💡First, I need to understand the problem. There are two varieties of rice: one costs Rs. 5 per kg and the other costs Rs. 8.75 per kg. They are mixed in a ratio of 0.5, which means for every 1 kg of the more expensive rice, there is 0.5 kg of the cheaper rice.To find the cost of the mixture per kg, I'll use the concept of weighted average. I'll start by assigning variables to the costs and quantities of each type of rice.Let’s denote:- ( C_1 = 5 ) Rs/kg (cost of cheaper rice)- ( C_2 = 8.75 ) Rs/kg (cost of more expensive rice)- ( R = 0.5 ) (ratio of quantities, Quantity of ( C_1 ) / Quantity of ( C_2 ))Next, I'll express the quantities in terms of a common variable. If I assume the quantity of the more expensive rice (( C_2 )) is 1 kg, then the quantity of the cheaper rice (( C_1 )) would be ( 0.5 ) kg based on the given ratio.Now, I'll calculate the total cost of the mixture by multiplying the cost per kg by the quantity for each type of rice and then summing them up.[text{Total Cost} = (C_1 times text{Quantity of } C_1) + (C_2 times text{Quantity of } C_2)][text{Total Cost} = (5 times 0.5) + (8.75 times 1) = 2.5 + 8.75 = 11.25 text{ Rs}]Then, I'll find the total weight of the mixture by adding the quantities of both types of rice.[text{Total Weight} = text{Quantity of } C_1 + text{Quantity of } C_2 = 0.5 + 1 = 1.5 text{ kg}]Finally, to determine the cost of the mixture per kg, I'll divide the total cost by the total weight.[text{Cost per kg} = frac{text{Total Cost}}{text{Total Weight}} = frac{11.25}{1.5} = 7.5 text{ Rs}]Therefore, the cost of the mixture per kg is Rs. 7.5.

💡Okay, so I have this problem about an arithmetic sequence. Let me try to figure it out step by step. The problem says that {a_n} is an arithmetic sequence, and we know two specific terms: a_2 is 0, and a_4 is -2. We need to find the maximum value of the sum of the first n terms, which is denoted as S_n or f(n). First, I remember that in an arithmetic sequence, each term is obtained by adding a common difference, d, to the previous term. So, the general formula for the nth term of an arithmetic sequence is a_n = a_1 + (n-1)d, where a_1 is the first term.Given that a_2 = 0, I can plug that into the formula. So, a_2 = a_1 + (2-1)d = a_1 + d = 0. That gives me one equation: a_1 + d = 0. Similarly, a_4 is given as -2. Using the same formula, a_4 = a_1 + (4-1)d = a_1 + 3d = -2. So now I have another equation: a_1 + 3d = -2.Now I have a system of two equations:1. a_1 + d = 02. a_1 + 3d = -2I can solve this system to find a_1 and d. Let me subtract the first equation from the second to eliminate a_1. So, (a_1 + 3d) - (a_1 + d) = -2 - 0Simplifying that: a_1 + 3d - a_1 - d = -2Which becomes: 2d = -2Therefore, d = -1.Now that I know d is -1, I can substitute back into the first equation to find a_1. From equation 1: a_1 + d = 0So, a_1 + (-1) = 0Which means a_1 = 1.Alright, so the first term is 1, and the common difference is -1. That means each subsequent term decreases by 1. Let me write out the first few terms to visualize the sequence:a_1 = 1a_2 = 0a_3 = -1a_4 = -2a_5 = -3... and so on.So, the sequence starts at 1, then goes to 0, then negative numbers. Now, the problem asks for the maximum value of S_n, which is the sum of the first n terms. I remember that the sum of the first n terms of an arithmetic sequence is given by S_n = n/2 * (2a_1 + (n-1)d) or also S_n = n*(a_1 + a_n)/2.Let me use the second formula because I already know a_n in terms of n. Since a_n = a_1 + (n-1)d, and we know a_1 = 1 and d = -1, so a_n = 1 + (n-1)*(-1) = 1 - (n - 1) = 2 - n.So, a_n = 2 - n. Therefore, the sum S_n can be written as:S_n = n*(a_1 + a_n)/2 = n*(1 + (2 - n))/2 = n*(3 - n)/2.So, S_n = (3n - n^2)/2. Alternatively, I can write this as S_n = (-n^2 + 3n)/2. This is a quadratic function in terms of n, and since the coefficient of n^2 is negative (-1/2), the parabola opens downward, meaning the vertex is the maximum point.To find the maximum value of S_n, I can find the vertex of this parabola. The vertex occurs at n = -b/(2a) for a quadratic function in the form an^2 + bn + c. In this case, a = -1/2 and b = 3/2.Wait, hold on, let me make sure. The quadratic is S_n = (-n^2 + 3n)/2, so in standard form, it's (-1/2)n^2 + (3/2)n + 0. So, a = -1/2, b = 3/2.Therefore, the vertex is at n = -b/(2a) = -(3/2)/(2*(-1/2)) = -(3/2)/(-1) = (3/2)/1 = 3/2.Hmm, so n = 1.5. But n has to be a positive integer because it's the number of terms. So, the maximum occurs around n = 1.5, but since n must be an integer, we check n = 1 and n = 2.Let me compute S_1 and S_2.S_1 = (3*1 - 1^2)/2 = (3 - 1)/2 = 2/2 = 1.S_2 = (3*2 - 2^2)/2 = (6 - 4)/2 = 2/2 = 1.So, both S_1 and S_2 are equal to 1. Wait, but let me check S_3 just to be thorough.S_3 = (3*3 - 3^2)/2 = (9 - 9)/2 = 0/2 = 0.And S_4 = (3*4 - 4^2)/2 = (12 - 16)/2 = (-4)/2 = -2.So, indeed, after n=2, the sum starts decreasing. Alternatively, I can think about the terms of the sequence. Since the terms are decreasing by 1 each time, starting from 1, then 0, then negative numbers. So, adding the first term gives 1, adding the second term gives 1 + 0 = 1, adding the third term gives 1 + 0 + (-1) = 0, and so on. So, the maximum sum occurs at n=1 or n=2, both giving a sum of 1.Therefore, the maximum value of f(n) is 1.**Final Answer**The maximum value of ( f(n) ) is boxed{1}.

💡First, I need to simplify the given expression ( k = sqrt{2}(sqrt{5} + sqrt{3})(sqrt{5} - sqrt{3}) ).I notice that the terms ( (sqrt{5} + sqrt{3}) ) and ( (sqrt{5} - sqrt{3}) ) form a difference of squares. Using the formula ( (a + b)(a - b) = a^2 - b^2 ), I can simplify this part of the expression.Calculating the difference of squares:[(sqrt{5} + sqrt{3})(sqrt{5} - sqrt{3}) = (sqrt{5})^2 - (sqrt{3})^2 = 5 - 3 = 2]Now, the expression for ( k ) becomes:[k = sqrt{2} times 2 = 2sqrt{2}]Next, I need to approximate the value of ( 2sqrt{2} ) to determine the closest integer. Knowing that ( sqrt{2} ) is approximately 1.414, multiplying this by 2 gives:[2sqrt{2} approx 2 times 1.414 = 2.828]The value 2.828 is closer to 3 than to 2. Therefore, the integer closest to ( k ) is 3.

💡Alright, so I've got this geometry problem here with two squares, ABCD and DEFG, both having equal side lengths. There's an angle given, angle DCE, which is 70 degrees, and I need to find the value of y. The options are 120, 160, 130, 110, and 140. Hmm, okay, let me try to visualize this.First, since ABCD and DEFG are both squares with equal side lengths, that means all their sides are equal. So, AB = BC = CD = DA and DE = EF = FG = GD. Also, all the angles in a square are right angles, so each angle is 90 degrees.Now, the angle given is angle DCE, which is 70 degrees. Let me try to figure out where point E is located. Since DEFG is a square, and D is a common vertex between both squares, E must be connected to D. So, point E is a corner of the square DEFG, and it's connected to point C, which is a corner of square ABCD.So, triangle DCE is formed by points D, C, and E. Since both DC and DE are sides of the squares, they must be equal in length. That makes triangle DCE an isosceles triangle with DC = DE. In an isosceles triangle, the angles opposite the equal sides are equal. So, angle DCE is 70 degrees, which is at point C. That means the angle at point E, which is angle DEC, should also be 70 degrees because they are opposite the equal sides DC and DE.Now, the sum of the angles in any triangle is 180 degrees. So, if we know two angles in triangle DCE, we can find the third one. Let me write that down:Angle DCE = 70 degreesAngle DEC = 70 degreesSo, angle CDE = 180 - 70 - 70 = 40 degrees.Okay, so angle CDE is 40 degrees. Now, I need to find the value of y. I'm not exactly sure where y is located in the diagram, but since it's a common problem setup, I can make an educated guess. It's likely that y is an angle formed around point D, perhaps between the sides of the two squares.Since both ABCD and DEFG are squares, the sides DC and DE are both equal and at right angles to their adjacent sides. So, if I consider point D, it's connected to points C and E, which are both at 90-degree angles from their respective squares.Wait, but angle CDE is 40 degrees, which is the angle between DC and DE. So, if I imagine standing at point D, looking towards C, and then turning towards E, that's a 40-degree turn. Now, around point D, the total angles should add up to 360 degrees because it's a full circle.So, if I consider all the angles around point D, they should sum up to 360 degrees. Let's break it down:1. The angle inside square ABCD at point D, which is angle ADC. Since it's a square, this is 90 degrees.2. The angle inside square DEFG at point D, which is angle GDE. Again, since it's a square, this is also 90 degrees.3. The angle CDE, which we found to be 40 degrees.4. The remaining angle, which is y.So, adding these up: 90 (ADC) + 90 (GDE) + 40 (CDE) + y = 360 degrees.Let me write that equation:90 + 90 + 40 + y = 360Adding those numbers together: 90 + 90 is 180, plus 40 is 220. So,220 + y = 360Subtracting 220 from both sides:y = 360 - 220y = 140So, y is 140 degrees. Let me just double-check to make sure I didn't miss anything. The key steps were recognizing that triangle DCE is isosceles with two equal sides, calculating the third angle as 40 degrees, and then using the fact that the sum of angles around a point is 360 degrees to find y. It all seems to add up correctly.I think I'm confident with this answer. It's one of the options provided, specifically option E, which is 140. So, I'll go with that.**Final Answer**boxed{E}

💡Okay, so I have this problem here with two propositions, p and q, and I need to figure out which of the judgments A, B, C, D is correct. Let me try to break it down step by step.First, let's look at proposition p: For all real numbers x, 2x² + 2x + 1/2 is less than 0. Hmm, that sounds a bit strange because quadratic functions usually have a parabola shape, and depending on the coefficient, they open upwards or downwards. Since the coefficient of x² is 2, which is positive, this parabola opens upwards. That means the function has a minimum point. So, if the minimum value is less than 0, then p would be true. Otherwise, if the minimum is greater than or equal to 0, p is false.To find the minimum value, I can complete the square for the quadratic expression. Let's see:2x² + 2x + 1/2.First, factor out the coefficient of x² from the first two terms:2(x² + x) + 1/2.Now, to complete the square inside the parentheses, take half of the coefficient of x, which is 1/2, square it, which is 1/4, and add and subtract it inside:2(x² + x + 1/4 - 1/4) + 1/2.This simplifies to:2[(x + 1/2)² - 1/4] + 1/2.Distribute the 2:2(x + 1/2)² - 2*(1/4) + 1/2.Which is:2(x + 1/2)² - 1/2 + 1/2.So, the -1/2 and +1/2 cancel out, leaving:2(x + 1/2)².Since (x + 1/2)² is always non-negative, multiplying by 2 keeps it non-negative. Therefore, 2(x + 1/2)² is always greater than or equal to 0. So, the expression 2x² + 2x + 1/2 is always greater than or equal to 0 for all real x. That means proposition p, which claims it's less than 0 for all x, is false.Alright, so p is false. That means the negation of p, which is ¬p, would be true. So, looking at the options, C says ¬p is false. But since ¬p is actually true, C is incorrect. So, I can eliminate option C.Now, let's look at proposition q: There exists a real number x₀ such that sin x₀ - cos x₀ equals √2. Hmm, okay. I remember that expressions like a sin x + b cos x can be rewritten using the amplitude-phase form. Specifically, a sin x + b cos x can be written as R sin(x + φ), where R is the amplitude and φ is the phase shift.Let me try to rewrite sin x₀ - cos x₀. So, that's like 1*sin x₀ + (-1)*cos x₀. So, a = 1 and b = -1.The amplitude R is calculated as sqrt(a² + b²) = sqrt(1 + 1) = sqrt(2). So, R is sqrt(2). Therefore, sin x₀ - cos x₀ can be written as sqrt(2) sin(x₀ + φ), where φ is some phase shift.The maximum value of sin(x₀ + φ) is 1, so the maximum value of sqrt(2) sin(x₀ + φ) is sqrt(2). Therefore, the expression sin x₀ - cos x₀ can indeed reach sqrt(2). So, there exists an x₀ such that sin x₀ - cos x₀ equals sqrt(2). Therefore, proposition q is true.So, q is true, which means the negation of q, ¬q, is false. Looking at the options, D says ¬q is false, which is correct because ¬q is indeed false. So, D is correct.Let me just recap:- Proposition p is false because the quadratic is always non-negative.- Proposition q is true because the expression can reach sqrt(2).- Therefore, ¬p is true (so C is wrong), and ¬q is false (so D is correct).So, the correct judgment is D.**Final Answer**boxed{D}

💡To determine the frequency of the sample data falling within the interval ((0,40]), we need to sum the frequencies of all groups that fall within this interval and then express this sum as a proportion of the total sample size.1. **Identify the relevant groups:** - The interval ((0,40]) includes the following groups: - ((0,10]) - ((10,20]) - ((20,30]) - ((30,40])2. **Sum the frequencies of these groups:** - Frequency of ((0,10]): 12 - Frequency of ((10,20]): 13 - Frequency of ((20,30]): 24 - Frequency of ((30,40]): 15 [ 12 + 13 + 24 + 15 = 64 ]3. **Calculate the proportion:** - Total sample size: 100 - Proportion of data in ((0,40]): [ frac{64}{100} = 0.64 ]