answer：Alright, so I have this quadratic equation: ( x^2 - 108x + k = 0 ). The problem says both roots of this equation are prime numbers, and I need to find how many possible values of ( k ) there are. Hmm, okay, let me think through this step by step.First, I remember that for a quadratic equation of the form ( x^2 + bx + c = 0 ), the sum of the roots is ( -b ) and the product of the roots is ( c ). So in this case, the equation is ( x^2 - 108x + k = 0 ), which means the sum of the roots is ( 108 ) and the product is ( k ).Let me denote the two prime roots as ( p ) and ( q ). So, according to Vieta's formulas:1. ( p + q = 108 )2. ( p cdot q = k )Since both ( p ) and ( q ) are primes, I need to find pairs of prime numbers that add up to 108. Once I have those pairs, I can calculate ( k ) for each pair and see how many distinct values of ( k ) there are.Now, I know that except for the prime number 2, all other primes are odd. So, if I have two primes adding up to 108, which is an even number, there are two possibilities:1. Both primes are odd.2. One prime is 2, and the other is even.But wait, the only even prime number is 2. So, if one of the primes is 2, the other one must be 108 - 2 = 106. Is 106 a prime number? Let me check.106 divided by 2 is 53, which is a prime number. So, 106 is not a prime because it has divisors other than 1 and itself. Therefore, 106 is composite. So, the pair (2, 106) doesn't work because 106 isn't prime.Now, considering the other possibility where both primes are odd. If I add two odd numbers, the result is even, which is consistent because 108 is even. So, in this case, both primes are odd, and their sum is 108.So, I need to find all pairs of prime numbers ( p ) and ( q ) such that ( p + q = 108 ). Each such pair will give me a unique value of ( k = p cdot q ). Therefore, the number of such pairs will determine the number of possible values of ( k ).Let me list out the primes less than 108 and see which pairs add up to 108. But that might take a while. Maybe there's a smarter way.I know that for any even number greater than 2, the Goldbach conjecture states that it can be expressed as the sum of two primes. Although this conjecture hasn't been proven for all even numbers, it's been verified up to very large numbers, so for 108, it should hold. So, there should be at least one pair of primes adding up to 108.But I need to find all such pairs. Let me try to find them.Starting from the smallest prime, 3:- 3 + 105 = 108. Is 105 prime? 105 is divisible by 5 and 7, so no.- Next prime: 5. 5 + 103 = 108. Is 103 prime? Yes, 103 is a prime number. So, (5, 103) is a valid pair.Next prime: 7. 7 + 101 = 108. Is 101 prime? Yes, 101 is prime. So, (7, 101) is another pair.Next prime: 11. 11 + 97 = 108. 97 is prime. So, (11, 97) is a pair.Next prime: 13. 13 + 95 = 108. 95 is not prime (divisible by 5 and 19). So, discard.Next prime: 17. 17 + 91 = 108. 91 is not prime (7*13). So, discard.Next prime: 19. 19 + 89 = 108. 89 is prime. So, (19, 89) is a pair.Next prime: 23. 23 + 85 = 108. 85 is not prime (5*17). Discard.Next prime: 29. 29 + 79 = 108. 79 is prime. So, (29, 79) is a pair.Next prime: 31. 31 + 77 = 108. 77 is not prime (7*11). Discard.Next prime: 37. 37 + 71 = 108. 71 is prime. So, (37, 71) is a pair.Next prime: 41. 41 + 67 = 108. 67 is prime. So, (41, 67) is a pair.Next prime: 43. 43 + 65 = 108. 65 is not prime (5*13). Discard.Next prime: 47. 47 + 61 = 108. 61 is prime. So, (47, 61) is a pair.Next prime: 53. 53 + 55 = 108. 55 is not prime (5*11). Discard.Next prime: 59. 59 + 49 = 108. 49 is not prime (7*7). Discard.Next prime: 61. 61 + 47 = 108. We already have this pair as (47, 61). Since addition is commutative, this is the same pair.Similarly, primes beyond this will just mirror the pairs we've already found.So, the valid prime pairs that add up to 108 are:1. (5, 103)2. (7, 101)3. (11, 97)4. (19, 89)5. (29, 79)6. (37, 71)7. (41, 67)8. (47, 61)So, there are 8 distinct pairs of prime numbers that add up to 108. Each of these pairs will give a different value of ( k ) because the product of two different primes will be unique unless there's some overlap, which I don't think there is in this case.Let me verify that each product is unique:- 5 * 103 = 515- 7 * 101 = 707- 11 * 97 = 1067- 19 * 89 = 1691- 29 * 79 = 2291- 37 * 71 = 2627- 41 * 67 = 2747- 47 * 61 = 2867Yes, all these products are distinct. Therefore, each pair gives a unique value of ( k ).So, the number of possible values of ( k ) is 8.Wait a minute, earlier I thought about the pair (2, 106), but 106 isn't prime, so that doesn't count. So, all the valid pairs are the ones I listed above, which are 8 in total.Therefore, there are 8 possible values of ( k ).

answer：Okay, so I have this problem here about buying pineapples and watermelons. Let me try to understand what's being asked. It says that a pineapple costs Rs 7 each, and a watermelon costs a certain amount. I spent a total of Rs 38, and I bought 2 pineapples. The question is asking how much does each watermelon cost.Alright, first, I need to figure out how much money was spent on pineapples. Since each pineapple is Rs 7 and I bought 2, that should be 2 times 7, which is 14. So, Rs 14 was spent on pineapples.Now, the total amount spent was Rs 38. If I subtract the amount spent on pineapples from the total, that should give me the amount spent on watermelons. So, 38 minus 14 is 24. That means Rs 24 was spent on watermelons.But wait, the problem doesn't say how many watermelons I bought. It just says I bought some watermelons, but it doesn't specify the number. Without knowing how many watermelons I purchased, I can't determine the exact cost per watermelon. For example, if I bought 1 watermelon, it would cost Rs 24. If I bought 2 watermelons, each would cost Rs 12. If I bought 3, each would cost Rs 8, and so on.So, it seems like there's missing information here. To find out the cost per watermelon, I need to know how many watermelons were bought. Without that, I can't give a specific answer. Maybe the problem assumes that only one watermelon was bought, but it's not clear. If I have to make an assumption, I might say that if only one watermelon was bought, it would cost Rs 24. But I should probably mention that the number of watermelons isn't specified, so the cost per watermelon can't be determined definitively.Alternatively, maybe the problem expects me to express the cost per watermelon in terms of the number bought. For example, if I bought 'w' watermelons, then each would cost 24 divided by w. But again, without knowing 'w,' I can't provide a numerical answer.In summary, to solve this problem accurately, I need to know how many watermelons were purchased. Without that information, I can only say that Rs 24 was spent on watermelons, but the exact cost per watermelon remains unknown.

answer：Alright, so I've got this trigonometry problem here, and I need to solve it step by step. Let me read it again to make sure I understand what's being asked."Given that cos a = 1/7 and cos(a - β) = 13/14, with 0 < β < a < π/2,(1) Find the value of tan 2a;(2) Find the value of β."Okay, so we have two parts here. First, I need to find tan 2a, and second, I need to find the angle β. Both angles a and β are between 0 and π/2, which is good because that means all the trigonometric functions will be positive in this range.Starting with part (1): Find tan 2a.I remember that there's a double-angle formula for tangent, which is tan 2a = 2 tan a / (1 - tan² a). So, to find tan 2a, I first need to find tan a.Given that cos a = 1/7, I can find sin a using the Pythagorean identity: sin² a + cos² a = 1.So, sin² a = 1 - cos² a = 1 - (1/7)² = 1 - 1/49 = 48/49.Therefore, sin a = sqrt(48/49) = (4√3)/7.Now, tan a = sin a / cos a = (4√3)/7 divided by 1/7, which simplifies to 4√3.So, tan a = 4√3.Now, plugging this into the double-angle formula:tan 2a = 2*(4√3) / (1 - (4√3)²) = 8√3 / (1 - 16*3) = 8√3 / (1 - 48) = 8√3 / (-47) = -8√3 / 47.Wait, that gives me a negative value for tan 2a. But since a is between 0 and π/2, 2a would be between 0 and π. However, tan is positive in the first and third quadrants. Since 2a is less than π, it could be in the first or second quadrant. But tan is positive in the first and negative in the second. Hmm, so if tan 2a is negative, that would mean 2a is in the second quadrant, which is between π/2 and π. But since a is less than π/2, 2a is less than π, so it's possible.But let me double-check my calculations because I might have made a mistake.Starting again:cos a = 1/7, so sin a = sqrt(1 - (1/7)^2) = sqrt(48/49) = 4√3/7.tan a = sin a / cos a = (4√3/7) / (1/7) = 4√3.Then, tan 2a = 2 tan a / (1 - tan² a) = 2*(4√3) / (1 - (4√3)^2) = 8√3 / (1 - 16*3) = 8√3 / (1 - 48) = 8√3 / (-47) = -8√3 / 47.So, the calculation seems correct. The negative sign indicates that 2a is in the second quadrant, which makes sense because a is less than π/2, so 2a is less than π, and if a is greater than π/4, then 2a would be greater than π/2, placing it in the second quadrant where tangent is negative.Alright, so part (1) seems to be tan 2a = -8√3 / 47.Moving on to part (2): Find the value of β.Given that cos(a - β) = 13/14.I need to find β. Since we know cos(a - β), maybe I can use the cosine of difference identity.The cosine of difference identity is:cos(a - β) = cos a cos β + sin a sin β.We know cos(a - β) = 13/14, cos a = 1/7, sin a = 4√3/7.So, plugging in the known values:13/14 = (1/7) cos β + (4√3/7) sin β.Let me write that equation:(1/7) cos β + (4√3/7) sin β = 13/14.To simplify, multiply both sides by 7:cos β + 4√3 sin β = 13/2.Wait, 13/14 multiplied by 7 is 13/2? Let me check:13/14 * 7 = (13*7)/14 = 91/14 = 13/2. Yes, that's correct.So, cos β + 4√3 sin β = 13/2.Hmm, that seems a bit large because the maximum value of cos β + 4√3 sin β is sqrt(1 + (4√3)^2) = sqrt(1 + 48) = sqrt(49) = 7. So, 13/2 is 6.5, which is less than 7, so it's possible.But let me think about another approach. Maybe I can express cos(a - β) in terms of β.Alternatively, since I know cos(a - β) = 13/14, I can find sin(a - β) using the Pythagorean identity.sin(a - β) = sqrt(1 - (13/14)^2) = sqrt(1 - 169/196) = sqrt(27/196) = 3√3/14.Now, using the sine of difference identity:sin(a - β) = sin a cos β - cos a sin β.We have:3√3/14 = (4√3/7) cos β - (1/7) sin β.So, now we have two equations:1) cos β + 4√3 sin β = 13/2.Wait, no, earlier I had:From cos(a - β):cos β + 4√3 sin β = 13/2.And from sin(a - β):(4√3/7) cos β - (1/7) sin β = 3√3/14.Let me write them clearly:Equation 1: cos β + 4√3 sin β = 13/2.Equation 2: (4√3/7) cos β - (1/7) sin β = 3√3/14.Hmm, these are two equations with two variables, cos β and sin β. Maybe I can solve them simultaneously.Let me denote x = cos β and y = sin β.Then, Equation 1: x + 4√3 y = 13/2.Equation 2: (4√3/7) x - (1/7) y = 3√3/14.Let me multiply Equation 2 by 7 to eliminate denominators:4√3 x - y = 3√3/2.So, Equation 2 becomes: 4√3 x - y = 3√3/2.Now, we have:Equation 1: x + 4√3 y = 13/2.Equation 2: 4√3 x - y = 3√3/2.Let me solve Equation 2 for y:From Equation 2: 4√3 x - y = 3√3/2.So, y = 4√3 x - 3√3/2.Now, substitute this expression for y into Equation 1:x + 4√3*(4√3 x - 3√3/2) = 13/2.Let me compute the terms inside:4√3*(4√3 x) = 4√3 * 4√3 x = 16*3 x = 48x.4√3*(-3√3/2) = -4√3*3√3/2 = -12*3/2 = -36/2 = -18.So, Equation 1 becomes:x + 48x - 18 = 13/2.Combine like terms:49x - 18 = 13/2.Add 18 to both sides:49x = 13/2 + 18 = 13/2 + 36/2 = 49/2.So, x = (49/2)/49 = 1/2.So, x = cos β = 1/2.Therefore, cos β = 1/2.Since β is between 0 and π/2, β must be π/3 because cos(π/3) = 1/2.So, β = π/3.Let me verify this solution.If β = π/3, then cos β = 1/2, sin β = √3/2.Now, let's check Equation 1:cos β + 4√3 sin β = 1/2 + 4√3*(√3/2) = 1/2 + 4*3/2 = 1/2 + 6 = 6.5 = 13/2. That checks out.Now, check Equation 2:(4√3/7) cos β - (1/7) sin β = (4√3/7)*(1/2) - (1/7)*(√3/2) = (2√3/7) - (√3/14) = (4√3/14 - √3/14) = 3√3/14. That also checks out.So, β = π/3 is correct.Wait a second, but earlier in part (1), I found tan 2a = -8√3 / 47. But since a is less than π/2, 2a is less than π, and if a is greater than π/4, then 2a is greater than π/2, making tan 2a negative, which is consistent with our result.But let me just make sure that a is indeed greater than π/4.Given that cos a = 1/7, which is approximately 0.1429. Since cos π/4 = √2/2 ≈ 0.7071, and 1/7 ≈ 0.1429 is much smaller, so a is actually greater than π/4 because cosine decreases as the angle increases in the first quadrant. So, a is greater than π/4, making 2a greater than π/2, hence tan 2a is negative. That makes sense.So, summarizing:(1) tan 2a = -8√3 / 47.(2) β = π/3.I think that's it. Let me just recap the steps to make sure I didn't skip anything.For part (1):1. Given cos a = 1/7, find sin a using Pythagorean identity.2. Calculate tan a = sin a / cos a.3. Use double-angle formula for tangent to find tan 2a.For part (2):1. Given cos(a - β) = 13/14, use cosine of difference identity to set up an equation involving cos β and sin β.2. Find sin(a - β) using Pythagorean identity.3. Use sine of difference identity to set up another equation.4. Solve the system of equations for cos β and sin β.5. Determine β based on the values of cos β and sin β.Everything seems to check out, so I'm confident in the answers.

answer：First, I need to determine the intensity of the original red paint. Let's denote this intensity as ( I ) (where ( I ) is a percentage).We know that a fraction of the original paint, specifically ( frac{2}{3} ), is replaced with a 30% solution of red paint. After this replacement, the new intensity of the paint mixture is 40%.To model this situation, I'll set up an equation that represents the mixture of intensities. The equation will consider the remaining original paint and the added 30% solution:[(1 - f) cdot I + f cdot 30% = 40%]Substituting the given fraction ( f = frac{2}{3} ) into the equation:[left(1 - frac{2}{3}right) cdot I + frac{2}{3} cdot 30% = 40%]Simplifying the equation:[frac{1}{3} cdot I + frac{2}{3} cdot 0.30 = 0.40][frac{1}{3} cdot I + 0.20 = 0.40]Subtracting 0.20 from both sides:[frac{1}{3} cdot I = 0.20]Multiplying both sides by 3 to solve for ( I ):[I = 0.20 cdot 3][I = 0.60]Therefore, the intensity of the original red paint is 60%.

answer：Alright, so I've got this multiple-choice question here, and I need to figure out which statement is correct. Let's break it down step by step.First, let's look at Statement A: "When a fair coin is tossed 10 times, it will definitely land heads up 5 times." Hmm, okay. I know that when you toss a fair coin, each toss has a 50% chance of landing heads and a 50% chance of landing tails. But does that mean that if I toss it 10 times, it will always land heads up exactly 5 times? That doesn't sound right. I think it's more about probability and expectation. On average, you might expect around 5 heads out of 10 tosses, but it's not guaranteed. Sometimes you could get more heads, sometimes fewer. So, Statement A seems incorrect because it's not a certainty.Next, Statement B: "The sun rises from the west tomorrow" is an impossible event. Okay, the sun rising from the west... Well, from what I know, the sun rises in the east and sets in the west because of the Earth's rotation. So, for the sun to rise from the west would mean something has drastically changed in the Earth's rotation or the sun's position, which isn't going to happen tomorrow. So, this event is impossible. That makes Statement B correct.Moving on to Statement C: "A basketball player shoots once from the free-throw line and scores" is a certain event. Hmm, a certain event means it will always happen. But even the best basketball players don't make every free throw. There's always a chance they might miss, right? So, it's not a certain event; it's more of a probable event. Therefore, Statement C is incorrect.Lastly, Statement D: "a is a real number, |a|geqslant 0" is a random event. Wait, the absolute value of any real number is always greater than or equal to zero. That's a mathematical fact. It's not random; it's always true. So, labeling this as a random event doesn't make sense. Statement D is incorrect.So, after going through all the statements, only Statement B seems to be correct. It's an impossible event because the sun rising from the west goes against the natural order of things based on our understanding of Earth's rotation.

answer：First, I recognize that the mean of the three numbers is 6, which means the total sum of the numbers is ( 3 times 6 = 18 ).Since the median is also 6, the middle number in the ordered set must be 6. Let's denote the three numbers as ( a ), 6, and 10, where ( a ) is the smallest number.To find ( a ), I set up the equation:[a + 6 + 10 = 18]Simplifying this, I get:[a + 16 = 18]Subtracting 16 from both sides gives:[a = 2]Now, the three numbers are 2, 6, and 10. The range is the difference between the largest and smallest numbers:[10 - 2 = 8]