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find $\overline{\mathbf{u}}, \operatorname{Re}(\mathbf{u}), \operatorname{Im}(\mathbf{u}),$ and $\|\mathbf{u}\|$. $$\mathbf{u}=(2-i, 4 i, 1+i)$$

So in this problem were asked to take them Applause. Transform of this function here T minus one square multiplied by you to the T minus one. And the first thing that we want to do is want to equate this to the form f of T minus a times U of T minus a So we can first identify what are is so looking over here we can identify pretty quickly at a is equal to what now? The second thing that you want to dio is you want to take a look at this Now let's call this GFT. This is the function that you want to take a look. Loss transform up. We want to equate this to this part G to is equal to f T minus a But what we want to do is we want to get to capital. After this, they look lost transform of that. And to do that, we have to add a to both sides inside here. So t plus a is equal to f t. And now we can get to the lip gloss transport. So taking a look at what g of t equals g o t equals t minus one squared. We can say that G of t plus a is equal to t plus a minus one squared and knowing that a equals one, we can cancel both of those out. And then we get that g of t plus A is equal to t squared. Great. So now we can go ahead and put it into our love loss transform bracket. So we have t squared multiplied by U of T minus one. We want to take the lip loss transform of that and what I like to do when I'm taking the plasterers from something that looks like this is take a look at this first because we can No, or we know really quickly what that turns out to be. But that turns out to be the form e to the negative A s a being one. So we just have e to the negative s. We're going to multiply that by the LaPlace transform a T square now t squared. Any time you have t race to the end, the LaPlace transform is in over s to the end, plus one. So taking deal. Applause Transfer of T square. We can get the to put it in the numerator. And then we have s to the two plus one s to the three. And our LaPlace transforms, finished and simplifying. Putting it all together. We have to e to the negative s all over. Yes, race to the third and there is our laplace transform.

Okay, So for this problem, we're after taking up loss transform. But this function here, you know, what we want to do is we want to said it to this form F t minus a multiplied by U of T minus a And now we can identify what are a is So to do that we look at this part and really quickly we can see that a is equal to what? The next thing that you want to dio is You want to take this function, call it G of T. And you want to equate that with this court here? So we have g t. It's equal to f of T minus A in our overall goal is to get the applause transform of this function. So what we need to do here is we need to add a inside both of these arguments, so g of t plus a is equal to if of tea. And then from there we can get toilet glass transport. So looking at what g of T equals G t, we know that is equal to T minus one squared. That brings us to G of T plus A is equal to T plus a minus one squared and we know that a equals one. So this term in this term end up getting canceled and then we have g of t plus A is equal to just t square. So now that we have that equated, we can go ahead and set it up. Incirlik loss transform brackets. So we have little applause. Transform of t squared multiplied by you of T minus one in the first thing would I like to do when I see something like this is I take this part first, since this is easier part Teoh transform and we know that that transforms to e to the minus A s a being one here. So we just have e to the minus ass and we're gonna multiply that by the transform of T Square. Now, if you have tables handy, you could look up the LaPlace transform of something that is in the form of tea to the end. And that goes into n factorial over s to the end, plus one. So we take the LaPlace transform a T square, we can see that are too turns into two factory, which is just 22 times one. And that's over. As to the to bless one, So that's over s to race in the third and simplifying. We get to e to the negative s all over s race is 1/3 and that is the LaPlace transform. Uh, T minus one squared of times. You teammates What?

And this question were given the age N C n is equal to, we have to find Yes. We know date. N C N is equal to in sectorial, divided by and minus and factorial into and victoria. Yeah. So you cannot. This is in fact a really went by In my tradition to zero. Factorial. In fact area infertile. Infertile can slow and we are left with one of our satisfaction which is going to one. So the answer is that and different option is option number two, which is a quarter to one. This is the answer. This question. Thank you for what can we do?

Find a solution for this special occasion. Find a characteristic region. So you look at the occasion. The characters to the creation is actually just or in my dream. You, we have are are landless people. Choose three three. We have our solution in the form life people to you No one you could lend one key keeping zero. Well, you people first. Power lines like the one He what? Maybe the second color in the land. One Get submission. Wife people too. One You could accuse you. Well, you she Kiki squared. You could