💡First, I need to determine the set ( M ) by solving the inequality ( x^2 - 3x - 4 < 0 ).I'll start by factoring the quadratic expression:[x^2 - 3x - 4 = (x - 4)(x + 1)]So, the inequality becomes:[(x - 4)(x + 1) < 0]To find the interval where this inequality holds true, I'll analyze the sign of the product:- When ( x < -1 ), both factors are negative, so the product is positive.- When ( -1 < x < 4 ), one factor is positive and the other is negative, so the product is negative.- When ( x > 4 ), both factors are positive, so the product is positive.Therefore, the solution to the inequality is:[-1 < x < 4]So, the set ( M ) is:[M = {x mid -1 < x < 4}]Next, the set ( N ) is given as:[N = {-2, -1, 0, 1, 2}]To find the intersection ( M cap N ), I'll identify the elements in ( N ) that fall within the interval ( -1 < x < 4 ):- ( -2 ) is less than ( -1 ), so it's not included.- ( -1 ) is equal to the lower bound, so it's not included.- ( 0 ), ( 1 ), and ( 2 ) are all within the interval.Therefore, the intersection ( M cap N ) is:[M cap N = {0, 1, 2}]The correct answer is:boxed{D}

💡Okay, so I have this problem about "homogeneous functions." From what I understand, these are functions that have the same analytical expression and the same range but different domains. The specific question is asking how many such functions exist where the expression is y = x² and the range is {1, 2}.First, let me break down what this means. The analytical expression is fixed as y = x², which is a parabola opening upwards. The range is {1, 2}, meaning that the output values of the function can only be 1 or 2. So, for this function, whenever I plug in an x-value, the result should be either 1 or 2.Now, since the range is {1, 2}, I need to figure out what x-values will give me these y-values. For y = 1, solving x² = 1 gives x = ±1. Similarly, for y = 2, solving x² = 2 gives x = ±√2. So, the possible x-values that satisfy the range are -1, 1, -√2, and √2.These are the only x-values that can be in the domain because any other x-value would result in a y-value outside of {1, 2}, which contradicts the given range. Therefore, the domain of each function must be a subset of {-1, 1, -√2, √2}.But not just any subset—since the range must include both 1 and 2, the domain must include at least one x-value that maps to 1 and at least one x-value that maps to 2. So, the domain can't be just {-1, 1} because that would only give a range of {1}, and similarly, it can't be just {-√2, √2} because that would only give a range of {2}. It has to include at least one from each pair.Now, for each of these pairs, {-1, 1} and {-√2, √2}, I can choose to include none, one, or both of the elements. But since we need at least one from each pair to cover the entire range, we have to exclude the cases where we include none from either pair.For the first pair, {-1, 1}, the possible choices are:1. Include neither -1 nor 1.2. Include only -1.3. Include only 1.4. Include both -1 and 1.Similarly, for the second pair, {-√2, √2}, the possible choices are:1. Include neither -√2 nor √2.2. Include only -√2.3. Include only √2.4. Include both -√2 and √2.However, as I mentioned earlier, we can't include neither from either pair because that would result in an incomplete range. So, for each pair, we have three valid choices: include only one, include the other, or include both.Therefore, for the first pair, we have 3 choices, and for the second pair, we also have 3 choices. Since these choices are independent of each other, the total number of possible domains is 3 multiplied by 3, which is 9.So, there are 9 different homogeneous functions that satisfy the given conditions.Wait, let me double-check that. If I consider all possible combinations, for each pair, I have 3 options, and since there are two pairs, it's 3^2 = 9. Yeah, that makes sense.But just to be thorough, let me list them out:For the first pair {-1, 1}:1. Only -12. Only 13. Both -1 and 1For the second pair {-√2, √2}:1. Only -√22. Only √23. Both -√2 and √2Now, combining each choice from the first pair with each choice from the second pair:1. Only -1 and only -√22. Only -1 and only √23. Only -1 and both -√2 and √24. Only 1 and only -√25. Only 1 and only √26. Only 1 and both -√2 and √27. Both -1 and 1 and only -√28. Both -1 and 1 and only √29. Both -1 and 1 and both -√2 and √2That's 9 different combinations, each resulting in a different domain but the same analytical expression and range. So, yes, 9 seems correct.I think I've covered all the possibilities and made sure that each combination includes at least one x-value from each pair to ensure the range is {1, 2}. Therefore, the number of homogeneous functions is 9.

💡First, I need to determine the number of students Beth had at the end of her final year. Starting with 150 students in the 10th grade, 30 more students joined the following year. This increases the total number of students to 180.In the final year, 15 students left the class. Subtracting these 15 students from the previous total of 180 students results in 165 students remaining at the end of the final year.

💡Okay, so I have this problem where I need to figure out the best selling price to maximize profit. Let me try to break it down step by step.First, the purchase price is 80 yuan per item. That means each item costs me 80 yuan to buy. If I sell it at 90 yuan, I make a profit of 10 yuan per item. And right now, I can sell all 400 items at that price. So, my current profit is 10 yuan multiplied by 400 items, which is 4000 yuan.But the problem says that for every 1 yuan increase in price, the sales volume decreases by 20 items. Hmm, so if I raise the price by 1 yuan to 91 yuan, I can only sell 380 items. If I raise it by another yuan to 92 yuan, I sell 360 items, and so on. I need to find the price that will give me the maximum profit.Let me think about how to model this. Maybe I can let x be the number of yuan I increase the price. So, the selling price becomes 90 + x yuan. Then, the number of items I can sell decreases by 20x, so it becomes 400 - 20x items.The profit per item is the selling price minus the purchase price, which is (90 + x) - 80 = 10 + x yuan per item. So, the total profit would be the profit per item multiplied by the number of items sold. That gives me:Total Profit = (10 + x) * (400 - 20x)Let me write that out as a function:P(x) = (10 + x)(400 - 20x)I can expand this to make it easier to work with. Let's do that:P(x) = 10*400 + 10*(-20x) + x*400 + x*(-20x)P(x) = 4000 - 200x + 400x - 20x²P(x) = 4000 + 200x - 20x²Hmm, that's a quadratic equation, and since the coefficient of x² is negative (-20), the parabola opens downward, which means the vertex is the maximum point.To find the maximum profit, I need to find the vertex of this parabola. The x-coordinate of the vertex in a quadratic equation ax² + bx + c is given by -b/(2a). In this case, a = -20 and b = 200.So, plugging in those values:x = -200 / (2 * -20) = -200 / (-40) = 5So, x is 5. That means I should increase the price by 5 yuan. Therefore, the selling price should be 90 + 5 = 95 yuan.Let me double-check to make sure I didn't make any mistakes. If I set the price at 95 yuan, then the number of items sold would be 400 - 20*5 = 400 - 100 = 300 items. The profit per item would be 95 - 80 = 15 yuan. So, total profit is 15 * 300 = 4500 yuan.If I check a price increase of 4 yuan, the selling price would be 94 yuan, selling 400 - 80 = 320 items. Profit per item is 14 yuan, so total profit is 14 * 320 = 4480 yuan, which is less than 4500.If I check a price increase of 6 yuan, the selling price would be 96 yuan, selling 400 - 120 = 280 items. Profit per item is 16 yuan, so total profit is 16 * 280 = 4480 yuan, which is also less than 4500.Okay, so it seems like 95 yuan is indeed the price that gives the maximum profit. I think I did everything correctly. The key was setting up the profit function correctly and then using the vertex formula to find the maximum.Another way to think about it is that the profit function is quadratic, and since it opens downward, the maximum occurs at the vertex. Calculating the vertex using -b/(2a) gave me the exact point where the profit is maximized. It's interesting how the relationship between price and quantity sold affects the total profit. Increasing the price too much reduces the number of items sold, which can lower the total profit, while keeping the price too low might not maximize the profit either. Finding the sweet spot is crucial.I also considered testing values around the calculated x to ensure that 95 yuan is indeed the maximum. Since both 94 and 96 yuan resulted in lower profits, it reinforces that 95 yuan is the optimal price.In summary, by setting up the profit equation, expanding it, and finding the vertex, I determined that increasing the price by 5 yuan to 95 yuan per item will maximize the profit.

💡Alright, so I have this problem: I need to find the equation of a line that passes through the point M(3, -1) and bisects the chord of the parabola y² = 4x. Hmm, okay. Let me break this down step by step.First, I remember that a chord of a parabola is a line segment joining two points on the parabola. So, if the line I'm looking for bisects this chord, it means that it passes through the midpoint of the chord. That makes sense. So, essentially, the line I need is the one that goes through M(3, -1) and is the midpoint of some chord of the parabola y² = 4x.Now, how do I approach this? I think I need to consider the general equation of a line passing through M(3, -1). Let me denote the slope of this line as m. So, the equation of the line can be written as:y - (-1) = m(x - 3)Simplifying that, it becomes:y + 1 = m(x - 3)Or,y = m(x - 3) - 1Okay, so that's the general form of the line. Now, this line intersects the parabola y² = 4x at two points, say A and B. The midpoint of AB should lie on the line we're trying to find, which is the same line passing through M(3, -1). So, if I can find the midpoint of AB and set it equal to M(3, -1), that should give me the necessary condition to solve for m.Wait, actually, no. The midpoint of AB is not necessarily M(3, -1). Instead, the line passes through M(3, -1) and bisects the chord AB, meaning M is the midpoint of AB. So, M(3, -1) is the midpoint of AB. That changes things. So, A and B are points on the parabola, and their midpoint is M(3, -1). So, I can use this information to find the equation of the line AB, which is the chord, and then find the equation of the bisecting line.But wait, the problem says the line passes through M(3, -1) and bisects the chord. So, does that mean the line is the bisector of the chord AB, meaning it passes through the midpoint M? Or is M the midpoint? I think it's the latter. The line bisects the chord, so it must pass through the midpoint of the chord. Since the line passes through M(3, -1), that point must be the midpoint of the chord.So, M(3, -1) is the midpoint of AB. Therefore, if I can find the coordinates of A and B on the parabola such that their midpoint is M(3, -1), then the line AB is the chord, and the line we're looking for is the bisector, which is the line passing through M(3, -1) and perpendicular to AB? Wait, no. The bisector of a chord in a parabola is actually the line that passes through the midpoint and is perpendicular to the chord if it's a diameter, but in this case, it's just a bisecting line.Wait, maybe I'm overcomplicating. Let me think again. The line we need passes through M(3, -1) and bisects the chord AB of the parabola. So, it's the line that passes through M and is the bisector of AB, meaning it passes through the midpoint of AB, which is M. So, essentially, the line we're looking for is the line that passes through M(3, -1) and is the bisector of the chord AB, meaning it's the line that contains the midpoint M and is perpendicular to AB if it's a diameter, but in this case, it's just a bisecting line, not necessarily perpendicular.Wait, no, in a parabola, the diameter (which is the bisector of a set of parallel chords) is not necessarily perpendicular to the chords. So, maybe I need to find the equation of the line that passes through M(3, -1) and is the bisector of the chord AB, meaning it passes through the midpoint M and has a certain slope related to the parabola.Alternatively, perhaps I can use the concept that the equation of the chord bisected at a point (h, k) on the parabola is given by T = S1, where T is the tangent equation and S1 is the value at (h, k). Wait, but M(3, -1) is not necessarily on the parabola. Let me check: plugging x=3, y=-1 into y²=4x gives (-1)²=1, which is not equal to 4*3=12. So, M is not on the parabola. Therefore, the chord is bisected at M, which is outside the parabola.So, in that case, the equation of the chord bisected at (h, k) is given by T = S1, but since (h, k) is not on the parabola, S1 ≠ 0. Wait, let me recall the formula. For a parabola y² = 4ax, the equation of the chord bisected at (h, k) is given by:ky - 2a(x + h) = k² - 4ahWait, is that correct? Let me verify. For the parabola y² = 4ax, the equation of the chord bisected at (h, k) is:ky - 2a(x + h) = k² - 4ahYes, that seems right. So, in our case, the parabola is y² = 4x, so a = 1. Therefore, the equation becomes:ky - 2(x + h) = k² - 4hGiven that the midpoint is M(3, -1), so h = 3, k = -1. Plugging these into the equation:(-1)y - 2(x + 3) = (-1)² - 4*3Simplify:- y - 2x - 6 = 1 - 12- y - 2x - 6 = -11Multiply both sides by -1 to make it neater:y + 2x + 6 = 11Simplify:2x + y + 6 - 11 = 02x + y - 5 = 0So, the equation of the chord bisected at M(3, -1) is 2x + y - 5 = 0.But wait, the problem says "the line that passes through the point M(3, -1) and bisects the chord of the parabola y²=4x." So, is this line the bisector, or is it the chord? I think it's the bisector, meaning the line that bisects the chord, which is the same as the line that contains the midpoint of the chord. But in this case, we derived the equation of the chord itself, not the bisector.Wait, no. The chord bisected at M(3, -1) is the line 2x + y - 5 = 0. But the bisector of the chord would be the line that passes through M and is perpendicular to the chord if it's a diameter, but in this case, since M is not on the parabola, it's just the line that bisects the chord, which is the same as the chord itself? No, that doesn't make sense.Wait, perhaps I'm confusing terms. The line that bisects the chord is the line that passes through the midpoint of the chord. So, in this case, the line we're looking for is the line that passes through M(3, -1) and is the bisector of the chord AB. But since M is the midpoint, the line AB is the chord, and the bisector is the line that passes through M and is perpendicular to AB if it's a diameter, but in this case, it's just the line that passes through M and bisects AB, which is the same as the line AB itself.Wait, no. The bisector of a chord is the line that passes through the midpoint and is perpendicular to the chord if it's a diameter. But in this case, since M is not on the parabola, it's just a point outside, so the bisector would be the line that passes through M and is the set of midpoints of all chords parallel to a certain direction.Wait, maybe I need to think differently. Let me recall that for a parabola, the equation of the chord bisected at a point (h, k) is given by T = S1, where T is the tangent at (h, k) if (h, k) is on the parabola, but since (h, k) is not on the parabola, it's the equation of the chord.Wait, no, the equation T = S1 is for the chord bisected at (h, k). So, in our case, since M(3, -1) is not on the parabola, we can still use this formula to find the equation of the chord bisected at M.So, using the formula for the chord bisected at (h, k):For parabola y² = 4ax, the equation is:ky - 2a(x + h) = k² - 4ahAs I did earlier, plugging in h=3, k=-1, a=1:(-1)y - 2(x + 3) = (-1)² - 4*1*3Simplify:- y - 2x - 6 = 1 - 12- y - 2x - 6 = -11Multiply both sides by -1:y + 2x + 6 = 11Simplify:2x + y - 5 = 0So, the equation of the chord bisected at M(3, -1) is 2x + y - 5 = 0.But the problem asks for the equation of the line that passes through M(3, -1) and bisects the chord of the parabola. So, is this line the bisector, or is it the chord? I think it's the bisector, meaning the line that bisects the chord, which is the same as the line that contains the midpoint of the chord. But in this case, the chord itself is 2x + y - 5 = 0, and the bisector would be the line that passes through M and is perpendicular to the chord if it's a diameter, but since M is not on the parabola, it's just the line that bisects the chord, which is the same as the chord itself.Wait, no. The bisector of a chord is the line that passes through the midpoint and is perpendicular to the chord if it's a diameter, but in this case, since M is not on the parabola, it's just the line that bisects the chord, which is the same as the chord itself. So, the line we found, 2x + y - 5 = 0, is the chord bisected at M(3, -1). Therefore, the line that passes through M and bisects the chord is this line itself.But wait, the problem says "the line that passes through M(3, -1) and bisects the chord of the parabola." So, it's the line that passes through M and bisects the chord, meaning it's the line that contains the midpoint of the chord, which is M. So, the line we found is the chord, but the bisector would be a different line, perhaps the axis of the parabola or something else.Wait, no. The bisector of a chord is the line that passes through the midpoint and is perpendicular to the chord if it's a diameter, but in this case, since M is not on the parabola, it's just the line that bisects the chord, which is the same as the chord itself. So, I think the line we found, 2x + y - 5 = 0, is the chord bisected at M(3, -1), and since the line passes through M and bisects the chord, it's the line we're looking for.Wait, but let me double-check. If I take the line 2x + y - 5 = 0 and see where it intersects the parabola y² = 4x, I should get two points A and B whose midpoint is M(3, -1).Let me solve the system:y² = 4x2x + y - 5 = 0 => y = 5 - 2xSubstitute y into the parabola equation:(5 - 2x)² = 4x25 - 20x + 4x² = 4x4x² - 24x + 25 = 0Let's solve this quadratic equation:x = [24 ± sqrt(576 - 400)] / 8x = [24 ± sqrt(176)] / 8x = [24 ± 4*sqrt(11)] / 8x = [6 ± sqrt(11)] / 2So, x = 3 + sqrt(11)/2 and x = 3 - sqrt(11)/2Corresponding y values:y = 5 - 2x = 5 - 2*(3 ± sqrt(11)/2) = 5 - 6 ∓ sqrt(11) = -1 ∓ sqrt(11)So, points A and B are:A: (3 + sqrt(11)/2, -1 - sqrt(11))B: (3 - sqrt(11)/2, -1 + sqrt(11))Now, let's find the midpoint of A and B:x-coordinate: [ (3 + sqrt(11)/2) + (3 - sqrt(11)/2) ] / 2 = (6)/2 = 3y-coordinate: [ (-1 - sqrt(11)) + (-1 + sqrt(11)) ] / 2 = (-2)/2 = -1So, the midpoint is indeed M(3, -1). Therefore, the line 2x + y - 5 = 0 is the chord bisected at M(3, -1), and since it passes through M and bisects the chord, it is the line we're looking for.Wait, but the problem says "the line that passes through M(3, -1) and bisects the chord of the parabola." So, it's the line that passes through M and bisects the chord, which is the same as the chord itself. Therefore, the equation is 2x + y - 5 = 0.Alternatively, if the problem is asking for the bisector line, which is different from the chord, but in this case, since M is the midpoint, the line we found is both the chord and the bisector. So, I think the answer is 2x + y - 5 = 0.Wait, but let me think again. If the line passes through M and bisects the chord, it means that for any chord of the parabola, this line would pass through its midpoint. But in this case, we're only considering one specific chord bisected at M. So, the line we found is the specific chord bisected at M, but the bisector line that bisects all chords parallel to a certain direction would be the diameter of the parabola, which has a different equation.Wait, no. The diameter of a parabola is the locus of the midpoints of all chords parallel to a given direction. So, if we consider all chords parallel to a certain line, their midpoints lie on the diameter. But in this problem, we're only considering one specific chord bisected at M(3, -1). So, the line we found is the specific chord, not the diameter.Therefore, the line that passes through M(3, -1) and bisects the chord of the parabola y²=4x is the line 2x + y - 5 = 0.Wait, but let me check the slope. The chord has a slope of -2 (from the equation 2x + y - 5 = 0, which can be written as y = -2x + 5). The slope of the parabola at the midpoint M(3, -1) can be found by differentiating y² = 4x, which gives 2y dy/dx = 4, so dy/dx = 2/y. At M(3, -1), dy/dx = 2/(-1) = -2. So, the slope of the tangent at M is -2, which is the same as the slope of the chord. That makes sense because the chord is bisected at M, so the tangent at M is parallel to the chord.Wait, but if the tangent at M has the same slope as the chord, that would mean that M is the midpoint of the chord, and the chord is parallel to the tangent at M. That seems correct.So, in conclusion, the line that passes through M(3, -1) and bisects the chord of the parabola y²=4x is 2x + y - 5 = 0.

💡First, I need to determine the relationship between the number of men and the time it takes to complete the work. Since more men can complete the work faster, there is an inverse relationship between the number of men and the number of days required.Let’s denote the number of men in the first group as ( x ).According to the problem, ( x ) men can complete the work in 25 days. This means that the work done by one man in one day is ( frac{1}{25x} ).When 20 men work on the same piece, they can complete it in 15 days. This means that the work done by 20 men in one day is ( frac{1}{15} ).Since the total work remains the same in both scenarios, I can set up the equation:[frac{1}{25x} times 25 = frac{1}{15} times 20]Simplifying both sides:[frac{25}{25x} = frac{20}{15}][frac{1}{x} = frac{4}{3}]Solving for ( x ):[x = frac{3}{4}]This result doesn't make sense in the context of the problem because the number of men should be a whole number. Therefore, I need to re-examine the setup of the equation to ensure it accurately reflects the relationship between the number of men and the time taken to complete the work.