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MNWHIA LANeNNRALRCA6) IalAG congrnt c8? LrelaunFad th volux 0 andd19) Svppk Ital AABC = Anry If MLA-1 _ _L1-1 ,| Ual nx?Zo)...

Question

MNWHIA LANeNNRALRCA6) IalAG congrnt c8? LrelaunFad th volux 0 andd19) Svppk Ital AABC = Anry If MLA-1 _ _L1-1 ,| Ual nx?Zo)

MNWHIA LANeNNRA LRCA 6) IalAG congrnt c8? Lrelaun Fad th volux 0 andd 19) Svppk Ital AABC = Anry If MLA-1 _ _L1-1 ,| Ual nx? Zo)



Answers

If $\left(\begin{array}{rr}7 & -6 \\ 8 & -7\end{array}\right)^{2008}=\left(\begin{array}{ll}7 & -6 \\ 8 & -7\end{array}\right)^{2010}$, then $\left(\begin{array}{rr}7 & -6 \\ 8 & -7\end{array}\right)^{2009}$ is (1) $\left(\begin{array}{ll}7 & -6 \\ 8 & -7\end{array}\right)^{2008}+\left(\begin{array}{ll}7 & -6 \\ 8 & -7\end{array}\right)^{2010}$ (2) $\frac{1}{2}\left[\left(\begin{array}{rr}7 & -6 \\ 8 & -7\end{array}\right)^{2010}-\left(\begin{array}{ll}7 & -6 \\ 8 & -7\end{array}\right)^{2008}\right]$ (3) $\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ (4) $\left(\begin{array}{rr}7 & -6 \\ 8 & -7\end{array}\right)$

Statement who is correct? Because n. c. -1. Let's n. c. r. is actually same as endless one. c. r. It's a very important for me actually it's very easy to prove. So what we do is we take the left hand side. The plant is in factorial by n minus R plus one factorial into ar minus one factorial. And this is in fact a well by end minus our factory into R. Factorial. So what will take common? So let's take common first the right length. In fact well by end minus R. Plus one in two and minus R. Factorial into our minus on factory in plus N. Factorial by end minus our factory into our into our minus one fatto. So then let's take common. We can take common in factorial by n minus R. Factorial into our -1 factorial in Bracket. What we get is one by n minus R. Plus one plus one bite. Uh huh. So let's take calcium in the bracket. So it becomes in factorial by N minus R. Factoring into ar minus two factorial. So when you take and see him you need to be in place one do not need to be in minus R. Plus one into art. But what is in fact a leading to N plus one? It is N plus one factorial divided by what is N minus R. Factory. And doing minus or plus one is n minus R. Plus one factory. In our into our minus one factories are factories. So which is nothing But and plus one C. R. So statement to is correct. Now let's use this formula and find the value of 15 c. four plus 15 C. Five plus 16 C six. 15 C four plus 15 to 5 will be 16 C. Five. Because in this and his 15 R. S. Endless one cr 16 five. Now that is getting added to 16 6 now here in in 60 RS six, so it should be 17 C six, but right hand side is actually given us 17 67 17 C six is not same as 17 C seven. In fact it is same as 1711. Alright, so statement one is wrong and statement tweets Okay?

Okay, so we're asked to solve this differential equation right here. Friends. So this is a linear second order differential equation. So the solution is given by the homogeneous plus the particular solution. Okay? The particular solution. We're gonna have to find that through the variation of parameters, because this right hand side is not an exponential polynomial or anything nice like that. Okay. All right. So was the homogeneous problem. What's the solution? Is that so? Why it satisfies this equals zero. The characteristic equation gives us this which factors as our minus three squared is equal zero. So we have to repeated roots, okay? And so this means that the solution is given by this, Okay, Right. Because we need to have too linearly independent solutions. So this right here is. Okay, well, let me just backtrack. We found the homogeneous equation, right? We found the homogeneous function. But what about the particular solution? The particular solution is given by V one y one plus b two. Why? To where Why Went on y two are the two linearly independent solutions to the homogeneous problem. Right? And V one and V two are given in the book in terms of Wyland. And why, too, Right? So this right here is why one in this right here is why, too. These are the two linearly independent solutions to the homogeneous equation. All right, so we figured out the what we figure out, why wouldn't why? To what is V one and V two? So V one, they give this on page 189. It's given by this formula right here. Yeah. Oops, f. Why one? But we're W D t. Where a is the coefficient on the second derivative. That's just one. So I can get rid of that. Right? F Is this function on the right hand side? Okay, so we've got that. Why one is e to the three t and why two is equal to each of the three t. All right and f is equal to eat to the three t over T cubes. You can check that, right? And w sorry, w I need to give what w is w is why one y two prime minus y one. Prime y two, right. This is my notation just to make it look needs her. Okay, So the first step is what is w Let's simplify and figure out what w is So w is so w is why one y two prime minus y one prime y two, right? And so got speeds of three t times and then differentiating this you get three t e to the three. T plus heat the three t minus three e to the three t t e to the three t this simplifies to eat the 60. Okay, so let's figure out what V one is first. What is the one? So v one waas If you look here, the integral simplifies to it's minus uh, why two over W Okay, so let's plug everything in. This is he to the three. T divided by t cubes times t e to the three. T divided by each the 60. Okay, so this simplifies to minus integral on over t squared. Okay, Not is simply one by t, plus some constant of integration. C one. Now let's do V two me to waas integral f y one divided by W T. C. Okay, so f was need to the three t divided by t cubes. Why one was beat to the treaty and w waas each of the 60. So simplifying all this, we get this interval right here. And so evaluating the integral we get this, then this is some second concept of integration. Okay, Now I can set C one and C two equals zero, So let me get rid of that. And so we get what we wouldn't be too. Are me? One is one over tea. And me too. Is this Okay? So let's go back and figure out what? Why? P is right. We want to figure out what the particular solution is. Well, let's plug everything in. So this is one by t times e to the three C plus minus won over t one over to t squared times. Well, this was t eats the three t and so this simplifies to one over to C E to the three t. So we figured out the particular solution, and we've already got the homogeneous solution and so I can write down finally what the solution is It's given by a plus b t e to the three t right. This is the homogeneous plus the particular solution, which we found out to be. This. Okay, All right. And we're done

So here we are given the metrics that is is equal to 8979 and -8. So we need to find out Here to support 2007. Hollande was okay, so here first let's find out what is a square. So a square is a call to in the way that is Will multiply 82 times. So this aren't doing matrix multiplication. We get a square physical too. Hi now we'll find out what is a Q acute physical too. Yes, we're into Yeah. And we know that esseker is good. I that is I which is same as it. So here we can observe that if the power of age is even number then we'll get high and if the power of age or number then we'll get So in this case 2007 is an order number, Therefore 8 to the Port Townsend will be equal to it. Therefore we need to find out what is eight of about 2017. It's in words which is same as a and was. So yeah, it was his equal one x 80- busy. So 80 negative 64 -63. That is plus 63 into will interchange the diagonals and we'll change the sign of The other two element. So this is same as eight, negative seven nine and -8, which is equal to Yeah. Therefore Here to about 2007 and was is equal to it.

In this problem, first time writing the accent. So just look at it carefully. Six came for f e CN six plus or three plus city as to so it will react to give the product ID six get to the FECN six plus six. He who add as the product. Therefore, according to the given option, option B age, correct and set absent. Be correct. Answer for this problem. I hope you understand the solution of this problem.


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