💡Okay, so I have this problem where I need to find the range of m that satisfies the inequality f(2m - 1) + f(3 - m) > 0, where f(x) = 3x + sin x. The options are A: m > -2, B: m > -4, C: m < -2, D: m < -4. Hmm, let me try to figure this out step by step.First, I remember that f(x) is a function given as 3x + sin x. I need to analyze its properties because that might help me solve the inequality. Let me see, is this function odd or even? Well, an odd function satisfies f(-x) = -f(x). Let me check:f(-x) = 3(-x) + sin(-x) = -3x - sin x. On the other hand, -f(x) = -(3x + sin x) = -3x - sin x. So, yes, f(-x) = -f(x), which means f(x) is indeed an odd function. That's good to know.Next, I think about whether the function is increasing or decreasing. To determine that, I can look at its derivative. The derivative of f(x) with respect to x is f'(x) = 3 + cos x. Since cos x ranges between -1 and 1, the derivative f'(x) will range between 3 - 1 = 2 and 3 + 1 = 4. So, f'(x) is always positive because the smallest it can be is 2, which is still greater than zero. Therefore, f(x) is an increasing function on the entire real line. That's an important property because it tells me that if a > b, then f(a) > f(b).Now, back to the inequality: f(2m - 1) + f(3 - m) > 0. Since f is an odd function, I know that f(-x) = -f(x). Maybe I can use this property to simplify the inequality. Let me try to rewrite the inequality using this.I can write f(3 - m) as f(-(m - 3)) because 3 - m = -(m - 3). So, f(3 - m) = f(-(m - 3)) = -f(m - 3) because f is odd. Therefore, the inequality becomes:f(2m - 1) + (-f(m - 3)) > 0Which simplifies to:f(2m - 1) - f(m - 3) > 0Or:f(2m - 1) > f(m - 3)Since f is an increasing function, if f(a) > f(b), then a > b. So, I can translate this inequality into:2m - 1 > m - 3Now, I need to solve this inequality for m. Let's subtract m from both sides:2m - 1 - m > -3Which simplifies to:m - 1 > -3Then, add 1 to both sides:m > -2So, the solution is m > -2. Looking back at the options, that's option A. Hmm, let me just double-check my steps to make sure I didn't make any mistakes.1. I recognized that f(x) is odd because f(-x) = -f(x). That seems correct.2. I found the derivative f'(x) = 3 + cos x, which is always positive, so f is increasing. That makes sense because 3 + cos x is always at least 2, so it's always increasing.3. I rewrote f(3 - m) as -f(m - 3) using the odd function property. That seems right.4. I then set up the inequality f(2m - 1) > f(m - 3) and, since f is increasing, concluded that 2m - 1 > m - 3. That logic holds.5. Solving 2m - 1 > m - 3 gives m > -2, which is option A.I don't see any errors in my reasoning. So, I think the correct answer is A: m > -2.**Final Answer**boxed{A}

💡First, I need to find the lateral surface area of the cylinder. The formula for the lateral surface area of a cylinder is (2pi r h), where (r) is the radius and (h) is the height. Given that the height (h) is (4r), I can substitute this into the formula.Next, I'll calculate the surface area of the sphere. The formula for the surface area of a sphere is (4pi R^2), where (R) is the radius of the sphere.According to the problem, these two areas are equal. So, I'll set the lateral surface area of the cylinder equal to the surface area of the sphere:[2pi r times 4r = 4pi R^2]Simplifying the left side:[8pi r^2 = 4pi R^2]To solve for (R/r), I'll divide both sides of the equation by (4pi):[2r^2 = R^2]Taking the square root of both sides gives:[R = rsqrt{2}]Finally, dividing both sides by (r) yields the ratio:[frac{R}{r} = sqrt{2}]

💡Okay, so I have this quadratic equation: (2x^2 - (sqrt{3} + 1)x + m = 0). The roots of this equation are given as (sin theta) and (cos theta), where (theta) is between 0 and (2pi). I need to find three things: 1. The value of the expression (frac{sin theta}{1 - cot theta} + frac{cos theta}{1 - tan theta}).2. The value of (m).3. The two roots of the equation and the value of (theta).Alright, let me start by recalling some properties of quadratic equations. For a quadratic equation (ax^2 + bx + c = 0), the sum of the roots is (-b/a) and the product is (c/a). So, in this case, the sum of the roots (sin theta + cos theta) should be equal to ((sqrt{3} + 1)/2), and the product (sin theta cdot cos theta) should be equal to (m/2).Let me write that down:1. (sin theta + cos theta = frac{sqrt{3} + 1}{2})2. (sin theta cdot cos theta = frac{m}{2})Okay, so that's the starting point. Now, moving on to part (1), which is evaluating the expression (frac{sin theta}{1 - cot theta} + frac{cos theta}{1 - tan theta}).Hmm, that looks a bit complicated. Maybe I can simplify it step by step. Let me first recall that (cot theta = frac{cos theta}{sin theta}) and (tan theta = frac{sin theta}{cos theta}). So, substituting these into the expression:[frac{sin theta}{1 - frac{cos theta}{sin theta}} + frac{cos theta}{1 - frac{sin theta}{cos theta}}]Let me simplify each term separately. Starting with the first term:[frac{sin theta}{1 - frac{cos theta}{sin theta}} = frac{sin theta}{frac{sin theta - cos theta}{sin theta}} = frac{sin^2 theta}{sin theta - cos theta}]Similarly, the second term:[frac{cos theta}{1 - frac{sin theta}{cos theta}} = frac{cos theta}{frac{cos theta - sin theta}{cos theta}} = frac{cos^2 theta}{cos theta - sin theta}]So, now the expression becomes:[frac{sin^2 theta}{sin theta - cos theta} + frac{cos^2 theta}{cos theta - sin theta}]Wait a second, the denominators are negatives of each other. So, (cos theta - sin theta = -(sin theta - cos theta)). Let me rewrite the second term:[frac{cos^2 theta}{cos theta - sin theta} = -frac{cos^2 theta}{sin theta - cos theta}]So, now the entire expression is:[frac{sin^2 theta}{sin theta - cos theta} - frac{cos^2 theta}{sin theta - cos theta}]Since they have the same denominator, I can combine them:[frac{sin^2 theta - cos^2 theta}{sin theta - cos theta}]Hmm, the numerator is a difference of squares. I remember that (a^2 - b^2 = (a - b)(a + b)), so applying that here:[frac{(sin theta - cos theta)(sin theta + cos theta)}{sin theta - cos theta}]The (sin theta - cos theta) terms cancel out, leaving:[sin theta + cos theta]Oh, that's nice! So, the entire expression simplifies to (sin theta + cos theta). From equation (1), we already know that (sin theta + cos theta = frac{sqrt{3} + 1}{2}). Therefore, the value of the expression is (frac{sqrt{3} + 1}{2}).Alright, that takes care of part (1). Now, moving on to part (2), finding the value of (m). From equation (2), we have:[sin theta cdot cos theta = frac{m}{2}]So, if I can find (sin theta cdot cos theta), I can solve for (m). To find this product, I can use the identity that relates ((sin theta + cos theta)^2) to (sin^2 theta + cos^2 theta + 2 sin theta cos theta). Since (sin^2 theta + cos^2 theta = 1), this gives:[(sin theta + cos theta)^2 = 1 + 2 sin theta cos theta]We already know (sin theta + cos theta = frac{sqrt{3} + 1}{2}), so let's square that:[left(frac{sqrt{3} + 1}{2}right)^2 = frac{(sqrt{3})^2 + 2 cdot sqrt{3} cdot 1 + 1^2}{4} = frac{3 + 2sqrt{3} + 1}{4} = frac{4 + 2sqrt{3}}{4} = frac{2 + sqrt{3}}{2}]So, we have:[frac{2 + sqrt{3}}{2} = 1 + 2 sin theta cos theta]Subtracting 1 from both sides:[frac{2 + sqrt{3}}{2} - 1 = 2 sin theta cos theta]Simplifying the left side:[frac{2 + sqrt{3} - 2}{2} = frac{sqrt{3}}{2} = 2 sin theta cos theta]Therefore:[sin theta cos theta = frac{sqrt{3}}{4}]Plugging this back into equation (2):[frac{sqrt{3}}{4} = frac{m}{2}]Solving for (m):[m = frac{sqrt{3}}{4} times 2 = frac{sqrt{3}}{2}]So, (m = frac{sqrt{3}}{2}).Now, onto part (3), finding the two roots of the equation and the value of (theta). The quadratic equation is:[2x^2 - (sqrt{3} + 1)x + frac{sqrt{3}}{2} = 0]To find the roots, I can use the quadratic formula:[x = frac{(sqrt{3} + 1) pm sqrt{(sqrt{3} + 1)^2 - 4 cdot 2 cdot frac{sqrt{3}}{2}}}{2 cdot 2}]Let me compute the discriminant first:[D = (sqrt{3} + 1)^2 - 4 cdot 2 cdot frac{sqrt{3}}{2}]Calculating each part:First, ((sqrt{3} + 1)^2):[(sqrt{3})^2 + 2 cdot sqrt{3} cdot 1 + 1^2 = 3 + 2sqrt{3} + 1 = 4 + 2sqrt{3}]Next, (4 cdot 2 cdot frac{sqrt{3}}{2}):[4 cdot 2 cdot frac{sqrt{3}}{2} = 4 cdot sqrt{3} = 4sqrt{3}]So, the discriminant (D) is:[4 + 2sqrt{3} - 4sqrt{3} = 4 - 2sqrt{3}]Hmm, that's interesting. So, the discriminant is (4 - 2sqrt{3}). Let me see if that can be simplified or expressed as a square.Let me compute (4 - 2sqrt{3}). Maybe it's equal to ((sqrt{3} - 1)^2):[(sqrt{3} - 1)^2 = (sqrt{3})^2 - 2 cdot sqrt{3} cdot 1 + 1^2 = 3 - 2sqrt{3} + 1 = 4 - 2sqrt{3}]Yes! So, (D = (sqrt{3} - 1)^2). Therefore, the square root of the discriminant is (sqrt{3} - 1).Now, plugging back into the quadratic formula:[x = frac{(sqrt{3} + 1) pm (sqrt{3} - 1)}{4}]Let's compute both roots.First, the root with the plus sign:[x_1 = frac{(sqrt{3} + 1) + (sqrt{3} - 1)}{4} = frac{2sqrt{3}}{4} = frac{sqrt{3}}{2}]Second, the root with the minus sign:[x_2 = frac{(sqrt{3} + 1) - (sqrt{3} - 1)}{4} = frac{2}{4} = frac{1}{2}]So, the two roots are (frac{sqrt{3}}{2}) and (frac{1}{2}).Now, since the roots are (sin theta) and (cos theta), we have two possibilities:1. (sin theta = frac{sqrt{3}}{2}) and (cos theta = frac{1}{2})2. (sin theta = frac{1}{2}) and (cos theta = frac{sqrt{3}}{2})Let's analyze both cases.Case 1: (sin theta = frac{sqrt{3}}{2}) and (cos theta = frac{1}{2})We know that (sin theta = frac{sqrt{3}}{2}) occurs at (theta = frac{pi}{3}) and (theta = frac{2pi}{3}) in the interval ([0, 2pi)). Similarly, (cos theta = frac{1}{2}) occurs at (theta = frac{pi}{3}) and (theta = frac{5pi}{3}).The common solution is (theta = frac{pi}{3}).Case 2: (sin theta = frac{1}{2}) and (cos theta = frac{sqrt{3}}{2})Here, (sin theta = frac{1}{2}) occurs at (theta = frac{pi}{6}) and (theta = frac{5pi}{6}). (cos theta = frac{sqrt{3}}{2}) occurs at (theta = frac{pi}{6}) and (theta = frac{11pi}{6}).The common solution is (theta = frac{pi}{6}).Therefore, the possible values of (theta) are (frac{pi}{3}) and (frac{pi}{6}).Wait, let me double-check that. If (theta = frac{pi}{3}), then (sin theta = frac{sqrt{3}}{2}) and (cos theta = frac{1}{2}), which matches case 1. Similarly, if (theta = frac{pi}{6}), then (sin theta = frac{1}{2}) and (cos theta = frac{sqrt{3}}{2}), which matches case 2. So, both are valid solutions.Therefore, the two roots are (frac{sqrt{3}}{2}) and (frac{1}{2}), and the corresponding angles are (theta = frac{pi}{3}) and (theta = frac{pi}{6}).Let me just recap:1. The expression simplifies to (sin theta + cos theta), which is (frac{sqrt{3} + 1}{2}).2. (m) is found to be (frac{sqrt{3}}{2}).3. The roots are (frac{sqrt{3}}{2}) and (frac{1}{2}), corresponding to (theta = frac{pi}{3}) and (theta = frac{pi}{6}).I think that covers all parts of the problem. I should make sure I didn't make any calculation errors, especially when computing the discriminant and simplifying the expression. Let me quickly verify the discriminant calculation:Discriminant (D = (sqrt{3} + 1)^2 - 4 cdot 2 cdot frac{sqrt{3}}{2})Calculates to:(4 + 2sqrt{3} - 4sqrt{3} = 4 - 2sqrt{3}), which is correct, and that's equal to ((sqrt{3} - 1)^2), so the square root is indeed (sqrt{3} - 1). So, the roots calculation is correct.Also, the expression simplification steps seem correct, leading to (sin theta + cos theta), which is given. So, all steps check out.**Final Answer**(1) (boxed{dfrac{sqrt{3} + 1}{2}})(2) (boxed{dfrac{sqrt{3}}{2}})(3) The two roots are (boxed{dfrac{sqrt{3}}{2}}) and (boxed{dfrac{1}{2}}), and the values of (theta) are (boxed{dfrac{pi}{3}}) and (boxed{dfrac{pi}{6}}).

💡First, I need to determine the total number of students who received an A in at least one of the subjects. I start by adding the number of students who received an A in literature to those who received an A in science:10 (Literature A's) + 18 (Science A's) = 28.However, this sum counts the students who received A's in both subjects twice. To correct this, I subtract the number of students who received A's in both subjects:28 - 6 = 22.This means that 22 students received an A in at least one subject. Finally, I subtract this number from the total number of students to find out how many did not receive an A in either subject:40 (Total students) - 22 (Students with at least one A) = 18.Therefore, 18 students did not receive an A in either subject.

💡First, I need to determine the number of hens bought. I know that there are 3 pigs and some hens, and the total cost is Rs. 1200.The average price of a hen is Rs. 30, so the total cost for the hens is 30 multiplied by the number of hens, which I'll denote as H.The average price of a pig is Rs. 300, and since there are 3 pigs, the total cost for the pigs is 300 multiplied by 3, which equals Rs. 900.Now, I can set up the equation for the total cost:30H + 900 = 1200To find the number of hens, I'll subtract 900 from both sides of the equation:30H = 300Finally, I'll divide both sides by 30 to solve for H:H = 10So, 10 hens were bought.

💡First, I need to determine the sum of the interior angles of a regular octagon. I recall that the formula for the sum of interior angles of any polygon is ( S = 180(n-2) ) degrees, where ( n ) is the number of sides.Since an octagon has 8 sides, I substitute ( n = 8 ) into the formula:[S = 180(8-2) = 180 times 6 = 1080 text{ degrees}]Next, to find the measure of each individual interior angle in a regular octagon, I divide the total sum of the interior angles by the number of sides:[text{Measure of each interior angle} = frac{1080}{8} = 135 text{ degrees}]Therefore, each interior angle of a regular octagon measures (boxed{135^circ}).