answer:a) Arithmetic Sequences Count 1. **Define the sequence terms and constraints:** Let the five drawn numbers in the arithmetic sequence be (a_1, a_2, a_3, a_4, a_5). By definition, for an arithmetic sequence: [ a_5 = a_1 + 4d ] where (d) is the common difference. We have the constraint: [ a_1 + 4d leq 90 implies 4d leq 89 implies d leq frac{89}{4} < 23 ] Since (d) must be an integer, it follows that: [ d leq 22 ] 2. **Evaluate (a_1)’s range for each (d):** The smallest value for (a_1) is 1, and (a_1) must satisfy: [ a_1 leq 90 - 4d ] Therefore, (a_1) can take values in the range: [ 1 leq a_1 leq 90 - 4d ] This gives (90 - 4d) different possible values for (a_1) for each (d). 3. **Summing permissible values for all (d):** The possible values of (d) range from 1 to 22, and for each (d), the sequence count is (90 - 4d). Thus, the total number of different outcomes is the sum: [ sum_{d=1}^{22} (90 - 4d) ] 4. **Calculate the summation:** The terms form an arithmetic series with the first term (86) (when (d = 1)) and the last term (2) (when (d = 22)): [ 86, 82, 78, ldots, 6, 2 ] The number of terms in this series is 22. The sum of an arithmetic series is: [ S = frac{n}{2} (a + l) ] where (n) is the number of terms, (a) is the first term, and (l) is the last term. [ S = frac{22}{2} (86 + 2) = 11 times 88 = 968 ] Conclusively, the arithmetic sequences: [ boxed{968} ] b) Geometric Sequences Count 1. **Define the sequence terms and constraints:** Let the five drawn numbers in the geometric sequence be (a_1, a_2, a_3, a_4, a_5). By definition, for a geometric sequence: [ a_5 = a_1 q^4 ] where (q) is the common ratio. We need: [ a_1 q^4 leq 90 ] 2. **Consider integer values for (q):** (q) must be greater than 1 (since the numbers are distinct) and (q < 4) to satisfy the constraint: [ a_1 q^4 leq 90 ] - For (q = 2): [ a_1 times 2^4 = a_1 times 16 leq 90 implies a_1 leq frac{90}{16} approx 5.625 ] Thus, (a_1) can be 1, 2, 3, 4, or 5, giving 5 possible sequences. - For (q = 3): [ a_1 times 3^4 = a_1 times 81 leq 90 implies a_1 leq frac{90}{81} approx 1.111 ] Thus, (a_1) can only be 1, giving 1 possible sequence. 3. **Consider non-integer rational values for (q):** Let (q = frac{p}{s}) where ((p, s) = 1) and (s > 1), then: [ a_1 left(frac{p}{s}right)^4 = a_1 frac{p^4}{s^4} leq 90 implies a_1 s^4 leq 90 implies s^4 leq frac{90}{a_1} ] And similarly, (p^4 leq frac{90}{a_1}). - Considering ((p, s)) pairs within these constraints: The feasible pair is ( (p, s) = (3, 2)), giving: [ a_1 = 16, a_5 = 81 ] As 16, 24, 36, 54, 81 forms a geometric sequence. Conclusively, including previously identified sequences: [ 5 + 1 + 1 = 7 ] The geometric sequences: [ boxed{7} ]

answer:Since the original data set x_{1}, x_{2}, x_{3}, x_{4}, x_{5} has a mean of 2 and variance of frac{1}{3}, we can determine the mean and variance of the new data set 3x_{1}-2, 3x_{2}-2, 3x_{3}-2, 3x_{4}-2, 3x_{5}-2 by using properties of mean and variance with respect to linear transformations. **Step 1: Calculating the new mean** The mean of the new data set can be calculated by applying the transformation to the original mean: overline{x}_{new} = 3 cdot 2 - 2 = 4 **Step 2: Calculating the new variance** Variance is affected by the scale of the transformation but not by shifts (constant addition/subtraction). Therefore, the new variance can be calculated by scaling the original variance by the square of the scale factor: S^{2}_{new} = 3^2 cdot frac{1}{3} = 3 Therefore, the mean and variance of the new data set are 4 and 3, respectively. Thus, the answer is: overline{x}_{new} = boxed{4} S^{2}_{new} = boxed{3} This solution demonstrates the application of properties and formulas pertaining to the mean and variance of data. The problem tests the understanding of how to calculate mean and variance and the importance of carefully applying the properties of these measures.

answer:**Analysis** This problem tests our understanding of the operations of derivatives and their geometric meaning. It is a basic question. **Solution** We have F'(x) = f'(x) sin x + f(x) cos x (according to the product rule of differentiation). Then, substituting the given values, we have: F'(frac{pi}{4}) = f'(frac{pi}{4}) sin frac{pi}{4} + f(frac{pi}{4}) cos frac{pi}{4} = 4 cdot frac{sqrt{2}}{2} + 2 cdot frac{sqrt{2}}{2} = 3sqrt{2}. Thus, the slope of the tangent line to the graph of the function F(x) = f(x) cdot sin x at x = frac{pi}{4} is boxed{3sqrt{2}}.

answer:The angle you are referring to is called the "dihedral angle" of a cube. It is the angle between any two adjacent faces of the cube. However, the measure of the dihedral angle of a cube is not 60 degrees; it is actually 90 degrees, because the faces of a cube are perpendicular to each other. If you are talking about the angle formed by the diagonals of two adjacent faces within the same face (i.e., the angle between the face diagonals where they meet at a vertex of the cube), that would be 45 degrees, because the diagonals of a square (which is what each face of a cube is) bisect the right angles at the corners of the square. If you are referring to the angle between a face diagonal and a space diagonal of the cube (where they meet at a vertex), that angle is indeed boxed{60} degrees. This angle does not have a specific name like the dihedral angle, but it is the angle between a diagonal of a face (which is a line in the plane of the face) and a diagonal that cuts through the interior of the cube (a space diagonal).