💡Okay, so I have this complex number problem to solve. The question is about finding which quadrant the point corresponding to the complex number ( z = frac{4i}{(1 - i)^2} + i^{2019} ) is located in the complex plane. The options are the first, second, third, or fourth quadrant. Let me try to break this down step by step.First, I remember that complex numbers can be represented in the form ( a + bi ), where ( a ) is the real part and ( b ) is the imaginary part. The position of the point ( (a, b) ) in the complex plane determines the quadrant. So, I need to simplify the given expression for ( z ) and find its real and imaginary parts.Looking at the expression, it has two parts: ( frac{4i}{(1 - i)^2} ) and ( i^{2019} ). I should handle each part separately and then combine them.Starting with the first part: ( frac{4i}{(1 - i)^2} ). To simplify this, I think I need to compute ( (1 - i)^2 ) first. Let me expand that.( (1 - i)^2 = (1)^2 - 2 times 1 times i + (i)^2 = 1 - 2i + i^2 ).I know that ( i^2 = -1 ), so substituting that in:( 1 - 2i + (-1) = 1 - 2i - 1 = (1 - 1) - 2i = 0 - 2i = -2i ).So, ( (1 - i)^2 = -2i ). That simplifies the denominator. Now, the first part becomes:( frac{4i}{-2i} ).Hmm, let me compute that. The ( i ) in the numerator and denominator should cancel out, right?( frac{4i}{-2i} = frac{4}{-2} times frac{i}{i} = -2 times 1 = -2 ).Wait, is that correct? Because ( frac{i}{i} = 1 ), so yes, it's just ( -2 ). So, the first part simplifies to ( -2 ).Now, moving on to the second part: ( i^{2019} ). I remember that the powers of ( i ) cycle every 4. So, ( i^1 = i ), ( i^2 = -1 ), ( i^3 = -i ), ( i^4 = 1 ), and then it repeats. So, to find ( i^{2019} ), I can divide 2019 by 4 and find the remainder.Let me compute ( 2019 div 4 ). 4 times 504 is 2016, so 2019 - 2016 = 3. So, the remainder is 3. Therefore, ( i^{2019} = i^{4 times 504 + 3} = (i^4)^{504} times i^3 = 1^{504} times (-i) = -i ).So, ( i^{2019} = -i ).Now, combining both parts, ( z = -2 + (-i) = -2 - i ).So, in the complex plane, this corresponds to the point ( (-2, -1) ). Now, I need to figure out which quadrant this point is in.Quadrants in the complex plane are similar to the standard Cartesian plane:- First quadrant: both real and imaginary parts positive.- Second quadrant: real part negative, imaginary part positive.- Third quadrant: both real and imaginary parts negative.- Fourth quadrant: real part positive, imaginary part negative.Looking at our point ( (-2, -1) ), both the real part (-2) and the imaginary part (-1) are negative. Therefore, this point is in the third quadrant.Let me just double-check my steps to make sure I didn't make a mistake.1. Calculated ( (1 - i)^2 ) correctly: expanded it, substituted ( i^2 = -1 ), got -2i. That seems right.2. Divided 4i by -2i: canceled out the i's, got -2. That makes sense.3. Calculated ( i^{2019} ): found the remainder when 2019 is divided by 4, which is 3, so ( i^{2019} = i^3 = -i ). That seems correct.4. Combined both parts: -2 - i, which is (-2, -1). Both negative, so third quadrant.I think that's all correct. So, the answer should be the third quadrant, which is option C.**Final Answer**boxed{C}

💡Okay, so I have this function g(x) = 5x - 7, and I need to find the value of 'a' where g(a) equals zero. Hmm, let me think about how to approach this. First, I remember that when a function equals zero, it's called the root or the zero of the function. So, I need to find the value of 'a' that makes g(a) = 0. That means I should set up the equation 5a - 7 = 0 and solve for 'a'. Alright, let's write that down: 5a - 7 = 0. Now, to solve for 'a', I need to isolate it on one side of the equation. I'll start by getting rid of the -7 on the left side. To do that, I can add 7 to both sides of the equation. So, adding 7 to both sides: 5a - 7 + 7 = 0 + 7. Simplifying that, the -7 and +7 on the left side cancel each other out, leaving me with 5a = 7. Now, I need to solve for 'a', which means I need to get rid of the coefficient 5 that's multiplying 'a'. To do this, I'll divide both sides of the equation by 5. Dividing both sides by 5: (5a)/5 = 7/5. On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving just 'a'. So, a = 7/5. Wait, let me double-check that. If I plug a = 7/5 back into the original function, does it equal zero? Let's see: g(7/5) = 5*(7/5) - 7. The 5 and the denominator 5 cancel out, so that's 7 - 7, which equals 0. Yep, that checks out. So, the value of 'a' that makes g(a) = 0 is 7/5. I think that's the correct answer.

💡First, I need to determine the largest divisor of 462 that is also a factor of 231. To do this, I'll start by finding the prime factors of both numbers.For 231:- 231 divided by 3 is 77.- 77 divided by 7 is 11.- 11 is a prime number.So, the prime factors of 231 are 3, 7, and 11.Next, for 462:- 462 divided by 2 is 231.- 231 divided by 3 is 77.- 77 divided by 7 is 11.- 11 is a prime number.So, the prime factors of 462 are 2, 3, 7, and 11.Now, I'll identify the common prime factors between 231 and 462, which are 3, 7, and 11.To find the greatest common divisor (GCD), I'll multiply these common prime factors together:3 × 7 × 11 = 231.Therefore, the largest divisor of 462 that is also a factor of 231 is 231.

💡First, I need to determine the complement of set ( A ) relative to the universe ( U ). The complement of ( A ), denoted as ( A^c ), consists of all elements in ( U ) that are not in ( A ).Given:- ( U = {1, 2, 3, 4, 5} )- ( A = {1, 2, 3} )Calculating the complement:[A^c = U setminus A = {4, 5}]Next, I need to find the intersection of ( A^c ) and set ( B ). The intersection of two sets includes only the elements that are common to both sets.Given:- ( B = {1, 4} )Calculating the intersection:[A^c cap B = {4, 5} cap {1, 4} = {4}]Therefore, the intersection of the complement of ( A ) relative to ( U ) and set ( B ) is ({4}).

💡First, I identify the amount of dog food consumed by each dog. The first dog eats 1.5 cups daily.Next, the second dog eats twice as much as the first dog. So, I calculate 1.5 cups multiplied by 2, which equals 3 cups.Then, the third dog eats 2.5 cups more than the second dog. Adding 2.5 cups to the second dog's consumption, I find that the third dog eats 5.5 cups.Finally, I sum up the food consumed by all three dogs: 1.5 cups + 3 cups + 5.5 cups equals a total of 10 cups.Therefore, Hannah should prepare 10 cups of dog food each day for her three dogs.

💡Okay, so I need to figure out what binom{150}{1} is. Hmm, I remember that binomial coefficients are used in combinations, right? They tell us how many ways we can choose a certain number of items from a larger set without considering the order. The formula for combinations is binom{n}{k} = frac{n!}{k!(n-k)!}, where "n" is the total number of items, and "k" is the number of items we want to choose.In this case, n is 150 and k is 1. So plugging those numbers into the formula, I get:binom{150}{1} = frac{150!}{1!(150-1)!}Simplifying the denominator, 150 minus 1 is 149, so it becomes:frac{150!}{1! cdot 149!}Now, I know that 1! is just 1, so that part is easy. The tricky part is dealing with the factorials. Factorials can get really big, but maybe there's a way to simplify this without calculating the entire thing.I recall that 150! means 150 multiplied by 149 multiplied by 148 and so on, all the way down to 1. Similarly, 149! is 149 multiplied by 148 and so on down to 1. So if I write out 150!, it's 150 times 149!. That means I can rewrite the numerator as 150 times 149!.So now the expression becomes:frac{150 times 149!}{1 times 149!}Oh, wait a minute! The 149! in the numerator and the denominator cancel each other out because they're the same. So that leaves me with:frac{150}{1}Which is just 150. So, binom{150}{1} equals 150.Let me double-check to make sure I didn't make a mistake. If I think about combinations, choosing 1 item out of 150 should just be 150 different ways, right? Because for each item, there's only one way to choose it. So yeah, that makes sense. There are 150 ways to choose 1 item from 150 items.I guess another way to think about it is that the binomial coefficient binom{n}{1} is always equal to n because there are n ways to choose 1 item from n items. So in this case, since n is 150, the answer is 150.I don't think I made any mistakes here. The logic seems straightforward, and the simplification of the factorials makes sense. It's always good to remember that factorials can often be simplified by canceling out common terms, which makes the calculations much easier.So, to summarize, binom{150}{1} is calculated by plugging into the combination formula, simplifying the factorials, and realizing that it just equals 150. That feels right, and I can't think of any reason why that wouldn't be the case. I think I'm confident with this answer.