💡First, I need to determine B's contribution to the capital based on the given profit-sharing ratio and the interest rates on both A's and B's investments.A starts the business with Rs. 3500 and invests for the entire year. B joins after 5 months, so B's investment period is 7 months. The profit is divided in the ratio 2:3, meaning A's share is 2 parts and B's share is 3 parts.Next, I'll calculate the effective capital for both A and B, including the interest accrued on their investments. For A, the interest is 10% per year on Rs. 3500, which amounts to Rs. 350. Therefore, A's effective capital is Rs. 3850.For B, the interest is 8% per year, but since B's investment period is only 7 months, the interest is calculated for 7/12 of a year. Let B's contribution be ( B_c ). The interest on B's contribution is ( 0.08 times frac{7}{12} times B_c ), which simplifies to approximately ( 0.0467 times B_c ). Therefore, B's effective capital is ( B_c times 1.0467 ).Given the profit-sharing ratio of 2:3, the ratio of A's effective capital to B's effective capital should also be 2:3. Setting up the equation:[frac{3850}{B_c times 1.0467} = frac{2}{3}]Cross-multiplying and solving for ( B_c ):[3850 times 3 = 2 times B_c times 1.0467][11550 = 2.0934 times B_c][B_c = frac{11550}{2.0934} approx 5517.63]Therefore, B's contribution to the capital is approximately Rs. 5517.63.

💡First, I need to evaluate each term in the expression separately.For the term (-1^{2008}), since exponentiation takes precedence over the negation, it simplifies to (-(1^{2008})). As (1^{2008}) is 1, this term becomes (-1).Next, for the term ((-1)^{2009}), since the exponent 2009 is odd, ((-1)^{2009}) equals (-1).The term (1^{2010}) is straightforward because any positive power of 1 is 1.Lastly, for the term (-1^{2011}), similar to the first term, it simplifies to (-(1^{2011})), which is (-1).Now, I can sum up all the evaluated terms:[-1 + (-1) + 1 - 1 = -1 - 1 + 1 - 1 = -2]Therefore, the simplified value of the expression is (boxed{-2}).

💡Alright, so I have this problem about counting the number of ordered triples (A, B, C) where each set is a subset of {1,2,3,4,5,6,7,8}. The condition is that they are minimally intersecting, which means that the intersection of any two sets has exactly one element, and there's no element common to all three sets. I need to find the number of such triples and then find the remainder when that number is divided by 1000.Okay, let's break this down. First, the definition: minimally intersecting means |A ∩ B| = |B ∩ C| = |C ∩ A| = 1, and A ∩ B ∩ C = ∅. So, each pair of sets shares exactly one unique element, and no element is shared by all three sets.So, let's think about how to construct such triples. Each pair A and B must share exactly one element, say x. Similarly, B and C share exactly one element, say y, and C and A share exactly one element, say z. Since A ∩ B ∩ C is empty, x, y, and z must all be distinct. So, x, y, z are three distinct elements from the universal set {1,2,3,4,5,6,7,8}.Therefore, the first step is to choose these three distinct elements x, y, z. How many ways can we choose them? Well, from 8 elements, we can choose 3 distinct elements in 8 × 7 × 6 ways because for x, we have 8 choices, for y, 7 remaining choices, and for z, 6 remaining choices. So, that's 8 × 7 × 6 = 336 ways.Now, after choosing x, y, z, we need to assign the remaining elements of the universal set to the sets A, B, and C. The universal set has 8 elements, so after choosing x, y, z, we have 5 elements left. Let's denote these remaining elements as S = {a1, a2, a3, a4, a5}.Each of these remaining elements can be assigned to any of the sets A, B, C, or to none of them. Since the sets are subsets, an element can be in A, B, C, or none. So, for each element, there are 4 choices. Therefore, for 5 elements, the number of ways is 4^5.Calculating 4^5: 4 × 4 × 4 × 4 × 4 = 1024.So, for each choice of x, y, z, we have 1024 ways to assign the remaining elements. Therefore, the total number of ordered triples is 336 × 1024.Let me compute that: 336 × 1024. Hmm, 336 × 1000 is 336,000, and 336 × 24 is 8,064. So, adding them together, 336,000 + 8,064 = 344,064.So, the total number of minimally intersecting ordered triples is 344,064.Now, the problem asks for the remainder when this number is divided by 1000. So, I need to compute 344,064 mod 1000.To find the remainder when divided by 1000, we can look at the last three digits of the number. 344,064. The last three digits are 064, which is 64. So, 344,064 mod 1000 is 64.Therefore, the remainder is 64.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, choosing x, y, z: 8 × 7 × 6 = 336. That seems correct.Then, the remaining 5 elements each have 4 choices: 4^5 = 1024. Correct.Multiplying 336 × 1024: Let me compute it step by step.336 × 1000 = 336,000.336 × 24: Let's compute 336 × 20 = 6,720 and 336 × 4 = 1,344. Adding them together: 6,720 + 1,344 = 8,064.Adding 336,000 + 8,064 = 344,064. Correct.Then, 344,064 divided by 1000. The remainder is the last three digits: 064, which is 64. So, 64 is the remainder.Yes, that seems correct.So, the final answer is 64.**Final Answer**boxed{64}

💡Okay, so I have this problem here: If (m - some number) is a factor of m² - 5m - 24, then what is that number? Hmm, I need to find the number that, when subtracted from m, makes (m - number) a factor of the quadratic expression m² - 5m - 24.Alright, let's start by recalling what it means for something to be a factor of a quadratic expression. If (m - a) is a factor of m² - 5m - 24, then when we plug in m = a into the quadratic, it should equal zero. That's the Factor Theorem, right? So, if I substitute m = a into m² - 5m - 24, the result should be zero.So, let's write that out: a² - 5a - 24 = 0. Now, I need to solve this equation for a. This is a quadratic equation in terms of a, so I can try to factor it or use the quadratic formula. Let me see if it factors nicely.Looking for two numbers that multiply to -24 and add up to -5. Hmm, let's list the pairs of factors of -24:- 1 and -24 (sum: -23)- 2 and -12 (sum: -10)- 3 and -8 (sum: -5)- 4 and -6 (sum: -2)Oh, there we go! 3 and -8 multiply to -24 and add up to -5. So, the quadratic factors as (a - 8)(a + 3) = 0. Therefore, the solutions are a = 8 and a = -3.But wait, the problem is asking for the number such that (m - number) is a factor. So, if a = 8, then (m - 8) is a factor. Similarly, if a = -3, then (m - (-3)) or (m + 3) is a factor.But the question specifies (m - some number), which implies that the number is positive because it's subtracted. So, the number we're looking for is 8.Let me double-check by expanding (m - 8)(m + 3) to make sure it equals m² - 5m - 24.Multiplying (m - 8)(m + 3):- First: m * m = m²- Outer: m * 3 = 3m- Inner: -8 * m = -8m- Last: -8 * 3 = -24Adding those up: m² + 3m - 8m - 24 = m² - 5m - 24. Yep, that matches the original quadratic. So, (m - 8) is indeed a factor, and the number is 8.I think that's it. The number is 8.

💡Alright, so I've got this logic problem here, and I need to figure out which option is correct. Let's see... The problem says:"If the proposition 'p or q' is true, and the proposition 'p and q' is false, then ( )"And then there are four options: A, B, C, and D. I need to determine which one is correct based on the given conditions.First, let's break down what each of these propositions means. 'p or q' being true means that at least one of p or q is true. So, either p is true, or q is true, or both are true. On the other hand, 'p and q' being false means that it's not the case that both p and q are true at the same time. So, if 'p and q' is false, then at least one of p or q must be false.Now, let's try to figure out the possible truth values of p and q based on these two conditions.1. **Case 1: p is true.** - If p is true, then for 'p or q' to be true, q can be either true or false. - But since 'p and q' is false, if p is true, then q must be false. Because if p is true and q were also true, then 'p and q' would be true, which contradicts the given condition.2. **Case 2: p is false.** - If p is false, then for 'p or q' to be true, q must be true. - In this case, 'p and q' would be false because p is false, which aligns with the given condition.So, from these two cases, we can deduce the possible truth values:- If p is true, then q is false.- If p is false, then q is true.Now, let's look at the options provided:**A: Both propositions p and q are false.**- From our analysis, if p is false, q must be true. So, both cannot be false. Therefore, A is incorrect.**B: Both propositions p and q are true.**- If both were true, then 'p and q' would be true, which contradicts the given condition. So, B is incorrect.**C: The truth values of proposition p and the proposition "not q" are different.**- Let's see. If p is true, then q is false, so "not q" is true. Therefore, p and "not q" are both true, so their truth values are the same.- If p is false, then q is true, so "not q" is false. Therefore, p and "not q" are both false, so their truth values are the same.- In both cases, p and "not q" have the same truth values. So, C is incorrect because it states they are different.**D: The truth values of proposition p and the proposition "not q" are the same.**- As we saw in the analysis for option C, in both cases, p and "not q" have the same truth values. Therefore, D is correct.So, after carefully analyzing each case and evaluating the options, the correct answer is D.

💡First, I need to determine the remainder when 433124 is divided by 17. This will help me understand how much more is needed to reach the next multiple of 17.I'll start by performing the division of 433124 by 17. Through calculation, I find that 17 multiplied by 25478 equals 433126. Subtracting this from 433124 gives a remainder of -2. However, since remainders should be positive, I'll add 17 to -2, resulting in a remainder of 15.Now, to make 433124 divisible by 17, I need to add the difference between 17 and the remainder. This means adding 2 to 433124.Finally, adding 2 to 433124 gives 433126, which is exactly divisible by 17 (17 × 25478 = 433126). Therefore, the smallest number that must be added is 2.