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MY NOTESAskyour TeachehMOfataelIu4PalntsltDOTAILSTANAPCALcnR9 4 5.004FetU...

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MY NOTESAskyour TeachehMOfataelIu4PalntsltDOTAILSTANAPCALcnR9 4 5.004FetU

MY NOTES Askyour Teacheh MOfatael Iu4Palntslt DOTAILS TANAPCALcnR9 4 5.004 FetU



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Ramu had 13 notes in the denominations of Rs 10, Rs 50 and Rs 100 . The total value of the notes with him was Rs 830. He had more of Rs 100 notes than that of Rs 50 notes with him. Find the number of Rs 10 notes with him. (1) 4 (2) 3 (3) 2 (4) 5

Uh huh. Okay. We are going to be taking these numbers in front of us and putting them into a format Where they are multiplied by powers of 10. This is otherwise known as scientific notation. So in order to make scientific notation happened, there needs to be one digit in front of the decimal. Otherwise we'll be multiplying Powers of 10 by powers of 10. So if it's just one digit multiplied by 10 then the factors will only change by 10. So 1.156. That is really easy. There's already one dead one digit in front of the decimal. So order to make this scientific notation We write 1.156 and multiply that by something that doesn't change it. Which if we multiply it by 10 it will change it. But if we multiply it by 10 to the zero anything to the zero is 11.156 times 10 to the zero or the first 10 to the zero is 1 Gives us 1.156 21.8. In this case we have to move the decimal over. This will then give us 2.18. But that's a whole different number than 21.8. Unless we multiplied by some power of 10. in this case it's times 10 to the first notice that the number of decimals I moved in this case one is the same as the number in the exponent and if I move it to the left, I got a positive exponent every single time. Let's see if you recognize a pattern here. Now I'm going to move this decimal To the right three places which gives me 6.8 Times 10 to the it's gonna be three. Yes, but I moved the decimal to the right this time that gives me a negative exponent. So I move the decimal to the right, I get negative, move it to the left, I get positive. 328.65. I'm going to move the decimal to the left twice. Giving me three 0.2, make some room here 3.2 86 five Times 10 to the you guessed it positive too. Now this next one 0-1 nine I moved the decimal to the right once Gives me 2.1 nine and I still put times 10. Mhm. In this case though it's now to the first but it's actually the negative first. This last one it's a nice hole round number. So we can assume the decimals right here Which means I'll then move the decimal to the left 12 places giving me four 0.4 four Times 10. In this case to the second. And that scientific notation you move the decimal until there's one digit in front of the decimal and you move it how No for every space you move it you add an exponent you can move to the left positive. If you move to the right, it's negative

If we want to evaluate the equation above, we can use the binomial theorem written ingredient. When the equation is in this form, we can see from each individual term that X in the binomial theorem corresponds toe one for a in the binomial theorem corresponds 23 over four. An end in the binomial theorem corresponds to five. We know that the binomial theorem is used to expand equations of the form X plus a the end. So we know that this entire equation here can be rewritten as 1/4 plus 3/4 to the fifth. This equals one to the fifth, which just equals one.

All right. So for the following question, we have a very long equation that we need to find a numerical value outs. I'm gonna radio now. It's may take a little bit. It's it is very long. So obviously as we can see, this is already a binomial expansion, and hopefully we all know the formula for that. But if lot luckily, this question so long that, um, you can probably find it by telling done writing this whole thing out, you know, we're gonna persevere. Write it all out. It's just easier when you write it out. It's a better way of learning. Should always right your questions out whenever doing it. The sword is a doozy. Still going a couple more lines and we're gonna be good. Some of you can probably already kind of guess the rest of them or not. Guess story. No, the rest of them a pattern or formula wherever we like to call it should be a four. Uh, and this is the last line. So all this we need to figure out what it equals. So we know that from the binomial theorem, whenever we're given an X plus a why to be X to the Y, and it could be equal to the sun of and J is equal starts at zero of an over J and then we have X to the n minus j. And then why Teoh the J So when we actually expand that out in just purely, um, mathematical form, it looks like this. So and over zero x to the end, plus and over one x to the n minus one. Why plus and over to extra the war and Linus to why? To the power of to and then plus, And it goes on essentially until you reach end over end. So in this case, we're gonna look for our aunts, RJ's and our, um, our xer wise and our ends essentially. So from the first term, you can see that it's gonna be end over zero and X to the end. So in this case, I was gonna point some things out. I think that five is our and it's our end, Um, and then from the first formula, and then this right here, the 1/4 is gonna be our X. And then, um, for the next one, we have that we once again have our end of five and then one is the next step in the formula. We once again have this as X still, and and now we have this which is actually gonna be our Why. So now I'm just gonna write out everything about we have. We have X is equal to 1/4. Why is equal to 3/4 and we have that end is equal to five. So the formula that would have started this binomial theorem would look like this 1/4, plus 3/4 to the power of five, and that's actually going to be a formula. So now, since there's no variables in it, we could just solve for the number in this case. So 1/4 plus 3/4 is actually is going to be one. It's gonna be one to the power five, and obviously that equals one. So therefore the equation slash binomial expansion. I'm just going right. B E for short is equal to one. And we found that by looking at the formula that were given and kind of reverse engineering to get it back to the formula that we know so we could solve for a numerical value

Over by a We have noted for over five when it's one or eight. Still, it's fine. R l c d To the mind these two where L C. D. Is composed of these two factions were just five and eight. It is too time or four times to, which is equal to two times two times two. He's our prime numbers, and so are these. Combined them with it five times, two times, two times two. Which gives us 40 for LCD. Okay, so these two factions are getting to get multiply by some term to get a denominator of 40. In this case for five, we have already divided by pi, which gives me AIDS. And for eats, we have five. We're fine. Okay, Now, combining these two, you get negative. See? Eight times four, which is negative. 32 over 40 minus 5/40. Which gives me a negative 3 7/40 Don't report be. We have native for over five times. One over eights. Okay, well, we just multiply it. It's through their four times. One is like before and five times it is 40. Now we know that 40 is four times 10 so we can simplify for and forth and read yet negative 1/10 as our solution


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