answer：Okay, so I have this triangle ABC, and I know the lengths of two sides and the cosine of one angle. Let me write down what I know:- Side a is opposite angle A, and a = 2.- Side c is opposite angle C, and c = 3.- The cosine of angle B is 1/4.I need to find two things: first, the length of side b, and second, the value of sin(2C). Hmm, okay, let's tackle the first part first.For part (1), finding side b. I remember there's something called the Law of Cosines which relates the sides of a triangle to the cosine of one of its angles. The formula is:b² = a² + c² - 2ac cos BLet me plug in the values I know into this formula. So, a is 2, c is 3, and cos B is 1/4.So, b² = (2)² + (3)² - 2*(2)*(3)*(1/4)Calculating each part step by step:First, 2 squared is 4, and 3 squared is 9. So, 4 + 9 is 13.Next, 2*2*3 is 12, and 12*(1/4) is 3.So, putting it all together: b² = 13 - 3 = 10.Therefore, b is the square root of 10. That seems straightforward. So, b = √10.Alright, that was part (1). Now, moving on to part (2), which is to find sin(2C). Hmm, okay. I remember that sin(2θ) is equal to 2 sin θ cos θ. So, if I can find sin C and cos C, I can compute sin(2C).First, I need to find angle C. Since I know sides a, b, and c, maybe I can use the Law of Cosines again to find cos C.The Law of Cosines formula for angle C is:cos C = (a² + b² - c²) / (2ab)I already know a, b, and c. Let me plug in the values:cos C = (2² + (√10)² - 3²) / (2*2*√10)Calculating each part:2 squared is 4, (√10) squared is 10, and 3 squared is 9.So, 4 + 10 - 9 equals 5.The denominator is 2*2*√10, which is 4√10.So, cos C = 5 / (4√10)Hmm, that's a fraction with a radical in the denominator. Maybe I should rationalize the denominator to make it simpler.Multiplying numerator and denominator by √10:cos C = (5√10) / (4*10) = (5√10)/40 = √10 / 8Okay, so cos C is √10 / 8.Now, I need to find sin C. I know that sin² C + cos² C = 1, so I can solve for sin C.sin² C = 1 - cos² Ccos² C is (√10 / 8)² = 10 / 64 = 5 / 32So, sin² C = 1 - 5/32 = 27/32Taking the square root of both sides, sin C = √(27/32)Simplify √27 and √32:√27 = 3√3, and √32 = 4√2So, sin C = (3√3) / (4√2)But again, there's a radical in the denominator. Let's rationalize it:Multiply numerator and denominator by √2:sin C = (3√3 * √2) / (4*2) = (3√6) / 8Okay, so sin C is 3√6 / 8.Now, going back to sin(2C) = 2 sin C cos C.Plugging in the values:sin(2C) = 2 * (3√6 / 8) * (√10 / 8)Multiply the numerators and denominators:Numerator: 2 * 3√6 * √10 = 6√60Denominator: 8 * 8 = 64Simplify √60. √60 can be simplified as √(4*15) = 2√15So, 6√60 = 6*2√15 = 12√15Therefore, sin(2C) = 12√15 / 64Simplify the fraction by dividing numerator and denominator by 4:12 ÷ 4 = 3, 64 ÷ 4 = 16So, sin(2C) = 3√15 / 16Alright, that seems to be the answer. Let me just recap to make sure I didn't make any mistakes.For part (1), I used the Law of Cosines to find b, which came out to √10. That seems right because when I plug the numbers back in, it satisfies the equation.For part (2), I found cos C using the Law of Cosines, then used the Pythagorean identity to find sin C. Then, using the double-angle formula, I calculated sin(2C) as 3√15 / 16. That seems consistent.I think I did everything correctly. I double-checked each step, and the simplifications look good. So, I feel confident about these answers.**Final Answer**(1) The value of ( b ) is (boxed{sqrt{10}}).(2) The value of ( sin 2C ) is (boxed{dfrac{3sqrt{15}}{16}}).

answer：Okay, so I need to figure out the range of the function ( g(x) = -x^2 + 2x + 3 ) on the interval ([0, 4]). Hmm, let me think about how to approach this.First, I remember that the range of a function is the set of all possible output values (y-values) it can produce. Since this is a quadratic function, its graph is a parabola. The coefficient of ( x^2 ) is negative (-1), which means the parabola opens downward. That tells me the function has a maximum point at its vertex and it goes to negative infinity as ( x ) moves away from the vertex in both directions. But since we're only looking at the interval ([0, 4]), I don't have to worry about the whole real line, just this specific segment.To find the range on this interval, I think I need to find the maximum and minimum values of ( g(x) ) within ([0, 4]). Since the parabola opens downward, the vertex will give me the maximum value, and the minimum values will occur at the endpoints of the interval.Let me recall how to find the vertex of a parabola. The general form of a quadratic function is ( ax^2 + bx + c ), and the x-coordinate of the vertex is given by ( -frac{b}{2a} ). In this case, ( a = -1 ) and ( b = 2 ), so plugging those in:( x = -frac{2}{2(-1)} = -frac{2}{-2} = 1 )So the x-coordinate of the vertex is 1. To find the corresponding y-value, which is the maximum value of the function, I plug ( x = 1 ) back into ( g(x) ):( g(1) = -(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4 )Alright, so the maximum value is 4 at ( x = 1 ).Now, I need to find the minimum value on the interval ([0, 4]). Since the parabola opens downward, the function decreases as we move away from the vertex in both directions. So, on the interval from 0 to 1, the function is increasing, and from 1 to 4, it's decreasing. Therefore, the minimum value should occur at one of the endpoints, either at ( x = 0 ) or ( x = 4 ).Let me calculate ( g(0) ):( g(0) = -(0)^2 + 2(0) + 3 = 0 + 0 + 3 = 3 )And ( g(4) ):( g(4) = -(4)^2 + 2(4) + 3 = -16 + 8 + 3 = -5 )So, ( g(0) = 3 ) and ( g(4) = -5 ). Comparing these two, the minimum value is -5 at ( x = 4 ).Therefore, the function ( g(x) ) on the interval ([0, 4]) takes all values from -5 up to 4. So the range is ([-5, 4]).Wait, let me double-check my calculations to make sure I didn't make any mistakes. Calculating ( g(1) ) again: ( -1 + 2 + 3 = 4 ), that seems right. ( g(0) = 3 ), straightforward. ( g(4) = -16 + 8 + 3 = -5 ), that also checks out.Just to visualize, the parabola peaks at (1, 4) and then goes down to (4, -5). So yes, the lowest point on the interval is -5, and the highest is 4. Therefore, the range should be all real numbers between -5 and 4, inclusive.I think that's it. So the range is ([-5, 4]), which corresponds to option D.

answer：First, I need to determine the total number of crayon packs Michael will have after purchasing the additional packs. He currently has 4 packs and wants to buy 2 more, which totals 6 packs.Next, I know that the combined value of all 6 packs is 15. To find the cost of one pack, I can set up the equation 6x = 15, where x represents the cost of one pack.Finally, I solve for x by dividing both sides of the equation by 6, resulting in x = 15/6, which simplifies to x = 2.5. Therefore, one pack of crayons costs 2.50.

answer：To determine the total number of girls in the school, we can use the information from the stratified sample.1. **Calculate the proportion of girls in the sample:** [ text{Proportion of girls} = frac{103}{200} = 0.515 ]2. **Apply this proportion to the total number of students:** [ text{Total number of girls} = 0.515 times 2000 = 1030 ]Therefore, the total number of girls in the school is **1030**.

answer：First, I need to calculate the compound interest on Rs. 4000 for 2 years at 10% per annum using the compound interest formula.Next, I'll determine the simple interest, which is half of the calculated compound interest.Then, using the simple interest formula, I'll solve for the number of years by plugging in the known values of the principal, rate, and simple interest.Finally, I'll perform the calculations to find the number of years the money was placed on simple interest.

answer：Okay, so I have this problem here about an odd function f(x) that's increasing on all real numbers. I need to figure out the relationship between three values: a, b, and c. Let me break it down step by step.First, let's recall what an odd function is. An odd function satisfies the property f(-x) = -f(x) for all x in its domain. That means if I plug in a negative value into the function, it's the same as taking the negative of the function at the positive value. Also, the function is increasing, which means as x increases, f(x) also increases.Now, let's look at the definitions of a, b, and c:- a = -f(log₂(1/5))- b = f(log₂4.1)- c = f(2^{0.8})I need to compare these three values. Since f is increasing, the larger the input, the larger the output. So, if I can figure out the order of the inputs, I can determine the order of a, b, and c.Let's start by simplifying each input:1. For a: The input is log₂(1/5). I know that log₂(1/5) is the same as log₂(5^{-1}) which is -log₂5. So, log₂(1/5) = -log₂5. Since f is odd, f(-log₂5) = -f(log₂5). Therefore, a = -f(log₂5) = f(-log₂5) = -f(log₂5). Wait, that seems a bit confusing. Let me write it out:a = -f(log₂(1/5)) = -f(-log₂5) because log₂(1/5) = -log₂5. Since f is odd, f(-log₂5) = -f(log₂5). So, substituting back, a = -(-f(log₂5)) = f(log₂5). Okay, so a simplifies to f(log₂5).2. For b: The input is log₂4.1. I need to figure out what this is approximately. Since 2² = 4, log₂4 = 2. So, log₂4.1 is just a bit more than 2. Maybe around 2.03 or something. Not exact, but I know it's slightly greater than 2.3. For c: The input is 2^{0.8}. I need to calculate this. 2^{0.8} is the same as e^{0.8 ln2}. Let me approximate ln2 ≈ 0.693. So, 0.8 * 0.693 ≈ 0.554. Then, e^{0.554} ≈ 1.74. So, 2^{0.8} ≈ 1.74.Now, let's list the inputs:- For a: log₂5. Let's calculate that. Since 2² = 4 and 2³ = 8, log₂5 is between 2 and 3. Specifically, log₂5 ≈ 2.32.- For b: log₂4.1 ≈ 2.03.- For c: 2^{0.8} ≈ 1.74.So, the inputs in order from smallest to largest are: c (≈1.74) < b (≈2.03) < a (≈2.32).Since f is an increasing function, the larger the input, the larger the output. Therefore, f(c) < f(b) < f(a). But wait, a is f(log₂5), which is f(a_input), and c is f(2^{0.8}), which is f(c_input). So, actually, the outputs should follow the same order as the inputs.But hold on, a is f(log₂5), which is the largest input, so f(a) is the largest. Then b is next, and c is the smallest. So, c < b < a.Wait, but let me double-check. The problem defines a as -f(log₂(1/5)). Earlier, I simplified that to f(log₂5). Is that correct?Yes, because log₂(1/5) = -log₂5, and since f is odd, f(-log₂5) = -f(log₂5). Therefore, -f(log₂(1/5)) = -(-f(log₂5)) = f(log₂5). So, a is indeed f(log₂5).So, to recap:- a = f(log₂5) ≈ f(2.32)- b = f(log₂4.1) ≈ f(2.03)- c = f(2^{0.8}) ≈ f(1.74)Since f is increasing, the order of the outputs is the same as the order of the inputs. Therefore, c < b < a.Looking back at the options:A: a < b < cB: b < a < cC: c < b < aD: c < a < bSo, the correct answer should be C: c < b < a.Wait, but let me make sure I didn't mix up anything. The function is increasing, so larger inputs give larger outputs. So, since 1.74 < 2.03 < 2.32, then f(1.74) < f(2.03) < f(2.32), which means c < b < a. Yes, that seems right.I think I got it. So, the answer is C.