A Level Maths Tuition Lesson 15 Proof Exam Practice!

foreign foreign foreign hmm hey guys welcome back uh to another one of the a-level revision sessions so today what we're going to be looking at is proof so this won't include uh proofs by contradiction because that's actually an A2 topic but we'll go through some of the other types of proofs that they can ask you so things are proved by exhaustion and things like that um but essentially they can it's a very big part of a level maths where they can just ask you to prove certain statements of true or false so uh without further Ado let's just uh jump straight into it so the first one that comes up is prove that x squared minus 4X plus 7 is positive for all values of X so in terms of proofs a lot of students find them really really tricky because it's hard to prove a statement you have to really think critically so what my hope is in this lesson and then uh the later proof lesson that will come out is that I will be able to go through a bunch of these different scenarios so you're as prepared as humanly possible to prove any statement because of course they can change this completely right they can even change it well the equation anyway so if I want to prove that a quadratic is above a certain value I'm thinking of an inequality basically what I want to try and prove is that x squared minus 4X plus 7 is indeed greater than zero right so the best way to do this is actually to use completing the square because completing the square will give you the minimum point of a quadratic so we're looking for the Min point which means we're going to be using completing the square okay because again if this minimum point positive then that means it's positive for all values of X right because it the smallest value is positive so let's have a go at that uh completing the square it's not too bad hopefully you guys remember that so you open bracket you put your X and then you just put plus b over 2 so in this case b is minus four so minus four divided by 2 is minus two make that squared and then minus minus two so B over 2 squared again and then plus seven so there we have x minus 2 and then we have minus four seven and lastly that should be squared we have x minus 2 squared plus 3. now if I have a look at this this gives me my minimum point which is 2 3 which is above zero therefore it must be positive for all values of X right so that's my Min therefore positive for all values of x and if you were to draw this out you would indeed find that that is true and again um so some students have said like can you draw the graph out and I've said okay if you were to draw the graph out how would you know it stops above the X above the x-axis and then say well completing the square and I say well why not just do the completing the square bit if I now draw the graph it's just excessive right it's just extra steps so let's go on to the next one so uh this is disproved statement N squared minus n plus 3 is a prime number for all values of n so this is quite tricky so let's have a think of different ways to do it well is there a way to algebraically show a prime number um as far as a level matter is concerned the answer is no because it's actually very very difficult to predict prime numbers uh funnily enough it's one of the big problem in maths at the moment so we can't do it algebraically so the best thing to do is just Sub in numbers well if I have a look at this can I think of any number off the top of my head that makes it Prime um well that's a bit tricky well I notice here that since I'm adding three right and here I'm going to be minusing a number what if I made these two things cancel out right like if this minus n and plus 3 cancel out I'm left with N squared now N squared can't be prime right but there's a square number isn't going to be prime so let's think if I want to make minus n and plus 3 cancel out then minus n should equal 3 so n should equal well in this case plus n should equal whatever three is but we want to make it minus right so minus n is 3. so if we Sub in n equals 3 then we get and again you have to actually go through these steps of showing the whole thing right and then that crosses out that crosses out and you're left with nine which is not Prime not prime square number right so and if you really want you can write nine is equal to three times three right but you do need to write you can't just write not prime you have to kind of specify a bit more like why is it not prime okay even if it does seem a bit obvious so again the way that I solved that is I had a look and I said okay I need to disprove that it's a prime number well if I look at it is there any values of n that I can sub in well if this n and this three cancel then I'm just left with N squared which is obviously not prime because it's a square number same thing as if I had 2N right like if this was 2N here obviously the rest would be a bit weird but again an even number can't be prime either except two so that is the best way to do it how else can you do it well what you can do is just keep subbing in value so something n equals 0 1 2 3 4 until you get to a number that isn't Prime in this case you'd have to do it three times to get three however if you just use a calculator to find the value of N and then just write down the one that gives you a prime number right so for example if you wanted to use a calculator let's quickly open ours up what I could do is I could just start subbing in values like this so just do like minus and then plus three and then just keep going okay n equals zero doesn't work well let's try n equals one and again bear in mind you can do this a lot faster by hand well that gives me three as well if that doesn't work okay that's a prime numbers that doesn't work I don't know why I decided to definitely and then obviously when we Sub in three we know it works right now how long does that take me it took me maybe 30 seconds 45 seconds it's a two mark question so it's more than in its capability so if you can't view it straight away just start doing stuff and just write down the one that works okay so this is where things get a bit more interesting prove that the sum of two consecutive odd numbers is a multiple of four so this is where you do need to know some stuff about algebra and proofs in general so what you need to do is you need to know how to write down generic even numbers and generic odd numbers so an even number can just if you want to do this for all even numbers you would sub in 2N have a think about why if I double any integer I get an even number right so if I want to prove something for all values of N and it's to do with even numbers I just Sub in 2N if it's odd numbers then how do I get it well an odd number is just an even number plus one so it's just 2N plus one okay so if I'm talking about odd numbers I'll sub in 2N Plus 1. now the only extra thing here is that to prove that the sum of two consecutive odd numbers the word consecutive means that they're next to each other so for example it could be one and three right or five and seven because those are odd numbers that are next to each other so what I'm going to do is I'm going to say 2N plus 1 is my first odd number and my second odd number would be just 2N plus 3 because to get to the next odd number in all of these I have to add two add two and again pick any two odd numbers that are next to each other and you always add two to get to that so now all we need to do is add these together so 2N plus 2N is 4N 1 plus 3 is 4. and now we need to prove it's a multiple of four so to prove something's a multiple of four or that it's even or whatever all you need to do is factorize it out so if I can pull out a 4 right it is a multiple of four because I can now divide this whole thing by four right and get a whole number that's how you prove it if you want to prove something's even you factorize out a two you're going to prove something as a multiple of three factorize that a three and so on and so forth prove that X Plus y all squared does not equal x squared plus y squared so again this is one of those things where you're going to have to prove this by just subbing stuff in okay now there are other ways to do this as well so for example if I were to expand the brackets I can show that these aren't equal to each other not always anyway right that's another way to do it or you can do something called disproof counter example right which is where we're going to sub in a value where this doesn't work right or you can expand so again how do we decide well if we Sub in one and one right X is one y is one we have one plus one is two two squared is 4. 1 squared plus 1 squared doesn't equal four so there you go just pick any two numbers right the only time this works is If X and Y is zero or X or Y is zero so if X and Y is zero or just X is zero just y zero then this actually does work right but it doesn't work for any actual values of X and Y that aren't zero so all we have to do here is we can say x is one y is you can say Y is two we can say Y is one doesn't really matter so we have one plus two squared that gives me nine and on the other side of this we have 1 squared plus two squared which equals one plus four which equals five now 9 does not equal 5 so it can't be true okay the next type is probably honestly it's not the hardest proof so if you have a look here all integers all values blah blah blah um a part a this is not the most fun proof okay um I would say proof by contradiction is the hardest but this is the most boring this is called proof by exhaustion and it's called that because you're going to be exhausted when you finish it okay now in all seriousness when you exhaust something means you run out okay so it says here prove that this thing is prime for all integers between one and five now the reason why this is important is we're going to exhaust all the possibilities we're going to take all of the integers between one and five so we're going to sub in N is one n is two n is three n is four n is five once we show all of them Prime we've exhausted the possibilities there is no other possibility it has to be prime has to be true so let's get to it so seven n equals one we have one squared plus one plus eleven one squared is one plus one is two plus eleven is thirteen which is prime and yeah this is exhaustion very painful very annoying but it's also probably the easiest so long as you want you don't rush anything you aren't being a bit silly so that's four plus two is six plus 11 is 17 which is indeed Prime and equals three three squared is 9 plus 3 is 12 plus 11 is 23 which is Prime again try not fall asleep while you're doing it um 4 squared plus 4 plus 11 4 squared is 16 plus 4 is 20 plus 11 is 31 which is indeed premium 25 30 41 which is prime and this is called proof by exhaustion like I said because you're going to be exhausted after the fact it was a great joke I just wanted to repeat it to make myself feel better and then Part B is a classic proof by counter example okay just like we've done before prove that this is not primed for all values of n right so pick a value that makes it not prime now there is one good thing right is one to five we know Prime so don't Sub in one to five again so what you'd want to do is just if you again it depends on if you can work it out or not right I can see something which I don't know if you'd see straight away but I've gone up in my square numbers right so you know 6 squared 36 that might not work and so on but if I look at 10 squared that's a hundred plus ten plus eleven that gives me 121 which I know is 11 squared um in terms of being able to spot that I can't really tell you how I worked that out it just kind of stared me right in the face so again if you really are unsure you can always just go back to the classic method of doing this and yet it's boring uh I've got nothing else to add but you can keep going until you find a number that is not that is not Prime and again keep in mind if you're on your calculator it doesn't actually take that long to do it um I know that 10 works so can we just degree so in n equals 10 we have 10 squared Plus 10. Plus 11. equals 121 not Prime because 121 equals 11 times 11. so again proof for exhaustion both of these are kind of exhaustion to a degree um it's boring it's tedious but to be honest how difficult was that like I've shown you on your calculator you just keep adding one and keep going keep going keep going keep going keep going it's not really that difficult so yeah boring but it's in my opinion three marks I've also just realized that three plus two is not six interesting anyways approved by exhaustion yay okay that the sum of two even positive integers is less than 10 sorry less than 10 is also even so basically what we need to do is we need to sub in every combination of even numbers so if we look at the even numbers below 10 it's 2 4 6 and 8 we need to do two plus four six eight four plus six and eight six plus eight and we're done let's get started okay two plus four two plus six two plus a or plus I like to write these out first because I don't want to miss one um that would be absolutely tragic and six plus a on the bright side this is if you get these wrong but again please be careful like please um getting these wrong is tragic in your actual exam um cool is also even well even again how long did that take me it's boring but that's three marks that should technically take me three minutes and I I whittled through that in probably 30 seconds 45 seconds um not too bad a bit tedious but it is what it is let's have a look-see If I multiply a number by two and add five the result is always greater than the original number okay so let's think of some cases okay so what we have to do is say whether it's always true sometimes true or never true right so let's think of something right now my mind immediately went to positive numbers right now if I do if I take a positive number times it by 2 and add five it's always going to be bigger right so again we can pick a number right like n equals three then we have three times two plus five that's three six eleven so in that case it's true right so when you want to test for all values here's a few things that you should do first of all you should try zero positive number negative number right those are the first three things you check so if I type in zero right I get zero times two is zero add five obviously it's greater I picked a random positive number and it works um so now the problem with this is always true I have to prove algebraically that's a bit tricky so let's see if it works for negative numbers let's pick negative three right well some of you may have already realized that it doesn't actually work minus three times two is minus six plus five is minus one which is uh actually in that case it's true so what we need to do I don't know why I wrote false is in that case it's true but what if I made if I looked at that it's not much bigger so what if I made this like minus 10 right minus 20 plus 5 is -15 it's false so I don't need to go any further now right because it's not always true and it's not never true so it's sometimes true so this is actually the best possible result the reason for that is for you to say it's always or never true you have to algebraically prove it for every single number period decimals everything right so you have to do it algebraically so the the best outcome is that it's sometimes true because you just need to pick one case for True one case for false that's it so again have a look think about it piece it together other buzzwords okay uh prove that 2N plus three all squared minus two n minus three all squared is a multiple of six for all values of n so again when whenever you see the word multiple of six we want to show that we can factorize a six out that is the end goal for this okay so let's do this let's expand some brackets shall we so 2N squared is going to be 4N squared they're going to be plus 6n plus 6n which is 12 n plus nine now this I'm going to put in leave in Brackets because the negative sign is going to mess things up and again I've seen students of all ages and abilities mess things like this up squared and then it's minus 12 n and then plus 9 still right and now do the negative right because you know be careful right you've got four marks right that's a long time take your time it's okay you don't need to Rush there is no rush then that's minus four N squared plus 12 n is nine so now we can go through minus nine and plus nine cancel the minus four N squared and the plus one squared so we're left with 24 n which I can indeed factorize out a six I've been a bit sloppy actually I should really write equals so it is a factor a sorry a multiple of six because we can factorize out six right which means we can divide by six okay so again these proofs aren't too bad they're nice and quick and short yeah and three or four marks and realistically like I've done it in four lines sure but I mean it doesn't take you four minutes it's very quick and you can save some marks but again it's one of those topics where you have to think and thinking is painful thinking hurts thinking leads to mistakes but it's one of those topics that you you can reason it through and a lot of other topics in maths you you don't need to apply reasoning that much right like if I give you an integration question or I give you an indices question you just need to follow steps that you've known with proofs there aren't really any steps I can teach you I can teach you some tricks some different ways of thinking and through practice you'll get better at it but at the end of the day it is down to a little bit of logic which is scary uh prove that the sum of the squares of two consecutive odd integers is always two more than a multiple of eight yeah that sounds like a fun expression doesn't it so multiple of eight again we're thinking factorize eight so I prove that the sum of the squares of two consecutive odd integers well we know how to do consecutive odd integers right yeah now it says sum so I want to put a plus between them the squares okay so all we need to do is put big old Square signs there and again expand show that easy peasy lemon squeezy 4N squared plus 4N plus one plus four N squared plus 6n plus nine gather like terms we have eight N squared plus ten n Plus 10. okay so I need to factorize out an eight and then also I'm going to have a plus two on the end now you might see I can't really factorize out and eight in this particular case um hang on I just realized that should be a 12 and so this should be a 16mm okay so I can factorize out an 8 from the first two terms right okay but that means I'm left with 10 more so what I can do instead is I'm going to do something very cheeky but completely mathematically allowed I'm gonna break it up into an eight and a two because if I have the eight then uh I can factorize it out so I'm going to factorize that an eight and also you can see the plus two so there are two two ways to do this in order to factorize out on eight I needed to turn ten into a number that um kind of lets lets me factorize out of 10. um let me see factorize on eight sorry and also have I need two more so I need a plus two so turning into an eight plus two makes a lot of sense so now we have n plus N squared sorry plus two n plus one plus two so that is exactly as they said it is a um multiple a two more than a multiple of eight right okay next again this is one of the final types of proof they can ask you again keep in mind there is proof by contradiction as well which is going to be a whole separate video because proof by contradiction is huge it's massive so this is probably around half of the proofs that they can ask you the different types of proof the next half is just contradiction and all of it um but I do have an exact formula for that and it um it's good it will help you out a lot so we have to prove that N squared plus 7n plus 15 is greater than n plus 3 for all values of n so let's write out the question one more time what we need to do here is what we're going to do is completing the square because completing the square gives me the minimum of any quadratic I like and this is indeed a quadratic so what I need to do is move everything over to one side so we have N squared 7n minus n gives me the 6n 15 minus 3 gives me 12 I was about to say 13 that would have been embarrassing now we do completing the square so we have n plus 3 squared minus 3 squared plus 12 is greater than zero and then all we need to do is do the n plus three squared minus 9 plus 12 is 3 greater than zero so now the Min point is just um I mean actually to be honest you don't really want to call it a minimum point because it's not actually a set of coordinates in this case so all we need to do is say that the smallest value is when n equals minus three in order to get rid of that bracket and if we Sub in n equals minus three what you get is you get minus three plus three squared plus three which is equal to 3 which is indeed greater than zero so that would be the best way to do it so I had to think it doesn't really make sense to call it a minimum point because we're not talking about coordinates so I'll leave it as that but it still gives me the minimum of the function so it doesn't actually matter what letter is in there uh so again that is going to be the end for the um the proof topic for now but again it does come back in contradiction so from next week onwards we're actually going to start doing statistics and the mechanics okay so I know that's something that a lot of you've been asking for so keep an eye out for that but other than that I hope you have a wonderful evening