💡Alright, so I've got these four propositions about necessary and sufficient conditions, and I need to figure out which ones are true. Let me go through each one step by step.Starting with Proposition 1: "The necessary and sufficient condition for vector **b** to be collinear with a nonzero vector **a** is that there exists exactly one real number λ such that **b** = λ**a**." Hmm, okay. Collinear vectors mean they lie on the same line, right? So, if **b** is collinear with **a**, there should be some scalar λ that scales **a** to get **b**. That makes sense. But wait, does it have to be exactly one λ? Well, if **a** is a nonzero vector, then for **b** to be collinear, λ has to be unique because if there were two different λs, say λ1 and λ2, such that **b** = λ1**a** and **b** = λ2**a**, then λ1 would have to equal λ2. So, yeah, there's exactly one λ. That seems correct. So, I think Proposition 1 is true.Moving on to Proposition 2: "The necessary and sufficient condition for a, b, c to form a geometric sequence is b² = ac." Okay, a geometric sequence means each term is multiplied by a common ratio. So, if a, b, c are in geometric sequence, then b = a*r and c = b*r = a*r². So, b² would be (a*r)² = a²*r², and ac would be a*(a*r²) = a²*r². So, indeed, b² = ac. But wait, is that the only condition? What if a or c is zero? If a is zero, then b and c would also have to be zero, right? But if a is zero, then b² = ac becomes 0 = 0, which is always true, but does that mean a, b, c are in a geometric sequence? Well, technically, if a is zero, then b and c have to be zero too, so it's a trivial geometric sequence. But sometimes, in some contexts, people might consider non-zero terms for a geometric sequence. So, does the proposition hold in general? It says "necessary and sufficient condition," so if a, b, c are in a geometric sequence, then b² = ac, and if b² = ac, then they are in a geometric sequence. But if a is zero, then b and c have to be zero, which is a geometric sequence with common ratio undefined, but still, it's a sequence. So, maybe it's still true. Hmm, I'm a bit uncertain here. Maybe I'll come back to this one.Proposition 3: "Two events being mutually exclusive is the necessary and sufficient condition for these two events to be complementary." Okay, mutually exclusive means that they can't happen at the same time, so their intersection is empty. Complementary events mean that one is the negation of the other, so their union is the entire sample space, and their intersection is empty. So, if two events are complementary, they are mutually exclusive, but are all mutually exclusive events complementary? No, because mutual exclusivity doesn't necessarily mean their union is the entire sample space. For example, if I roll a die, the event of getting a 1 and the event of getting a 2 are mutually exclusive, but they aren't complementary because their union isn't the entire sample space. So, mutual exclusivity is necessary for complementarity, but it's not sufficient. Therefore, Proposition 3 is false.Proposition 4: "The necessary and sufficient condition for the function y = x² + bx + c to be an even function is b = 0." Okay, an even function satisfies y(x) = y(-x) for all x. So, let's test that. If y(x) = x² + bx + c, then y(-x) = (-x)² + b*(-x) + c = x² - bx + c. For y(x) to be equal to y(-x), we need x² + bx + c = x² - bx + c. Subtracting x² and c from both sides, we get bx = -bx, which implies 2bx = 0 for all x. The only way this can be true for all x is if b = 0. So, yes, b must be zero. Therefore, Proposition 4 is true.Going back to Proposition 2, I was a bit unsure. Let me think again. The condition b² = ac is necessary for a, b, c to be in a geometric sequence, but is it sufficient? Suppose we have numbers where b² = ac, does that necessarily mean they form a geometric sequence? Let's say a = 1, b = -1, c = 1. Then b² = (-1)² = 1, and ac = 1*1 = 1, so b² = ac. But is 1, -1, 1 a geometric sequence? Yes, because each term is multiplied by -1. So, that works. What if a = 0? Then b² = 0*c = 0, so b must be 0, and c can be anything? Wait, no, because if a = 0, then to form a geometric sequence, b must be 0, and c must be 0 as well. So, if a = 0, then b and c must also be 0 for it to be a geometric sequence. But if someone says a, b, c with a = 0 and b² = ac, then c can be anything, but actually, c must be 0 as well. So, in that case, the condition b² = ac is not sufficient because if a = 0, c can be non-zero, but then it's not a geometric sequence. Wait, no, if a = 0, then b must be 0, and c must be 0. So, if a = 0, and b² = ac, then b = 0, and c can be anything? Hmm, no, if a = 0, then ac = 0 regardless of c, but for a, b, c to be a geometric sequence, c must be b*r, but if a = 0 and b = 0, then c can be anything, but in a geometric sequence, if a = 0, then all terms must be 0. So, if a = 0, b = 0, but c ≠ 0, then it's not a geometric sequence. So, in that case, b² = ac is not sufficient because c could be non-zero even though a and b are zero, which would not form a geometric sequence. Therefore, Proposition 2 is false because the condition b² = ac is not sufficient in all cases, especially when a = 0.So, summarizing:1. True2. False3. False4. TrueTherefore, the true propositions are 1 and 4.

💡Okay, so I have this sequence where the first term is 1, and each subsequent term is twice the previous term plus one. I need to find a general formula for the nth term. Hmm, let me try to figure this out step by step.First, let's write down the given information:- The first term, ( a_1 = 1 ).- The recurrence relation is ( a_n = 2a_{n-1} + 1 ) for ( n > 1 ).So, starting from ( a_1 = 1 ), let's compute the next few terms to see if I can spot a pattern.- ( a_2 = 2a_1 + 1 = 2*1 + 1 = 3 )- ( a_3 = 2a_2 + 1 = 2*3 + 1 = 7 )- ( a_4 = 2a_3 + 1 = 2*7 + 1 = 15 )- ( a_5 = 2a_4 + 1 = 2*15 + 1 = 31 )Wait a minute, these numbers look familiar: 1, 3, 7, 15, 31... Each of these is one less than a power of 2. Specifically:- ( 2^1 - 1 = 1 )- ( 2^2 - 1 = 3 )- ( 2^3 - 1 = 7 )- ( 2^4 - 1 = 15 )- ( 2^5 - 1 = 31 )So, it seems like ( a_n = 2^n - 1 ). Let me test this hypothesis with the recurrence relation.Assume ( a_{n-1} = 2^{n-1} - 1 ). Then,( a_n = 2a_{n-1} + 1 = 2(2^{n-1} - 1) + 1 = 2^n - 2 + 1 = 2^n - 1 ).Yes, that works! So, the formula ( a_n = 2^n - 1 ) satisfies both the initial condition and the recurrence relation.But wait, how did I come up with that formula in the first place? I mean, I noticed the pattern, but is there a more systematic way to derive it?Let me think. This is a linear recurrence relation of the form ( a_n + c = 2(a_{n-1} + c) ). If I can find a constant ( c ) such that the equation simplifies, that might help.Let's set ( b_n = a_n + c ). Then,( b_n = a_n + c = 2a_{n-1} + 1 + c = 2(a_{n-1} + c) + 1 - 2c ).To make this a simple geometric sequence, we need the extra constants to cancel out. So,( 1 - 2c = 0 ) => ( c = 1/2 ).Wait, but ( c ) should be an integer if possible, right? Maybe I made a mistake here. Let me try again.Alternatively, perhaps I can solve the recurrence relation using the method for linear nonhomogeneous recurrence relations.The general solution is the sum of the homogeneous solution and a particular solution.First, solve the homogeneous equation ( a_n - 2a_{n-1} = 0 ). The characteristic equation is ( r - 2 = 0 ), so ( r = 2 ). Thus, the homogeneous solution is ( a_n^{(h)} = A*2^n ).Next, find a particular solution. Since the nonhomogeneous term is a constant (1), we can try a constant particular solution ( a_n^{(p)} = B ).Substitute into the recurrence:( B = 2B + 1 ) => ( -B = 1 ) => ( B = -1 ).Therefore, the general solution is ( a_n = A*2^n - 1 ).Now, apply the initial condition ( a_1 = 1 ):( 1 = A*2^1 - 1 ) => ( 1 = 2A - 1 ) => ( 2A = 2 ) => ( A = 1 ).Thus, the general formula is ( a_n = 2^n - 1 ).Okay, that confirms my earlier hypothesis. So, even though I initially noticed the pattern by computing the first few terms, using the method for solving linear recurrence relations gives me the same result in a more systematic way.I think I've got it. The general term is ( 2^n - 1 ).

💡First, I need to determine the amount of milk and water in each cup. Since both cups have the same volume, I'll assume the volume of each cup is ( V ).In the first cup, milk fills ( frac{3}{5} ) of the cup, so the volume of milk is ( frac{3}{5}V ). The remaining volume, ( frac{2}{5}V ), is filled with water.In the second cup, milk fills ( frac{4}{5} ) of the cup, so the volume of milk is ( frac{4}{5}V ). The remaining volume, ( frac{1}{5}V ), is filled with water.Next, I'll calculate the total amount of milk and water when the two mixtures are combined. The total milk is ( frac{3}{5}V + frac{4}{5}V = frac{7}{5}V ). The total water is ( frac{2}{5}V + frac{1}{5}V = frac{3}{5}V ).Finally, to find the ratio of water to milk in the mixture, I'll divide the total water by the total milk:[text{Ratio of water to milk} = frac{frac{3}{5}V}{frac{7}{5}V} = frac{3}{7}]

💡Alright, let's tackle this problem step by step. So, we have a function ( f: mathbb{R} to mathbb{R} ) that satisfies the equation ( f(x^2) f''(x) = f'(x) f'(x^2) ) for all real ( x ). We're given that ( f(1) = 1 ) and ( f'''(1) = 8 ). We need to find ( f'(1) + f''(1) ).First, let's write down what we know:1. The functional equation: ( f(x^2) f''(x) = f'(x) f'(x^2) ).2. ( f(1) = 1 ).3. ( f'''(1) = 8 ).Our goal is to find ( f'(1) + f''(1) ).Since the equation holds for all real ( x ), it must hold when ( x = 1 ). Let's substitute ( x = 1 ) into the equation:( f(1^2) f''(1) = f'(1) f'(1^2) ).Simplifying, since ( 1^2 = 1 ):( f(1) f''(1) = f'(1) f'(1) ).We know ( f(1) = 1 ), so:( 1 cdot f''(1) = [f'(1)]^2 ).Therefore:( f''(1) = [f'(1)]^2 ).So, we have a relationship between ( f''(1) ) and ( f'(1) ). Let's denote ( a = f'(1) ) and ( b = f''(1) ). Then, from the above, we have:( b = a^2 ).So, if we can find ( a ), we can find ( b ), and then compute ( a + b ).Now, we need another equation to relate ( a ) and ( b ). We have ( f'''(1) = 8 ). To get another equation, we can differentiate the original functional equation with respect to ( x ) and then substitute ( x = 1 ).Let's differentiate both sides of the equation ( f(x^2) f''(x) = f'(x) f'(x^2) ) with respect to ( x ).Using the product rule on both sides:Left side: ( frac{d}{dx}[f(x^2) f''(x)] = f'(x^2) cdot 2x cdot f''(x) + f(x^2) cdot f'''(x) ).Right side: ( frac{d}{dx}[f'(x) f'(x^2)] = f''(x) cdot f'(x^2) + f'(x) cdot f''(x^2) cdot 2x ).So, putting it all together:( 2x f'(x^2) f''(x) + f(x^2) f'''(x) = f''(x) f'(x^2) + 2x f'(x) f''(x^2) ).Now, let's substitute ( x = 1 ) into this differentiated equation:Left side: ( 2 cdot 1 cdot f'(1^2) f''(1) + f(1^2) f'''(1) = 2 f'(1) f''(1) + f(1) f'''(1) ).Right side: ( f''(1) f'(1^2) + 2 cdot 1 cdot f'(1) f''(1^2) = f''(1) f'(1) + 2 f'(1) f''(1) ).Simplifying both sides:Left side: ( 2 a b + 1 cdot 8 = 2ab + 8 ).Right side: ( b a + 2 a b = ab + 2ab = 3ab ).So, we have:( 2ab + 8 = 3ab ).Subtracting ( 2ab ) from both sides:( 8 = ab ).But we already know that ( b = a^2 ), so substituting:( 8 = a cdot a^2 = a^3 ).Therefore:( a^3 = 8 ).Taking the cube root of both sides:( a = 2 ).So, ( f'(1) = 2 ).Then, since ( b = a^2 ):( b = 2^2 = 4 ).So, ( f''(1) = 4 ).Therefore, ( f'(1) + f''(1) = 2 + 4 = 6 ).Wait, let me double-check the differentiation step because sometimes signs or coefficients can be tricky.When differentiating the left side: ( frac{d}{dx}[f(x^2) f''(x)] ), we have:- The derivative of ( f(x^2) ) is ( f'(x^2) cdot 2x ).- The derivative of ( f''(x) ) is ( f'''(x) ).So, by the product rule, it's ( f'(x^2) cdot 2x cdot f''(x) + f(x^2) cdot f'''(x) ). That seems correct.On the right side: ( frac{d}{dx}[f'(x) f'(x^2)] ).- The derivative of ( f'(x) ) is ( f''(x) ).- The derivative of ( f'(x^2) ) is ( f''(x^2) cdot 2x ).So, by the product rule, it's ( f''(x) cdot f'(x^2) + f'(x) cdot f''(x^2) cdot 2x ). That also seems correct.Substituting ( x = 1 ), we get:Left side: ( 2 cdot f'(1) cdot f''(1) + f(1) cdot f'''(1) = 2ab + 8 ).Right side: ( f''(1) cdot f'(1) + 2 cdot f'(1) cdot f''(1) = ab + 2ab = 3ab ).So, ( 2ab + 8 = 3ab ) leads to ( 8 = ab ), which with ( b = a^2 ) gives ( a^3 = 8 ), so ( a = 2 ). Then ( b = 4 ). So, ( a + b = 6 ).Therefore, the final answer is 6.**Final Answer**boxed{6}

💡Alright, let's tackle this problem step by step. We have two distinct lines, ( a ) and ( b ), and two distinct planes, ( alpha ) and ( beta ). We need to determine which of the statements A, B, C, or D is true.**Statement A: If ( alpha parallel beta ) and ( a subset alpha ), then ( a parallel beta ).**Okay, so if two planes are parallel, any line in one plane should be parallel to the other plane. Let me visualize this. Imagine two parallel sheets of paper (planes ( alpha ) and ( beta )). If I draw a line on one sheet, that line shouldn't intersect the other sheet because the sheets themselves never meet. So, the line should be parallel to the second plane. That makes sense. I think Statement A is true.**Statement B: If ( a subset alpha ), ( b subset beta ), and ( alpha perp beta ), then ( a perp b ).**Hmm, this one is trickier. If two planes are perpendicular, does that mean every line in one plane is perpendicular to every line in the other plane? I don't think so. For example, take the floor and a wall as two perpendicular planes. If I draw a line along the base of the wall (which is in the wall plane) and a line on the floor that's parallel to the wall, those two lines aren't perpendicular. They could be parallel or intersect at some angle other than 90 degrees. So, Statement B doesn't seem to hold true in all cases. I think B is false.**Statement C: If ( a parallel alpha ) and ( a parallel b ), then ( b parallel alpha ).**Alright, if line ( a ) is parallel to plane ( alpha ), that means ( a ) doesn't intersect ( alpha ) and lies in a direction that's parallel to it. If another line ( b ) is parallel to ( a ), then ( b ) should also be parallel to ( alpha ), right? Because if ( a ) doesn't intersect ( alpha ), and ( b ) is going in the same direction as ( a ), ( b ) shouldn't intersect ( alpha ) either. So, I think Statement C is true.Wait a second, is there a scenario where ( b ) could lie on ( alpha ) instead of being parallel? If ( a ) is parallel to ( alpha ), ( b ) being parallel to ( a ) could either be parallel to ( alpha ) or lie on ( alpha ). But the statement says "then ( b parallel alpha )", which might not always be the case if ( b ) could lie on ( alpha ). Hmm, so maybe Statement C isn't necessarily true. I need to think more about this.If ( a parallel alpha ), then ( a ) doesn't intersect ( alpha ) and isn't lying on ( alpha ). If ( a parallel b ), then ( b ) could be parallel to ( alpha ) or lie on ( alpha ). But if ( b ) lies on ( alpha ), it's not necessarily parallel; it's just part of the plane. So, the conclusion that ( b parallel alpha ) isn't always valid. Therefore, Statement C is false.**Statement D: If ( a parallel alpha ) and ( b parallel alpha ), then ( a parallel b ).**Okay, so both lines ( a ) and ( b ) are parallel to the same plane ( alpha ). Does that mean they have to be parallel to each other? I don't think so. For example, imagine plane ( alpha ) is the floor. Line ( a ) could be a line on a wall running north-south, and line ( b ) could be another line on a different wall running east-west. Both are parallel to the floor, but they aren't parallel to each other; they're actually perpendicular. So, Statement D is false.**Summary:**- **A:** True- **B:** False- **C:** False- **D:** FalseSo, the only true statement is A.

💡First, I recognize that there are 8 identical books that need to be distributed between two locations: the school library and the students.Since the books are identical, the order in which they are distributed doesn't matter. This means that the problem can be approached by determining the number of ways to split the 8 books into two groups: one group in the library and the other group with the students.To find the total number of ways to distribute the books without any restrictions, I consider that each book has two choices: either it goes to the library or it stays with the students. Therefore, for 8 books, the total number of possible distributions is (2^8 = 256).However, the problem specifies that there must be at least one book in the library and at least one book with the students. This means we need to exclude the two scenarios where all books are in one location:1. All 8 books are in the library.2. All 8 books are with the students.Each of these scenarios represents only 1 way, so we subtract them from the total number of distributions:[256 - 2 = 254]Therefore, there are 254 different ways to distribute the 8 identical books such that at least one book is in the library and at least one book is with the students.