💡First, I need to determine the selling price for each of the three articles based on their respective cost prices and desired profit percentages.For the first article, the cost price is ₹500, and the desired profit is 20%. To calculate the profit amount, I multiply the cost price by the profit percentage: 20% of ₹500 equals ₹100. Adding this profit to the cost price gives the selling price: ₹500 plus ₹100 equals ₹600.Next, for the second article, the cost price is ₹750, and the desired profit is 25%. Calculating the profit: 25% of ₹750 is ₹187.50. Adding this to the cost price results in a selling price of ₹750 plus ₹187.50, which equals ₹937.50.Finally, for the third article, the cost price is ₹1000, and the desired profit is 30%. The profit amount is 30% of ₹1000, which is ₹300. Adding this to the cost price gives a selling price of ₹1000 plus ₹300, totaling ₹1300.By following these calculations, I can determine the appropriate selling prices for each article to achieve the desired profit margins.

💡Okay, so I have this problem about a sequence where each term is defined based on the previous term, and it's supposed to have a period of 2008. Hmm, that sounds a bit complicated, but let me try to break it down.First, the sequence is given by ( a_1 > 0 ) and for ( n geq 2 ), ( a_n = frac{a_1 + a_{n-1}}{1 - a_1 a_{n-1}} ). The smallest positive period is 2008, which means that after 2008 terms, the sequence repeats itself. So, ( a_{2009} = a_1 ), ( a_{2010} = a_2 ), and so on.Looking at the recursion formula, it reminds me of the tangent addition formula. I remember that ( tan(A + B) = frac{tan A + tan B}{1 - tan A tan B} ). So, maybe I can model this sequence using tangent functions. Let me try that.Let's assume that ( a_n = tan theta_n ) for some angle ( theta_n ). Then, substituting into the recursion formula, we get:( tan theta_n = frac{tan theta_1 + tan theta_{n-1}}{1 - tan theta_1 tan theta_{n-1}} )Which simplifies to:( tan theta_n = tan(theta_1 + theta_{n-1}) )So, this suggests that ( theta_n = theta_1 + theta_{n-1} ). That means each angle is just the previous angle plus ( theta_1 ). So, the sequence of angles is an arithmetic sequence:( theta_n = n theta_1 )Therefore, ( a_n = tan(n theta_1) ).Now, since the sequence has a period of 2008, that means ( a_{2008 + 1} = a_1 ). So, ( a_{2009} = a_1 ). Substituting into our tangent formula:( tan(2009 theta_1) = tan(theta_1) )When does ( tan alpha = tan beta )? It happens when ( alpha = beta + kpi ) for some integer ( k ). So, we have:( 2009 theta_1 = theta_1 + kpi )Simplifying this, we get:( 2008 theta_1 = kpi )So,( theta_1 = frac{kpi}{2008} )Now, since ( a_1 = tan(theta_1) ), we have:( a_1 = tanleft(frac{kpi}{2008}right) )But we need to make sure that this is the smallest period. That means that ( k ) should be chosen such that ( theta_1 ) is the smallest angle that satisfies the periodicity condition. So, ( k ) should be such that ( frac{k}{2008} ) is in the lowest terms, meaning ( k ) and 2008 are coprime.Also, since ( a_1 > 0 ), ( tan(theta_1) ) must be positive. The tangent function is positive in the first and third quadrants. However, since ( theta_1 ) is a small positive angle (as ( k ) is between 1 and 2007), we can focus on the first quadrant.Therefore, ( k ) must be an integer such that ( 1 leq k leq 1003 ) (since beyond that, the angles would start repeating in the third quadrant, but we want the smallest positive period).So, putting it all together, ( a_1 = tanleft(frac{kpi}{2008}right) ) where ( k ) is an integer between 1 and 1003 that is coprime with 2008.Wait, but the problem asks for the value of ( a_1 ). It doesn't specify a particular ( k ), just that the period is 2008. So, does that mean there are multiple possible values for ( a_1 )? Or is there a specific one?I think it's multiple, but the problem might be expecting the general form. Alternatively, maybe there's a specific ( k ) that makes it the minimal period. Since 2008 is the minimal period, ( k ) must be such that ( frac{k}{2008} ) doesn't reduce further, so ( k ) and 2008 are coprime.So, the value of ( a_1 ) is ( tanleft(frac{kpi}{2008}right) ) where ( k ) is an integer between 1 and 1003 that is coprime with 2008.But let me verify if this makes sense. If ( k ) and 2008 are coprime, then ( theta_1 = frac{kpi}{2008} ) is such that the period is exactly 2008. If ( k ) and 2008 had a common divisor, say ( d ), then the period would be ( frac{2008}{d} ), which is less than 2008, contradicting the minimal period condition.Therefore, ( k ) must be coprime with 2008. So, the value of ( a_1 ) is ( tanleft(frac{kpi}{2008}right) ) with ( k ) coprime to 2008 and ( 1 leq k leq 1003 ).Wait, but the problem says "determine the value of ( a_1 )", implying a specific value. Maybe I missed something.Looking back, the problem states that the sequence has the smallest positive period of 2008. So, ( a_{2008 + 1} = a_1 ), but ( a_{n} ) for ( n < 2008 ) should not equal ( a_1 ). Therefore, the period is exactly 2008, not a divisor of it.So, ( theta_1 = frac{kpi}{2008} ) must be such that ( k ) is coprime with 2008, as otherwise, the period would be shorter.Therefore, ( a_1 = tanleft(frac{kpi}{2008}right) ) where ( k ) is coprime with 2008 and ( 1 leq k leq 1003 ).But the problem asks for the value of ( a_1 ). It might be expecting a specific value, but since there are multiple possible ( k ), maybe it's expressed in terms of ( tan ) with ( k ) coprime to 2008.Alternatively, perhaps the minimal period implies that ( k = 1 ), but I'm not sure. Let me think.If ( k = 1 ), then ( theta_1 = frac{pi}{2008} ), and the period would be 2008 because ( tan(2009 cdot frac{pi}{2008}) = tan(pi + frac{pi}{2008}) = tan(frac{pi}{2008}) ). So, yes, that works.Similarly, for other ( k ) coprime with 2008, the period would still be 2008 because the angle wouldn't repeat before that.So, perhaps the answer is ( a_1 = tanleft(frac{pi}{2008}right) ), but I'm not entirely sure if it's specifically ( k = 1 ) or any ( k ) coprime with 2008.Wait, the problem says "the smallest positive period of 2008", so it's possible that ( a_1 ) can be any such tangent value with ( k ) coprime to 2008. But maybe the problem expects the principal value, which would be ( k = 1 ).Alternatively, perhaps the answer is expressed in terms of ( tan ) with ( k ) coprime to 2008, but I'm not sure.Wait, let me check the initial assumption. I assumed ( a_n = tan theta_n ), but maybe I should have considered a different trigonometric function or a different substitution.Alternatively, maybe using hyperbolic functions? But the recursion formula is similar to tangent addition, so I think tangent is the right approach.Another thought: since the period is 2008, which is even, and 2008 factors into 8 * 251, so 2008 = 8 * 251. Therefore, the coprime ( k ) would be numbers not divisible by 2 or 251.But I'm not sure if that affects the value of ( a_1 ).Wait, maybe the problem expects a specific value, but since it's not given, perhaps the answer is ( tanleft(frac{pi}{2008}right) ).Alternatively, maybe the answer is ( tanleft(frac{pi}{2008}right) ), but I'm not entirely certain.Wait, let me think again. If ( k ) is coprime with 2008, then ( theta_1 = frac{kpi}{2008} ), and ( a_1 = tan(theta_1) ). So, the value of ( a_1 ) is ( tanleft(frac{kpi}{2008}right) ) where ( k ) is an integer between 1 and 1003 that is coprime with 2008.But the problem says "determine the value of ( a_1 )", so maybe it's expecting a specific value, but since there are multiple possibilities, perhaps it's expressed in terms of ( tan ) with ( k ) coprime to 2008.Alternatively, maybe the answer is ( tanleft(frac{pi}{2008}right) ), but I'm not sure.Wait, another approach: since the period is 2008, the sequence repeats every 2008 terms. So, ( a_{2009} = a_1 ), which implies ( tan(2009 theta_1) = tan(theta_1) ). So, ( 2009 theta_1 = theta_1 + kpi ), leading to ( 2008 theta_1 = kpi ), so ( theta_1 = frac{kpi}{2008} ).To have the minimal period, ( k ) must be such that ( frac{k}{2008} ) is in lowest terms, i.e., ( k ) and 2008 are coprime. Therefore, ( a_1 = tanleft(frac{kpi}{2008}right) ) where ( k ) is coprime with 2008.So, the answer is ( a_1 = tanleft(frac{kpi}{2008}right) ) with ( k ) coprime to 2008 and ( 1 leq k leq 1003 ).But the problem asks for "the value of ( a_1 )", not "all possible values". So, maybe it's expecting the general form, but I'm not sure.Alternatively, perhaps the answer is ( tanleft(frac{pi}{2008}right) ) because that's the smallest positive angle, but I'm not certain.Wait, another thought: since the period is 2008, and the sequence is defined by ( a_n = frac{a_1 + a_{n-1}}{1 - a_1 a_{n-1}} ), which is similar to the tangent addition formula, the sequence is essentially adding ( theta_1 ) each time in the tangent function.Therefore, to have the period 2008, ( 2008 theta_1 ) must be an integer multiple of ( pi ), but not a multiple that would make the period smaller.So, ( 2008 theta_1 = kpi ), so ( theta_1 = frac{kpi}{2008} ), and ( k ) must be such that ( frac{k}{2008} ) is in lowest terms, meaning ( k ) and 2008 are coprime.Therefore, ( a_1 = tanleft(frac{kpi}{2008}right) ) where ( k ) is coprime with 2008.So, the answer is ( a_1 = tanleft(frac{kpi}{2008}right) ) with ( k ) coprime to 2008.But the problem says "determine the value of ( a_1 )", so maybe it's expecting the general form, but I'm not sure.Alternatively, perhaps the answer is ( tanleft(frac{pi}{2008}right) ), but I'm not certain.Wait, maybe I should check if ( k = 1 ) is coprime with 2008. 2008 factors into 2^3 * 251, so 1 is coprime with any number, including 2008. So, ( k = 1 ) is valid.Therefore, ( a_1 = tanleft(frac{pi}{2008}right) ) is a valid solution.But are there other solutions? Yes, any ( k ) coprime with 2008 would work, but the problem might be expecting the principal solution, which is ( k = 1 ).So, I think the answer is ( a_1 = tanleft(frac{pi}{2008}right) ).But to be safe, I'll go with that.

💡Okay, so I have this problem here that says: "A function that satisfies 'for any real numbers x, y within its domain, f = f(x) + f(y)' can be ( )". Then there are four options: A is f(x) = x squared, B is f(x) = 2 to the power of x, C is f(x) = log base 2 of x, and D is f(x) = e raised to the natural log of x. I need to figure out which of these functions satisfies the given condition.First, let me make sure I understand the condition correctly. It says that for any real numbers x and y within the domain of the function, f equals f(x) plus f(y). Hmm, that wording is a bit confusing. Wait, does it mean that f(x * y) equals f(x) + f(y)? Because if f is a function of x and y, then f(x, y) = f(x) + f(y). But the way it's written is a bit unclear. It says "for any real numbers x, y within its domain, f = f(x) + f(y)". Maybe it's saying that f(x * y) = f(x) + f(y). That would make sense because it's relating the function's value at the product of x and y to the sum of the function's values at x and y individually.So, if that's the case, then we're looking for a function f such that f(x * y) = f(x) + f(y) for all x and y in the domain of f. That rings a bell. I think logarithmic functions have this property. Because one of the logarithm properties is that log(a * b) = log(a) + log(b). So, if f is a logarithmic function, then f(x * y) would equal f(x) + f(y). Let me check the options.Option A is f(x) = x squared. Let's test if this satisfies f(x * y) = f(x) + f(y). So, f(x * y) would be (x * y)^2, which is x squared times y squared. On the other hand, f(x) + f(y) is x squared plus y squared. These two are not equal unless x or y is zero, which isn't generally true. So, A doesn't satisfy the condition.Option B is f(x) = 2^x. Let's test this. f(x * y) would be 2^(x * y). f(x) + f(y) is 2^x + 2^y. These two expressions are not equal in general. For example, if x = 1 and y = 1, f(x * y) = 2^(1*1) = 2, and f(x) + f(y) = 2 + 2 = 4, which are not equal. So, B doesn't work either.Option C is f(x) = log base 2 of x. Let's test this. f(x * y) would be log base 2 of (x * y). Using the logarithm property, this is equal to log base 2 of x plus log base 2 of y, which is f(x) + f(y). So, yes, this satisfies the condition. That seems promising.Option D is f(x) = e raised to the natural log of x. Let me simplify that. e raised to ln(x) is just x, because e and ln are inverse functions. So, f(x) = x. Let's test this. f(x * y) would be x * y. On the other hand, f(x) + f(y) is x + y. These are not equal unless one of them is zero, which isn't generally true. So, D doesn't satisfy the condition either.Therefore, the only function that satisfies f(x * y) = f(x) + f(y) is option C, which is the logarithmic function. So, I think the answer is C.Wait, just to make sure I didn't make a mistake. Let me go through each option again quickly.A: f(x) = x^2. As I saw before, f(x * y) = (x * y)^2, which is x^2 * y^2, not x^2 + y^2. So, no.B: f(x) = 2^x. f(x * y) = 2^(x * y), which is not equal to 2^x + 2^y. For example, x = 2, y = 3: f(6) = 64, f(2) + f(3) = 4 + 8 = 12. Not equal.C: f(x) = log2(x). f(x * y) = log2(x * y) = log2(x) + log2(y) = f(x) + f(y). Perfect.D: f(x) = e^(ln x) = x. f(x * y) = x * y, which is not equal to x + y. So, no.Yep, definitely C. I think I got it right.**Final Answer**boxed{C}

💡Okay, so I have this problem about a hyperbola centered at the origin. It says it opens either horizontally or vertically, and it passes through three points: (0, -2), (3, 4), and (2, s). I need to find s squared. Hmm, okay, let me think about how to approach this.First, I remember that the standard form of a hyperbola centered at the origin can be either (frac{x^2}{a^2} - frac{y^2}{b^2} = 1) if it opens horizontally, or (frac{y^2}{b^2} - frac{x^2}{a^2} = 1) if it opens vertically. So, I need to figure out which one it is.Looking at the points it passes through, one of them is (0, -2). If I plug that into the horizontal opening hyperbola equation, I get (frac{0^2}{a^2} - frac{(-2)^2}{b^2} = 1), which simplifies to (-frac{4}{b^2} = 1). That would mean (-4/b^2 = 1), so (b^2 = -4). But that doesn't make sense because b squared can't be negative. So, that must mean the hyperbola doesn't open horizontally. Therefore, it must open vertically.So, the equation should be (frac{y^2}{b^2} - frac{x^2}{a^2} = 1). Now, let's use the point (0, -2) to find b squared. Plugging in x=0 and y=-2, we get:[frac{(-2)^2}{b^2} - frac{0^2}{a^2} = 1 implies frac{4}{b^2} = 1 implies b^2 = 4]Great, so now we know that (b^2 = 4). So, the equation becomes:[frac{y^2}{4} - frac{x^2}{a^2} = 1]Now, we need to find (a^2). For that, we can use another point that the hyperbola passes through, which is (3, 4). Let's plug x=3 and y=4 into the equation:[frac{4^2}{4} - frac{3^2}{a^2} = 1 implies frac{16}{4} - frac{9}{a^2} = 1 implies 4 - frac{9}{a^2} = 1]Subtracting 4 from both sides gives:[- frac{9}{a^2} = -3]Multiplying both sides by -1:[frac{9}{a^2} = 3]Then, solving for (a^2):[a^2 = frac{9}{3} = 3]Wait, hold on, that doesn't seem right. Let me check my steps again. So, plugging in (3,4):[frac{16}{4} - frac{9}{a^2} = 1 implies 4 - frac{9}{a^2} = 1]Subtract 4:[- frac{9}{a^2} = -3]Multiply both sides by -1:[frac{9}{a^2} = 3]Then, (a^2 = 9 / 3 = 3). Hmm, that still gives me (a^2 = 3). But wait, if (a^2 = 3), then the equation is:[frac{y^2}{4} - frac{x^2}{3} = 1]Is that correct? Let me test it with the point (3,4):[frac{16}{4} - frac{9}{3} = 4 - 3 = 1]Yes, that works. So, (a^2 = 3). Okay, so the equation of the hyperbola is:[frac{y^2}{4} - frac{x^2}{3} = 1]Now, we need to find the value of s when x=2. So, plug x=2 and y=s into the equation:[frac{s^2}{4} - frac{2^2}{3} = 1 implies frac{s^2}{4} - frac{4}{3} = 1]Let me solve for (s^2). First, add (frac{4}{3}) to both sides:[frac{s^2}{4} = 1 + frac{4}{3} = frac{3}{3} + frac{4}{3} = frac{7}{3}]Then, multiply both sides by 4:[s^2 = frac{7}{3} times 4 = frac{28}{3}]Wait, that's different from what I thought earlier. Let me double-check my steps. So, plugging x=2:[frac{s^2}{4} - frac{4}{3} = 1]Adding (frac{4}{3}):[frac{s^2}{4} = 1 + frac{4}{3} = frac{7}{3}]Multiplying by 4:[s^2 = frac{28}{3}]Hmm, so s squared is 28/3. But wait, earlier I thought it was 44/7. Did I make a mistake somewhere?Let me go back. When I used the point (3,4), I got (a^2 = 3). Then, plugging x=2, I get (s^2 = 28/3). But in the initial thought process, I thought (a^2 = 7), which led to (s^2 = 44/7). So, where did I go wrong?Wait, in the initial thought process, I might have miscalculated. Let me redo the calculation for (a^2).Given the point (3,4):[frac{4^2}{4} - frac{3^2}{a^2} = 1 implies frac{16}{4} - frac{9}{a^2} = 1 implies 4 - frac{9}{a^2} = 1]Subtract 4:[- frac{9}{a^2} = -3]Multiply both sides by -1:[frac{9}{a^2} = 3 implies a^2 = frac{9}{3} = 3]Yes, that's correct. So, (a^2 = 3). Therefore, when plugging x=2:[frac{s^2}{4} - frac{4}{3} = 1 implies frac{s^2}{4} = frac{7}{3} implies s^2 = frac{28}{3}]So, s squared is 28/3. Therefore, the answer should be (boxed{dfrac{28}{3}}).Wait, but in the initial thought process, I thought (a^2 = 7), which was incorrect. So, I need to correct that. The correct value is (a^2 = 3), leading to (s^2 = 28/3).Let me confirm once more. The hyperbola equation is (frac{y^2}{4} - frac{x^2}{3} = 1). Plugging in (3,4):[frac{16}{4} - frac{9}{3} = 4 - 3 = 1]Correct. Plugging in (2, s):[frac{s^2}{4} - frac{4}{3} = 1 implies frac{s^2}{4} = frac{7}{3} implies s^2 = frac{28}{3}]Yes, that seems consistent. So, I think the correct answer is 28/3.

💡Okay, so I have this problem about a regular tetrahedron and its inscribed sphere. The question is asking me to find a real number 'a' such that the height of the regular tetrahedron is 'a' times the radius of its inscribed sphere. The options given are A: 2, B: 3, C: 4, D: 5.First, I remember that in the case of an equilateral triangle, the height is three times the radius of its inscribed circle. The problem mentions this as an analogy, so I think the approach for the tetrahedron might be similar but extended into three dimensions.Let me recall some properties of a regular tetrahedron. A regular tetrahedron has four triangular faces, all of which are equilateral triangles. It's a three-dimensional shape where all edges are equal in length. The inscribed sphere (inradius) touches all the faces of the tetrahedron from the inside.I need to relate the height of the tetrahedron to the radius of its inscribed sphere. Maybe I can use the concept of volume here because the inradius is related to the volume and the surface area.I remember that for a regular tetrahedron, the volume can be calculated in two ways: one using the height and the base area, and another using the inradius and the total surface area.Let me denote:- V as the volume of the tetrahedron,- S as the area of one face,- h as the height of the tetrahedron,- r as the radius of the inscribed sphere,- A as the total surface area.Since all faces are equilateral triangles, the total surface area A is 4 times the area of one face, so A = 4S.The volume of a tetrahedron can be expressed as:V = (1/3) * base area * height = (1/3) * S * h.On the other hand, the volume can also be expressed in terms of the inradius. I think the formula is similar to the two-dimensional case but extended to three dimensions. In 2D, the area is equal to the inradius times the semiperimeter. In 3D, the volume should be equal to the inradius times the surface area divided by 3. So:V = (1/3) * A * r.Since both expressions equal the volume, I can set them equal to each other:(1/3) * S * h = (1/3) * A * r.Simplifying this, the (1/3) cancels out on both sides:S * h = A * r.But I know that A = 4S, so substituting that in:S * h = 4S * r.Now, I can divide both sides by S (assuming S ≠ 0, which it isn't for a tetrahedron):h = 4r.So, the height h is equal to 4 times the inradius r. Therefore, the real number 'a' is 4.Let me double-check my reasoning to make sure I didn't make a mistake. I used the volume formulas for a tetrahedron, related the height to the inradius through the surface area, and correctly substituted A = 4S. The algebra seems straightforward, and the result makes sense since in 2D it was 3, and in 3D, it's a bit higher, which is 4.I think that's correct. So, the answer should be C: 4.**Final Answer**boxed{C}

💡To determine the probability that the product of eight die rolls is even, we first recognize that the product will be even if at least one of the rolls is even. A standard die has three even numbers (2, 4, 6) and three odd numbers (1, 3, 5), so the probability of rolling an even number on a single roll is (frac{1}{2}), and the probability of rolling an odd number is also (frac{1}{2}).The complementary probability that all eight rolls are odd (and thus the product is odd) is calculated as:[left(frac{1}{2}right)^8 = frac{1}{256}]Therefore, the probability that the product is even is:[1 - frac{1}{256} = frac{255}{256}]