answer:To find the horizontal asymptote, we consider the behavior of the function as x approaches infinity. We begin by simplifying the function: 1. The degrees of the polynomials in the numerator and the denominator are both 2. Divide every term by x^2: [ y = frac{6x^2 - 4}{4x^2 + 6x - 3} = frac{6 - frac{4}{x^2}}{4 + frac{6}{x} - frac{3}{x^2}} ] 2. As x to infty, the terms frac{4}{x^2}, frac{6}{x}, and frac{3}{x^2} approach 0. Thus, the expression simplifies to: [ y approx frac{6}{4} = 1.5 ] 3. Therefore, the horizontal asymptote of the graph is y = boxed{1.5}.

answer:This question mainly examines the application of the definition of an arithmetic sequence and is considered a basic problem. The definition method is the fundamental approach to determine (or prove) whether a sequence {a_n} is an arithmetic sequence. The steps are as follows: (1) Calculate the difference a_{n+1}-a_n; (2) Transform the difference equation; (3) If a_{n+1}-a_n is a constant that does not depend on n, then the sequence {a_n} is an arithmetic sequence; if a_{n+1}-a_n is not a constant but an algebraic expression related to n, then the sequence {a_n} is not an arithmetic sequence. Applying these steps to the sequence {b_n}: (1) Calculate the difference b_{n+1}-b_n = (3a_{n+1}+4) - (3a_n+4); (2) Transform the difference equation to 3(a_{n+1}-a_n); (3) Since a_{n+1}-a_n=d is a constant, it follows that 3(a_{n+1}-a_n)=3d is also a constant. Therefore, the sequence {b_n} is an arithmetic sequence with a common difference of 3d. The final answer is boxed{text{Yes, }{b_n}text{ is an arithmetic sequence.}}

answer:Let's denote the probability of rolling a red side as ( P(R) ) and the probability of rolling a non-red side as ( P(NR) ). Since there are exactly 3 red sides out of 10, the probability of rolling a red side is ( P(R) = frac{3}{10} ). Consequently, the probability of rolling a non-red side is ( P(NR) = frac{7}{10} ). Kumar rolls the die a certain number of times, and it lands with a red side up for the first time on the third roll. This means that the first two rolls were not red, and the third roll was red. The probability of this sequence of events is the product of the probabilities of each individual event, since each roll is independent: [ P(NR) times P(NR) times P(R) = left(frac{7}{10}right)^2 times frac{3}{10} ] Given that the probability of this happening is 0.147, we can set up the equation: [ left(frac{7}{10}right)^2 times frac{3}{10} = 0.147 ] [ left(frac{49}{100}right) times frac{3}{10} = 0.147 ] [ frac{147}{1000} = 0.147 ] This confirms that the probability calculation is correct. Since the die lands with a red side up for the first time on the third roll, Kumar rolls the die exactly boxed{3} times.

answer:1. **Substitute and Simplify**: Let x^2 = u. The expression x^4 - 65x^2 + 64 becomes u^2 - 65u + 64. 2. **Factorize the Quadratic**: [ u^2 - 65u + 64 = (u - 1)(u - 64) ] This factorization is derived because the roots of the quadratic equation u^2 - 65u + 64 = 0 are: [ u = frac{-b pm sqrt{b^2 - 4ac}}{2a} = frac{65 pm sqrt{65^2 - 4 cdot 64}}{2} = frac{65 pm sqrt{4096 - 256}}{2} = frac{65 pm 60}{2} ] yielding u = 1 and u = 64. 3. **Determine the Interval for Negativity**: The expression (u - 1)(u - 64) is negative when 1 < u < 64. 4. **Identify Perfect Squares**: We find the perfect square values of u that fall between 1 and 64: 4, 9, 16, 25, 36, 49, 64. 5. **Count the Corresponding x Values**: - u = 4 Rightarrow x = pm 2 - u = 9 Rightarrow x = pm 3 - u = 16 Rightarrow x = pm 4 - u = 25 Rightarrow x = pm 5 - u = 36 Rightarrow x = pm 6 - u = 49 Rightarrow x = pm 7 - u = 64 Rightarrow x = pm 8 This results in 2 times 7 = 14 valid x values. Conclusion: There are 14 integers x for which x^4 - 65x^2 + 64 is negative. The final answer is boxed{textbf{(D)} 14}