2026 05 11 - Maths F5 - FINAL May 2026 P2 prep session

追加: 2026-05-12

[00:00:06]Okay.

[00:00:08]All right, guys.

[00:00:12]Right.

[00:00:19]One, two, three, four.

[00:00:23]Cool. Okay. So, like I said, what we trying to do is get some circle geometry and some bearings done.

[00:00:33]So, what we'll do is we do the circle geometry first, then the bearings, right?

[00:00:38]Okay.

[00:00:41]Um, okay.

[00:00:56]All right. So, this is G 22, not five. God damn it.

[00:01:09]Cool. Right. Nice. So, let me actually put that to print for you guys first.

[00:01:13]So, um 1 2 3 4. Cool.

[00:01:19]Current page range. That's 22 to 23 2 to 2.

[00:01:51]Cool.

[00:01:53]Vivian, how you doing?

[00:01:58]Sure.

[00:02:00]All right. So, they want to find X, they want to find Y, and they want to find Z.

[00:02:04]Each of them has two marks, right? Okay, cool. So, let's copy and paste the relevant part.

[00:02:12]Why can you open?

[00:02:50]Good morning.

[00:03:05]>> Yeah.

[00:03:07]from from Monday to Wednesday it's just P Thursday from Thursday to Monday cuz right Yeah.

[00:03:34]Sorry.

[00:03:47]All right. Okay, folks. So, let me um sort out all the sort outs there ASAP cuz we're going to do four of these questions. Four bearings, four matrices, and four vectors. I don't know that that will happen.

[00:04:04]Okay, this is so for Jan 2022. So, anybody who following along, we're just doing certain pieces, right? We did from Jan 2026 come back to May 2022. Full papers already. So we just decided, hey, you know, we did May 2019 earlier was a nice challenging paper and we're just picking out the last piece. We did question eight from Jan 2022 to Jan 209, I think, or 2020, including May 2022.

[00:04:31]May 2021, sorry. Um, but we I think we'll bypass the May and just do the JS, but I'll see. Right. Okay. So, without further ado, let's um get into it.

[00:04:41]Right.

[00:04:42]Nice. So diagram below shows a circle center right with points PQR and S P QR and S lying on the circumference.

[00:04:54]RP is a diameter. Okay, cool.

[00:05:00]Hold on just now. Wait a second.

[00:05:03]Yeah, RP is a diameter. A diameter cuts a circle in half. And if you build a triangle on set diameter, the angle at the circumference is 90°. So we have a 90° angle there. Um, one second. Let me just check a message here real quick. Right. So we have a 90° angle at that. Well, the whole angle is 90°. Yeah.

[00:05:26]Um, AB is a tangent to the circle. So AB is a tangent to the circle.

[00:05:36]All right, cool. Okay, where was I?

[00:05:54]Cool. Now, we know that. Sorry, let me get a slightly heavier weight on that point there so it'll stand out. That that pink line is a radius and the angle between tangent and radius is 90°. So we have a 90 on that side, 90 on that side.

[00:06:08]Okay.

[00:06:09]Now we have three angles. Hold on.

[00:06:15]So we have a 54° angle down here. We have a 3x, a 2x, and a 1 x. Okay, cool.

[00:06:21]No problem there. So they want us to find angle x, angle y, and angle z.

[00:06:26]Right? So I didn't put in the requirements, but that's what they want us to find with the reason. And there's two marks each. Okay? Now, right away, I could tell you one time without any hesitation at all that angle angles Y and Z are equal because they follow what I like to call the butterfly rule, right? Angles in the same segment standing on the same arc or built on the same chord are equal. So once you find angle Y, you find angle Z or once you find angle Z, you find angle Y. So the question the quest is now to find that angle then.

[00:06:59]All right. So there we go. All right. So with angle.

[00:07:05]So yeah, the first one. Hold on, let me just pull it up.

[00:07:09]Right, they ask for angle X first. Now there's no necessity to go in order, but most times they kind of they kind of sequence them. So you need whatever you did before to do what you have to do thereafter, right? But not always.

[00:07:19]Sometimes you could find certain ones kind of out of sequence. Just make sure they give you reasons. All right. Okay, cool. So the angle between tangent and radius. So I'm going to go back to this one. angle between tangent and radius is 90°. All right? So, in other words, um this angle here is 90. And what I want you to realize is that these these two angles, the 2x and the 3x correspond to that 90, right? Cuz if you look at it from right all here to here, right? So, that's 2x and 3x.

[00:08:00]Right. So, uh I guess I'll start with this color.

[00:08:04]Doesn't really matter.

[00:08:06]All right. So, for angle x, so we could start by saying that 2x + 3x is equal to 90°. And if you want, you can cite the reason there or if you prefer, finish your working, solve for x, and then go back and state your reason after. Right? So, 2x + 3x is 5x.

[00:08:26]And if 5x is 90, x is equal to 90 / 5.

[00:08:31]So x is equal to 18°. Right? And again the reason here, right, is angle between tangent and radius is 90°.

[00:08:48]>> Uhhuh.

[00:08:53]>> Monday, Tuesday, Wednesday, next week.

[00:08:54]So that's the 18th, 19th and 20th.

[00:08:58]All right, we'll confirm in the chat what times if you want the morning, the triple session in the morning or if you want to do after the evening.

[00:09:19]All right. So that's how we know that 2x + 3x is 90. That means 5x is 90. and then to find 90 sorry to find x and divide 90 by 5. Okay, cool.

[00:09:29]Now we're going we're going again. Right now to find angle Y, we could do something kind of not similar per say, right? Um let me just actually fill in the whole thing there. Right.

[00:09:46]Okay, cool. Um let me actually blank off a couple extra lines here. Blank off this one.

[00:10:02]Okay. So the rule I'm going to use now is um the one that I had discovered actually within the last 10 years. So the angle between a tangent and a chord is equal to the angle in the alternate segment. So sometimes called the alternate segment theorem, right? So a chord as you should know is any straight line that goes from one point on the circumference of a circle straight to the other um other side right from one point in the circumference to the next point. Now it doesn't have to pass through the center right to be a chord.

[00:10:35]It could pass it could go from right here to right here uh right here to right here. Right? It could pass through the center. Right? In other words this this diameter is a cord. So the rule could apply here and it actually does.

[00:10:48]All right. So right now actually to find actually no sorry I'm not going to use that to find y. My bad there. Now you could you could um but the easier way to find y sorry since we know x is 18°. Oop sorry I had to go back down here. Right.

[00:11:09]And we know that this is 90° because the angle in a semicircle is 90°. What's the sum of the interior angles in a any triangle?

[00:11:19]180. So if we have 18 and 90, could we find y? Yes, we could. Yes, we can.

[00:11:28]Okay, cool. So to find y.

[00:11:33]So y + 90 + 18 is equal to 180°, right? So I'll put a little one here. Well, I'll put a little one here and a little two here.

[00:11:46]I'll show you why this cuz there are two different reasons, right? So, of course, 90 + 18 will give us 108°.

[00:11:56]And therefore, to find y, all we have to do is subtract 108 from both sides of the equation.

[00:12:08]Hold on.

[00:12:10]Nearly there. Okay. Right.

[00:12:14]180. Sorry, Mr. Cool. Honestly, honestly, just send a message.

[00:12:21]10 seconds. Good.

[00:12:53]and for Zeno Z is equal to Y to the interior opposite angles.

[00:13:05]>> So wait, is the interior.

[00:13:16]Okay. So, this is exterior.

[00:13:35]Am I thinking it like thinking it too much? Like my >> right and this one like I showed earlier is the angles in the same segment.

[00:13:51]Right.

[00:13:55]standing on the same arc are equal.

[00:14:11]Right? Now, another way you could have found it is if you were to find So, since since um x is 18, 6x would have been 108, would it? Yeah, which means that this angle has been 72 because the angle on a straight line is 180, right? So if you have 10 108 on that side, you subtract 180 in here, you get 72. And yeah, so that this would have been the alternate segment to that exterior and this is the alternate segment to this exterior angle here.

[00:14:44]Well, angle between tangent and cord, not exterior. Sorry, it is. No, no, it's not totally. All right, listen. I sent Anna something real quick because I think she no Anastasia I come back after the help for PO. Okay, cool. Good to see you. Hey 45 boy.

[00:15:09]Pretty good. That's why I couldn't press that earlier cuz there's a big crack on my bloody screen now.

[00:15:21]Okay. All right. In class people, are we okay with this item here? All three angles. Any questions? Anybody?

[00:15:27]Anything?

[00:15:29]All right. Cool. All right. Okay. Um Elliot, just give him a little thumbs up for our G. Just tell them. You could always unmute and tell them because it's just us.

[00:15:43]>> Yeah.

[00:15:43]>> All right. Cool.

[00:15:46]One sec. I'm trying to find something I'll send to this guy.

[00:16:18]Okay, cool. All right, so that's 2022.

[00:16:40]Yeah, that's bottom thing. All right, cool. So, let me pull up the Jan 2021 paper. Pull up the circle jump. All right, so there's an error in this this particular one, right? So, hold on. They want ECD.

[00:16:52]All right, the 25 28, right?

[00:17:00]Where is the dash?

[00:17:02]Oh.

[00:17:20]All right.

[00:17:22]Somebody over here.

[00:17:24]Watch your phone.

[00:17:31]Hey, what?

[00:18:21]What? What? Wait, what happened there?

[00:18:29]Okay, cool. And then one more for Jonathan, right?

[00:18:38]28.

[00:18:45]Okay.

[00:18:54]across it might be. I can't guarantee. I don't really patronize it that much, so I can't tell you for sure.

[00:19:16]>> All right, cool.

[00:19:31]I have to service every time pressure.

[00:19:52]Okay. So, first thing to do on the diagram, I want you to please that that angle on the kind of top right by F is not one. Whatever they have there is actually 112. There was an error in the paper.

[00:20:11]>> Yeah.

[00:20:13]Pen here.

[00:20:18]up here.

[00:20:33]Okay, cool. So, I didn't write down what the question wanted. So, let me do that very quickly. They want ECD.

[00:20:40]Whoa. Don't want to do all that.

[00:20:48]Right. C EG and CGF.

[00:20:57]So E C D is this angle here.

[00:21:02]C EG was the next one.

[00:21:07]C EG and then CGF.

[00:21:12]All right. So, we have three angles to find them. So, what I'll do is I'll give you a minute or two if you want a little head start. All right. But what we do know is well, EFD is 106.

[00:21:33]Yeah. No, that's not 106, was it? Yeah.

[00:21:36]No, it's actually 112.

[00:21:51]All right, we know that this is a this is the center, right? So, and they tell us that um AE is a what you call it is a tangent.

[00:22:06]No, I don't want red. I would like it black and yellow, please. Thanks.

[00:22:13]And it's actually parallel to this other lineup. But I want to know why not green. Why not?

[00:22:23]Oops. A little too crooked to autocorrect straight.

[00:22:28]And they tell us that they're parallel.

[00:22:29]The tangent AE B is parallel to CD. So those two lines are parallel. So we know with parallel lines we have scope for F angles and C angles or corresponding and alternating angles and we also know that this line EG is a diameter right so and a diameter of course can be cut into two radi Okay.

[00:23:11]Oh yes.

[00:23:22]help. So if I draw a line like this.

[00:23:31]So yeah, you should be able to see the kind of backward Z being formed there, right? It's it's kind of it's going literally backwards.

[00:23:40]And we know with Z angles, the angles and the kind of corners are equal. Those are alternating angles which means that angle X is actually equal to 68 because of alternating angles.

[00:23:55]So X is equal to 68° because of alternating or Z angles.

[00:24:12]All right.

[00:24:16]Now, don't forget this piece here. Hold on.

[00:24:22]This piece here is also a radius. And this is a tangent. And the angle between tangent and radius is 90°.

[00:24:32]So this whole angle here inclusive of 68 and y is 90°.

[00:24:42]So therefore to find y right so angle between tangent and radius Yes.

[00:25:10]So therefore, y is equal to 90°us 60, which is 22°.

[00:25:19]Oh, lord child.

[00:25:30]112.

[00:25:32]Say that again. Sorry.

[00:25:37]>> 112 - 68.

[00:25:41]Okay. Right. So you're you're saying that this angle here is 112 because of the angles.

[00:25:45]>> No.

[00:25:51]>> Well, I'm not sure what the reason is.

[00:25:52]You going to tell me what was the reason for 1/2 minus the 68 like like I said if you follow the Z right this angle this whole angle here hold give me one more second I'm going put a little extra line here right right so you have a you have a next kind of Z angle going on there a kind of elongated so this whole angle here is 112 I could agree with that now 112 - 60 will give us this whole angle here and then divide by two but then what's the rational for divide by Yeah, but how you know that that this here is double y, right? I mean, it turns out that it is interestingly enough, right? The other other thing actually how you're going to have found why is that this is 90° here. How do I know that? Because look, you have you have a f angle. So this 90° sorry there. So this is 90 and that's 90 as well.

[00:26:56]All right. So if you have a 90° here 90 plus the 68 is 158. 158 from 180 which is the sum of the interior angles of a triangle.

[00:27:06]150 minus sorry 180us 150 will give us 122. Sorry 22. Mhm.

[00:27:13]Alter Robin Hood. Yep. X is 60. That's correct. Alternating. Good. Yeah. Yeah.

[00:27:20]And then to find the next one, the minus 112. Uh-huh.

[00:27:24]Okay. Okay. Very good.

[00:27:28]Yeah. Right. So, we have one more angle to find which is the angle at the top which is angle Z.

[00:27:38]Right. Now, that's an interesting one to find.

[00:27:47]So, what I'm thinking is if we actually take a look at the Hold on. Well, I I want to get rid of the colors and I like the colors any So, okay, here I'll copy and paste it, right? Okay, I'll get rid of the colors now cuz I'll I'll put in some different colors here. Um, yeah. So so the yellow shape is a cyclic quadrilateral and the opposite angles of a cyclic quadrilateral are supplementary not so so that means they add to 180. So if we could find this whole this whole angle here we could subtract it from 180 to get the Z on the top there. Now again there probably other ways to find this right. Okay. So how do we find this angle here?

[00:28:47]Right? So we could if it please you all right if we extend this line here right we can see that this angle here is also 112 because of the parallel lines and the f angles right which makes this angle here also equal to 68 because the angle on a straight line is 180° and then this being 90 we can find that this is 22 which is the same as across here interestingly enough. Okay. And if we have 22 and 22 that's 44. And to find Z it will just be 180 minus the 44 which is 136.

[00:29:37]That's a lot for two marks though. I feel like there should be an easy way to find that.

[00:29:45]That's 60 there. That's 60 there.

[00:29:52]Don't know the angles with that. That's just angle Z.

[00:29:57]The 2290.

[00:29:59]Okay. Right. I'm seeing another way.

[00:30:06]Right. So another way to find it if you're curious about it is like this.

[00:30:12]So we will need to acknowledge that um this line here is a radius sorry a diameter a di a diameter cut a circle in half not so so if you build a triangle on set diameter the angle at the circumference will be what? 90°. All right cool. So if you focus on the triangle now right you have sorry you have 22 here you have 90 here so we could find this piece of it all right no and then we still have to go here cuz even even though this is this is also 90 we have to show that this is equal to 22 as well all right seems like a lot of work for a little bit of marks. But anyhow, yeah, so that's how we will play it.

[00:31:13]Okay, cool. So let's do that quickly.

[00:31:18]All right. So, right, angle Z is equal to 180 minus the 22 + 22 is 44. So, you get 13°. This is because um opposite angles of a cyclic chord uh supplementary which means that the adapter 180° I think this is May 2021. Hold on. Just know.

[00:32:22]Yeah, this was in May. Sorry, I put June. All right, my bad.

[00:32:29]All right, small thing.

[00:32:32]Okay. Yeah. Um, let me see that triangle. That triangle we didn't know. We could find that angle there and that angle there. Yeah. Okay. So, we could find that this was this was 68° because again angle on a straight line is 180, right? Which mean this is 22 here. Um, this is 90. This is 68 which means this is 22 here. Yep.

[00:33:02]Yeah. So, we could do that as well. And 22 + 22 subtract from 180, which is exactly what we did here. All right.

[00:33:10]Okay. All right. Elliot, talk to me. You feel okay with that one?

[00:33:16]All right. Cool. All right. And CL people, are you okay with X, Y, and Z?

[00:33:21]Any further questions? Anything to clarify?

[00:33:24]Good. All right. Okay.

[00:33:58]Oh my go. Can I not just write an M?

[00:34:01]What is wrong with this thing?

[00:34:08]I don't even know anymore.

[00:34:16]Cool.

[00:34:18]Lovely. Cool. All right.

[00:34:21]Zan. Okay.

[00:34:26]26 to 29 for your uncle.

[00:34:54]All right. So, that's printing.

[00:34:59]Wait, no, not you.

[00:35:08]Right.

[00:35:15]D A C Q and R. Okay, cool. Why don't you put it P then? Oh, I have a period.

[00:35:32]Okay.

[00:35:55]So again we find PAC.

[00:35:59]So we define PQR. Long story short 90.

[00:36:11]All right. And each of them is two mark except for the first one. Cool. All right. Wasn't let me just hand out the um hand copies in class. But anybody who could see I want to take a start by all means. Uh Chanel this one is the Jan 2021 but just just the circle geometry right we're doing after this one one more circle geometry then switch into bearing/ trigonometry don't seem like Okay, the first thing you want to know is why is angle A, B, C 90°? So, let's take a read of the information first, right? In the diagram below, A, B, C, and D are points on the circumference of a circle with center O.

[00:37:32]A O C and B O D are diameters of the circle. So, a O C and B O D are diameters. A diameter cut a circle in half. Half a circle is called a semicircle. If you build a triangle on that diameter, the angle at the circumference is 90°. So, the reason for the first piece there is because the angle in a semicircle is 90°.

[00:38:00]Uh, and then also A, B, and D are parallel.

[00:38:11]Right. So you have a arrow there. Arrow there. So you have Z angles on that side and this side.

[00:38:21]Okay.

[00:38:36]So ABC is 90° because the angle in a semicircle.

[00:38:53]Then they want um B uh B A. So they want 2 P then they want Q and then they want R. Right. I think we can see quite easily that based on the shape of the diagram.

[00:39:05]Right. Um right these two angles will be equal. So once you find 2 P you're going to find that and then R R would actually be equal to Q because well look you have Z angles here and this is a radius that's a radius and you have a triangle which is isocles so the two base angles will be equal okay so find the values you have the reasons find the values Right.

[00:40:59]So if we focus on this triangle here with oop sorry where we know the angle here is 90° and there are two other angles in the triangle. So again what's the sum of the interior angles in a triangle 180. So right so for angle um so the first one is b a c they actually want the value of 2 p they don't just want p right so we know that 2 p + 3 p + the 90° is equal to 180 because the angles in a triangle add up to 180. So 2 P + 3 P is 5 P + 90 is equal to 180 which means that 5 P is 180 - 90. So 5 P itself is equal to 90° which means P is equal to 9 sorry not 99 90 / 5 which is 18° which means that 2P is equal to 36° right um angles in a triangle sum to 180° Peace.

[00:42:44]Why is my scroll not working?

[00:42:48]No.

[00:42:53]Okay.

[00:42:56]All right. So, we just said that we know if you follow your shape.

[00:43:00]Oh, they're actually open this time.

[00:43:02]>> Yeah. They follow your shape. We know that 2 P and Q are equal because angles in the same segment standing on the same arc are equal.

[00:43:21]That's angle BDC.

[00:43:24]Angle Q is equal to 2B which is 36°.

[00:44:03]Right? And then we said R. So because uh because this is an isocles triangle because or again you just follow the angles, right? Because this is a radius and has a radius and you build a triangle there. It's an isoclesles triangle and the two base angles are necessarily equal or alternating or angle. I think I'll go with that one.

[00:44:56]Miss R say do you have any link for paper pass account one? Uh no and they gave me any recent ones if you need to DM me and um I'll see what how I can help this by the post in the chat publicly.

[00:45:19]Uh I'm not seeing comments now. out of the app.

[00:45:33]XB68 X the previous one. Yeah.

[00:45:42]Okay.

[00:45:44]Awesome. Alter not not alternative.

[00:45:47]Alternate not alternative.

[00:45:49]Okay. Cool. All right.

[00:45:55]Okay. Ladies and gents in class, are we okay with what's on your screen there for the circle geometry question? Any further queries?

[00:46:04]Elliot. Elliot, you good there?

[00:46:09]Yeah. Okay, cool. Um, so that was what 2021. So I we'll just do the 2020 Jan.

[00:46:16]Um, and then we had a we'll have a shift to the other thing we want to do, right?

[00:46:20]So 2020 Jan. What's 2020 looking like?

[00:46:24]Plant is nice too, huh? Yeah.

[00:46:29]That one open. Okay.

[00:46:46]Okay. So, say it is 83.

[00:47:12]Turn this way.

[00:47:59]Right.

[00:48:08]There.

[00:48:32]All right. So, you can always take a little stud if you want.

[00:49:30]Okay.

[00:49:35]>> Circle shown below has center O and the points A, B, C, and D lying on the circumference A, B C, and D. All right.

[00:49:42]Straight line passes through the points A and B. Straight line pass through the points A and B. Cool.

[00:49:50]Angle CBD is 49. C B D 49° O AB O A B is 37. Okay.

[00:50:03]Um Right. Write down the mathematical names for the for the straight lines BC and OA.

[00:50:15]BC O A.

[00:50:24]So BC is a chord and O A is a radius.

[00:50:42]Now, interestingly enough, this this O A, this O B is also a radius, which makes that triangle and okay, I want you to find X and Y.

[00:51:26]Watch.

[00:52:49]slept in one bed and I'm going to give you guys a tour.

[00:53:27]Huh?

[00:53:42]>> For sure.

[00:53:44]>> I understand.

[00:53:45]>> I know what I could feel.

[00:53:51]Yeah.

[00:53:54]Hey from YouTube, >> right? Write down the mathematical name.

[00:54:01]Okay. So then they wanted angle X and angle Y. So again because the two green lines are radi, we know that the angle down in the corner there also has to be what?

[00:54:11]37°.

[00:54:13]Right? And we also Right. So let's let's focus on one thing at a time. Right. So angle X would simply have to be what?

[00:54:21]The sum, right? So the sum of angle X actually no I need to go. Yeah. So X + 37 + 37 would be 180, right?

[00:54:37]That's good.

[00:54:43]All right. So x + 74 is equal to 180. So therefore x is equal to 180° minus 74 which is 106.

[00:54:54]Right? Now couple reasons here two reasons specifically right about the so you have one there and you have a two right. Two reason one is because base right base base angles of an isosles triangle are equal and two angles in a triangle sum to 180.

[00:55:52]Okay. Right.

[00:55:54]And for angle Y. Well, again, we know that um hold on.

[00:56:03]This line here is a diameter. And if you build a triangle on that diameter, right? The angle at the circumference is 90°.

[00:56:21]So again, y + 49 + 90 is 180°. So therefore, to find y, you subtract those the sum of those two from the 180. All right. So, y + 139 is equal to 180° which is y is equal to 180 minus 139 is 41°.

[00:57:06]All right. And for reason one, angle in the semicircle is 90.

[00:57:34]And reason two same as across here.

[00:57:51]All right. Okay.

[00:57:54]Right. So that's about was that 22 21 and 20. So that's four questions in about an hour. 15 minutes of question is all right.

[00:58:12]Okay.

[00:58:15]People in class, how we feeling about our stuff so far? Any questions? this use.

[00:58:23]All right.

[00:58:33]All right. And then we go with this one.

[00:58:36]>> Yeah.

[00:58:37]>> All right. Cool.

[00:58:41]All right. So, folks, we're switching gears now. So, we did some circle geometry. Admittedly, the questions weren't as interesting as some of the ones we did before um with the more recent exams, but we did those right from Jan 2026. Come back. All right. So, now we're going to switch gears and we're going to deal with some bearings questions. How about that?

[00:59:01]Huh? No.

[00:59:09]>> Uhhuh.

[00:59:12]>> The news. Yeah.

[00:59:14]>> What is it?

[00:59:24]>> To do what?

[00:59:27]>> Going to do that.

[00:59:40]I heard Which news?

[00:59:49]>> Really?

[00:59:51]>> That that is most interesting.

[01:00:07]This is allegations that Right.

[01:00:13]So I I gave you guys print outs here already, right?

[01:00:29]>> The draw.

[01:00:50]Jesus attempt.

[01:01:22]But she is French.

[01:01:30]>> All right.

[01:01:31]>> Okay. So, the Jan 2022, right? This is what it looks like. I g I when I printed out the circle geometry, I printed out the whole um the whole question nine. So yeah, so you should have this question.

[01:01:42]If not, let me know. I'll put it out quickly. Let me know fast. Right. Okay.

[01:01:46]So diagram below shows straight roads connecting towns L, M, N, and R. L M N and R straight lines, right? Straight roads. LR is equal to 18.

[01:02:01]LN is equal to 12. and MN is equal to 10. Okay.

[01:02:07]R LN is 25.

[01:02:10]L MN is 88 and I seen a 50° there.

[01:02:17]So the 50° appears to be the bearing of M from L.

[01:02:23]Okay. Calculate angle M LN. So they want the first thing is this angle here.

[01:02:30]Okay. Now, if we need the whole the whole angle here, we could just add up 50 and 25 and subtract. But we don't have that information.

[01:02:38]>> All right.

[01:02:56]All right. And then let's grab it from Did I? Yeah. There you go.

[01:03:20]Cool.

[01:03:22]Straight roads connecting.

[01:03:25]All right. MLN cool.

[01:03:29]So this angle here. So that particular angle lies in a non-right angle triangle.

[01:03:42]And interestingly enough we have two other lengths and one angle right which is opposite one of the lengths. So this is a perfect um situation to use um what you call it sign rule.

[01:04:01]Okay. Okay. So to use sign rule, what do we need? So we're looking for an angle. So let's just call it angle L, right? Which means this side here across here is small L and this side across there is small M.

[01:04:16]So sin L / L is equal to sin M sorry right over M.

[01:04:39]All right. So m right all right so sin m to solve for that you can multiply by l on both sides. Okay so let's let's plug in the values now sin m is 88° all over 12 by 10.

[01:04:59]So plug them in your calculator.

[01:05:07]So 832 and therefore to find L what we have to do is find the sign inverse of832 do the other dot.

[01:05:26]So about 56.4° 4°.

[01:05:32]Let's just call it 56°, right to nearest degree.

[01:05:46]All right. So, you have to analyze the diagram to know whether using S rule, cosine rule or something else.

[01:05:52]Right? Yeah, we'll stick with whole numbers for now.

[01:05:58]That's the first part.

[01:06:07]What's the next part of the question?

[01:06:08]The distance n R.

[01:06:11]Okay.

[01:06:24]Okay.

[01:06:46]All right. So, that's a perfect candidate for the cosine rule. You want a length in a triangle. you have the other two adjacent sides and the included angle the angle between the two sides you have that's a straight shot at cosine rule right so let's just call that this is this is also a opposite angle a different L so that's small L right um this would be small r across here this would be small n across here you could always use the the capital letters for the vertices you could do nr and lr and ln if you want right so The length L 2 is equal to N 2 + R 2 - 2 N R um by the cosine of angle L which in this case is 25°. Yeah.

[01:07:38]All right. So n All right. So 18^ 2 + 12^ 2 - 2 by 18 by 12 cossine 25 right + 2 by 18 by 12 by 25 cosine close brackets equals 76.47 47 that those Yeah. Okay. All right.

[01:08:17]Therefore to find L you have to find the square root of that same set 76.47.

[01:08:27]So 8.7 so approximately 9. So to 9 km and there was one more part of the question if I'm not mistaken the bearing of R from L All right.

[01:09:45]bearing of R.

[01:09:49]So R from L means you want the clockwise angle from the north line at the start point to the line connecting the two points.

[01:10:06]Right? So all we're really missing is um this piece here which we found in the first part of the question. That's so what was that? 56°.

[01:10:19]Okay, cool. There you go.

[01:10:29]So what we have to do is add together the three angles. So that's 50 + 56 + 25. So you get 131° uh yeah with raven or even yeah doing this since 9 and up since 5. I went to bed at 12 and lonely yesterday. So yes, definitely tired.

[01:10:58]All right. So wasn't actually a hard bearings question. You just had to understand what they asking and how to find it and yeah. So we didn't have any big set of extra work to do there thankfully. All right. How you feeling with that one boy?

[01:11:13]>> All right. Nice. In class people, how are we feeling with that one? So far so good. Okay. Cool. So we're going again 2021. Right. So I gave you guys the question already, right? So it's just for me to kind of copy and paste it. All right, give me a second.

[01:12:02]Oh, the way. One second.

[01:12:46]Come on. Give me my menu.

[01:12:51]Oh my god. All right. Home. Where's Where's my menu? Where's my paste?

[01:13:03]Okay, hold on.

[01:13:09]No.

[01:13:13]Okay.

[01:13:22]All right. Fine. You know what?

[01:13:24]We'll just do it as is.

[01:13:35]Okay, let's um let's zoom in a little bit and then I also have to kind of blow this one up too. Whoa, cool.

[01:13:49]I have three pieces. Okay, cool. So, I think I will copy and paste that three times.

[01:13:57]Wait, no.

[01:14:04]Okay.

[01:14:05]From a harbor H, the bearing of two ships Q and R uh 69° and 151 respectively. Okay. Hold on. Hold on. Bearing of Q from H is 69°. Okay. The bearing of R from H is 151. So that whole angle there is 151.

[01:14:29]Right. Um Q is 175 from H and R is 242.

[01:14:33]Complete the diagram above to show the information given. Okay. So I want you to do that piece for me and then find QR.

[01:14:49]Yeah.

[01:15:08]chance in that.

[01:17:07]Okay. All right. So let's let's start here and see how well we're doing, right? Okay, cool. So we we want to put in um so Q is 175. So we can put 175 here, right? 175 km.

[01:17:27]Now the whole angle like I said that we were talking about is 151, right? So 151°.

[01:17:37]Uh any other pieces of info once? No.

[01:17:40]Okay, cool. Okay, then calculate QR.

[01:17:43]Okay, hold on one second. Like I want to put that inside here. All right, cool.

[01:17:56]Now to find QR, right? That's a straight coine ru because now I have the 151 and the 69.

[01:18:04]So I could actually find this angle here by simply subtracting. So 150 - 70 is 80. So I think it's 82° 82 right so that so this is the age across here. This is small Q. This is small R.

[01:18:36]Right? So to find QR which is small H.

[01:18:39]So small H squ is equal to Q ^ 2 + R 2 - 2 QR cossine 82°.

[01:18:53]Right? So 242^ 2 + 1 75^ 2 - 2 by 242 by 175 cossine 2 242 + 175^ 2us 2 by 242 by 175 by 82 cosine 77 401 and if I square it that so 278 approximately So 278 km to the nearest kilometer.

[01:19:59]All right. We okay with 278 somewhere around. So 278.2.

[01:20:05]All right.

[01:20:08]How are we feeling that one?

[01:20:13]>> All right. Now this one, this next piece here is quite interesting. Calculate how far due south ship R is ship R of the harbor R. How far due south of harbor R is ship R. Okay. So due south means maybe it's the same diagram.

[01:20:46]So basically they wanted to kind of create a right angle triangle going on here and you might say well how are we going to find anything inside of there? Well actually it's easier than you think because when you continue when you continue a north line straight down you have a straight line not so and the angle on that straight line is equal to 180°. So if if the piece up to here is 151, to find the remainder, all you have to do is subtract.

[01:21:19]Oops, that is a way too heavy point.

[01:21:25]That's 29°.

[01:21:28]So this is all we need because we have a right angle triangle. This length is opposite the right angle, which makes it the high.

[01:21:36]That would be the opposite side relative to the angle we have. We are neither looking for nor do we have the opposite side. But we do want the third side which is the adjacent side.

[01:21:48]Which means we could use basic soc. And which ratio uses adjacent and hypotenuse?

[01:21:57]Cosine.

[01:21:59]So cosine of 29 is equal to the adjacent length which is the how far south divided by the hypotenuse of 242.

[01:22:08]So what we're going to do here to solve for the adjacent side is multiply by the denominator of 242.

[01:22:28]So, we have about 212 right to nearest kilometer.

[01:22:54]All right. And that's the question I didn't need.

[01:22:58]other copies of it.

[01:23:04]Oh, here no problems. All right, in class people, any questions, problems, issues, or are you okay with that one there?

[01:23:16]Yeah. All right.

[01:23:19]How you feeling about um that last piece there with the adjust?

[01:23:26]All right, cool.

[01:23:27]>> Yeah, >> I have to go now.

[01:23:29]>> Yeah, man. All right. Well, good luck tomorrow. Let me know how it goes and I'll see you maybe tomorrow evening for Admax.

[01:23:35]>> N tomorrow.

[01:23:36]>> All right, no worries. Here, I'll send a poll in the chat. There's nobody coming.

[01:23:39]I'll just fool myself and take a rest and we could pick up from the next well whenever next. All right. Later.

[01:24:31]Okay. All right.

[01:24:35]Cool.

[01:24:38]People count too. Uh that Saturday.

[01:24:41]Sorry.

[01:24:43]Yeah. Robin, we win again.

[01:24:46]Fundamentals. Nah, sorry. I can't do fundamentals now. But check my YouTube channel. Pretty sure I did from last year.

[01:24:57]But again if you want to know a bearing fundamental you have two points A and B.

[01:25:08]The bearing of B from A is the clockwise angle between two lines. The north line at the start point which is A and the line joining two points. That angle there is a bearing.

[01:25:29]Okay. So, we're going to do 2020 journal.

[01:25:49]Oh, 31.

[01:25:51]Wait, where is it? Oh, hold on one.

[01:25:54]Sorry.

[01:26:11]Oh, yeah. This one was a serious one.

[01:26:13]Yeah, I kind of glad we do this one. All right, cool.

[01:26:23]It wasn't as hard as I initially thought once I kind of was able to settle on and approach it from a a slightly calmer perspective. What? No.

[01:26:55]Okay. uh from a haba h bearing of two buoys S and Q 185. So H or sorry it was locking H. So H is that that starting point there in the middle here right the bearing of S is 185. So that whole thing there is 185 and then the bearing of Q is 311. So this whole angle here is 311. Okay, cool.

[01:27:30]Um, Q is 5.4 km from H and S is 3.5 km.

[01:27:49]Okay. And the first thing they want on the diagram below which shows the sketch information insert the value of the marked angle QHS.

[01:28:01]So QHS Hey yourself, what are you doing? Right, it's from here to here to here. So what they're asking for essentially is the value of the angle here, the yellow angle.

[01:28:17]Right?

[01:28:22]Now, if we know that the external piece here is 185 and then the whole angle is 311, finding the yellow piece becomes pretty easy. All we have to do is what?

[01:28:35]Subtract.

[01:28:42]So QHS is equal to 3 11 - 185.

[01:28:52]So that's 115 + 126 Okay.

[01:29:15]Now the next piece of the question that I didn't get to copy. So they want QS the distance between the two boys. Okay.

[01:29:23]Let's do one thing at a time. So QS and then the bearing of S from Q. So QS of course is this line here.

[01:29:35]Okay. That's a straight cosine rule cuz you have the two other adjacent sides and we just found at the angle in the middle here is what? 126.

[01:29:46]Yeah. So that's straight cosine rule.

[01:29:54]All right. So that's small h that's small q that's small s.

[01:30:01]So h 2 is equal to q ^2 + s^ 2 - 2 qs cossine h.

[01:30:17]So 3.5^ 2 + 5.4^ 2 - 2 by 3.5 by 5.4 4 cosine 126.

[01:30:36]Okay. So, plug and play.

[01:30:53]So, 63.6 blah blah blah.

[01:31:00]Therefore, h is equal to the square root of says 63. So that's almost 64, right?

[01:31:06]So it's going to be pretty close to 8.

[01:31:10]So 7.97. Even if we wanted to go to one decimal place because all the other lengths are one decimal place, we'll get when you round off eight All right.

[01:31:48]Sorry.

[01:31:50]Okay.

[01:31:53]All right. In class before, we okay with that one there? All right. Now, we still have one more piece to do. So, that was the distance. Now, we want the bearing of S from Q. This is an interesting one.

[01:32:05]So, we want the bearing of S from Q. So the angle we want right hold on. Um so the bearing we want of S from Q is this whole angle here.

[01:32:35]Right? Now we could find the angle inside. So like this like this piece here we could find that external piece to me is a little tricky but there was there was something easy about it. The h what was the thing I saw the other day that made it easy. All right. Well sometimes things don't jump out. So you have to kind of play around and experiment and look for it. Right.

[01:33:00]Okay. Cool.

[01:33:02]So to find that angle on the inside, we could use hold on one second.

[01:33:08]Yeah.

[01:33:10]So we could use cosine rule and reverse it or we could use sine rule, right?

[01:33:14]Which probably is the best option. But the better option are going to be honest. But either one should give you the same value. Sorry, one sec.

[01:33:26]All right.

[01:33:32]Yeah. Oh, no. Okay. I saw the easy thing to do. All right. Cool. Cool. All right.

[01:33:36]So, let me show you that piece one time.

[01:33:38]So, let let's let's call the internal portion here um X and let's call the exter. So, okay. X here and Y here.

[01:33:53]Okay.

[01:33:55]So, try this. If we were to extend this line here, right, we would create F and Z angle cuz of course you have a a par two parallel lines here. So the angle the angle here is the same as Y. But we don't really have that. But the thing is we could actually find another angle that could be helpful, which is this angle here.

[01:34:24]How can we find that? Because the sum of the angles at a point. So the full circle is how many degrees? 360. All right. And if we have 311, then this angle here becomes 360us 311, which is 49°, right?

[01:34:49]which makes this angle here 180 minus 49° and 131 because of there being co-terior angles.

[01:35:04]Cool. And now to find angle the x portion of it that's just sign rule right. So you could say uh sin x over 3.5 is equal to sin 126 over e.

[01:35:32]So sin x is equal to sin 126 / 8 by 3.5.

[01:35:38]Okay. Plug it up.

[01:35:44]Oh 3539 and therefore x would be equal to the sin inverse of 0.3 539.

[01:36:08]So about 20 about 21°.

[01:36:16]So for the full bearing is the sum of those two 131 + 21.

[01:36:48]Yeah, I remember that question being harder the first time I tried. I don't know why. I just couldn't see what I what I had to see the the the big the big 311 angling and the two pieces. For some reason, I didn't see how to use the extra um 49 is 49. Yeah, 49 degrees there.

[01:37:07]Yeah, I was trying to figure out how to find that like real bad.

[01:37:13]Okay, so that's 2021. So I think we have time for the Jan 2020 and possibly the Jan 2019 and then that will be punto finale unfortunately. No, unless you want to hold off on bearings and take a little browser some matrices for transformations. Y'all tell me what you want to do. Stick with the bearings.

[01:37:31]Yeah. All right, we're going.

[01:37:41]That's a nice question.

[01:37:43]All right, cool. 2011 had practically an identical question, by the way.

[01:37:49]Dr. Boom.

[01:38:21]Okay, one sec. Let me just clean up some stuff here.

[01:38:35]All right. So, they want W and they also want bearing of P from Q and distance RP. Okay, cool.

[01:38:54]Okay. Diagram below not drawn to scale shows the route of a ship cruising from Palm City to Katon Keton and into Rivertown. So from basically what they're saying is from P to Q to R.

[01:39:06]Okay, no problem. The bearing of P of Q from P is 133 as you could see here. Q from P and the angle PQR is 56°. Okay, cool. No problem. Calculate. So the first thing they want is angle W. Okay, cool.

[01:39:23]So we could use the same technique we used just now. And what I mean by that is you're extending this line here.

[01:39:33]Right? Now why are we doing that?

[01:39:35]Because look, you have one, two, three sections, right? one, two, three segments into which that that full 180° angle is cut because that's the angle in a straight line. The angle in a straight line is 180°. Uh aesthetic boss, sorry the vectors I had time for that tonight, boy in about 10 minutes. I done I was I on since 9:00 is one, so I kind of Yeah, but you can check the YouTube channel. I might have some stuff from previous right now. So if you if you could find this angle here then we could put that with the 56 and then subtract from 180 right now to find that again straight line. So the the angle there is 180° right and we could actually find this piece of it here because if you notice this and this was two north lines which are mutually parallel and the green line is the transversal cutting them which gives us gives rise to f and z angles.

[01:40:32]So this is 133° here, which makes this angle 180 minus 133, which is 47°. Hold on one second.

[01:40:54]47.

[01:41:05]Right? Now that means that the angle W right. So w + 56 + 47 is equal to 180° because the angle on a straight line is 180°. Now 56 and 47 is 103.

[01:41:36]And to find W, we'll simply subtract 103 from both sides of the equation. So W will be equal to 77°.

[01:41:47]Okay, next thing we got they want the bearing of P from Q.

[01:41:54]Oh, so they want a reverse bearing.

[01:41:55]Okay, so in that case, you just have to add 180 to it, right? That's a shortcut, right? That's something we you should know from um early early in bearing how to find what I call the reverse bearing.

[01:42:13]>> Huh? plus the 133 and I'll show you why.

[01:42:17]Right.

[01:42:19]So, give me one sec.

[01:42:28]Okay.

[01:42:30]What they want? They want the bearing of P from Q. Yes. Bearing of P from Q is a clockwise angle that starts at Q cuz it's from Q. Yes.

[01:42:42]So it starts at Q and goes clockwise until it hits the line joining the two points. We're not counting this portion down here. Right now the thing is that that is already broken up into sorry into two pieces.

[01:43:00]Done. Done.

[01:43:01]>> Yeah.

[01:43:03]>> All right. No worries.

[01:43:05]>> Okay. Cool.

[01:43:07]All right. So this piece here is 133° and the piece on the next side is the angle in a straight line which is what?

[01:43:17]180°. So all we have to do is add 133 and 180. Right?

[01:43:25]So bearing of P from Q is equal to 180 + the 133 which gives us 313°.

[01:43:45]Right? And the last piece that I want is the distance RP.

[01:43:52]So RP is assistance here.

[01:44:06]So the orange line. Okay, cool. So that's a straight shot at cosine rule, right? Because we have the two the the 210, the green and the blue, the 210 and 290, the two are the adjacent sides and the included angle, the 56°.

[01:44:23]So this this will be so this angle that's small Q that's side Q that's side P and that's side R.

[01:44:34]Okay cool.

[01:44:36]So Q ^2 is equal to P ^ 2 + R 2 - 2 P R cosine Q.

[01:44:50]I feel out of space this cuz it's 210.

[01:44:53]But that's 210^ 2 + 290^ 2 - 2 by sorry 2x 210 by 290 by cosine 56°.

[01:45:15]Okay, cool. So plug and play.

[01:45:21]210^ 2 + 290^ 2 minus open brackets 2 by 210 by 290 by the cosine of 56.

[01:45:33]So 6090 dot dot dot. So therefore Q is the square root of said 6090 dot dot dot So 245 the nearest kilometer.

[01:46:07]Didn't even need this.

[01:46:19]All right, guys. We have five minutes again. What are you all feeling? One one more for the road or no like this man seriously going again.

[01:46:30]Then we just take a look at 2019. If it's easy enough, we'll just take a skip through. If not, then we don't. Oh, they give you angles of elevation, depression. Nah, I good. I good.

[01:46:41]I think I'm good with that.

[01:46:49]And then I have to move 12 again.

[01:46:54]>> 12 night.

[01:46:55]>> 12 night.

[01:46:57]>> Yeah. Yeah. But it's just it's just tonight.

[01:47:09]>> Yeah. All right.

[01:47:11]Nobody on the um >> on the WhatsApp on the >> thing.

[01:47:41]All right.

[01:47:43]>> All right. So, ladies and gents, tomorrow when you're going the exam, keep your head up. First of all, get a good night tonight, please. I know it's English B and maths. I had to feel prepared. If you don't get enough sleep, you are taking a risk that I don't suggest you take, but it's your choice.

[01:48:01]Right. You're going on everybody. Take care. Um, so I'll be in touch with the P and M.

[01:48:07]Right. P should probably start next week Monday, but I know it's about exam. So, check in the chat.

[01:48:15]>> All right. Good luck.

[01:48:20]>> I hear you.

[01:48:22]>> Okay. Bye-bye.

[01:48:24]>> All right. Take care. Good luck tomorrow. You know how it goes.

[01:48:31]>> All right.

[01:48:32]>> Okay.

[01:48:34]YouTube, Tik Tok is just me and you, boy.

[01:48:38]>> Okay, I I'm Yeah. on about actually no halftime, but still >> have two of them on the inside.

[01:49:04]Dear Mario, Daniel guys, I I on 9 this morning 9. I good. I can't do anymore. I really Sorry. Right here. What to do?

[01:49:13]Here what to do. Hold on. My YouTube channel should have some replays from last year. I'm pretty sure I did some statistics, some clustered statistics questions just like how I was doing things. Right.

[01:49:26]All right. Okay. Later. Functions.

[01:49:29]Functions. They can check for functions too. But guys, yeah, my eyes right now hitting I see them blurry. So I I I hate to stop, right? But um I wish everybody the best tomorrow. I hope the exam I hope it comes well. I hope it's a good paper. Um but of course everybody's definition of good differs, right guys?

[01:49:51]So I do I do apologize that I can't facilitate your requests, right? But check my channel some stuff there. All right folks.

[01:50:00]So if we do math from tomorrow evening or soap pasta 6 we might um hey saf I'm not logging off again. Yeah. Um from past six to just a single session where my time table.

[01:50:27]It's so hard to believe only one week really pass.

[01:50:33]Yeah, last week was really the only week so far apart from orals and thing.

[01:50:40]Yeah, this is where we are. Yeah, it was. Yeah, one week. I mean, Friday before dog dog.

[01:50:50]Well, this actual dog I'm talking about, by the way.

[01:50:55]I seen it.

[01:51:01]Yeah, the Mario. Sorry about that. Yeah.

[01:51:07]Um uh once yeah once I um once I once you see me exit here just check in about 5 minutes or less dog inside. Dog inside Wolfie this dog listening.

[01:51:28]Oh yeah. I wasn't able to have class here. That was Mother's Day thingies.

[01:51:42]Yeah, I saw that.

[01:51:48]>> Right. Yeah, guys. I'm a you know, two kids have their um paper to tomorrow.

[01:51:55]Yeah. So, adm Wednesday evening as well. Unless the kids want to come early in the day, I could do that too and take the evening off. That would be nice. I have bio and it Thursday, so I don't know who coming to class Wednesday. It'll be Friday. So yeah. Well, yeah. Thursday obviously I have class if people didn't buy on it.

[01:52:15]And yeah, Saturday I had to do a double.

[01:52:16]Sunday I to do a double cuz the exam is Monday. So 2 4 6 8 10.

[01:52:22]Yeah. Might get more than 60.

[01:52:25]Yeah. Hey. So yeah, you're good.

[01:52:30]Yeah.

[01:52:32]Yeah. Oh yeah. And and and as of Monday, we jamming some um paper ones for POE.

[01:52:40]I know Kim and Econ is the next day. So I know plenty of people going to be missing. And then Spanish paper one is on the Wednesday and his shoot on the same day as paper one. So I know some people not going to be in the mix.

[01:52:51]Anyhow, and then right after that, well, I know physics is the Friday. So I know who come to class Thursday to study for for maths but not everybody does physics but I'll do like a single session here if anything yeah double Saturday or double Sunday I find I play for that Sunday.

[01:53:21]Yeah. And then after that h yeah from the Saturday to the to the end thing there is admats down the line.

[01:53:32]I don't do hard math before. Oh fitting keep accounting. Oh god pressure. All right me doing math is 9 past 9. So it's past 8 now. I mean I took a couple of hours in between here and there as you all know but that I had to hit my bed. any YouTube.

[01:53:53]Um, thanks for the support. We're at 45,000 subscribers currently, right? I appreciate it.

[01:54:01]Yeah, hopefully I could put out some more content when things ease up a little bit and yeah, boost it boost it up. Anyhow, YouTube um, Inservitress Unknown. Thanks very much. Yeah, YouTube are out. So you guys have get a good night's sleep. Don't stay up too late and make sure you take your time in the exam tomorrow. Right?

[01:54:26]Don't get flustered. Do what you could do. Do what you know how to do. Right? A mark a mark is a mark is a mark. If question one give in trouble, leave it and go on. But come back and do what you could. Right? Try not to leave anything blank on the paper if you could manage it. But do not let what you cannot do stop you from doing what you can do.

[01:54:44]Right? have sometimes some I going through papers they be one and some of them theories I don't know I said I go look it up right yeah random no problem yeah my pleasure there guys do play with your sleep is important anyhow I will yes tomorrow.