💡Alright, so I have this problem about an odd function ( f(x) ) that's defined on the entire real line. It has some periodic properties, and I need to find its expression on the interval [1,5] and determine the range of a real number ( a ) such that the set ( A = { x mid f(x) > a, x in mathbb{R} } ) is not empty. Hmm, okay, let's break this down step by step.First, the function is odd, which means ( f(-x) = -f(x) ) for all ( x ). That's a useful property because it tells me the function is symmetric about the origin. Also, it's given that ( f(x + 2) = -f(x) ). That seems like a functional equation that relates the value of the function at ( x + 2 ) to its value at ( x ). Maybe this can help me find the function's behavior outside the interval where it's explicitly defined.It's also given that for ( -1 leq x leq 1 ), ( f(x) = x^3 ). So, I know the function on this interval, and since it's odd, I can extend it to the interval ( -1 leq x leq 1 ) by reflecting it over the origin. Wait, actually, since ( f(x) ) is already defined as ( x^3 ) on ( -1 leq x leq 1 ), and it's odd, that definition is consistent with ( f(-x) = -f(x) ). So, I don't need to do anything extra for negative values in this interval.Now, the first part asks for the expression of ( f(x) ) when ( x in [1, 5] ). Let's see. I know ( f(x) ) on ( [-1, 1] ), and I have this functional equation ( f(x + 2) = -f(x) ). Maybe I can use this to extend ( f(x) ) beyond ( x = 1 ).Let me think about how to use ( f(x + 2) = -f(x) ). If I let ( x = t - 2 ), then ( f(t) = -f(t - 2) ). So, this tells me that the function at ( t ) is the negative of the function at ( t - 2 ). That seems like a recursive relation. Maybe I can use this to express ( f(x) ) in terms of ( f(x - 2) ), and so on.Let me try to find ( f(x) ) on the interval [1, 3]. If ( x ) is in [1, 3], then ( x - 2 ) is in [-1, 1]. Since I know ( f(x) ) on [-1, 1], I can use the functional equation to find ( f(x) ) on [1, 3].So, for ( x in [1, 3] ), ( x - 2 in [-1, 1] ). Therefore, ( f(x) = -f(x - 2) ). But ( f(x - 2) ) is just ( (x - 2)^3 ) because ( x - 2 ) is in [-1, 1]. So, ( f(x) = - (x - 2)^3 ) for ( x in [1, 3] ).Okay, that takes care of [1, 3]. Now, what about [3, 5]? For ( x in [3, 5] ), ( x - 2 in [1, 3] ). But I just found ( f(x) ) on [1, 3], so I can use the functional equation again.So, for ( x in [3, 5] ), ( f(x) = -f(x - 2) ). But ( x - 2 in [1, 3] ), and ( f(x - 2) = - (x - 2 - 2)^3 = - (x - 4)^3 ). Therefore, ( f(x) = - [ - (x - 4)^3 ] = (x - 4)^3 ).So, putting it all together, on [1, 3), ( f(x) = - (x - 2)^3 ), and on [3, 5], ( f(x) = (x - 4)^3 ). That should be the expression for ( f(x) ) on [1, 5].Wait, let me double-check. For ( x = 1 ), ( f(1) = - (1 - 2)^3 = - (-1)^3 = - (-1) = 1 ). But from the original definition, ( f(1) = 1^3 = 1 ). So, that matches. For ( x = 3 ), from the left side, ( f(3) = - (3 - 2)^3 = -1 ). From the right side, ( f(3) = (3 - 4)^3 = (-1)^3 = -1 ). So, it's continuous at 3, which is good.Similarly, at ( x = 5 ), ( f(5) = (5 - 4)^3 = 1 ). But wait, if I use the functional equation again, ( f(5) = -f(3) = -(-1) = 1 ). That also matches. So, it seems consistent.Okay, so part (1) is done. Now, part (2) is about finding the range of ( a ) such that ( A = { x mid f(x) > a, x in mathbb{R} } ) is not empty. So, ( A ) is non-empty means there exists at least one ( x ) such that ( f(x) > a ). Therefore, ( a ) must be less than the maximum value of ( f(x) ).But wait, what's the maximum value of ( f(x) )? Let's think about it. Since ( f(x) ) is defined as ( x^3 ) on [-1, 1], the maximum there is 1, at ( x = 1 ). On [1, 3], it's ( - (x - 2)^3 ). The maximum of this function on [1, 3] would be at ( x = 1 ), which is 1, and it decreases to -1 at ( x = 3 ). On [3, 5], it's ( (x - 4)^3 ), which goes from -1 at 3 to 1 at 5.So, the function oscillates between 1 and -1 every 2 units. Wait, actually, since ( f(x + 2) = -f(x) ), the function has a period of 4 because ( f(x + 4) = f(x) ). So, it's periodic with period 4, oscillating between 1 and -1.Therefore, the maximum value of ( f(x) ) is 1, and the minimum is -1. So, for ( A ) to be non-empty, ( a ) must be less than 1 because if ( a ) is less than 1, there will be points where ( f(x) > a ). If ( a ) is equal to 1, then ( f(x) > a ) would only be true at points where ( f(x) > 1 ), but since the maximum is 1, there are no such points. Similarly, if ( a ) is greater than 1, there are no points where ( f(x) > a ).Wait, but hold on. The function is periodic and oscillates between 1 and -1. So, the maximum value is 1, and it's achieved at multiple points: ( x = 1, 5, 9, ) etc., and similarly at negative counterparts because it's odd. So, the supremum of ( f(x) ) is 1, and the infimum is -1.Therefore, for ( A ) to be non-empty, ( a ) must be less than 1. If ( a ) is equal to 1, ( A ) would be empty because ( f(x) ) never exceeds 1. If ( a ) is greater than 1, it's also empty. So, the range of ( a ) is all real numbers less than 1.But wait, let me think again. Since ( f(x) ) is defined for all real numbers, and it's periodic, it keeps oscillating between 1 and -1. So, for any ( a < 1 ), there will be intervals where ( f(x) > a ). For example, near the peaks at ( x = 1, 5, 9, ) etc., the function reaches 1, so just below those points, it's slightly less than 1, but still greater than any ( a < 1 ).Therefore, the range of ( a ) is ( (-infty, 1) ).Wait, but is that correct? Let me consider ( a = 0 ). Then ( A ) would be all ( x ) where ( f(x) > 0 ). Since ( f(x) ) is positive on intervals like ( (0, 2) ), ( (4, 6) ), etc., ( A ) is definitely non-empty. Similarly, if ( a = 0.5 ), ( A ) would include points near 1, 5, etc., where ( f(x) ) is greater than 0.5. So, yeah, as long as ( a < 1 ), ( A ) is non-empty.If ( a = 1 ), then ( f(x) > 1 ) would never be true because the maximum of ( f(x) ) is 1. Similarly, for ( a > 1 ), it's also never true. So, the range of ( a ) is indeed all real numbers less than 1.Okay, so summarizing:1. For ( x in [1, 5] ), ( f(x) ) is defined piecewise: - ( f(x) = - (x - 2)^3 ) for ( x in [1, 3) ) - ( f(x) = (x - 4)^3 ) for ( x in [3, 5] )2. The range of ( a ) such that ( A ) is non-empty is ( (-infty, 1) ).I think that's it. Let me just visualize the function to make sure I didn't make any mistakes. From ( x = -1 ) to 1, it's ( x^3 ). Then from 1 to 3, it's the negative of ( (x - 2)^3 ), which would be a downward cubic curve from 1 to -1. Then from 3 to 5, it's ( (x - 4)^3 ), which goes from -1 back up to 1. This creates a wave-like pattern that repeats every 4 units. Yeah, that makes sense with the functional equation ( f(x + 4) = f(x) ).So, I think my reasoning is correct.**Final Answer**1. The expression for ( f(x) ) on ( [1, 5] ) is: [ f(x) = begin{cases} -(x - 2)^3 & text{if } 1 leq x < 3, (x - 4)^3 & text{if } 3 leq x leq 5. end{cases} ] 2. The range of ( a ) is ( (-infty, 1) ).So, the final answers are:1. (boxed{f(x) = begin{cases} -(x - 2)^3 & text{for } 1 leq x < 3, (x - 4)^3 & text{for } 3 leq x leq 5. end{cases}})2. (boxed{(-infty, 1)})

💡Alright, so I have this equation here: a over 2 equals b over 3, and both a and b are not zero. I need to figure out which of the transformations A, B, C, or D is incorrect. Hmm, okay, let's take it step by step.First, let's write down the given equation clearly:[frac{a}{2} = frac{b}{3}]I think the best way to approach this is to manipulate the equation and see which of the options doesn't match. So, starting with the given equation, maybe I can cross-multiply to get rid of the fractions. If I do that, I multiply both sides by 2 and 3, which gives me:[3a = 2b]Okay, so that's a simpler equation without fractions. Let's see how this relates to the options given.Looking at option A: (frac{a}{b} = frac{2}{3}). Hmm, if I take the equation I derived, (3a = 2b), and divide both sides by b, I get:[frac{3a}{b} = 2]Then, dividing both sides by 3:[frac{a}{b} = frac{2}{3}]So that matches option A. Good, so A is correct.Moving on to option B: (2a = 3b). Wait a second, from my earlier step, I have (3a = 2b). If I try to rearrange this to get 2a on one side, I would multiply both sides by 2/3, right?[2a = frac{4}{3}b]But that's not the same as (2a = 3b). So this seems off. Maybe I made a mistake? Let me double-check.Starting again from (3a = 2b), if I want to solve for a in terms of b, I divide both sides by 3:[a = frac{2}{3}b]Then, multiplying both sides by 2:[2a = frac{4}{3}b]Yeah, that's still not (2a = 3b). So option B doesn't seem to follow from the original equation. Maybe B is incorrect?But let's not jump to conclusions yet. Let's check the other options to be sure.Option C: (frac{b}{a} = frac{3}{2}). Starting from (3a = 2b), if I divide both sides by a, I get:[3 = frac{2b}{a}]Then, dividing both sides by 2:[frac{3}{2} = frac{b}{a}]So that's the same as (frac{b}{a} = frac{3}{2}), which matches option C. So C is correct.Option D: (3a = 2b). Well, that's exactly the equation I derived from the original equation by cross-multiplying. So D is definitely correct.So, going back to option B: (2a = 3b). From my calculations, I ended up with (2a = frac{4}{3}b), which is not the same as (2a = 3b). Therefore, option B is incorrect.Wait, but just to make sure, maybe there's another way to look at it. Let's try solving the original equation for a and then plug it into option B.From (frac{a}{2} = frac{b}{3}), solving for a:[a = frac{2}{3}b]Now, plug this into option B: (2a = 3b).Substituting a:[2 times frac{2}{3}b = 3b]Simplify the left side:[frac{4}{3}b = 3b]Hmm, that's not true unless b is zero, but the problem states that b is not zero. So this confirms that option B is incorrect.Just to recap:- A is correct because (frac{a}{b} = frac{2}{3}) follows from the original equation.- B is incorrect because substituting a in terms of b leads to a contradiction.- C is correct because (frac{b}{a} = frac{3}{2}) is consistent with the original equation.- D is correct because it's the direct result of cross-multiplying the original equation.So, after carefully checking each option, I'm confident that option B is the incorrect transformation.

💡Okay, so I have this integral to compute using Cauchy's integral formula. The integral is:int_{C} frac{e^{z^{2}}}{z^{2}-6 z} dzAnd there are three different contours given:1) C:|z-2|=12) C:|z-2|=33) C:|z-2|=5First, I need to recall what Cauchy's integral formula is. From what I remember, it's a formula in complex analysis that relates the value of a holomorphic function at a point inside a contour to the integral of the function around that contour. The formula is:int_C frac{f(z)}{z - z_0} dz = 2 pi i f(z_0)where f is analytic inside and on the contour C, and z_0 is a point inside C.So, to apply this formula, I need to see if the integrand can be expressed in the form frac{f(z)}{z - z_0}, where f(z) is analytic inside the contour, and z_0 is a singularity inside the contour.Looking at the integrand:frac{e^{z^{2}}}{z^{2}-6 z}I can factor the denominator:z^{2} - 6z = z(z - 6)So, the integrand becomes:frac{e^{z^{2}}}{z(z - 6)}This means the function has singularities at z = 0 and z = 6. These are the points where the denominator becomes zero, so the function is not analytic there.Now, I need to check for each contour whether these singularities lie inside the contour.Let's start with the first contour: C:|z-2|=1.1) **Contour C:|z-2|=1**This is a circle centered at z=2 with radius 1. So, the points inside this circle are those z such that the distance from z to 2 is less than 1.Let's check if z=0 is inside this circle. The distance from 0 to 2 is 2, which is greater than the radius 1. So, z=0 is outside this contour.Similarly, z=6 is at a distance of 4 from 2, which is also outside the contour.Therefore, neither of the singularities z=0 nor z=6 lies inside this contour.Since the integrand is analytic everywhere inside and on the contour, by Cauchy's theorem, the integral is zero.So, for the first contour, the integral is 0.2) **Contour C:|z-2|=3**This is a circle centered at z=2 with radius 3. So, the points inside this circle are those z such that the distance from z to 2 is less than 3.Check if z=0 is inside this circle. The distance from 0 to 2 is 2, which is less than 3. So, z=0 is inside this contour.Check if z=6 is inside this circle. The distance from 6 to 2 is 4, which is greater than 3. So, z=6 is outside this contour.Therefore, only z=0 is inside the contour.Now, to apply Cauchy's integral formula, I need to express the integrand in the form frac{f(z)}{z - z_0} where z_0 is inside the contour.Looking at the integrand:frac{e^{z^{2}}}{z(z - 6)} = frac{e^{z^{2}}}{z} cdot frac{1}{z - 6}But I need to write it as frac{f(z)}{z - z_0}. Let me see.Alternatively, I can perform partial fraction decomposition on the denominator:frac{1}{z(z - 6)} = frac{A}{z} + frac{B}{z - 6}Solving for A and B:Multiply both sides by z(z - 6):1 = A(z - 6) + B zSet z = 0:1 = A(-6) + B(0) implies A = -frac{1}{6}Set z = 6:1 = A(0) + B(6) implies B = frac{1}{6}So,frac{1}{z(z - 6)} = -frac{1}{6z} + frac{1}{6(z - 6)}Therefore, the integrand becomes:frac{e^{z^{2}}}{z(z - 6)} = -frac{e^{z^{2}}}{6z} + frac{e^{z^{2}}}{6(z - 6)}So, the integral becomes:int_{|z-2|=3} left( -frac{e^{z^{2}}}{6z} + frac{e^{z^{2}}}{6(z - 6)} right) dzNow, split the integral into two parts:-frac{1}{6} int_{|z-2|=3} frac{e^{z^{2}}}{z} dz + frac{1}{6} int_{|z-2|=3} frac{e^{z^{2}}}{z - 6} dzNow, let's evaluate each integral separately.First integral:-frac{1}{6} int_{|z-2|=3} frac{e^{z^{2}}}{z} dzHere, z_0 = 0, which is inside the contour. So, by Cauchy's integral formula:int_{C} frac{f(z)}{z - z_0} dz = 2 pi i f(z_0)Here, f(z) = e^{z^{2}}, so f(0) = e^{0} = 1.Therefore,int_{|z-2|=3} frac{e^{z^{2}}}{z} dz = 2 pi i cdot 1 = 2 pi iSo, the first integral is:-frac{1}{6} cdot 2 pi i = -frac{pi i}{3}Second integral:frac{1}{6} int_{|z-2|=3} frac{e^{z^{2}}}{z - 6} dzHere, z_0 = 6, but z=6 is outside the contour |z-2|=3 (distance from 6 to 2 is 4, which is greater than 3). Therefore, by Cauchy's theorem, this integral is zero.So, the second integral is:frac{1}{6} cdot 0 = 0Adding both integrals together:-frac{pi i}{3} + 0 = -frac{pi i}{3}Therefore, the integral over the contour |z-2|=3 is -frac{pi i}{3}.3) **Contour C:|z-2|=5**This is a circle centered at z=2 with radius 5. So, the points inside this circle are those z such that the distance from z to 2 is less than 5.Check if z=0 is inside this circle. The distance from 0 to 2 is 2, which is less than 5. So, z=0 is inside this contour.Check if z=6 is inside this circle. The distance from 6 to 2 is 4, which is less than 5. So, z=6 is also inside this contour.Therefore, both singularities z=0 and z=6 lie inside the contour.Now, to compute the integral, I can again use partial fractions as before:frac{1}{z(z - 6)} = -frac{1}{6z} + frac{1}{6(z - 6)}So, the integrand becomes:frac{e^{z^{2}}}{z(z - 6)} = -frac{e^{z^{2}}}{6z} + frac{e^{z^{2}}}{6(z - 6)}Therefore, the integral becomes:int_{|z-2|=5} left( -frac{e^{z^{2}}}{6z} + frac{e^{z^{2}}}{6(z - 6)} right) dzSplit the integral into two parts:-frac{1}{6} int_{|z-2|=5} frac{e^{z^{2}}}{z} dz + frac{1}{6} int_{|z-2|=5} frac{e^{z^{2}}}{z - 6} dzNow, evaluate each integral separately.First integral:-frac{1}{6} int_{|z-2|=5} frac{e^{z^{2}}}{z} dzHere, z_0 = 0, which is inside the contour. So, by Cauchy's integral formula:int_{C} frac{f(z)}{z - z_0} dz = 2 pi i f(z_0)Here, f(z) = e^{z^{2}}, so f(0) = e^{0} = 1.Therefore,int_{|z-2|=5} frac{e^{z^{2}}}{z} dz = 2 pi i cdot 1 = 2 pi iSo, the first integral is:-frac{1}{6} cdot 2 pi i = -frac{pi i}{3}Second integral:frac{1}{6} int_{|z-2|=5} frac{e^{z^{2}}}{z - 6} dzHere, z_0 = 6, which is inside the contour. So, by Cauchy's integral formula:int_{C} frac{f(z)}{z - z_0} dz = 2 pi i f(z_0)Here, f(z) = e^{z^{2}}, so f(6) = e^{6^{2}} = e^{36}.Therefore,int_{|z-2|=5} frac{e^{z^{2}}}{z - 6} dz = 2 pi i cdot e^{36}So, the second integral is:frac{1}{6} cdot 2 pi i e^{36} = frac{pi i e^{36}}{3}Adding both integrals together:-frac{pi i}{3} + frac{pi i e^{36}}{3} = frac{pi i (e^{36} - 1)}{3}Therefore, the integral over the contour |z-2|=5 is frac{pi i (e^{36} - 1)}{3}.Wait, but in the second part, when I had |z-2|=3, I got -frac{pi i}{3}, and in the third part, I have frac{pi i (e^{36} - 1)}{3}. That seems consistent because in the third contour, both singularities are inside, so both contribute to the integral.Let me just double-check my partial fractions:Starting with:frac{1}{z(z - 6)} = frac{A}{z} + frac{B}{z - 6}Multiplying both sides by z(z - 6):1 = A(z - 6) + B zSetting z = 0:1 = A(-6) implies A = -frac{1}{6}Setting z = 6:1 = B(6) implies B = frac{1}{6}Yes, that's correct. So, the partial fractions are correct.Also, when applying Cauchy's formula, for each singularity inside the contour, we get a term. So, in the third case, both z=0 and z=6 are inside, so both contribute, whereas in the second case, only z=0 is inside.Therefore, the results seem consistent.So, summarizing:1) For |z-2|=1, the integral is 0.2) For |z-2|=3, the integral is -frac{pi i}{3}.3) For |z-2|=5, the integral is frac{pi i (e^{36} - 1)}{3}.I think that's all. I don't see any mistakes in the reasoning. Maybe I should check if the function e^{z^2} is analytic everywhere, which it is, since the exponential function is entire. So, no issues there.Also, the partial fractions were correctly applied, and the distances from the center of the contour to the singularities were correctly calculated.So, I think the answers are correct.

💡Okay, so I have this probability problem here about tossing a fair coin four times. Each toss can result in either heads or tails, each with a probability of 1/2. They define a sequence {a_n} where each term is 1 if it's heads and -1 if it's tails. Then, S_n is the sum of the first n terms of this sequence. The question is asking for the probability that S_4 is greater than 0. The options are A: 1/16, B: 1/4, C: 5/16, D: 1/2.Hmm, let me try to understand this. So, S_4 is the sum of four terms, each of which is either 1 or -1. So, S_4 can range from -4 to 4, right? Because if all four are tails, S_4 would be -4, and if all four are heads, it would be 4.But we need the probability that S_4 is greater than 0. So, that means S_4 can be 1, 2, 3, or 4. Wait, but actually, since each term is either 1 or -1, the sum S_4 must be an even number? Or is that not necessarily the case?Wait, no. Let's see: each a_n is either 1 or -1, so each term is odd. The sum of four odd numbers. Let me recall: the sum of an even number of odd numbers is even. So, four odd numbers added together will give an even number. So, S_4 must be even. Therefore, S_4 can be -4, -2, 0, 2, 4.So, the possible values for S_4 are -4, -2, 0, 2, 4. So, S_4 > 0 would mean S_4 is either 2 or 4. So, we need the probability that S_4 is 2 or 4.Alternatively, since S_4 is the sum of four independent random variables each taking values 1 and -1 with probability 1/2, this is equivalent to a simple symmetric random walk after four steps. The probability that the walk is positive after four steps.Alternatively, another way to think about it is in terms of the number of heads and tails. Since each head contributes +1 and each tail contributes -1, the sum S_4 is equal to (number of heads) - (number of tails). Let me denote the number of heads as H and the number of tails as T. Since we have four tosses, H + T = 4. Therefore, S_4 = H - T = H - (4 - H) = 2H - 4.So, S_4 = 2H - 4. We want S_4 > 0, so 2H - 4 > 0 => 2H > 4 => H > 2. So, H must be greater than 2, meaning H can be 3 or 4.Therefore, the probability that S_4 > 0 is equal to the probability that we get 3 or 4 heads in four tosses.Since each toss is independent and the probability of heads is 1/2, this is a binomial distribution problem. The probability of getting exactly k heads in n tosses is C(n, k) * (1/2)^k * (1/2)^{n - k} = C(n, k) * (1/2)^n.So, for n = 4, the probability of getting exactly 3 heads is C(4, 3) * (1/2)^4, and the probability of getting exactly 4 heads is C(4, 4) * (1/2)^4.Calculating these:C(4, 3) is 4, so 4 * (1/16) = 4/16.C(4, 4) is 1, so 1 * (1/16) = 1/16.Adding these together: 4/16 + 1/16 = 5/16.So, the probability is 5/16, which is option C.Wait, let me just make sure I didn't make a mistake. So, S_4 > 0 implies H > 2, so H = 3 or 4. Calculated the probabilities for H=3 and H=4, added them up, got 5/16. That seems right.Alternatively, maybe I can think of all possible outcomes. Since each toss is independent, there are 2^4 = 16 possible outcomes. Each outcome is equally likely with probability 1/16.Now, how many of these outcomes result in S_4 > 0? As we saw, that's when H=3 or H=4.Number of outcomes with H=3: C(4, 3) = 4.Number of outcomes with H=4: C(4, 4) = 1.Total favorable outcomes: 4 + 1 = 5.Therefore, the probability is 5/16, which is the same as before.Alternatively, I can list all possible outcomes and count them, but that might take too long. But for four tosses, it's manageable.Each outcome is a sequence of H and T. Let me list all 16:1. HHHH2. HHHT3. HHTH4. HTHH5. THHH6. HHTT7. HTHT8. HTTH9. THHT10. THTH11. TTHH12. HTTT13. THTT14. TTHT15. TTTH16. TTTTNow, let's compute S_4 for each:1. HHHH: 4 heads, S_4 = 42. HHHT: 3 heads, 1 tail, S_4 = 3 - 1 = 23. HHTH: same as above, 3 heads, 1 tail, S_4 = 24. HTHH: same, 3 heads, 1 tail, S_4 = 25. THHH: same, 3 heads, 1 tail, S_4 = 26. HHTT: 2 heads, 2 tails, S_4 = 07. HTHT: same, 2 heads, 2 tails, S_4 = 08. HTTH: same, 2 heads, 2 tails, S_4 = 09. THHT: same, 2 heads, 2 tails, S_4 = 010. THTH: same, 2 heads, 2 tails, S_4 = 011. TTHH: same, 2 heads, 2 tails, S_4 = 012. HTTT: 1 head, 3 tails, S_4 = -213. THTT: same, 1 head, 3 tails, S_4 = -214. TTHT: same, 1 head, 3 tails, S_4 = -215. TTTH: same, 1 head, 3 tails, S_4 = -216. TTTT: 0 heads, 4 tails, S_4 = -4Now, let's count how many have S_4 > 0. Looking at the list:1. S_4 = 4 > 02. S_4 = 2 > 03. S_4 = 2 > 04. S_4 = 2 > 05. S_4 = 2 > 06. S_4 = 07. S_4 = 08. S_4 = 09. S_4 = 010. S_4 = 011. S_4 = 012. S_4 = -213. S_4 = -214. S_4 = -215. S_4 = -216. S_4 = -4So, the first five outcomes have S_4 > 0. That's 5 outcomes. So, 5 out of 16, which is 5/16. So, that's consistent with my earlier calculation.Therefore, the probability is 5/16, which is option C.I think that's solid. I approached it in two different ways: using the binomial probability formula and also by enumerating all possible outcomes. Both methods gave me the same answer, so I feel confident that 5/16 is correct.**Final Answer**boxed{C}

💡Okay, so I need to compute the fourth power of this matrix: (begin{pmatrix} 2 & -2 1 & 1 end{pmatrix}). Hmm, that sounds a bit intimidating at first, but I think if I break it down step by step, it'll make more sense. First, I remember that to compute a matrix raised to a power, I can do it by multiplying the matrix by itself multiple times. Since it's the fourth power, I guess I need to multiply the matrix by itself four times. But that might take a while, so maybe there's a smarter way to do it. Maybe I can compute the square first and then square the result? That might save some steps. Let me try that.So, let's start by computing the square of the matrix. The matrix is:[A = begin{pmatrix} 2 & -2 1 & 1 end{pmatrix}]To compute (A^2), I need to multiply matrix (A) by itself. I remember that matrix multiplication is done row by column. So, the element in the first row and first column of the resulting matrix is obtained by multiplying the elements of the first row of the first matrix by the corresponding elements of the first column of the second matrix and then adding them up. Similarly, the element in the first row and second column is obtained by multiplying the first row of the first matrix by the second column of the second matrix, and so on.Let me write that out step by step.First, the element at position (1,1) of (A^2) is:[(2 times 2) + (-2 times 1) = 4 + (-2) = 2]Next, the element at position (1,2) of (A^2) is:[(2 times -2) + (-2 times 1) = -4 + (-2) = -6]Moving on to the second row. The element at position (2,1) is:[(1 times 2) + (1 times 1) = 2 + 1 = 3]And the element at position (2,2) is:[(1 times -2) + (1 times 1) = -2 + 1 = -1]So putting it all together, the square of matrix (A) is:[A^2 = begin{pmatrix} 2 & -6 3 & -1 end{pmatrix}]Okay, that wasn't too bad. Now, I need to compute (A^4), which is ((A^2)^2). So, I need to square the matrix (A^2) that I just found. Let's denote (B = A^2 = begin{pmatrix} 2 & -6 3 & -1 end{pmatrix}), so I need to compute (B^2).Again, I'll use the same method for matrix multiplication. Let's compute each element of (B^2).Starting with the element at position (1,1):[(2 times 2) + (-6 times 3) = 4 + (-18) = -14]Next, the element at position (1,2):[(2 times -6) + (-6 times -1) = -12 + 6 = -6]Now, the element at position (2,1):[(3 times 2) + (-1 times 3) = 6 + (-3) = 3]And finally, the element at position (2,2):[(3 times -6) + (-1 times -1) = -18 + 1 = -17]Putting these together, the matrix (B^2) is:[B^2 = begin{pmatrix} -14 & -6 3 & -17 end{pmatrix}]So, since (B = A^2), then (B^2 = (A^2)^2 = A^4). Therefore, the fourth power of matrix (A) is:[A^4 = begin{pmatrix} -14 & -6 3 & -17 end{pmatrix}]Wait a second, let me double-check my calculations to make sure I didn't make any mistakes. It's easy to mix up signs or multiplication steps when dealing with negative numbers.Starting with (A^2):- (1,1): 2*2 + (-2)*1 = 4 - 2 = 2 ✔️- (1,2): 2*(-2) + (-2)*1 = -4 - 2 = -6 ✔️- (2,1): 1*2 + 1*1 = 2 + 1 = 3 ✔️- (2,2): 1*(-2) + 1*1 = -2 + 1 = -1 ✔️Looks good. Now, (B^2 = (A^2)^2):- (1,1): 2*2 + (-6)*3 = 4 - 18 = -14 ✔️- (1,2): 2*(-6) + (-6)*(-1) = -12 + 6 = -6 ✔️- (2,1): 3*2 + (-1)*3 = 6 - 3 = 3 ✔️- (2,2): 3*(-6) + (-1)*(-1) = -18 + 1 = -17 ✔️Everything checks out. So, I'm confident that the fourth power of the original matrix is indeed (begin{pmatrix} -14 & -6 3 & -17 end{pmatrix}).I wonder if there's another way to compute this, maybe by diagonalizing the matrix or using eigenvalues. That might be a more advanced method, but it could be interesting to explore. However, since I already have the answer using straightforward multiplication, and it seems correct, I think this is sufficient for now.Another thought: maybe I can compute (A^3) and then multiply by (A) to get (A^4). Let me see if that gives the same result as (A^2) squared.First, compute (A^3 = A^2 times A). We have (A^2 = begin{pmatrix} 2 & -6 3 & -1 end{pmatrix}) and (A = begin{pmatrix} 2 & -2 1 & 1 end{pmatrix}).Compute each element of (A^3):- (1,1): 2*2 + (-6)*1 = 4 - 6 = -2- (1,2): 2*(-2) + (-6)*1 = -4 - 6 = -10- (2,1): 3*2 + (-1)*1 = 6 - 1 = 5- (2,2): 3*(-2) + (-1)*1 = -6 - 1 = -7So, (A^3 = begin{pmatrix} -2 & -10 5 & -7 end{pmatrix}).Now, compute (A^4 = A^3 times A):- (1,1): (-2)*2 + (-10)*1 = -4 - 10 = -14- (1,2): (-2)*(-2) + (-10)*1 = 4 - 10 = -6- (2,1): 5*2 + (-7)*1 = 10 - 7 = 3- (2,2): 5*(-2) + (-7)*1 = -10 - 7 = -17So, (A^4 = begin{pmatrix} -14 & -6 3 & -17 end{pmatrix}), which matches the result I got earlier by squaring (A^2). That’s a good consistency check.I guess both methods work, and it's reassuring that they give the same result. It might be more efficient to compute (A^2) and then square it rather than computing each power step by step, especially for higher exponents. But for the fourth power, either way is manageable.Just to recap, the steps I took were:1. Compute (A^2) by multiplying (A) by itself.2. Then, compute (A^4) by squaring (A^2).3. Alternatively, compute (A^3) by multiplying (A^2) by (A), and then compute (A^4) by multiplying (A^3) by (A).4. Verified both methods give the same result.I think that covers it. I don't see any mistakes in my calculations, and both approaches confirm the same answer. So, I'm pretty confident that the fourth power of the matrix is correct.**Final Answer**The fourth power of the matrix is (boxed{begin{pmatrix} -14 & -6 3 & -17 end{pmatrix}}).

💡Okay, so I have this problem here: "What is the probability that a randomly drawn positive factor of 120 is less than 9?" And there are multiple-choice answers: (A) 1/16, (B) 7/16, (C) 3/8, (D) 1/2, (E) 5/16. Hmm, I need to figure this out step by step.First, I remember that to find the probability, I need two things: the number of favorable outcomes and the total number of possible outcomes. In this case, the total number of possible outcomes is the total number of positive factors of 120. The favorable outcomes are the factors of 120 that are less than 9.So, let me start by finding all the positive factors of 120. To do that, I think I should first find the prime factorization of 120. I remember that prime factorization is breaking down a number into its prime number components. Let me try that.Starting with 120, I can divide by 2: 120 ÷ 2 = 60. Then 60 ÷ 2 = 30, and 30 ÷ 2 = 15. Now, 15 isn't divisible by 2, so I move to the next prime number, which is 3. 15 ÷ 3 = 5. Then, 5 is a prime number itself. So, putting it all together, the prime factorization of 120 is 2^3 * 3^1 * 5^1.Now, to find the total number of positive factors, I can use the formula where you add 1 to each of the exponents in the prime factorization and then multiply them together. So, for 120, that would be (3+1)*(1+1)*(1+1) = 4*2*2 = 16. So, there are 16 positive factors of 120 in total.Alright, so the total number of possible outcomes is 16. Now, I need to find how many of these factors are less than 9. Let me list all the factors of 120 and then count how many are less than 9.To list the factors, I can pair them up. Starting from 1 and 120: 1*120=120. Then 2*60=120, 3*40=120, 4*30=120, 5*24=120, 6*20=120, 8*15=120, 10*12=120. Wait, let me make sure I have all the factors:1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.Let me count them: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Yep, that's 16 factors. Now, I need to pick out the ones less than 9.Looking at the list: 1, 2, 3, 4, 5, 6, 8. Let's see, 1 is less than 9, 2 is, 3 is, 4 is, 5 is, 6 is, 8 is. What about 10? 10 is not less than 9, so we stop there. So, the factors less than 9 are: 1, 2, 3, 4, 5, 6, 8. That's 7 factors.So, the number of favorable outcomes is 7, and the total number of possible outcomes is 16. Therefore, the probability is 7/16.Wait, let me double-check to make sure I didn't miss any factors or count incorrectly. Let me list the factors again and mark which are less than 9:1 - yes2 - yes3 - yes4 - yes5 - yes6 - yes8 - yes10 - no12 - no15 - no20 - no24 - no30 - no40 - no60 - no120 - noYep, that's 7 factors less than 9. So, 7 favorable out of 16 total. So, the probability is 7/16.Looking back at the options, (B) is 7/16, so that should be the answer.I think that's solid. I don't see any mistakes in my reasoning. I found the prime factors, calculated the total number of factors, listed them all out, counted the ones less than 9, and then formed the probability. Everything checks out.**Final Answer**boxed{B}