💡Okay, so I've got this problem here: Given that (2^a = 5^b = m) and (frac{1}{a} + frac{1}{b} = 2), I need to find the value of (m). The options are A. (sqrt{10}), B. 10, C. 20, D. 100.Alright, let's break this down step by step. First, I see that both (2^a) and (5^b) equal (m). That means (2^a = m) and (5^b = m). So, both expressions equal the same (m). Maybe I can express (a) and (b) in terms of (m) using logarithms. Since (2^a = m), taking the logarithm base 2 of both sides gives (a = log_2 m). Similarly, (5^b = m) implies (b = log_5 m). So, (a) and (b) are logarithms of (m) with different bases.Now, the equation given is (frac{1}{a} + frac{1}{b} = 2). Let me substitute (a) and (b) with the expressions I just found. So, (frac{1}{log_2 m} + frac{1}{log_5 m} = 2).Hmm, I remember that (frac{1}{log_b a} = log_a b). So, applying that property, (frac{1}{log_2 m} = log_m 2) and (frac{1}{log_5 m} = log_m 5). Therefore, the equation becomes (log_m 2 + log_m 5 = 2).Combining the logarithms, since (log_m 2 + log_m 5 = log_m (2 times 5) = log_m 10), the equation simplifies to (log_m 10 = 2).Now, what does (log_m 10 = 2) mean? It means that (m^2 = 10), because by definition, if (log_b a = c), then (b^c = a). So, (m^2 = 10), which implies (m = sqrt{10}) or (m = -sqrt{10}). But since (m) is an exponential result ((2^a) and (5^b) are always positive), (m) must be positive. Therefore, (m = sqrt{10}).Looking back at the options, A is (sqrt{10}), so that should be the correct answer.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from (2^a = m) and (5^b = m), I correctly expressed (a) and (b) in terms of logarithms. Then, substituting into (frac{1}{a} + frac{1}{b} = 2), I used the reciprocal logarithm property correctly to get (log_m 2 + log_m 5 = 2). Combining those logs into (log_m 10 = 2) is right, and solving for (m) gives (m = sqrt{10}). Yep, that all checks out.I don't think I made any errors here. It's straightforward once you remember the logarithm properties. So, I'm confident that the answer is A, (sqrt{10}).**Final Answer**boxed{A}

💡First, I need to understand what a counterexample is. A counterexample is a specific case where the hypothesis is true, but the conclusion is false, thereby disproving the proposition.The proposition states: "If ( a > b ), then ( a^2 > b^2 )." To find a counterexample, I need to find values of ( a ) and ( b ) such that ( a > b ) is true, but ( a^2 > b^2 ) is false.I will evaluate each option provided:**Option A**: ( a = -1 ), ( b = 0 )- Check if ( a > b ): ( -1 > 0 ) is false.- Since the hypothesis is false, this option does not serve as a counterexample.**Option B**: ( a = -1 ), ( b = -1 )- Check if ( a > b ): ( -1 > -1 ) is false.- Again, the hypothesis is false, so this is not a counterexample.**Option C**: ( a = -1 ), ( b = -2 )- Check if ( a > b ): ( -1 > -2 ) is true.- Now, check if ( a^2 > b^2 ): ( (-1)^2 = 1 ) and ( (-2)^2 = 4 ), so ( 1 > 4 ) is false.- Here, the hypothesis is true, but the conclusion is false. This is a valid counterexample.**Option D**: ( a = 2 ), ( b = 1 )- Check if ( a > b ): ( 2 > 1 ) is true.- Check if ( a^2 > b^2 ): ( 2^2 = 4 ) and ( 1^2 = 1 ), so ( 4 > 1 ) is true.- Both the hypothesis and conclusion are true, so this does not disprove the proposition.After evaluating all options, only **Option C** provides a valid counterexample where ( a > b ) is true, but ( a^2 > b^2 ) is false.

💡Okay, so I need to find the area of a parallelogram constructed on vectors a and b. The vectors are given as a = p - 3q and b = p + 2q. I also know the magnitudes of p and q, which are |p| = 1/5 and |q| = 1. Additionally, the angle between p and q is π/2, which means they are perpendicular to each other. Hmm, I remember that the area of a parallelogram formed by two vectors can be found using the cross product of those vectors. The formula is S = |a × b|, where S is the area. So, I need to compute the cross product of vectors a and b.Let me write down the vectors again:a = p - 3qb = p + 2qSo, the cross product a × b would be (p - 3q) × (p + 2q). I think I can expand this using the distributive property of cross products. Let me try that.Expanding the cross product:a × b = p × p + p × 2q - 3q × p - 3q × 2qSimplify each term:- p × p is zero because the cross product of any vector with itself is zero.- p × 2q is 2(p × q)- -3q × p is -3(q × p). But I remember that q × p is equal to -p × q, so this becomes -3(-p × q) which is 3(p × q)- -3q × 2q is -6(q × q). Again, the cross product of any vector with itself is zero, so this term is also zero.Putting it all together:a × b = 0 + 2(p × q) + 3(p × q) + 0 = 5(p × q)So, the cross product a × b is 5 times (p × q). Now, I need to find the magnitude of this cross product to get the area.The magnitude of the cross product |p × q| is equal to |p||q|sinθ, where θ is the angle between p and q. Since θ is π/2, sinθ is 1. Therefore, |p × q| = |p||q| = (1/5)(1) = 1/5.So, |a × b| = |5(p × q)| = 5|p × q| = 5*(1/5) = 1.Therefore, the area of the parallelogram is 1.Wait, let me double-check my steps to make sure I didn't make any mistakes. 1. Expanded the cross product correctly: yes, each term was handled properly.2. Simplified each term: yes, p × p and q × q are zero, and q × p became -p × q.3. Calculated |p × q| correctly: yes, since the angle is π/2, sinθ is 1, so it's just the product of the magnitudes.4. Multiplied by 5: yes, 5*(1/5) is indeed 1.I think everything checks out. So, the area should be 1.**Final Answer**The area of the parallelogram is boxed{1}.

💡Okay, so I've got this equation to solve: leftlfloorfrac{9x - 4}{6}rightrfloor = frac{12x + 7}{4}Hmm, the floor function can be tricky because it involves integers. Let me try to break this down step by step.First, I know that the floor function, denoted by [a] or ⎣a⎦, gives the greatest integer less than or equal to a. So, if I have ⎣y⎦ = m, then m is an integer and m ≤ y < m + 1.In this equation, the left side is the floor of (9x - 4)/6, and the right side is (12x + 7)/4. Since the floor function results in an integer, the right side must also be an integer. That gives me a clue: (12x + 7)/4 must be an integer. Let me denote that integer as m. So, I can write:frac{12x + 7}{4} = mWhere m is an integer. Solving for x, I get:12x + 7 = 4m 12x = 4m - 7 x = frac{4m - 7}{12}Okay, so x is expressed in terms of m. Now, since the left side of the original equation is the floor of (9x - 4)/6, which equals m, I can write:m leq frac{9x - 4}{6} < m + 1Substituting x from earlier into this inequality:m leq frac{9left(frac{4m - 7}{12}right) - 4}{6} < m + 1Let me simplify the middle part step by step. First, calculate 9*(4m - 7)/12:frac{9(4m - 7)}{12} = frac{36m - 63}{12} = 3m - frac{63}{12} = 3m - 5.25Wait, that might not be the best way to simplify. Let me try another approach. Let's compute the entire numerator first:9x - 4 = 9left(frac{4m - 7}{12}right) - 4 = frac{36m - 63}{12} - 4 = frac{36m - 63 - 48}{12} = frac{36m - 111}{12}So, the expression inside the floor function becomes:frac{36m - 111}{12 times 6} = frac{36m - 111}{72} = frac{12m - 37}{24}Wait, that doesn't seem right. Let me check my steps again.Starting from:frac{9x - 4}{6} = frac{9left(frac{4m - 7}{12}right) - 4}{6}Compute 9*(4m - 7)/12:frac{9(4m - 7)}{12} = frac{36m - 63}{12} = 3m - 5.25Then subtract 4:3m - 5.25 - 4 = 3m - 9.25Now, divide by 6:frac{3m - 9.25}{6} = frac{3m}{6} - frac{9.25}{6} = 0.5m - 1.541666...Hmm, this is getting messy with decimals. Maybe I should keep it as fractions.Let me try again:frac{9x - 4}{6} = frac{9left(frac{4m - 7}{12}right) - 4}{6} = frac{frac{36m - 63}{12} - 4}{6}Convert 4 to twelfths:4 = frac{48}{12}So,frac{frac{36m - 63 - 48}{12}}{6} = frac{frac{36m - 111}{12}}{6} = frac{36m - 111}{72} = frac{12m - 37}{24}Okay, so:m leq frac{12m - 37}{24} < m + 1Now, multiply all parts by 24 to eliminate the denominator:24m leq 12m - 37 < 24m + 24Let's split this into two inequalities:1. 24m ≤ 12m - 372. 12m - 37 < 24m + 24Solving the first inequality:24m ≤ 12m - 37Subtract 12m from both sides:12m ≤ -37Divide by 12:m ≤ -37/12 ≈ -3.0833Since m is an integer, m ≤ -4Now, the second inequality:12m - 37 < 24m + 24Subtract 12m from both sides:-37 < 12m + 24Subtract 24 from both sides:-61 < 12mDivide by 12:-61/12 ≈ -5.0833 < mSince m is an integer, m ≥ -5So combining both inequalities:-5.0833 < m ≤ -3.0833But m must be an integer, so m can be -5 or -4.Now, let's find x for each m.For m = -5:x = (4*(-5) - 7)/12 = (-20 - 7)/12 = -27/12 = -9/4For m = -4:x = (4*(-4) - 7)/12 = (-16 - 7)/12 = -23/12Now, we need to verify these solutions in the original equation to ensure they work.First, x = -9/4:Compute (9x - 4)/6:9*(-9/4) = -81/4-81/4 - 4 = -81/4 - 16/4 = -97/4Divide by 6:-97/4 /6 = -97/24 ≈ -4.0417Floor of that is -5.Now, compute (12x + 7)/4:12*(-9/4) = -27-27 + 7 = -20-20/4 = -5So, both sides equal -5. Good.Now, x = -23/12:Compute (9x - 4)/6:9*(-23/12) = -207/12 = -17.25-17.25 - 4 = -21.25Divide by 6:-21.25/6 ≈ -3.5417Floor of that is -4.Now, compute (12x + 7)/4:12*(-23/12) = -23-23 + 7 = -16-16/4 = -4Both sides equal -4. Good.So both solutions work.Therefore, the solutions are x = -9/4 and x = -23/12.

💡Okay, so I have this problem here: Given z equals 2 minus i, I need to find z times (the conjugate of z plus i). The options are A: 6 minus 2i, B: 4 minus 2i, C: 6 plus 2i, and D: 4 plus 2i. Hmm, let me think about how to approach this.First, I remember that the conjugate of a complex number z = a + bi is z conjugate = a - bi. So in this case, since z is 2 - i, the conjugate of z should be 2 + i, right? Let me write that down: z conjugate is 2 + i.Now, the problem is asking for z times (z conjugate plus i). So, I need to compute z multiplied by (z conjugate + i). Let me break this down step by step.First, let's find z conjugate plus i. Since z conjugate is 2 + i, adding i to that would be 2 + i + i. That simplifies to 2 + 2i. Okay, so z conjugate plus i is 2 + 2i.Now, I need to multiply z, which is 2 - i, by this result, 2 + 2i. So, it's (2 - i) multiplied by (2 + 2i). Let me set that up as a multiplication of two binomials.Using the distributive property, I can expand this out:(2 - i)(2 + 2i) = 2*2 + 2*2i - i*2 - i*2i.Let me compute each term:First term: 2*2 = 4.Second term: 2*2i = 4i.Third term: -i*2 = -2i.Fourth term: -i*2i = -2i².Now, let me combine these terms:4 + 4i - 2i - 2i².Combine like terms. The real parts are 4 and -2i², and the imaginary parts are 4i - 2i.So, for the imaginary parts: 4i - 2i = 2i.For the real parts: 4 and -2i². Wait, I remember that i² is equal to -1. So, -2i² is -2*(-1) = 2.Therefore, the real parts add up to 4 + 2 = 6.Putting it all together, the result is 6 + 2i.Looking back at the options, that's option C: 6 + 2i.Wait, let me double-check my steps to make sure I didn't make a mistake.Starting with z = 2 - i, so z conjugate is 2 + i. Then, z conjugate plus i is 2 + i + i = 2 + 2i. Multiplying z by that gives (2 - i)(2 + 2i). Expanding that:2*2 = 4,2*2i = 4i,-i*2 = -2i,-i*2i = -2i².Combine terms: 4 + 4i - 2i - 2i².Simplify: 4 + 2i - 2i².Since i² = -1, -2i² = 2. So, 4 + 2 + 2i = 6 + 2i.Yes, that seems correct. So, the answer should be C: 6 + 2i.

💡Okay, so I have this problem about a tourist attraction's profit. Let me try to understand it step by step. First, the problem says there's a fixed cost of 500 yuan per day. That means no matter how many people come, they have to pay 500 yuan every day. Then, each ticket is priced at 30 yuan. So, for each person who buys a ticket, they make 30 yuan. Now, the variable cost is directly proportional to the square root of the number of people buying tickets. Hmm, that means as more people come, the variable cost increases, but not linearly—it increases with the square root of the number of visitors. So, if x is the number of people, the variable cost would be something like k times the square root of x, where k is a constant we need to find.The problem also mentions that when the daily ticket sales reach 25, the attraction breaks even. Breaking even means that the total revenue equals the total cost, so profit is zero. That should help us find the constant k.Additionally, if the daily ticket sales exceed 100, they have to pay an extra 200 yuan for insurance. So, the cost structure changes a bit when x is more than 100.Alright, for part (1), we need to find the function relationship between profit y and the number of ticket buyers x.Let me start by setting up the profit function. Profit is usually revenue minus costs. Revenue is the number of tickets sold times the price per ticket, so that's 30x. Costs include the fixed cost of 500 yuan and the variable cost, which is k times the square root of x. So, the profit function should be:y = 30x - 500 - k√xNow, we know that when x is 25, the profit is zero because they break even. So, plugging in x = 25 into the equation:0 = 30*25 - 500 - k√25Calculating that:30*25 is 750. So, 750 - 500 is 250. Then, √25 is 5, so we have:0 = 250 - k*5Solving for k:250 = 5k k = 250 / 5 k = 50So, the variable cost is 50√x.Therefore, the profit function when x is less than or equal to 100 is:y = 30x - 50√x - 500But when x exceeds 100, they have to pay an additional 200 yuan for insurance. So, the total cost becomes 500 + 200 = 700, and the variable cost is still 50√x. So, the profit function for x > 100 is:y = 30x - 50√x - 700So, putting it all together, the function is piecewise:y = 30x - 50√x - 500, for x ≤ 100 y = 30x - 50√x - 700, for x > 100I think that's part (1) done.Now, moving on to part (2). The tourist attraction wants to avoid losses when the number of visitors reaches 20. They plan to increase the ticket price. We need to find the minimum amount each ticket must be increased by to ensure they don't lose money at x = 20.So, let's denote the new ticket price as m yuan. Then, the revenue becomes m*x, which is m*20 in this case. The costs remain the same: fixed cost is still 500, variable cost is 50√20, and since 20 is less than 100, there's no additional insurance fee.So, the profit equation when x = 20 is:Profit = m*20 - 500 - 50√20They want this profit to be at least zero:m*20 - 500 - 50√20 ≥ 0Let me compute 50√20. Since √20 is √(4*5) = 2√5, so 50√20 = 50*2√5 = 100√5.Given that √5 is approximately 2.24, so 100*2.24 = 224.So, the equation becomes:20m - 500 - 224 ≥ 0 20m - 724 ≥ 0 20m ≥ 724 m ≥ 724 / 20 m ≥ 36.2Since the ticket price must be an integer, we round up to the next whole number, which is 37 yuan.But wait, the original ticket price was 30 yuan. So, the increase needed is 37 - 30 = 7 yuan.Wait, hold on, the question says "the minimum amount that each ticket must be increased by." So, it's the increase, not the new price. So, if the new price is 37, the increase is 7 yuan.But let me double-check my calculations to make sure I didn't make a mistake.First, profit when x=20:Profit = m*20 - 500 - 50√20We have √20 = 2√5 ≈ 2*2.24 = 4.48So, 50√20 ≈ 50*4.48 = 224So, Profit ≈ 20m - 500 - 224 = 20m - 724Set this greater than or equal to zero:20m ≥ 724 m ≥ 724 / 20 m ≥ 36.2So, m must be at least 36.2, so 37 yuan. Therefore, the increase is 37 - 30 = 7 yuan.But wait, let me think again. The original price is 30, so if we set the new price to 37, the increase is 7. But is 37 the minimum integer that satisfies m ≥ 36.2? Yes, because 36 would be 36, which is less than 36.2, so 36 wouldn't be enough. So, 37 is the minimum integer.Therefore, the minimum increase is 7 yuan.But wait, let me check the profit with m=36.2:Profit = 36.2*20 - 500 - 224 = 724 - 724 = 0So, at m=36.2, profit is zero. So, since they want to avoid losses, they need profit ≥ 0. So, m must be at least 36.2. Since the ticket price must be an integer, they have to round up to 37. So, the increase is 7 yuan.Therefore, the answer is 7 yuan.Wait, but in the initial solution, the assistant said the minimum ticket price must be 37 yuan, which is correct, but the question asks for the minimum amount each ticket must be increased by, so 7 yuan. But in the initial solution, it was written as 37 yuan, which is the new price, not the increase.So, perhaps the initial solution was incorrect in that part. Let me check again.Original ticket price: 30 yuan.New ticket price: 37 yuan.Increase: 37 - 30 = 7 yuan.So, the minimum increase is 7 yuan.But in the initial solution, it was written as 37 yuan, which is the new price, not the increase. So, perhaps that was a mistake.Wait, let me check the problem statement again:"(2) The tourist attraction hopes to avoid losses when the number of visitors reaches 20. If the measure taken is to increase the ticket price, what is the minimum amount (rounded to the nearest integer) that each ticket must be increased by?"So, it's asking for the increase, not the new price. So, the correct answer should be 7 yuan.But in the initial solution, the assistant wrote:"Therefore, the minimum ticket price must be boxed{37} yuan."Which is incorrect because it's the new price, not the increase. The increase is 7 yuan.So, perhaps the initial solution had a mistake in part (2). So, I need to correct that.Therefore, the correct answer for part (2) is 7 yuan.But let me make sure I didn't make any mistakes in my calculations.Compute 50√20:√20 = 4.4721 (approximately 4.4721)50*4.4721 ≈ 223.605So, 20m - 500 - 223.605 ≥ 0 20m ≥ 723.605 m ≥ 723.605 / 20 m ≥ 36.18025So, m must be at least approximately 36.18, which rounds up to 37 yuan. Therefore, the increase is 37 - 30 = 7 yuan.Yes, that seems correct.So, summarizing:(1) The profit function is:y = 30x - 50√x - 500, for x ≤ 100 y = 30x - 50√x - 700, for x > 100(2) The minimum increase in ticket price is 7 yuan.But in the initial solution, part (2) was answered as 37 yuan, which is incorrect because it's the new price, not the increase. So, the correct answer should be 7 yuan.Wait, but in the initial solution, the assistant wrote:"m ≥ 25 + 5√5 ≈ 36.2, Therefore, the minimum ticket price must be boxed{37} yuan."Wait, where did 25 + 5√5 come from? Let me check.Wait, in the initial solution, the assistant wrote:"m ≥ 25 + 5√5 ≈ 36.2"But in my calculation, I had m ≥ 36.2, which is correct. But the assistant wrote 25 + 5√5, which is 25 + 5*2.24 = 25 + 11.2 = 36.2. So, that's correct. So, m ≥ 36.2, so m must be at least 37 yuan. But since the question asks for the increase, it's 37 - 30 = 7 yuan.So, the initial solution was correct in calculating m as 37, but the question asks for the increase, so the answer should be 7 yuan.Therefore, the initial solution had a mistake in part (2). The correct answer is 7 yuan.But in the problem statement, the initial solution was written as:"(2) Let the ticket price be m yuan. According to the problem, we have m × 20 - 50√20 - 500 ≥ 0 Thus, m ≥ 25 + 5√5 ≈ 36.2, Therefore, the minimum ticket price must be boxed{37} yuan."So, the assistant correctly calculated m as 37, but the question asks for the increase, so the answer should be 7 yuan.Therefore, the correct answer for part (2) is 7 yuan.But in the initial solution, the assistant answered 37 yuan, which is incorrect because it's the new price, not the increase.So, to correct that, the answer should be 7 yuan.But wait, let me check the problem statement again:"(2) The tourist attraction hopes to avoid losses when the number of visitors reaches 20. If the measure taken is to increase the ticket price, what is the minimum amount (rounded to the nearest integer) that each ticket must be increased by?"So, it's asking for the increase, not the new price. Therefore, the correct answer is 7 yuan.Therefore, the initial solution was incorrect in part (2). The correct answer is 7 yuan.But in the initial solution, the assistant wrote 37 yuan, which is the new price, not the increase.So, to conclude:(1) The profit function is:y = 30x - 50√x - 500, for x ≤ 100 y = 30x - 50√x - 700, for x > 100(2) The minimum increase in ticket price is 7 yuan.But in the initial solution, part (2) was answered as 37 yuan, which is incorrect. So, the correct answer is 7 yuan.Therefore, the final answer for part (2) is 7 yuan.