GCSE Maths Grade 7-9 Problem Solving Explained 1 - Gold Paper

Added: 2026-05-07

[00:00:00]Hey guys, welcome back to another video.

[00:00:01]So, I released a video somewhat recently titled how to actually study maths or something along those lines. And in that video, I mentioned that if you really want to get to the grades seven, eights and nines, you need to focus on problem solving. So, I said that 40% of the content which gets you almost to a grade six is just on processes. So, can you factoriize, can you complete the square, etc. And so I found these papers, these gold papers produced by Pearson themselves, which give you AO3, which is the problem solving element style questions. So this aimed at grade 7 to N. So if you are aiming, if you're on at least a five already, and again when I say at least a five, I mean you're closer to a six than a five, then these problem solving questions are what's going to get you to the grades seven, eight, and nines. So what I'm going to do is I'm going to go through this paper. It's only 10 questions or so, and I'm going to go through my own thought process. That way you can kind of learn from my thought process and enhance your problem solving skills which will get you to those top grades because again it's not just about watching someone do a question or knowing how to do a question. It's knowing why you do certain steps. Uh just so you know um it says it's not actually non-cal. That's what the asterisk is. It's only questions marked with an asterisk. But to be honest, I don't know if I'll show you me working out how to times two numbers because again, if you're aiming for these grades, you probably know how to times and divide any numbers on the planet, right? But let's just jump straight into it now. So, first question is they give you a big old cuboid and they give you some of the lengths and they say calculate the volume of the cuboid. So, how do I approach this in a problem solving way? First things first, I'd start from the end. Calculate the volume. How do I work out the volume?

[00:01:37]But what you need to do is you need to times together all of the dimensions of a cuboid, right? You need to times some people call it like the length, the width, and the height or whatever. I'm just going to call it XY Z because let's just keep it simple. Now, we're going to label what we know from the question.

[00:01:54]So, in the question, they tell me that AB is seven. So, AB is seven. So, that means we know that Y is seven. So, I'm just going to put a seven on top of the Y.

[00:02:02]And I'm going to take it off. They say that AF is equal to five. That's the height. Cool. So, I'm going to put that on top of the X. And FC is 15. Okay.

[00:02:14]Now, the first bit of problem solving I want to tell you about is, and this is the general idea behind every question that you'll ever do in any exam ever.

[00:02:21]Firstly, they never give you information you don't need. Ever. Every bit of information is important. The fact it's a cuboid is important. The fact that we have these lengths is important. So, this 15 cm is relevant to the question.

[00:02:34]So if you're stuck, what you can do is say, well, what information have I not used yet? And what can I do with that information?

[00:02:43]Secondly, they have to give you enough information to solve the question.

[00:02:46]Shocking, I know, right? So if you look at this, I need to work out the length zed.

[00:02:51]Now they've given me this 15 cm. What can I do with it? Well, if I look, they've given me the diagonal from F to C. If I draw a dotted line going from A to C, I get a big right angle triangle that looks like this, where this is A, F, and C. And if I label the sides, I get 15 and I get five. So, I could work out the length A to C. Again, I don't know how that's going to help me just yet, but if you kind of look ahead a little bit, A to Creates another triangle, which I'm going to put in, I don't know, green. That's let's say and if I look at this from the top down it gives me a triangle that looks like this. A c B where zed is the side I want and I know that this is seven. So now I'm starting to see what the question's asking me to do. If I work out AC, then I know AC in this green triangle, which means I can work out zed. And remember at the end I need to times together five, seven, and zed. There you go. That's how you do it.

[00:03:59]So for this one, you're going to do Pythagoras. We're going to do 15^ 2 - 5^ 2 because remember we already have the hypotenuse. So we're going to be subtracting. Since this isn't a non-calculate question, let's just do it. Um it might also be a decimal. Yeah, it's 10 <unk>2.

[00:04:14]And I'm going to save this as the actual answer, like the exact value just so I don't make a mistake later on. And again zed, well that's going to be square<unk> 10 <unk>2. Remember to put this in brackets, right? Because it's not just <unk>2. If I just write this 10 <unk>2 2, the only thing my calculator squares is the roo<unk>2. It's 10 <unk>2 all^ 2 - 7^ 2. So I'm going to do square<unk> answer 2 - 7^ 2, which gives me 151.

[00:04:47]That's my value for zed. So the final answer for the volume is going to be 5 * 7 *<unk> [clears throat] 151.

[00:04:58]So* 5 * 7 and the question says to give it to 3SF. 3SF that's 430 um cm cubed. Just to point out I will put the link to this in the description.

[00:05:13]uh in the mark scheme it says for some reason 431 but the answer you get is 430.08 08. So I think that might be a slight error um because the method is is identical. So again when I eventually put this up just be aware that the answer they put in the marking is 431 for some strange reason. Yeah that's how you do that first question. So again this is another geometry question. I think quite a few of the questions tend to be geometry with problem solving which kind of makes sense. Now again it says a solid cone. It gives you this. I often get asked about what they give you in the formula booklet and according to Pearson you don't really need to memorize many of the area of volume equations because they will give them.

[00:05:55]For example, with a cone, they can ask you to use it, but they have to give it to you in the question like they've done in this case. They give me the diameter is 24x, height is 16x. Cool. The curved surface area of the cone is 2160 pi. The volume of the cone is v pi. Find the value of v. Cool. So the volume of the cone, they tell me the volume of the cone equals a3 pi r 2 h. Now I know what the radius is.

[00:06:23]The radius is 12x. That's completely fine. But I and I do know what the um height is as well. So, okay, pi. We know that's going to be 12x^2* 16x.

[00:06:40]Now, the problem here is we don't know what x is, right? So, we're going to stop there for a second. But just so you know, subbing into this formula gets you a mark. And this is a five mark question in total. So, what 20% done without actually working anything out?

[00:06:54]Okay, so this information I've used kind of. It says the curved surface area of the cone. Remember, they wouldn't give it to me unless it was critical to the question. If I look over here, curved surface area of cone is pi r. So, let's just make those two things equal to each other quickly.

[00:07:11]And just to make my life easy, I'm going to divide both sides by pi.

[00:07:15]So, I have 12 xl= 2160. So, I'm thinking I could work out x if I knew what l was, right? I think I think I could do that.

[00:07:24]So, what we're going to do is we're just going to move this over to the other side. We can simplify in a second. Um, but again, there's no asterisk, which means we could just use a calculator to do 2160 over 12. Maybe make it a bit more simpler to us. So, 2160 / 12 is 180.

[00:07:41]Okay.

[00:07:43]Interesting. So, I want to work out X and I don't know L sadly.

[00:07:50]What else could I do? Well, L is the slanted height here, right? So, if I draw it on here, it is this length over here.

[00:07:59][clears throat and cough] Now, they've given me these two sides.

[00:08:03]Now, this kind of looks like it could be made into a right angle triangle where this would be the radius 12x because it goes from the center of the circle to the edge and this would be 16x.

[00:08:18]So I could work out an expression for L in terms of X and then I have two equations with X in them which means I can solve it. So that's the exact strategy. So L is equal to square<unk> 16^ 2 X^2 + 12^ 2 X^2.

[00:08:37]Just so you know uh you can do this because we're going to add them together and then when we square root it just becomes X again. So on my calculator I'm just going to write 16^2 + 12 2 which is 20 and then the roo<unk> of x^2 is x. So there we go. We have 180 /x = 20x. Uh divide both sides by 10.

[00:08:59]Divide both sides by two. So I get 9x = x which means 9 = x^2 which means x is equal to plus or minus 3. The reason why I'm putting plus or minus is because when you square root you get two answers. But in recent years, especially 2025, they use the fact that there's two answers a hell of a lot in their questions, like like an unbelievable amount. So we need to decide are there actually two values of X? Well, what is X? X is a length. Can a length be negative? No. So in this context specifically, X does equal three. But remember, you can't assume that. Okay.

[00:09:40]Okay, so now I know what x is. If we go all the way back to the very first thing we did, we can now work out the volume of the whole thing. And it says give it in terms of pi. So we don't need to actually put in pi into our calculator.

[00:09:51]In fact, I'm pretty sure you can do this whole question without a calculator. So it's interesting that this one isn't one of the ones that's um required to be non-calculated, but whatever, right? Let's take it as we leave it. And then 16 * 3.

[00:10:06]And yeah, again on my calculator, I'm not going to put pi in because they want it in not even in terms of pi. They just want a number that that's in front of pi, right? So we're going to have 1 over 3 open brackets 12 open bracket 3 close bracket close bracket squared open bracket 16 open bracket 3 close bracket close bracket. I get 2736 pi which means the letter in front V is equal to 27 2736 as your final answer for five marks not too bad I hopefully now on to the very first non-cal question so again I'm not going to use a calculator shockingly and we get this hemisphere now when I did this with my tuition class. It's really interesting seeing people make the same mistake. So, I did this with my year 11s at school and I did it with uh tuition class and there are people that made the same kind of mistake and I'm going to show you what that is in a second. Once again, I'm going to start from the bottom. It says work out the exact total surface area. What does exact mean? Exact means you could have a third, you don't work it out. You could have a fraction, you don't work it out. Or you could have it in terms of pi and things like that. So it means whatever you get on your calculator, I know it's non-cal, but whatever you get, you do not simplify.

[00:11:28]Do not work it out. So you can't work out thirds because it won't be exact.

[00:11:32]You can't work out pi because it won't be exact. You can't work out things like 1 over 3 or whatever because it won't be exact.

[00:11:38]Total surface area. Now they've given you a hint with the word total, and I'll come back to that in a second of the solid hemisphere. So let's think about it. Well, look at the equations they've given us. Volume of a sphere, surface area of a sphere. So logically if I half this surface area because it's a hemisphere that's going to be the first part right? So 4 pi r^ 2 over 2 which logically is just going to be 2 p<unk> r^ 2 right now is that it well no see when you cut a sphere in half if I take off the top I also expose this surface. So this formula here is only for the curved surface area of the sphere. It's just that for a sphere, the whole surface area is curved. But with a hemisphere, you also have this bit on the top as well. You expose that part.

[00:12:30]Think of it like cutting like an orange in half. You expose the inside flat circle on top. So, we need to actually add p<unk> r^ 2 as well. And that gives me 3 pi r^ 2. So, that's going to be the equation I sub into the second I figure out what r is. Cool. There you go. Now, I understand. Now, you know, I'm getting close to there. Okay, what else have they given me? The volume of the hemisphere is 250 over 3 pi.

[00:12:57]Volume of See what I mean? They never give you information you don't use. And if you just keep using that logic, it's not saying that math becomes easy, but you can at least scrape together half the marks or most of the marks to get into that next grade. Remember, you're not looking, you don't have to get 100%.

[00:13:13]So, again, it's a hemisphere. So, what we're going to do is we're going to half this volume. Now, volume is the space inside of a shape. So, when I half when I've cut it in half, I have half the volume. I don't need to add anything else. And that's going to be 2 over 3 pi r cubed.

[00:13:30]Why? Because again, 43 / 2. I'll show you nice and quick. You change them both into fractions. Keep flip change.

[00:13:40]Times it across. You get 4 over 6, which is the same as 2/3. If you'd like, when you divide a fraction by a whole number, you just divide the numerator or you times the denominator, whichever way you want to do it. And we know that equals 250 over 3 pi.

[00:13:57]So now think about what you can cancel.

[00:13:59]Well, there's a pi on both sides. I'm going to divide both sides by pi. I'm also going to times both sides by three because they're both being divided by three. Cool.

[00:14:10]Now, um, I'm going to write out one more time just so it gets a little less messy, but you could actually do the whole thing using the same strategy.

[00:14:16]Divide both sides by two. R cub= 125, which means R is equal to the cube root of 125, which is 5. Cool. Now, we can sub into that original equation. We get the surface area. I'm going to call it SA= 3 pi 5^ 2. Well, 5^ squar is 25. 3 * 25 is 75. So, we get 75 pi.

[00:14:46]So, I'm hoping that you're seeing that this is actually isn't too bad. And if you have someone going through the problem solving questions with you, you can kind of pick up on the strategies and things like that. Again, there's a link in the description or in the pinned comment if you want to join those tuition classes where this is all we're doing until the exams. Okay, so we have Thma spins a biased coin twice. The probability will come down on heads both times is 0.09. Calculate the probability that come down on heads both times. So although it's probability, we're still going to use the exact same I don't know techniques, right? So again, I'm going to write what do I actually want? Tails both times. So I'm going to do t times t. So in other words, I'm going to take the probability of getting tails and times it by the probability getting tails. Okay? And what have they given me? They're giving me the probability that it lands on heads both times. So h* h is equal to 0.09.

[00:15:40]So in other words, the probability of getting heads squared is 0.09, which means if I square root this, I get the probability of getting heads once. And what would that be actually? So square root 0.09 obviously 0.3. Now I'm not going to put the plus or minus because probabilities can't be negative. That it's nonsensical.

[00:16:04]Okay, so I have the probability of getting heads. How can I get the probability of getting tails? Well, if you think about it, how many outcomes do we have when we spin a coin or toss a coin? Well, you either have tails or heads.

[00:16:18]Now, that means that those two probabilities have to add up to one.

[00:16:21]It's guaranteed to either get tails or get heads. So, if I know that heads is 0.3, then I Whoops. Then I know that tails must be 1 minus 0.3 which is 0.7 which means my final answer is just 0.7 * 0.7 which is 0.49. I don't need a calculator for that one. The number of students that would not answer this is incredible because probability has kind of a a bad reputation for being very tricky. But you can basically make it algebra every single time, right? We don't deal with probability distributions like we do in Alevel maths. So it's just algebra really and just making equations. So I don't think that's too bad. Now this question genuinely is actually pretty tricky in terms of a conceptual level.

[00:17:06]If you want to do Alevel physics, what I'm about to do you actually use quite a lot which is the idea of a proportional relationship. So here it says a pendulum of length L has time period T and T is proportional directly proportional to the square root of L. So we now know that t is proportional to square roo which means we can write an equation like this which again already gets us a mark even though I haven't done anything right now they tell me the length of the pendulum is increased by 40% work out the percentage increase in the time period now this is quite tricky uh for a lot of people to get but essentially if I increase it by 40% that means we are now at 140% of the original value for the length Right? So if it was 100 cm, we're now at 140 cm as an example. As a multiplier, that's 1.4.

[00:17:59]So this is the same as me replacing the letter L with square<unk> 1.4 L, right?

[00:18:06]Because I've timesed it by 1.4.

[00:18:09]Now, if they only want the percentage increase, I don't need to work this out as a number. I'm not working out the time period. I'm working out the percentage increase. In other words, I'm working out the multiplier for the time.

[00:18:22]So, what I can do is I can just work out the square root of 1.4. So, if I work out the square of 1.4, which I'm going to use my calculator for because I want to, I get 1.18.

[00:18:37]Now if you look at this and compare it to the original equation, this time period is 1.18 times the original one because don't forget K root L is the original time period. What is that as a percentage increase? Well, that's 118% which means it's an 18% increase.

[00:19:01]And that's the answer.

[00:19:04]There you go. use that a lot in physics.

[00:19:06]Hopefully it makes some amount of sense.

[00:19:07]It is a tricky question. I'm not going to lie to you. Okay, so back to geometry again. Shocking. So we get a right angle triangle. X - 2 X. Perfect. All measurements are in cm. The area of the triangle is 2.5. Find the perimeter. So if we want the perimeter, we need to add up all three of these sides. I don't know this side though, so that's something I need to work out in the future. And I don't technically know these two sides either because it says X and it that's not a number. what it is, but we just don't know what the number is. So, what have they given me to help?

[00:19:35]Uh, the area of the triangle. Cool. Area of a triangle is half base times height.

[00:19:39]So, that means it's going to be x * x - 2 over 2 = 2.5.

[00:19:47]Now, we just need to solve this. It's going to be a quadratic. So, I would probably times both sides by two just to get rid of the fraction because everyone hates fractions.

[00:19:55]Equals 5. Expand the left side. x^2 - 2x = 5. So if you move this all over to one side, we get x^2 - 2x - 5 = 0. Now I'm pretty sure that doesn't factoriize, especially because they say three significant figures. That kind of gives me a bit of a hint that they might not factoriize either. So I'm just use the quadratic formula which you get given in the exam anyway. Now, with doing this, I've seen again grade seven, eight, and nine students make mistakes on this all the time because they don't use brackets, and it's the dumbest reason to lose marks. So, for example, especially when you get to a square, you put it in brackets. Minus 2^ 2 on a calculator gives you a different answer to brackets - 2^ 2. So you need to be really really really careful when you're doing this. In my opinion, anytime you substitute, you should be using brackets. That's the best advice I can give. We're going to get two values of x. So just shove it on the calculator now. Writing exactly what I've just written on the screen. I'm going to start with the plus one because um it feels more natural to me. -4 1 and -5 all over 2 * 1. I get 1 plus roo<unk> 6 which is 3.44. In fact, I'm going to leave it as one plus roo<unk> 6 because when they ask you to give your answer to 3sf. If you round x to three significant figures, what might happen is when you get to the end, you round again and you get a slightly different answer. So, I'm going to keep it as an exact value until the very end. And I'd suggest suggest you do the same.

[00:21:34]And the negative obviously I get 1 minus<unk> 6. However, something to note, the one on the right is less than zero. It's negative. Now, again, if you think about the lengths of a triangle, it should be positive. So, we're going to be using the positive value.

[00:21:50]So, what do we know? Well, we know that x - 2 is now 1 +<unk> 6 - 2, which well 1 - 2 is - 1, right? So, that's going to be roo<unk> 6. In fact, I'm going to replace it. R<unk> 6 - 1.

[00:22:04]This is 1 +<unk> 6. And now I can work out this side by just doing square root in brackets<unk> 6 - 1^ 2 + 1 +<unk> 6 all squared and yep let's just quickly do that. So square<unk> 6 - 1 all squar plus 1 whoops plus<unk> 6 all squared gives me<unk> 14. Again, notice I'm leaving everything as a precise value until the very end. So the perimeter now is I just add up these three things. So we have roo<unk> 6 - 1.

[00:22:43]So I'm write P is equal to roo<unk> 6 - 1 plus let's see the bottom side 1 plus roo<unk> 6 which means those two ones actually cancel. That's quite nice plus 14 which was the hypotenuse. So if we do that we get [clears throat and cough] add roo<unk> 16 sorry six not 16 plus roo<unk> 6 and to 3SF I get 8.64 and in the mark scheme it's between 8.63 and 8.65 with slap bang in the middle.

[00:23:16]So we get that mark and I think that was a a six mark question and honestly it's not that long really. It only took me uh four minutes to explain. So yeah it's not too bad. Not too bad. We have another non-calculated paper paper question. But to be honest, there aren't really any numbers in it. So that a fat lot how good a calculator would do anyway. So there are 10 pens in a box. X red pens. All other pens are blue. Jack takes at random two pens. Find an expression for probability takes one of each color. So that's going to be blue then red or red then blue. And we need to work an expression for both of those um expression expression for both of those probabilities. then add them together. Now, the thing that I would recommend, especially when it's algebra instead of numbers, is along the side when you're doing conditional probability like this, is to just write out how many red and blue pens you have.

[00:24:10]So, we know that there's 10 pens in total, there are x red pens, we know there's x red. Okay, I'm fine with that.

[00:24:16]The blue pens, well, it says all other pens are blue. So, that' be 10 minus x, right? Now, how would I come up with that if I was a bit stuck in the exam?

[00:24:26]Give a number for for red. So, if there's 10 pens in a bo box and there's three red pens, how would you work out how many blue pens there are? 10 minus three. Well, if there's five red pens, you'd say 10 - 5. So, it's just 10 minus whatever the red number is. That's it.

[00:24:42]Okay. So, and we know the total is 10, right? That's fine. So, what's the probability of picking out a blue? Well, there's 10 min - x blue pens out of a total of 10 pens. And that would be times well, the number of red pens.

[00:24:57]Well, if I take out one of the blues, I still have x red pens, but I only have nine pens in total now because I've taken one out. Cool. Um, just so you know, actually, I'll tell you this in a second. It's a very uh good trick that you can use. I'm going do the same thing for the other one. So I have x red pens out of out of 10 times well there's still 10 - x blue pens out of a total of nine. Now if you notice these two probabilities are exactly the same. In general just changing the order in which you do something doesn't change the probability of it happening. So you can literally just do the top one and double it which I had some students do which is very very clever. But let's just assume we weren't we didn't spot that right? So we get 10 - x * x over 90. And this is also 10 - x * x over 90. And we have to add these two together. Now adding these two together is quite interesting.

[00:25:57]If you did the trick that some of my students did spot, which is just tsing it by two, you actually get to the answer very quickly.

[00:26:07]But let's say you didn't know that.

[00:26:09]Well, I would literally add them together. I have two lots of the same thing which means I get 2 x 10 - x right because I have two lots of it so it's two times that then I can divide the top and bottom by by two and I get x 10 - x over 45. Now um I do get asked by some students can you do you have to expand it?

[00:26:37]I'm going to say no. You don't have to expand it, but sometimes they do, sometimes they don't. You still get the marks either way for again another five marks. Mark has made a clay model. He will now make a table that's mathematically similar. Remember, when it's similar, it means there's a scale factor between them. They give us two areas. Probably guessing what we need to do. Now, the interesting thing is they give us mass.

[00:26:59]Interesting, right? Well, because density is mass over volume.

[00:27:04]That means volume times density is equal to mass, right?

[00:27:11]Volume and mass are proportional. So, for example, if you have a a brick, right? If you get two bricks or one brick that's twice the size, it weighs twice as much. If I have a lump of copper and I get twice as much copper, it weighs twice as much. I know this is revolutionary right now, but my point is is the volume scale factor is the same scale factor for mass because if you make something twice as big, it gets twice as heavy.

[00:27:40]At least again in this in this context.

[00:27:43]So what we're trying to do here, it says we need to, you know, work out how many bags he needs to buy. Well, that depends on how much mass this model, like this real thing is going to need, this statue.

[00:27:55]Um I don't know who makes clay statues though. But anyways, so we need to work out how heavy this clay statue has to be. Well, if you remember, um we have so we have model. Yeah, I'm going to use model and uh I don't know statue and then then length area volume scale factors. They give us volumes which is six to two 53.5.

[00:28:17]So to get to volume, what we're going to do is we're going to first work out the area scale factor by dividing these two things. I I don't know how I don't know if it's going to be a very nice number.

[00:28:28]Probably not. It's not bad. 169 over4.

[00:28:32]Then if we square root this, we will get the length scale factor. And then I'm going to cube my answer in order to get the volume scale factor. So square root answer is 13 over2. So we're going to get 13 /2 all cubed, which I don't actually need a calculator for, but you know, why not? 2197 over 8.

[00:28:56]So if the model weighs 2 kilos, then the real thing is going to be 2 times this giant number.

[00:29:07]That's how much mass we're going to need for the real thing. So 549.25.

[00:29:15]That's a that's a lot of clay. Now sold in 10 kilo bags. So, we're going to do is divide it by 10 to get the number of bags, which is 54.925.

[00:29:24]So, because you can't go to like I don't know, you can't go and buy 0.925 of a bag means he's going to need 55 bags.

[00:29:33]So, yeah, if they ever ask you for mass, it just means the same thing as volume really in in that regard. Okay. Um, iteration is weird. It's a topic that a few students really find difficult as a process. Um, probably because of the notation, I imagine. But all this means is the population in the next year is just 1.05 times the population this year minus 250. So they give us it's a number of B's. They want it at the start of 2018. So for example, if we start with 2015, that's 9,500.

[00:30:06]To get the next year, all we're doing is we're subbing it into this formula here.

[00:30:11]So 1.05 5 * 9500 minus 250 and then we get the answer and we sub to the next one, the next one, the next one, right? Then we get 2017 and 2018.

[00:30:23]Now that's quite tedious, but this is to be fair three mark question. So you could do that. So find the answer here, sub it in again, get the next answer, sub it in again, get the next answer.

[00:30:33]There is a much faster way to do it. You take your starting value on your calculator and you just press equals.

[00:30:39]Right? So, shockingly, you get that result. The answer button is now 9,500.

[00:30:46]The good thing about the answer button is every time you get a new answer, it automatically updates. So, all I'm going to do on my calculator is I'm going to type this now.

[00:30:55]And this works for every iteration question where they ask you just to do it repeatedly. And every time I press equals, it's going to give me the next year. So, by the way, just so you know, with this question, you don't have to write out each year, but I'm going to just so you can follow along. So in 2016 it gives me 9712.5.

[00:31:14]I press equals again it gives me another value 993 5.625.

[00:31:21]One more time I get 10,169.906 going on. So that means how many B's are there? We're going to round it because you can't have a fraction of a B.

[00:31:32]10,170.

[00:31:34]They also accept 10,169 if you just want to truncate instead. Yeah, using the answer button makes that super super easy. It's not really problem solving, but it's still in here anyway. And for the final question, some coordinate geometry, which does tend to be quite tricky. And if you look through all the past papers, it does tend to be one of the last questions, especially on the non-cal paper, probably because most of it's done with fractions and stuff anyway. That's why you don't really need a calculator. So, it says here, find an equation of the line that passes through C. So this point here and is perpendicular to AB. So if I were to draw that looks like that.

[00:32:13]Now again start with what you know. I want an equation of a line which means we need y= mx + c. What two things do we need for an equation of a line? You need a gradient and you need a coordinate. Now when I say a coordinate I mean a coordinate that lies on that line. So on this blue line, do we have any coordinates? We have C. It says it passes through C. So the only thing I actually need is the gradient. So okay, what else have they actually told me? I've already used the fact that there's going to be a C. I know I'm going to use that second. But it tells me it's perpendicular to AB. So I must be able to use that to work out the gradient. How? Well, when you have two lines that are perpendicular, their gradients are negative reciprocals. So I'm going to work out the gradient of the line AB. So I'm going to call that N A.

[00:33:04]I'm going to do change in Y over change in X. So it's 4 minus 0.

[00:33:09]In fact, something that actually helped some of the students when I was going through this is labeling the points 04.

[00:33:15]And this would be - 2 0 right over 0 - - 2 which gives me 4 over 2 which means the gradient of AB is 2 which means the gradient of my line that I actually want is -1 over two. Remember you flip the fraction. So that's 2 over 1. Flip it.

[00:33:33]It's 1 over two. Change the sign.

[00:33:36]Sub it into this equation. We get y is equal to minus a half x plus c. and then use the only other piece of information they've given me, which is the coordinate when x is 5, y is - 1. So we have -1 = a half * 5 + c. So min -1 equals, you could do it as a decimal if you want, but I'm going to do it as a fraction plus c, which means c is equal to minus1 - 5 / 2.

[00:34:07]And which should give me the hang on a minute, that's minus a half. Oops.

[00:34:12]That be minus. That' be + 5 over2.

[00:34:15]Uh 1 is just the same as 2 over2. So we have - 2 over2 + 5 over2 which is 3 over2. So c is 3 over2. So we get y is equal to - a half x.

[00:34:28]That's bad. Plus 3 over2 for the final answer for four marks. Not too bad. If I show you the actual bit at the bottom, it tells you where they get the questions from. Just so you know, they are actual kind of past papers. Well, they don't give you like the other, you know, how I've done aiming for grade nine papers on the channel. In fact, I've done like six sets of them. They give you how many marks you need in order to get whatever grade. They don't tell you that sadly, but this is where the questions come from. Again, I'll put a link in the description so you can also check the mark scheme. Um, but yeah, hopefully you found that useful and I'll see you in the next