answer：Alright, so I have this logic problem here with four conclusions, and I need to figure out which ones are correct. Let me take it step by step.First, let me recall some basic logic concepts. I know that in logic, "sufficient condition" means that if the condition is met, then the outcome must happen, but the outcome could also happen in other ways. On the other hand, a "necessary condition" means that for the outcome to happen, the condition must be met, but meeting the condition doesn't guarantee the outcome.Okay, so let's look at each conclusion one by one.**Conclusion 1:** "p land q" being true is a sufficient but not necessary condition for "p lor q" to be true.Hmm, let's break this down. If both p and q are true, then certainly p lor q is true because in a logical OR, if at least one is true, the whole statement is true. So, if p land q is true, then p lor q must be true. That makes p land q a sufficient condition for p lor q.But is it necessary? Well, no. Because p lor q can be true even if only one of them is true. For example, if p is true and q is false, p lor q is still true. So, p land q being true isn't necessary for p lor q to be true. Therefore, conclusion 1 seems correct.**Conclusion 2:** "p land q" being false is a sufficient but not necessary condition for "p lor q" to be true.Wait, this seems a bit tricky. If p land q is false, does that mean p lor q must be true? Let's think. If p land q is false, it means that at least one of p or q is false. But p lor q being true requires at least one of them to be true. So, if p land q is false, it's possible that both p and q are false, which would make p lor q false. Therefore, p land q being false doesn't guarantee that p lor q is true. So, it's not a sufficient condition. Hence, conclusion 2 is incorrect.**Conclusion 3:** "p land q" being true is a sufficient but not necessary condition for "neg p" to be false.Alright, let's parse this. If p land q is true, then both p and q are true. So, neg p would be false because p is true. That means p land q being true does make neg p false, so it's a sufficient condition.Is it necessary? Well, neg p can be false even if p is true and q is false. So, p land q doesn't have to be true for neg p to be false. Therefore, it's not a necessary condition. So, conclusion 3 is correct.**Conclusion 4:** "neg p" being true is a sufficient but not necessary condition for "p land q" to be false.Okay, if neg p is true, that means p is false. If p is false, then regardless of q, p land q will be false because both need to be true for the conjunction to be true. So, neg p being true does make p land q false, making it a sufficient condition.Is it necessary? Well, p land q can be false even if p is true but q is false. So, neg p doesn't have to be true for p land q to be false. Therefore, it's not a necessary condition. So, conclusion 4 is correct.Wait, let me double-check conclusion 2 again. If p land q is false, does that mean p lor q is true? No, because p land q being false just means at least one is false, but p lor q being true requires at least one to be true. So, if both are false, p lor q is false. Therefore, p land q being false doesn't ensure p lor q is true. So, conclusion 2 is indeed incorrect.So, after going through each conclusion, I think conclusions 1, 3, and 4 are correct.

answer：Alright, so I have this problem where I need to exchange a 25-ruble bill using exactly eleven bills of denominations 1, 3, and 5 rubles. Hmm, okay, let's break this down step by step.First, I need to figure out how many of each bill I need. Let's denote the number of 1-ruble bills as x, the number of 3-ruble bills as y, and the number of 5-ruble bills as z. So, I have two main equations here:1. The total number of bills: x + y + z = 112. The total amount: 1*x + 3*y + 5*z = 25Alright, so I have two equations with three variables. That means I might need to find integer solutions that satisfy both equations. Since the denominations are 1, 3, and 5, and I need exactly eleven bills, I need to find non-negative integers x, y, and z that satisfy both equations.Let me start by trying to express one variable in terms of the others using the first equation. Let's solve for x:x = 11 - y - zNow, substitute this into the second equation:1*(11 - y - z) + 3*y + 5*z = 25Simplify this:11 - y - z + 3y + 5z = 25Combine like terms:11 + 2y + 4z = 25Now, subtract 11 from both sides:2y + 4z = 14I can simplify this further by dividing both sides by 2:y + 2z = 7Okay, so now I have y + 2z = 7. This equation relates y and z. Since y and z must be non-negative integers, I can try different values of z and see what y would be.Let's list possible values for z:1. If z = 0, then y = 72. If z = 1, then y = 53. If z = 2, then y = 34. If z = 3, then y = 15. If z = 4, then y = -1 (Wait, y can't be negative, so z can't be 4 or higher)So, the possible pairs (y, z) are:- (7, 0)- (5, 1)- (3, 2)- (1, 3)Now, for each of these pairs, I can find x using x = 11 - y - z.Let's calculate x for each:1. For (y=7, z=0): x = 11 - 7 - 0 = 4 So, (x, y, z) = (4, 7, 0)2. For (y=5, z=1): x = 11 - 5 - 1 = 5 So, (x, y, z) = (5, 5, 1)3. For (y=3, z=2): x = 11 - 3 - 2 = 6 So, (x, y, z) = (6, 3, 2)4. For (y=1, z=3): x = 11 - 1 - 3 = 7 So, (x, y, z) = (7, 1, 3)Alright, so I have four possible combinations:1. 4 one-ruble bills, 7 three-ruble bills, and 0 five-ruble bills.2. 5 one-ruble bills, 5 three-ruble bills, and 1 five-ruble bill.3. 6 one-ruble bills, 3 three-ruble bills, and 2 five-ruble bills.4. 7 one-ruble bills, 1 three-ruble bill, and 3 five-ruble bills.Let me verify each of these to make sure they add up to 25 rubles and use exactly 11 bills.1. 4*1 + 7*3 + 0*5 = 4 + 21 + 0 = 25 rubles. Total bills: 4 + 7 + 0 = 11. Correct.2. 5*1 + 5*3 + 1*5 = 5 + 15 + 5 = 25 rubles. Total bills: 5 + 5 + 1 = 11. Correct.3. 6*1 + 3*3 + 2*5 = 6 + 9 + 10 = 25 rubles. Total bills: 6 + 3 + 2 = 11. Correct.4. 7*1 + 1*3 + 3*5 = 7 + 3 + 15 = 25 rubles. Total bills: 7 + 1 + 3 = 11. Correct.All four combinations work. So, there are multiple ways to exchange the 25-ruble bill using exactly eleven bills of 1, 3, and 5 rubles.I think that's all the possible combinations since z can't be higher than 3 without making y negative, and we've covered all z from 0 to 3. So, these are the only solutions.

answer：Okay, so I have this problem here: I need to determine the sum of all prime numbers ( p ) for which there exists no integer solution in ( x ) to the congruence ( 5(6x + 2) equiv 3 pmod{p} ). Hmm, let me try to figure this out step by step.First, I should probably simplify the given congruence to make it easier to work with. The original congruence is ( 5(6x + 2) equiv 3 pmod{p} ). Let me expand this:( 5 times 6x + 5 times 2 equiv 3 pmod{p} )So that becomes:( 30x + 10 equiv 3 pmod{p} )Now, I can subtract 10 from both sides to isolate the term with ( x ):( 30x equiv 3 - 10 pmod{p} )Which simplifies to:( 30x equiv -7 pmod{p} )Hmm, okay. So now I have ( 30x equiv -7 pmod{p} ). I remember that for a linear congruence of the form ( ax equiv b pmod{m} ), a solution exists if and only if ( gcd(a, m) ) divides ( b ). In this case, ( a ) is 30, ( m ) is ( p ), and ( b ) is -7.Since ( p ) is a prime number, ( gcd(30, p) ) can only be 1 or ( p ) itself. If ( p ) divides 30, then ( gcd(30, p) = p ), otherwise, it's 1. So, if ( p ) is a prime factor of 30, then ( gcd(30, p) = p ), which would mean that ( p ) divides 30. Otherwise, ( gcd(30, p) = 1 ).Wait, so if ( gcd(30, p) = 1 ), then 30 has an inverse modulo ( p ), and the congruence ( 30x equiv -7 pmod{p} ) has a unique solution modulo ( p ). That means there is a solution for ( x ) in this case.On the other hand, if ( gcd(30, p) = p ), which happens when ( p ) is a prime factor of 30, then ( p ) must divide the constant term on the right side of the congruence for a solution to exist. The constant term here is -7. So, for the congruence to have a solution, ( p ) must divide -7. But since ( p ) is a prime number, it can only divide -7 if ( p ) is 7 or -7, but since primes are positive, ( p ) must be 7.Wait, hold on. Let me think again. If ( p ) divides 30, then ( gcd(30, p) = p ), and for the congruence ( 30x equiv -7 pmod{p} ) to have a solution, ( p ) must divide -7. So, ( p ) must be a prime divisor of 7, which is only 7 itself.But 7 is not a prime factor of 30, because 30 factors into 2, 3, and 5. So, if ( p ) is 2, 3, or 5, then ( gcd(30, p) = p ), and since ( p ) does not divide -7 (because 2, 3, and 5 do not divide 7), the congruence ( 30x equiv -7 pmod{p} ) has no solution.Therefore, the primes ( p ) for which there is no solution are 2, 3, and 5. Let me verify this.For ( p = 2 ): The congruence becomes ( 30x equiv -7 pmod{2} ). Since 30 is congruent to 0 modulo 2, and -7 is congruent to 1 modulo 2, so we have ( 0 times x equiv 1 pmod{2} ), which simplifies to ( 0 equiv 1 pmod{2} ). That's not possible, so no solution.For ( p = 3 ): The congruence becomes ( 30x equiv -7 pmod{3} ). 30 is congruent to 0 modulo 3, and -7 is congruent to 2 modulo 3. So again, ( 0 times x equiv 2 pmod{3} ), which simplifies to ( 0 equiv 2 pmod{3} ). Not possible, so no solution.For ( p = 5 ): The congruence becomes ( 30x equiv -7 pmod{5} ). 30 is congruent to 0 modulo 5, and -7 is congruent to 3 modulo 5. So, ( 0 times x equiv 3 pmod{5} ), which is ( 0 equiv 3 pmod{5} ). Again, not possible, so no solution.For ( p = 7 ): Let's check this as well. ( 30x equiv -7 pmod{7} ). 30 modulo 7 is 2, and -7 modulo 7 is 0. So, the congruence becomes ( 2x equiv 0 pmod{7} ). This implies ( x equiv 0 pmod{7} ), so there is a solution, namely ( x = 0 ).So, indeed, only for ( p = 2, 3, 5 ) there is no solution. Therefore, the primes we're looking for are 2, 3, and 5. Their sum is ( 2 + 3 + 5 = 10 ).I think that's it. I don't see any other primes that would cause the congruence to have no solution. Since primes larger than 5 don't divide 30, their gcd with 30 is 1, so 30 is invertible modulo those primes, meaning the congruence will have a solution.Yeah, I'm pretty confident now. The sum is 10.**Final Answer**The sum of all such prime numbers is boxed{10}.

answer：Okay, so I have this problem where I'm given two vectors, vector a and vector b. Vector a is (cosθ, 1) and vector b is (1, 3cosθ). The problem says that vector a is parallel to vector b, and I need to find the value of cos2θ. Alright, let me try to figure this out step by step.First, I remember that if two vectors are parallel, one must be a scalar multiple of the other. That means there exists some scalar k such that vector a = k * vector b. So, in terms of components, that would mean:cosθ = k * 1 ...(1)and1 = k * 3cosθ ...(2)Hmm, okay, so from equation (1), I can solve for k. If cosθ = k, then k = cosθ. Then, I can substitute this value of k into equation (2). Let me write that down:1 = k * 3cosθBut since k = cosθ, substitute that in:1 = cosθ * 3cosθSimplify that:1 = 3cos²θSo, 3cos²θ = 1Divide both sides by 3:cos²θ = 1/3Alright, so cos²θ is 1/3. Now, the question asks for cos2θ. I remember there are double-angle formulas for cosine, and one of them is:cos2θ = 2cos²θ - 1I think that's the one I can use here since I already have cos²θ. Let me plug in the value I found:cos2θ = 2*(1/3) - 1Calculate that:2*(1/3) is 2/3, so:cos2θ = 2/3 - 1Which is:cos2θ = -1/3Wait, that seems straightforward. Let me just double-check my steps to make sure I didn't make a mistake.1. I started by setting vector a equal to k times vector b because they are parallel.2. That gave me two equations: cosθ = k and 1 = 3k cosθ.3. Substituted k from the first equation into the second, leading to 1 = 3cos²θ.4. Solved for cos²θ, getting 1/3.5. Applied the double-angle formula for cosine, which is 2cos²θ - 1, substituting 1/3 in.6. Calculated and got -1/3.Everything seems to check out. I don't see any errors in my reasoning. So, I think the value of cos2θ is indeed -1/3.Just to be thorough, let me recall another double-angle formula for cosine to see if I can get the same result. Another formula is:cos2θ = 1 - 2sin²θBut I don't have sin²θ, so maybe I can find sin²θ from cos²θ. Since sin²θ + cos²θ = 1, then sin²θ = 1 - cos²θ = 1 - 1/3 = 2/3.So, plugging into the other formula:cos2θ = 1 - 2*(2/3) = 1 - 4/3 = -1/3Same result! Okay, that's reassuring. So, whether I use the formula in terms of cos²θ or sin²θ, I end up with cos2θ = -1/3. That gives me more confidence that my answer is correct.I guess another way to think about it is by considering the direction of the vectors. If vector a is parallel to vector b, they must point in the same or exactly opposite directions. So, their components must be proportional. That's essentially what I used in the beginning when I set vector a equal to k times vector b.Let me just visualize this for a second. Vector a has components (cosθ, 1), and vector b has components (1, 3cosθ). If they are parallel, their slopes must be equal. The slope of vector a is (1)/(cosθ), and the slope of vector b is (3cosθ)/1. So, setting these equal:(1)/(cosθ) = (3cosθ)/1Cross-multiplying:1 = 3cos²θWhich is the same equation I had before. So, that's another way to see why 3cos²θ = 1. This gives cos²θ = 1/3, leading to the same conclusion about cos2θ.I think I've approached this problem from multiple angles now: using the scalar multiple definition, using the double-angle formulas, and even considering the slopes of the vectors. All these methods point to the same answer, which is cos2θ = -1/3. I feel pretty confident that this is the correct solution.Just to recap:1. Vectors parallel implies one is a scalar multiple of the other.2. Set up the proportionality equations.3. Solved for cos²θ.4. Applied the double-angle formula.5. Verified using an alternative formula and by considering slopes.All steps led to the same result, so I think it's safe to say that cos2θ is -1/3.**Final Answer**The value of cos 2θ is boxed{-dfrac{1}{3}}.

answer：Alright, let's tackle these problems one by one. They all involve areas and perimeters of rectangles and squares, so I should remember the basic formulas for those. Starting with part (a): We have a rectangle with an area of 108 cm² and one side length of 12 cm. I need to find the perimeter. Okay, so the area of a rectangle is length times width. If one side is 12 cm, then the other side can be found by dividing the area by 12. Let me calculate that: 108 divided by 12 is 9. So, the sides are 12 cm and 9 cm. The perimeter is twice the sum of the length and width, so that would be 2 times (12 + 9). Adding those together gives 21, and multiplying by 2 gives 42 cm. That seems straightforward.Moving on to part (b): This one is a bit trickier. There's a rectangle divided into a white rectangle and a grey square with an area of 36 cm². I need to find the perimeter of the white rectangle. Hmm, so the grey square has an area of 36, which means each side is 6 cm because the square root of 36 is 6. Now, the entire figure is a rectangle with an area of 108 cm². If the grey part is a square of 6 cm sides, then the larger rectangle must have one side of 6 cm. Let me visualize this: the larger rectangle is split into two parts, one being the 6x6 square, and the other being the white rectangle. So, the length of the larger rectangle would be the same as the length of the white rectangle plus 6 cm. Wait, actually, if the grey square is 6x6, then the width of the larger rectangle is 6 cm. The area of the larger rectangle is 108 cm², so the length can be found by dividing 108 by 6, which is 18 cm. So, the larger rectangle is 18 cm long and 6 cm wide. The grey square takes up 6 cm of that length, so the white rectangle must be 18 - 6 = 12 cm long and 6 cm wide. Therefore, the white rectangle has sides of 12 cm and 6 cm. The perimeter would be 2 times (12 + 6), which is 2 times 18, giving 36 cm. That makes sense.Now, part (c): This is a square divided into two rectangles. The area of the square is 108 cm², so each side of the square is the square root of 108. Let me calculate that: the square root of 108 is approximately 10.39 cm, but maybe I can express it more precisely. 108 can be broken down into 36 times 3, so the square root is 6 times the square root of 3, which is approximately 10.39 cm. But wait, the problem says it's divided into two rectangles. I need to find the area of one of these rectangles. Since it's divided into two rectangles, they must have equal areas if it's divided equally, but the problem doesn't specify how it's divided. Hmm, maybe it's divided along the length or the width. If it's divided into two equal parts, each rectangle would have half the area of the square, so 54 cm² each. But I'm not sure if it's divided equally. The problem doesn't specify, so maybe I need to assume it's divided into two equal rectangles. Alternatively, perhaps the division is such that one rectangle is a square and the other is a rectangle, but that might not make sense. Wait, the problem says it's divided into two rectangles, so both are rectangles, but one could be a square if the division is along the middle. But without more information, I think the safest assumption is that it's divided into two equal rectangles, each with half the area of the square. So, each would have an area of 54 cm². But the problem doesn't ask for the area of the smaller rectangles; it just says the third figure is a square divided into two rectangles. Maybe it's asking for something else, like the perimeter or the dimensions. Wait, looking back, the original problem statement for part (c) is incomplete. It just says, "c) The third figure is a square, which she divided into two rectangles." It doesn't specify what to find. Maybe it's a typo or something missing. Assuming it's similar to the previous parts, maybe it's asking for the perimeter of one of the smaller rectangles. If that's the case, and if the square is divided into two equal rectangles, each would have dimensions of 10.39 cm by 5.195 cm (half of 10.39). But that seems messy. Alternatively, if the square is divided along integer dimensions, maybe the side length is such that it can be evenly divided. Wait, 108 is not a perfect square, so the side length is irrational. Maybe the division isn't equal. Alternatively, perhaps the square is divided into two rectangles with integer sides. Let me think. If the area is 108, and it's divided into two rectangles, maybe one has an area of 36 and the other 72, but that's just a guess. Without more information, it's hard to determine exactly what is needed for part (c). Maybe I need to assume it's divided into two equal rectangles, each with an area of 54 cm², and then find their dimensions or perimeter. If the square has a side length of 6√3 cm, then dividing it into two equal rectangles would mean each rectangle has a length of 6√3 cm and a width of (6√3)/2 = 3√3 cm. The perimeter of each rectangle would be 2 times (6√3 + 3√3) = 2 times 9√3 = 18√3 cm, which is approximately 31.18 cm. But I'm not sure if that's what the problem is asking for. Alternatively, if the division is such that one rectangle is a square and the other is a rectangle, but that would require the side length to be a multiple of the square's side, which might not be the case here. Given the ambiguity, I think the most logical assumption is that the square is divided into two equal rectangles, each with half the area, and then calculating their perimeter based on that. So, each rectangle would have an area of 54 cm², and since the original square has sides of 6√3 cm, each rectangle would have dimensions of 6√3 cm by 3√3 cm. Therefore, the perimeter would be 2*(6√3 + 3√3) = 18√3 cm. But I'm not entirely confident about this without more information. Maybe the problem expects a different approach or has specific details I'm missing. I'll proceed with this answer, but I might need to revisit it if more context is provided.

answer：Okay, so I have this trigonometry problem here. It says that angle a is in the second quadrant, and sin(a - π/6) equals 3/5. I need to find sin a. Hmm, okay, let me think about how to approach this.First, I remember that when dealing with angles in different quadrants, the signs of the trigonometric functions matter. Since a is in the second quadrant, that means between π/2 and π radians. So, in the second quadrant, sine is positive, and cosine is negative. That might be useful later.Now, the problem gives me sin(a - π/6) = 3/5. I need to find sin a. Maybe I can use some trigonometric identities to express sin a in terms of sin(a - π/6). Let me recall the sine addition formula. It goes like this:sin(A + B) = sin A cos B + cos A sin BBut in this case, I have sin(a - π/6). Maybe I can write sin a as sin[(a - π/6) + π/6]. That way, I can apply the addition formula. Let me try that.So, sin a = sin[(a - π/6) + π/6] = sin(a - π/6)cos(π/6) + cos(a - π/6)sin(π/6)Okay, that seems right. Now, I know sin(a - π/6) is 3/5, but I don't know cos(a - π/6). Since I know sin(a - π/6), I can find cos(a - π/6) using the Pythagorean identity. Remember, sin²θ + cos²θ = 1.So, let's compute cos(a - π/6). Let θ = a - π/6. Then sin θ = 3/5, so cos θ = ±√(1 - sin²θ) = ±√(1 - 9/25) = ±√(16/25) = ±4/5.Now, I need to figure out whether cos θ is positive or negative. Since θ = a - π/6 and a is in the second quadrant, let's see where θ would be.If a is in the second quadrant, then a is between π/2 and π. So, a - π/6 would be between π/2 - π/6 = π/3 and π - π/6 = 5π/6. So, θ is between π/3 and 5π/6, which is still in the second quadrant. In the second quadrant, cosine is negative. So, cos θ = -4/5.Alright, so cos(a - π/6) = -4/5. Now, let's plug everything back into the addition formula.sin a = sin(a - π/6)cos(π/6) + cos(a - π/6)sin(π/6)I know sin(a - π/6) is 3/5, cos(π/6) is √3/2, cos(a - π/6) is -4/5, and sin(π/6) is 1/2.So, substituting these values:sin a = (3/5)(√3/2) + (-4/5)(1/2)Let me compute each term separately.First term: (3/5)(√3/2) = (3√3)/10Second term: (-4/5)(1/2) = (-4)/10 = -2/5So, sin a = (3√3)/10 - 2/5Wait, I need to combine these terms. To subtract them, they should have the same denominator. 2/5 is equal to 4/10.So, sin a = (3√3)/10 - 4/10 = (3√3 - 4)/10Hmm, looking at the options, option C is (3√3 - 4)/10, which matches what I got. So, that should be the answer.But just to make sure I didn't make any mistakes, let me double-check my steps.1. I used the sine addition formula correctly: sin(A + B) = sin A cos B + cos A sin B. Applied to sin[(a - π/6) + π/6], which is sin a.2. Calculated cos(a - π/6) using the Pythagorean identity. Since a - π/6 is in the second quadrant, cosine is negative, so -4/5. That seems right.3. Plugged into the formula: (3/5)(√3/2) + (-4/5)(1/2). Calculated each term correctly: 3√3/10 and -4/10.4. Combined the terms: 3√3/10 - 4/10 = (3√3 - 4)/10.Yes, that all seems correct. So, I think option C is the right answer.**Final Answer**boxed{C}