answer:Let's analyze each statement step by step: **Statement A:** Given frac{a}{c^{2}} > frac{b}{c^{2}}, we can multiply both sides by c^{2} (assuming c^2 > 0 to avoid reversing the inequality) to obtain: [a > b] Thus, statement A is boxed{text{correct}}. **Statement B:** For xin(0,pi), we know sin x > 0. By applying the AM-GM inequality to sin x and frac{4}{sin x}, we get: [sin x + frac{4}{sin x} geq 2sqrt{sin x times frac{4}{sin x}} = 2sqrt{4} = 4] Equality occurs when sin x = frac{4}{sin x}, which implies sin^2 x = 4. However, since sin^2 x leq 1 for all x, equality cannot be achieved. Therefore, the minimum value of sin x + frac{4}{sin x} is strictly greater than 4, making statement B boxed{text{incorrect}}. **Statement C:** The original proposition p states there exists an xin mathbb{R} such that x^2 + 2x + 3 < 0. The correct negation of this statement, neg p, would be that for all xin mathbb{R}, x^2 + 2x + 3 geq 0. The given negation in the statement incorrectly uses a strict inequality, so statement C is boxed{text{incorrect}}. **Statement D:** To determine the probability of selecting three numbers that form a right-angled triangle, we first identify all possible combinations of three numbers from the set {1, 2, 3, 4, 5}, which are left(1,2,3right), left(1,2,4right), left(1,2,5right), left(1,3,4right), left(1,3,5right), left(1,4,5right), left(2,3,4right), left(2,3,5right), left(2,4,5right), and left(3,4,5right), totaling 10 combinations. Among these, only the combination (3, 4, 5) can form a right-angled triangle. Therefore, the probability is: [P = frac{1}{10}] Thus, statement D is boxed{text{correct}}. Hence, the correct options are boxed{A text{ and } D}.

answer:- **Senary System Mapping**: When each box can hold up to 5 balls, we map the system to base-6 (senary). - Just like before, each box corresponds to a digit in base-6 representation of the step number but can now hold 0 to 5 balls. - **Computation**: - Convert 2010 to base-6. - Divide 2010 by 6 repeatedly till the quotient is less than 6, recording remainders each time: - First division: 2010 div 6 = 335 remainder 0 - Second division: 335 div 6 = 55 remainder 5 - Third division: 55 div 6 = 9 remainder 1 - Fourth division: 9 div 6 = 1 remainder 3 - Final division (since quotient < 6): 1 - Base-6 representation: 13150_6 (read backwards from obtained remainders and final quotient). - **Summation of Digits**: Sum of the digits 1 + 3 + 1 + 5 + 0 = boxed{10}.

answer:The arithmetic sequence here starts with 35 books and decreases by 3 books per shelf, until there is only 1 book. 1. **Identifying the sequence**: The sequence is 35, 32, 29, ..., 1. The common difference d is -3. 2. **Finding the number of terms**: Using the formula for the nth term of an arithmetic sequence, a + (n-1)d = 1: [ 35 + (n-1)(-3) = 1 implies 35 - 3n + 3 = 1 implies 3n = 37 implies n = frac{37}{3} approx 12.33 ] Since n must be an integer, we need to recheck our calculations, as fractions of a shelf do not make sense. Instead, we use the exact formula and solve: [ 35 - 3(n-1) = 1 implies 3n = 35 implies n = 12 ] The last book count should be checked: 35 - 3 times 11 = 2, thus the sequence should end at 2 books, not 1. 3. **Calculating the sum**: Using the sum formula for an arithmetic sequence, frac{n}{2}(a + l): [ frac{12}{2}(35 + 2) = 6 times 37 = 222 ] Thus, the total number of books on the shelves is boxed{222}.

answer:To find the remainder when dividing x^3 + 3x^2 - 4 by x^2 + x - 2, we employ polynomial long division. 1. **Set up the division**: Divide x^3 + 3x^2 - 4 by x^2 + x - 2. 2. **Divide the leading terms**: The leading term of x^3 + 3x^2 - 4 is x^3, and the leading term of x^2 + x - 2 is x^2. Dividing x^3 by x^2 gives x. 3. **Multiply and subtract**: Multiply x by x^2 + x - 2, which yields x^3 + x^2 - 2x. Now subtract x^3 + x^2 - 2x from x^3 + 3x^2 - 4: [ (x^3 + 3x^2 - 4) - (x^3 + x^2 - 2x) = x^3 + 3x^2 - 4 - x^3 - x^2 + 2x = 2x^2 + 2x - 4. ] Since we need the remainder to have a degree less than the divisor (x^2 + x - 2 has degree 2), continued division is necessary. 4. **Next stage of division**: Divide the leading terms of the result, 2x^2 by x^2 gives 2. Multiply 2 by x^2 + x - 2 to get 2x^2 + 2x - 4. Subtract from 2x^2 + 2x - 4: [ (2x^2 + 2x - 4) - (2x^2 + 2x - 4) = 0. ] The remainder is 0, indicating exact division after the further steps. Thus, the remainder is 0. The final answer is boxed{text{(C)} 0}