Aiming for a Grade 9 Paper 3 Set 5 2024 Papers

Okay, we're doing some of the aiming for grade 9 questions. These are from the 2024 papers and just to give you some context This is what edXL produced So essentially this is our 37. The students that got grade nines on average scored 25 marks on These particular questions and then grade 8, 15 marks and so on So if you look at how big of a jump this is if you're aiming for a 7, 8 or 9 and you're currently stuck on Like a 6 or maybe below These are the exact questions you want to learn how to do in order to push yourself up because for example going from a 6 to a 7 That's only going up by what 4.3 marks. Let's say 5 marks on these questions That's not that bad and then so on going for the next grade above Remember that in GCSE maths for higher tier 40% of your marks are just based on processes So for example, if they ask you to do completely square, they ask you to do quadratic inequalities They ask you to solve an equation. That's what 40% of your marks are based on. On average, it's 45% For a grade 6. So being able to do just the processes doesn't even get you a grade 6 in practically every single year 60% is from problem-solving which is exactly where a lot of students lose the majority of their marks You know besides silly mistakes, of course So these questions here, the Aiming for Grade 9 questions are all of the problem-solving questions I mean, you know the drill by now. I've got like five sets of these papers already. So it's like the 15th video I've got on these. So let's just jump straight into the first question. So the very first question here We have a difference of two squares We know that because A and B are numbers and if you square them, they're both square numbers Which means all you need to do is square root both numbers and put them into brackets So if I square A and B we get A plus B and then here we get A minus B Not too bad But part B is a very interesting question. Show that 2 to the 40 minus 1 is the product of two consecutive odd numbers now this is one of those questions where part A really does link into part B because if I have a look at this I'm just going to zoom in for a second Both of these are square numbers 1 is a square number because 1 squared gives you 1 and the 2 to the 40 Well, I can break that down into 2 to the 20 squared Which means 2 to the 40 is also a square number In general if you can if there's an even power of a number then it's a square number for example 9 squared is obviously a square number 9 to the 56 is a square number because again 56 is an even power and this works for literally any number by the way So that is also going to be a square number So all we need to do again is square root both numbers and see what happens. This will give us one of the marks So again put our big brackets We have 2 to the power of 20 plus 1 and 2 to the power of 20 minus 1 And it says here show that it's the product so show that it multiplies two consecutive odd numbers So basically what it's saying is Show that these two are odd numbers that are consecutive So here's what I'm going to do If I write these numbers in order We have 2 to the power of 20 minus 1 and we're going to have 2 to the power of 20 plus 1 In the middle, we're going to have 2 to the power of 20, right? So imagine if you're going up right 2 to the power of 20 minus 1 then 2 to the power of 20 2 to the power of 20 plus 1 then 2 to the power of 20 plus 2 and so on This number in the middle is an even number because it's a power of 2 And these two are very clearly odd numbers because if you add or subtract 1 from an even number you always get an odd number And this is also an odd number So if you notice because these two odd numbers are the closest two odd numbers that you have so they're one after another It is the product of two consecutive odd numbers If you want you can just write that 2 to the power of 20 plus 1 These consecutive odd number odd number God I hate writing to 2 to the power of 20 minus 1 and that would be it for those two marks, but again fairly tricky question Circle has the equation x squared plus y squared equals 25 the point P at the corner is minus 3, 4 lies in the circle Alex says that the tangent to the circle at P crosses the x-axis at this point. Is he correct? Show how you get your answer. So I think here again like the way they address to the question They've shown you it. That's the tricky bit the way they've worded it So I'm going to explain what I would do in this situation If I'm looking at this, how can I show whether or not he's correct or not? well, the easiest thing to the first thing that comes to my mind anyway is to work out the equation of the tangent and find the x-intercept Right, because if I find the x-intercept if it's minus 8, he's correct. If it's anything else, he's incorrect So all we have to do here is literally find the equation of a tangent at this point to the circle So essentially the question is just find the equation of a tangent for 5 marks. So again, not really too bad It's just that the way they've dressed up. The question is a little bit trickier because they try and make it a proof But yes, all you're doing is the equation of a tangent One other thing I'm going to show you is how else do we know right? Well, first of all, if we look at the answer of the question, it's worth 4 marks Which means we're going to have to do quite a bit of work to get the answer It's not going to be something simple like subbing in the number or something like that So Minus 3 4. Okay, so that's going to be like over here somewhere that's my point P and The gradient that's where the gradient the tangent is going to look something like vaguely like this again But that's the way it is. It's just a sketch anyway So what do we do? Well now we can actually work out what the Equation of the tangent is these steps are always the same So to work out the equation of the tangent what you're going to do is you're going to work out the equation of the radius or the gradient of the radius and Then take the negative reciprocal because the tangent and the radius are always at 90 degrees remember when you work out equation of a tangent what you're working on is y equals mx plus C and What's true always with y equals mx plus C. You need two things one. You need at least one coordinate and Two you need a gradient now the coordinate they've given us minus 3 and 4 so they've given us this part It's just the gradient Well because the tangent it's always perpendicular to the radius so what I'm going to do is I'm going to in red do the gradient of the radius Remember it starts at 0 0 and it ends up at minus 3 4 so we're going to have 4 minus 0 over Minus 3 minus 0 which gives me minus 4 over 3 Which means the gradient of the actual tangent we call it MT Is the negative reciprocal so negative becomes positive and then you flip the fraction so it becomes 3 quarters Again, does that make sense? Well, my lines going upwards and the gradients positive. So, okay, that's probably correct Now something into y equals mx plus C. Well We have so far y equals 3 quarters of X plus C And then we can sub in the coordinate from here sub this into here. So y is equal to 4 and X is equal to minus 3 and Then all we're doing is solving so 4 equals by the way, this was a calculator paper This is specifically for the paper 3 of that year Well, they don't need it. It's 16. That's 25 over 4 Yeah, that's not right. And so we have our equation there y equals 3 quarters of X plus 25 over 4 Now again, what was the point of this question? Alex says the tangent crosses the x-axis at minus 8 0 So what we're going to do is I again, but it is multiple ways to do this, right? So I'm not saying this is the only way but I would sub in y equals 0 and find what the x-coordinate is So y equals 0 4 minus 25 over 4 from both sides minus 25 over 4 equals this I'm going to times both sides by 4 which basically just gets rid of the 4s and Then I'm going to divide by 3 so I get that X equals minus 25 over 3 Which obviously is not equal to minus 8. So Alex is wrong Alex for 4 months Hopefully that makes a bit of sense Is we have here that son and needs to draw the graph of y plus 3 to have X is 0 Less than equal to X less than equal to 4 if he draws this graph Zoom out quite a bit. It says write down one thing that Sana has done wrong now to be honest when I looked at this It's um, it's not the most obvious in the world. So some common things I get students to look at is Look at the axes for the starters, right? Are they labeled and are they consistent? So if you look what I mean by consistent is that they go up by the same amount each time with the same gaps So for example, if 10 small squares gives you 10 then 10 more square a small square should go up by another 10 Because sometimes they do like 10 30 40 50 something like that. This looks all perfect to me. Some of the x-axis again It does look like it's a different gap, but it's not it's a bit of an optical illusion So when I look at this, it's um, it's a smooth curve. It doesn't look too bad It's really awkward. The problem is this so To get the y-intercept you have to sub in x equals 0, right? Well if I sub in x equals 0 into here So when x equals 0 y equals 3 to the power of 0 which equals 1 Not 0 so it shouldn't actually cross here. It should cross out 1 which is literally like a single square up That's what she did wrong. She basically put the y-intercept in the wrong place Yeah, so again, do you think things like the shape of the graph you're looking at the axes and in this case We're looking at also what the x and y-intercepts are just to make sure that what she's done is correct okay, so transformations is normally I Think it's just people don't revise it that much and I kind of I get why but I'd recommend the day before each of your Papers. Well, for example, if it comes up in paper 1 You don't have to revise it paper 2 and 3 But you should just kind of go through and make sure you know how to do it Because honestly, they're not difficult and they're just free marks in my opinion. Although part B Yeah, part B is a bit tricky and the fact that it's only one mark is kind of nuts But let's start with this. So enlarge triangle P by scale factor minus one and a half So I'm literally gonna write those minus 1.5 again. It's a calculator paper So you could just use that the center enlargement minus 2 minus 1 so first step label the center enlargement At that point. I'm also well aware that a lot of people use different weight methods of enlarging stuff. I'm fine with that This is the way I do it. All I do is I count squares, right? So for example to get to this point I have to go 1 2 3 4 to the left and then 1 2 up and now what I'm gonna do is times both of those by 1.5, so 4 times 1.5 gives me 6 and 2 times 1.5 gives me 3 and because it's a negative it reverses the direction So this is gonna be 3 down and this is going to be 6 to the right and again you start from the center enlargement So we're gonna go again 1 2 3 4 5 6 and then 1 2 3 So that point there is now over here and you can repeat this You can repeat this for all three points or if you're I don't say a little bit clever because that's kind of like, you know making me sound better than I am But you can kind of tell what's going to be remember when you have a negative enlargement it flips the shape So this point up here I know is going to be below this point here and I can work out what the lengths are going to be, right? The length of this side is 4 4 times 1.5 Gives me 6 so I can actually draw a line going 6 down 1 2 3 6 right and By the way, I haven't reflected it yet for those of you saying like I've got it wrong or something. Give me two seconds This point here again 2 times 1.5 is 3 and again because this is going to the left It's going to go 3 to the right 1 2 3 now again This part the pit that I did wrong is that it should be attached to the other corner here, right? so we move that over like this and Then you can just join the two points because you know what a triangle looks like It's gonna always look the same and that is what's going to be Triangle cube so realistically you only need one point and then you can kind of tell what the rest of the shape looks like Again, if you have another way of doing it, that's fine. I don't think this is too crazy now Here's here's the tricky part right triangle P is transformed by a combined Transformation of a rotation of 90 degrees anti-clockwise about the origin followed by blah blah blah blah blah, right? This is like a lot of information Exactly one vertex of triangle P is invariant. So what that means is One of the core corners here is going to end up on its starting position so for example This point could end up back here or this point will end up back here or this point ends up back here Now if you notice it says exactly one So there's actually three different answers for this question One that makes the first corner the same one that makes the second and one makes the third because remember we're not including More than one invariant points, so if I just get rid of this for a second Obviously the one thing we have to do first is we have to do that slightly disgusting Rotation, so it says rotation 90 degrees anti-clockwise about the origin followed by translation. Cool So again, if we're rotating about the origin Now honestly, the easiest way to do this is to use tracing paper. So you put the tracing paper on here Sketch it out and then you're going to rotate 90 degrees anti-clockwise which will be going back up here somewhere. So I'm going to quickly do that and I'll come back to you with the answer for that part Okay, this might be a bit scuffed. I had to rotate the whole document so it might not be perfect for something along these lines and Again, as I mentioned that there's three different possibilities because one vertex being invariant Just means that one of the vertices is actually on the same place it started. So either Here goes to here. So what would that be? Well, we have to just count the squares, right? It goes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 that'd be a vector of minus 16 and then 1 down so For which one was down there for a second. That was embarrassing or you have this This one is probably the one I would have done the actual exams is it looks like it's the closest So 1 2 3 4 5 6 7 8 so minus 8 and minus 2 So again any of those would work that was a bit of a pain in the neck to do on a computer But hopefully you see the logic behind the question Okay, so 3d of Pythagoras slash trigonometry. It looks fairly. Okay, hopefully They give you a triangular prism, which is quite rare. Normally they give you some kind of cuboid, but it should be okay and It gives you a bunch of information So honestly, the first thing I would do is I'd start labeling this information on here because I don't know what any of it means Otherwise right until I can visualize it. So it says that's fourteen point seven centimeters Maybe BC is three point eight And MT is two point three. So let's try and squeeze that on there like that And it tells you that P is the point on a B. So along this thing here such that a P to P B is Five to two so it's gonna be it's gonna be closer to B than it is to cool and Calculate says the angle between TP and the base. So what I'm going to do first now makes you want to draw that The actual angle they want right? So come on water correct. Really? Fine, I suppose There we go. So if I extract this triangle out for a second TMP That was awful we try and draw an actual straight line this time There you go. T P they're asking me to work out this angle here because remember MP is along the base of the shape and then TP is the diagonal that they're talking about right the angle between TP and the base of the Prison, so we need to try and work out Two sides right now, the first one is we actually know what TM is is 2.3. So we actually only need to work out one side so after this some people get stuck and I fully understand that but Am I more likely to work out TP or MP? Let's look what TP is. It's It's kind of like I don't really know any information about it. But if I look at MP Well a B, I actually I know the length of a B and I know the ratio that gives me a P so I actually do know what a P is I can work that out really quickly and likewise CB is 3.8 But because the prison means it's the same throughout which means the base of the prison there CB should be the same as da So actually not that is also if I draw out to the triangle N a P I know that MA is half of this length. So 1.9 because again M is the midpoint of da and AP I can work that out because AP is 5 over 7 Hopefully okay with where I got that from because again AP is 5 parts out of a total of 7 parts along the line It's 5 plus 2. It's 5 sevenths of 14.7 Which means I can work out MP once I've worked out MP I can work out the angle So, although this question looks really complicated first of all remember all of the information is useful I've used the 3.8 I've used the ratio. I've used the length. I've used all of it and secondly If you start drawing stuff out and thinking about what you need Generally speaking that'll get you to the right answer This is a calculator paper. Thank God because I really can't be bothered to do this without calculator So that's that gives me 21 over 2 So MP is literally just going to be square root 1.9 squared plus that'll be 10.5 squared 21 over 2, right? so square root 1.9 squared plus answer squared gives me 10.67 and How do I work out the in fact I'm going to do dot dot dot because it goes on for a while How do we work out the angle now? Well if we label up our triangle, it's a right angle So that side's hypotenuse this side's opposite opposite the angle we want and this side's adjacent So O and A means it's going to be tan. So we have tan theta equals 2.3 over 10.67 dot dot dot again I'm using the answer button in my calculator to make sure I get the correct value. So inverse tan 2.3 over the answer gives me 12.16 degrees to one decimal place that's 12.2 degrees So yeah, um, I do like 3d Pythagoras and trig I know it's difficult to get started, but for four marks, we actually didn't really do that much work. So Do try and practice these questions a bit. I actually expect them to give you something like this this year because There was something in the practice papers Which was kind of similar to this where they gave you a random sequence And they asked you to prove that the consecutive numbers is a I think it was even in that case, right? But they gave you something you've never really heard of before. You should have heard of triangular numbers Don't get me wrong, but maybe you don't know the Algebra for it But they gave some random other sequence and they look quite complicated but same thing Think in terms of this proof of the sum of any two consecutive triangular numbers. Well, the first triangular number can just be this, right? That's fine. And then we're going to do a sum which means we're going to be adding something to it, right? Consecutive so what would the next triangular number be? Well, if I subbed in n equals 1 That'd give me the first triangular number and then when I sub in n equals 2 They give me the second then n equals 3 and n equals 4. So in other words, I just want n plus 1 Right. So instead of n I'm going to write n plus 1 And again, instead of n I'm writing n plus 1 and then plus 1 again because that's in the equation So basically I'm replacing n with n plus 1 all over 2 So what I'm going to do is in fact I'll take my time with this just so you guys can see the algebra because I know some people find that a little bit tricky We have n plus 1, n plus 2 over 2 We can add these fractions straight away, but I'm actually going to first expand it So we get n squared plus n over 2 for the first fraction The next one we get n squared n times n then we get plus 2n plus n. So that's 3n and then plus 2 over 2 Adding this all together. We get n squared plus n squared gives me 2n squared n plus 3n gives me 4n And 2 is just 2 all over 2 Divide that through we get n squared plus 2n plus 1. Now here's where some people might get a bit stuck But we've got two of the three marks, which is quite nice While they get stuck. Well, how do I show this as a square number? When you show something as a square number what you want to do is you want to show that you can write it Such that it's like x squared that's a square number or n squared or n to the 4 So, how can I write this expression as something squared? Well, if you're ever stuck have a look at what you have and try and factorize it So here we want two numbers that times the 1 but add to 2. Well, that's just 1 and 1 So in other words is n plus 1 n plus 1 So in other words is n plus 1 all Squared which is always a square number As shown. So again, honestly getting the first line of algebra, that's the hard bit the rest of it I think you guys are pretty clever. You guys can probably rearrange this and do all that fun stuff So honestly writing plans and scales like this is again something that hasn't come up very often But again, this is 2024 only two years ago But I don't know how many students that are revising this and this might catch people off guard So here it says on a plan a room in the house is for over 48 square centimeters work out the real area So the common mistake I would see is just doing 48 times 50 Remember scales like this are the length scale factor Because they're the scale you use when you're drawing your individual lines not when you're drawing your areas so in other words the length scale factor is equal to 50 which means the area scale factor has to be 50 squared So the first step which by the way 3 marks is actually a lot for this on the plan We have 48 centimeters squared. We want to work out the real area So if we just do 48 times 50 squared, we have the real area now There's a reason why they give you more marks than you might think is necessary. If that gives me 120,000 Centimeters squared. I'm gonna make sure that I leave that there the units haven't changed They want the answer in meters squared Now here's where some people might forget the unit conversion. So here's how I remember it To go from centimeters to meters everyone knows that you divide by 100 So to go from centimeters squared to meters squared you divide by a hundred Squared to go from centimeters cubed to meters cubed you divide by a hundred cubed. So literally my final answer is a hundred twenty thousand Divided by a hundred Squared and bear in mind. I know you might look at this and think well, that's really easy. That's kind of stupid Bear in mind this is what students got wrong in 2024, which I think was again a fairly hard paper So I have all the questions This is what grade eight and nine students got wrong Again, it's these minor little things. So make sure that makes sense to you If something makes sense, you'll remember it for longer. I don't expect you to sit at home like memorizing this Makes sense. You'll remember it for longer Okay, the graph of Y plus F of X is shown on the grid It's just like an N shape like a horseshoe on the grid scratch the graph Yeah graph transformations tends to be kind of doodoo as well. So here it's actually a combined transformation There's more than one thing going on, but take it step by step Anything inside the bracket is lying to you and it affects X So what do we mean by lying to you? We do the opposite. This is saying times by negative one So we're going to divide X by minus one. There's also the opposite of times in by minus one It's dividing by minus one. Now. Yes, those do the exact same thing, but just trust me, bro If it's outside of the bracket and by the brackets, I mean like the F of X brackets. It tells you the truth To do what it says and it affects Y. So here we're going to add three to the Y coordinates Now, how does this actually work on here? So I see some students trying to do some crazy stuff, right? They try and like do it all mentally or something. No, no, no, no. No, let's take it simple, right? I will pick probably two or three points and just move those first Right. So for example here the coordinates there are minus two and three so first of all Times the X by minus one that gives me positive two and the three But what was it saying saying add three so we're going to add three to three, which is six So instead of it being minus two three I'm going to go to two and six and that point now that part tip of the horseshoe here is now there And you repeat it for two other points. This one here is The corner minus six minus seven so again times the exponent by minus one because that's what we said that becomes six and Then it says we add three to the Y coordinates and minus seven add three is minus four if you're paranoid like I am Do it on a calculator. It's fine. So six minus four is here Again, last but not least and but the reason why I chose three points here is because there's three interesting points, right? The start point the end point and then the point where it turns so you might need one point You might need three points, whatever. It's fine That corner is two minus seven Again times the exponent by minus one minus two add three to that minus four Again just all because of this so step one write what the transformations are doing. That's step one Step two pick the points and do the transformation step three We're now going to draw a nice line to all of these in fact, that's rather than actually write this point down minus two minus four and Again, by the way, when you draw it, you don't have to be some kind of artist, right? Like they're not Testing your art skills, which is the only reason why I don't fail my GCSE exams for maths. And there you go Two marks is a bit rough. I'm not going to lie to you, but that's how you get those two marks If this question we have two triangles that are kind of overlapped with each other I mean to show the two triangles are similar Now the key thing here is similar right is we're not showing that they're congruent Okay, that's not what this is congruent means identical similar means that there is a Scale factor between them. Okay, so you can do this in one of two ways You can either show that all angles are the same Or you can show that one angle is Plus a scale factor. Okay And that's the really really really important because you actually can't show they're congruent because they're not congruent shapes So if you think of SAS or ASA or whatever, that's the mistake that you're going to make with this question Whenever it is similar It means there's a scale factor between them and you're either showing all of the angles and the triangle are the same So and whatever or you're showing that one angle is the same and there's a scale factor, so I would probably start by drawing these Triangles separately. They don't have to be super accurate But it's just to help you visualize and then I'm gonna try and draw one that looks vaguely similar to that Yeah identical, right? ABC and Now we can write down the sides. So for example, this one has size twenty one point six and thirty two and this will be thirty two plus twenty two is fifty four and The other one is twenty one point six plus fifty eight point four, which is eighty Now clearly they have they share an angle right they share angle BAC Which means that angle is the same in both triangles, right? So this angle here now in terms of the scale factor is actually not too bad All you need to do is divide the two pairs of sides so in this case fifty four divided by thirty two and you do the same thing for eighty over twenty one point six because again, those are the Complementary sides here and to show they give you the same number. So fifty four Right about thirty two gives me I was probably twenty seven over sixteen because I really don't want to do this and then eighty divided by twenty one point six is a hundred over twenty seven So those two sides aren't the actual ones we're doing so we're going to divide the other pairs of sides now So what you can do is also do eighty divided by thirty two as an example, but that will give us something a bit closer So eighty over thirty two And what's the other pair of sides we haven't divided yet fifty four over twenty one point six Is also five two, there you go There you go, and they also have the same scale factor If you're wondering why the other pair didn't work like the pair that looks like they should work It's because when you have this enlargement, bear in mind you can have an enlargement that Flips the shape right like we had earlier in the question so all that's happened is that the side AD is times by 2.5 and flips to give you AC and Then AE is times the flip speed of that. So just dividing the pairs the same scale factor same angle, which means they must be similar shapes Total of Y counts in a box There are X pink counters and five blue counters in said box. The rest of the counters are green in the ratio of X Y being 1 to 3 Cool, Freda takes two at random two counters from the box Finds X probability that it takes two counters of the same color. So I'd write out the combinations first So she could take pink and pink or blue than blue or green than green So there's a total of Y counters in the box and they also give me this so I would probably use this to start with What that tells me is that Y is three times bigger than X so I can write that Y is equal to 3 X So the total number of counters is just 3 X then the pink counters is X blue counters is 5 and that means the green would be 3 X Minus X minus 5 because remember we've already got X pink counters and 5 blue counters. So everything left over should be Green, I think the tricky part here was definitely the fact that it says the rest of the counts are green Then it gives you this ratio here But remember what ratios tell you if I put this on top of each other it might make it a bit clearer It means that Y has to be 3 times bigger than X so for every 1 X there's 3 Y's Right. So for example X was 2 then 3 times 2 would give you 6 here and then so on right? Cool. Now we can actually sub this all into here. So pink then pink that's going to be X over 3 X Times now because we've taken one pink counter out There'll be X minus 1 pink counters left to choose from so for example, there's 5 pink counters now There's only 4 right and the total also goes down by 1 I'm not going to work these out yet. I'm gonna work them out all at once blue That means there's going to be 5 over 3 X times again, there's one fewer blue counters of 4 over 3 X minus 1 and Green is that's probably the slightly trickier one 2 X minus 5 over 3 X Times 2 X minus 6 because remember we're minusing 1 from the number of green counters over 3 X minus 1 Now it's good that all of these will have the same denominator that's kind of intention so we have X X minus 1 Over and I'm not going to expand it yet I know that you could but I like to do everything all at once and this was 20 over 3 X 3 X minus 1 and That would be right 2 X minus 5 2 X minus 6 over 3 X 3 X minus 1 and all they've done because it says don't forget the 2 counts on the same color means we have to add up all of these once you get your Combinations you add up all of them to your final answer and then all they've done is expanded the top and expand at the bottom That's it. So if we add these all up because they all have the same denominators We can make it one big fraction right now. We don't have to wait Which is quite nice Which again is why another reason why I don't expand it straight away something might not spot that and we have X X minus 1 plus 20 plus 2 X minus 5 2 X minus 6 if we expand and simplify the top and bottom we will get our final answer for 5 marks So take it nice and slow. The first one would be X squared minus X and that'd be plus 20 The next one we have 2 X times 2 X which would be 4 X squared and you know what? I'm actually do this properly 2 X times minus 6 is minus 12 X then minus 5 times 2 X gives me minus 10 X and then minus 5 times minus 6 would be positive 30 and on the bottom we have 9 X squared Minus 3 X now again, it is possible that we're going to be simplifying this in a second But also maybe not it didn't actually say give it is the simplest form so I don't have to simplify it Let's start with the X squared. So we have X squared plus 4 X squared. So that gives me 5 X squared Cross off those two now. We have minus X minus 12 X minus 13 X minus another 10 X minus 23 X again, those two now and I like doing this if you do it really faintly the examiner won't think you marked it off or You can do it like a dot on top. Some people do that 20 plus 30 is 50 and The bottom remains exactly the same. So 5 marks not actually that bad, but you know Could be tricky All right. So for a bounds question, this is a little bit spooky But the good news is they're giving you the equation for this so we have to do is try and find out what each of these little parts are and So everything makes braces the cell there's materials The first thing I do in any bounds question without even reading the question is literally do the upper and lower bound for everything and you Always get marks for this Always so you might as well do it. So 190 to the nearest pound. So sorry to the nearest five pounds you need to be really careful here is It's not 195 and 185 right Instead you half this you do 2 pound 15 you add and subtract it here I'm gonna add on 2 pound 50, which is gonna be 192 pound 50 Because everything above that will round up to 5 to 195 and you subtract it. So 187 point 5 I know some students actually got that wrong even in a set one. So be careful with that Again, 875. Well now, you know, the first one is actually not too bad But eight seven seven point five and eight seven two point five And 72 hours the nearest hour again half it so it's gonna be half an hour at a point five since 2.5 hours or 71.5 hours Now we need to consider this This is the hourly rate of pay and this is the minimum hourly rate is to a pound 20 Determine if Ebony's hourly rate of pay was more than a pound 20 So logically here am I finding the upper or lower bound for her pay? Well, if I want to check if it's above a minimum It is definitely right. So even the lowest she could have got should be higher than a pound 20 I want to work out what the lowest hourly rate it could be So if we take again, whatever bounds we need to make this as small as humanly possible If that's above a pound 20, she definitely earns more than a pound 20 an hour So to do that, we need to make the selling price as low as possible and the cost of materials as much as possible Hopefully this logically makes sense, right? Like if if the materials cost more Then the hourly rate of pay goes down if the second price goes down and obviously the hourly rate of pay goes down as well But the other way to think of it is in terms of numerators to make the numerator as small as possible I take a smaller number and I subtract a bigger number from it and the time taken should also be the upper bound So if I just now look at these, this is materials so the selling price between the lower bound 872.5 And you subtract the materials Again, we're taking the upper bound of materials of 192.5 and the hourly rate we are making that as big as possible so 72.5 So just adding that into our calculator and we're all done So again, when you're thinking of bounds, be really really careful and try and use like some logic as well, right? Like it should it should make sense So the minimum she earned was £9.34 apparently Which is obviously bigger than £8.20. So yes, right. She definitely earned more than £8.20 For four marks, not bad. Now, this is such an interesting question Like column vectors, bear in mind is grade five. So it's foundation as well I don't know if they give you this on a foundation paper, but You know because it says parallel probably not because you don't need to know about parallel vectors in the foundation But this is a really interesting question And this is something that you do if any of you want to do A-level maths We actually use this a bit more in A-level maths and even A-level further. So it's actually a really really interesting question Before I even get started. I'm gonna think about what this actual statement means What does it mean when something's parallel? What it means is that this vector here should be a multiple of that vector Right, so for example, it could be 2 times this vector, it could be 5 times this vector, it could be minus 1 times this vector It could be a half of this vector. That's multiple. So for example 26 and 12 that's a parallel vector as an example So now I'm going to work out what this is So I'm going to times both of these by A because that's what you do with vectors, right? When you have a scale, scale it in front of it. Do the same thing with the other one And I'm going to add them together to make one big Let's call it a super vector. What the hell's wrong with me? One big vector. And again, we know that this should be Parallel. So in other words, it should be equal to n13,6. So I'm going to call it 13n And 6n Now it should be noted here, right? It's we're not working out the value of A and B. We can't. We're finding an expression so The only way they could give you this question, another way to give you this question is there's only one unknown So for example, they get rid of B That way you could work out A and they could say find the value of A or vice versa But here because there's three unknowns, we aren't going to solve it We're going to find expression for B in terms of A. So in other words B equals sunstuffed A So what do we know? Well, we know the top line should equal each other. So we have 2A Plus 8B equals 13n and we also know that 6A plus 2B should equal 6n So what don't what don't I care about? I don't care about n. So I'm going to eliminate n and to do that I need to make these the same so I think I'm going to simplify this one quickly by dividing by 2 to get 3A plus B equals 3n and Now how do I make n's the same? Well, I could times this one by Just times them by each other times it by 3 and times it by 13, which gives me 6A plus 24 39n and that gives me again. It should give me 33 times 13 is 39 plus 13B equals again 39n Now because these are the same I can make them equal to each other So I'm literally just going to write this equals this because again they both equal the same thing So I get 6A plus 24B equals 39A plus 13B and if you remember the question if we scroll back up find B in terms of A So I'm going to move the B's to the left side and the A's to the other side 24B minus 13B would give me 11B And 39C minus 6A would give me 33C So in other words B is equal to 3C How did that happen? Equals 3A and that's it for the final question Yeah, that's um a bit interesting. That was quite an interesting one, but hopefully you found that paper useful And yeah, I'll see you in the next one