💡Okay, so I have this problem here asking which of the given functions is monotonically increasing on the interval (0, +∞). The options are:A: y = 1/xB: y = x² - 2xC: y = 1 - xD: y = |x| - 1Alright, I need to figure out which one of these functions increases as x increases from 0 to infinity. Let me take each option one by one and analyze them.Starting with Option A: y = 1/x. Hmm, I remember that this is a hyperbola. When x increases, 1/x decreases. For example, when x is 1, y is 1; when x is 2, y is 0.5; when x is 3, y is about 0.333, and so on. So, as x gets larger, y gets smaller. That means this function is actually decreasing, not increasing. So, Option A is out.Moving on to Option B: y = x² - 2x. This looks like a quadratic function. Quadratics have a parabola shape, and since the coefficient of x² is positive, it opens upwards. The vertex of this parabola will be its minimum point. To find the vertex, I can use the formula x = -b/(2a). Here, a = 1 and b = -2, so x = -(-2)/(2*1) = 2/2 = 1. So, the vertex is at x = 1. Now, since the parabola opens upwards, the function decreases to the left of the vertex and increases to the right of the vertex. That means for x < 1, the function is decreasing, and for x > 1, it's increasing. But the interval we're looking at is (0, +∞). So, on this interval, the function decreases from x = 0 to x = 1 and then increases from x = 1 onwards. Since it's not increasing throughout the entire interval, just part of it, Option B isn't monotonically increasing on (0, +∞). So, B is also out.Next is Option C: y = 1 - x. This is a linear function with a slope of -1. Since the slope is negative, the function decreases as x increases. For example, when x increases by 1, y decreases by 1. So, clearly, this function is decreasing on the entire interval (0, +∞). Therefore, Option C is not the answer either.Finally, Option D: y = |x| - 1. Absolute value functions can sometimes be tricky, but let's break it down. For x > 0, |x| is just x, so the function simplifies to y = x - 1. This is a linear function with a slope of 1, which is positive. That means as x increases, y also increases. To double-check, let's plug in some values. When x = 1, y = 0; when x = 2, y = 1; when x = 3, y = 2, and so on. Each time x increases by 1, y increases by 1. So, this function is definitely increasing on the interval (0, +∞).Just to make sure I'm not missing anything, let's consider the behavior around x = 0. When x approaches 0 from the positive side, y approaches -1. As x increases, y increases without bound. There's no point where the function starts decreasing again. So, yes, it's consistently increasing.Therefore, after analyzing all the options, Option D is the only function that is monotonically increasing on the interval (0, +∞).

💡First, I recognize that ( a = 105 ) and ( a^3 = 21 times n times 45 times 49 ). My goal is to find the value of ( n ).I start by calculating ( a^3 ):[a^3 = 105^3 = 105 times 105 times 105]Next, I factorize each number in the equation:[21 = 3 times 7][45 = 3 times 3 times 5][49 = 7 times 7][105 = 3 times 5 times 7]Substituting these factorizations into the equation:[(3 times 5 times 7) times (3 times 5 times 7) times (3 times 5 times 7) = (3 times 7) times n times (3 times 3 times 5) times (7 times 7)]Simplifying both sides:[3^3 times 5^3 times 7^3 = 3^3 times 5 times 7^3 times n]To balance the equation, ( n ) must account for the missing factors:[n = 5^2 = 25]

💡Okay, so I have this problem here about a moving point M(x, y) and two fixed points F1(-a, 0) and F2(a, 0). The product of the slopes of the lines connecting M to F1 and F2 is a constant λ. The trajectory of M is a hyperbola with eccentricity √3, and I need to find the value of λ. The options are A: 2, B: -2, C: 3, D: √3.First, let me visualize this. There are two fixed points on the x-axis, symmetric about the origin, at (-a, 0) and (a, 0). The moving point M is somewhere in the plane, not on the x-axis because y ≠ 0. The product of the slopes from M to each fixed point is constant. That should give me an equation relating x and y, which is supposed to be a hyperbola.So, the slope from M(x, y) to F1(-a, 0) would be (y - 0)/(x - (-a)) = y/(x + a). Similarly, the slope from M(x, y) to F2(a, 0) is (y - 0)/(x - a) = y/(x - a). The product of these slopes is λ, so:(y/(x + a)) * (y/(x - a)) = λLet me write that as:(y^2) / ((x + a)(x - a)) = λSimplify the denominator:(x + a)(x - a) = x^2 - a^2So, the equation becomes:y^2 / (x^2 - a^2) = λMultiplying both sides by (x^2 - a^2):y^2 = λ(x^2 - a^2)Let me rearrange this:y^2 - λx^2 = -λa^2Hmm, this looks like a quadratic equation. Let me write it in standard form. If I divide both sides by -λa^2, I get:(y^2)/(-λa^2) + (x^2)/(a^2) = 1But wait, this doesn't look quite right because the standard form of a hyperbola is either (x^2/a^2) - (y^2/b^2) = 1 or (y^2/b^2) - (x^2/a^2) = 1. So, let me adjust the equation accordingly.Starting from y^2 = λx^2 - λa^2, which can be written as:y^2 - λx^2 = -λa^2Let me bring all terms to one side:λx^2 - y^2 = λa^2Divide both sides by λa^2:(x^2)/(a^2) - (y^2)/(λa^2) = 1Yes, that looks better. So, this is the standard form of a hyperbola centered at the origin, opening along the x-axis. The standard form is (x^2)/(a^2) - (y^2)/(b^2) = 1, where b^2 is equal to λa^2 in this case.So, from the standard hyperbola equation, we know that the relationship between a, b, and c (the distance from the center to each focus) is c^2 = a^2 + b^2. The eccentricity e is given by e = c/a.Given that the eccentricity e is √3, we can write:e = c/a = √3So, c = a√3But c^2 = a^2 + b^2, so:(a√3)^2 = a^2 + b^2Which simplifies to:3a^2 = a^2 + b^2Subtract a^2 from both sides:2a^2 = b^2But earlier, we found that b^2 = λa^2, so:λa^2 = 2a^2Divide both sides by a^2 (since a > 0, a^2 ≠ 0):λ = 2So, the value of λ is 2, which corresponds to option A.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Calculated the slopes correctly: y/(x + a) and y/(x - a). Their product is y^2/(x^2 - a^2) = λ.2. Rearranged to y^2 = λx^2 - λa^2.3. Rewrote in standard hyperbola form: (x^2)/(a^2) - (y^2)/(λa^2) = 1.4. Identified b^2 = λa^2.5. Used the relationship c^2 = a^2 + b^2, with c = a√3, leading to 3a^2 = a^2 + b^2.6. Solved for b^2 = 2a^2, so λa^2 = 2a^2 => λ = 2.Everything seems to check out. So, I think the answer is A: 2.**Final Answer**boxed{A}

💡First, determine the amount of flour Katie needs, which is 3 pounds.Next, calculate the amount of flour Sheila needs by adding 2 pounds to Katie's requirement: 3 + 2 = 5 pounds.Finally, add the amounts of flour needed by both Katie and Sheila to find the total: 3 + 5 = 8 pounds.

💡Alright, so I've got this equation to solve: 5 minus 3i times x equals negative 2 plus 6i times x. Hmm, okay, let me write that down to make it clearer. So, it's 5 - 3i x = -2 + 6i x. First off, I remember that when solving equations, I need to get all the terms with x on one side and the constants on the other. So, maybe I should start by moving the terms with x to one side. Let me see, if I add 3i x to both sides, that should help. Adding 3i x to both sides, I get: 5 = -2 + 6i x + 3i x. Wait, that simplifies to 5 = -2 + 9i x. Okay, that seems better. Now, I need to get rid of that -2 on the right side. So, I'll add 2 to both sides to balance it out. Adding 2 to both sides, I have: 5 + 2 = 9i x, which simplifies to 7 = 9i x. Alright, now I need to solve for x. To do that, I should divide both sides by 9i. So, x equals 7 divided by 9i. But wait, dividing by i can be a bit tricky. I remember that i is the square root of -1, and sometimes it's helpful to rationalize the denominator when dealing with complex numbers. To rationalize, I can multiply both the numerator and the denominator by the conjugate of the denominator. In this case, the denominator is 9i, so its conjugate is -9i. Let me try that. Multiplying both numerator and denominator by -i, I get: x equals (7 * -i) divided by (9i * -i). Simplifying the numerator, 7 times -i is -7i. In the denominator, 9i times -i is -9i squared. But wait, i squared is -1, so -9 times -1 is 9. So, putting it all together, x equals -7i divided by 9. That simplifies to x equals -7i over 9. Let me double-check my steps to make sure I didn't make any mistakes. Starting with 5 - 3i x = -2 + 6i x, I moved the 3i x to the right side by adding 3i x to both sides, resulting in 5 = -2 + 9i x. Then, I added 2 to both sides to get 7 = 9i x. Dividing both sides by 9i gives x = 7/(9i). To rationalize, I multiplied numerator and denominator by -i, leading to x = (-7i)/9. Hmm, that seems correct. I don't think I missed any steps, and the algebra checks out. So, the solution should be x equals -7i over 9.

💡Alright, so I have this problem about triangle ABC. I know that vertex B is at (3,2), and there's an altitude on side AB with the equation x - 2y + 2 = 0. I need to find the equation of line AB first, and then choose one of the given conditions to find the equation of line AC.Starting with part (1), finding the equation of line AB. I remember that an altitude of a triangle is perpendicular to the side it's drawn from. So, if I can find the slope of the altitude, I can determine the slope of AB since they are perpendicular.The given equation of the altitude is x - 2y + 2 = 0. Let me rewrite this in slope-intercept form (y = mx + b) to find its slope. x - 2y + 2 = 0 Subtract x and 2 from both sides: -2y = -x - 2 Divide both sides by -2: y = (1/2)x + 1So, the slope of the altitude is 1/2. Since AB is perpendicular to this altitude, its slope should be the negative reciprocal of 1/2, which is -2. Now that I have the slope of AB, which is -2, and I know it passes through point B(3,2), I can use the point-slope form to write the equation of AB. Point-slope form is y - y1 = m(x - x1). Plugging in the values:y - 2 = -2(x - 3)Simplify this:y - 2 = -2x + 6 Add 2 to both sides: y = -2x + 8To write this in general form (Ax + By + C = 0), I'll rearrange it:2x + y - 8 = 0So, the equation of line AB is 2x + y - 8 = 0.Moving on to part (2), I need to choose one of the given conditions to find the equation of line AC. I'll choose condition ①: The equation of the angle bisector of angle A is x + y - 2 = 0.First, I need to find the coordinates of point A. Since the angle bisector passes through point A, and we know the equation of AB, I can find point A by solving the system of equations consisting of AB and the angle bisector.The equations are:1. 2x + y - 8 = 0 (equation of AB)2. x + y - 2 = 0 (equation of the angle bisector)Let me solve this system. I'll subtract the second equation from the first to eliminate y:(2x + y - 8) - (x + y - 2) = 0 2x + y - 8 - x - y + 2 = 0 x - 6 = 0 x = 6Now, substitute x = 6 into the second equation to find y:6 + y - 2 = 0 y = -4So, point A is at (6, -4).Next, I need to find the equation of line AC. To do this, I might need another point on line AC. Since AC is another side of the triangle, and we know point A, we need another point to define it. However, we don't have point C yet.Given that the angle bisector of angle A is x + y - 2 = 0, this line should pass through point A and divide the angle at A into two equal angles. I think I can use the concept of reflection here. The idea is that the angle bisector reflects one side over the other.So, if I reflect point B over the angle bisector, the reflected point should lie on line AC. Let's find the reflection of point B(3,2) over the angle bisector x + y - 2 = 0.To find the reflection of a point over a line, I can use the formula for reflection over a line ax + by + c = 0. The formula is:x' = x - 2a(ax + by + c)/(a² + b²) y' = y - 2b(ax + by + c)/(a² + b²)But first, let me write the angle bisector equation in standard form: x + y - 2 = 0, so a = 1, b = 1, c = -2.Now, plug in point B(3,2):Compute ax + by + c: 1*3 + 1*2 - 2 = 3 + 2 - 2 = 3Now, compute the reflection:x' = 3 - 2*1*(3)/(1² + 1²) = 3 - 6/2 = 3 - 3 = 0 y' = 2 - 2*1*(3)/(1² + 1²) = 2 - 6/2 = 2 - 3 = -1So, the reflection of point B over the angle bisector is point B'(0, -1).Since B' lies on line AC, and we know point A(6, -4), we can find the equation of line AC by finding the line passing through A(6, -4) and B'(0, -1).First, find the slope (m) between A(6, -4) and B'(0, -1):m = (y2 - y1)/(x2 - x1) = (-1 - (-4))/(0 - 6) = (3)/(-6) = -1/2Now, using point-slope form with point A(6, -4):y - (-4) = -1/2(x - 6) y + 4 = -1/2x + 3 Multiply both sides by 2 to eliminate the fraction: 2y + 8 = -x + 6 Bring all terms to one side: x + 2y + 2 = 0So, the equation of line AC is x + 2y + 2 = 0.Let me double-check my calculations to make sure I didn't make any mistakes. Starting from the reflection of point B over the angle bisector:Point B(3,2). The angle bisector is x + y - 2 = 0.Using the reflection formula:x' = x - 2a(ax + by + c)/(a² + b²) = 3 - 2*1*(3 + 2 - 2)/(1 + 1) = 3 - 2*(3)/2 = 3 - 3 = 0y' = y - 2b(ax + by + c)/(a² + b²) = 2 - 2*1*(3 + 2 - 2)/(1 + 1) = 2 - 2*(3)/2 = 2 - 3 = -1That seems correct. So, point B' is indeed (0, -1).Then, finding the slope between A(6, -4) and B'(0, -1):m = (-1 - (-4))/(0 - 6) = (3)/(-6) = -1/2That's correct.Using point-slope form:y - (-4) = -1/2(x - 6) y + 4 = -1/2x + 3 Multiply by 2: 2y + 8 = -x + 6 Rearranged: x + 2y + 2 = 0Yes, that looks right.So, I think I've done everything correctly. The equation of line AB is 2x + y - 8 = 0, and the equation of line AC is x + 2y + 2 = 0.