statements are TRUE?P: If f∈L1(ℝ), then F is continuous.Q: If f∈L1(ℝ) and lim|x|→∞f(x) exists, then the limit is zero.R: If f∈L1(ℝ), then f is bounded.S: If f∈L1(ℝ) is uniformly continuous, then lim|x|→∞f(x) exists and equals zero.

Question 69. Let S = {z ∈ ℂ∶ |z|=1} with the induced topology from ℂ and let f:[0,2] → S be defined as f(t) = e2πit. Then, which of the following is TRUE?