This video provides comprehensive solutions to the CSEC Mathematics January 2026 Paper 2 exam, covering key topics including fraction operations, base number systems (converting between bases using place values), estimation techniques, ratio and proportion problems, simple interest calculations, algebraic expansion and simplification, inequality solving, Pythagorean theorem applications, rotational symmetry and polygon angles, similar triangles and trigonometry, coordinate geometry, pie charts and probability, map scales and area calculations, perimeter and circumference problems, pattern recognition, function composition and inverses, geometric transformations, bearings and scale drawings, vectors and matrices, and transformation matrices. The instructor demonstrates step-by-step problem-solving approaches for each question type, emphasizing key concepts like place values for base conversions, the importance of reversing inequality signs when multiplying/dividing by negatives, and the relationship between exterior angles and interior angles in polygons.
CSEC Maths Class May 10, 2026 - January 2026 Paper 2 SolutionHinzugefügt:
All right, guys. Ready?
People online can hear me and see the screen.
>> Yes, sir.
>> All right. Good.
So, just going to work through this paper.
Let me get my calculator.
So this is the January 2026 exam paper.
So this is the very last exam C set was this past January. So I guess it's a good gauge of the the standard and the level of exam not necessarily the question that you'll get but the standard should generally be the same on the paper that you're going to all right so work through this work through this paper there are one or two questions um person asking about this paper um that we might want to call special questions because they are rare rarely set all right I See if I can take some time and talk about this. So there's a base question here. I think somebody asked me about a rotational symmetry question as I get to those question best way possible. So it's stuff that again that we have done in the class before but maybe you don't remember exam paper. All right. So let's go.
So the first question is asking us to express this thing this thing as a single fraction in lowest term. So this is a calculator question really.
All right it's really a calculator question. It is 47.
It is 4 7 / 12 or 12 is same as 12 / 1. So by 121 is to multiply by 1 / 12. So it becomes 4 7 * 1 / 12.
I invert and multiply. The 12 is same as 12 / 1.
So this can be simplified. 4 this goes one time. 4 goes three times.
And so it would work out to 1 * 1 is 1 and 7 * 3 is 21.
And we can test that answer by putting the same thing in the calculator here.
4 over 7 / 12 as a fraction. It is 1 / 21 right here.
can see it 1 21 right so this answer is correct 1 21 this answer for part a part one part two now is this base question so this number is base four you can see the four there that mean base four this is base seven the best way to do base question is to use place values place values I don't want us to know what is the difference in the three.
Calculate the difference in the value that is underlined. So you need to know the value of this three in base four.
You need to know the value of this three in base 7. And you're going to see the difference between them. Now they did not tell us what base you must put the difference in. So we're going to put the difference in base 10. And when you put the difference in base 10, you have to say it's base 10. The difference is base 10. So the thing to do is get this value in base 10. what the value is in base 10. Get this value in base 10 and then tell the difference. All right. Now, the first thing we need to do now is what are is to basically use the place values for this one and the place values for this one. Not it's not complicated. All right. So, let me show you now. So, we have those two numbers. This is part two. So, we have um let's do down here.
Shift this up.
and put this right here. So the part two we have 3 2 0 1 to base 4 and we're comparing that with um 6 351 to base 7. Now I'm going to put on the place values here for you. Place values.
place values.
The place values first place value in base 4 is four to the 0. First place value is always to the zero. So this number has base four. So this place value is 4 to the 0.
So this number is 1 * 4 to the 0. So it's value one. This place value the second one is 4 ^ 1. So this four this zero has a value of 0 * 4 1.
This place value is 4^ 2ar.
So this 2 is really 2 * 4 2. That's the value of this two right now. 2 * 4 2.
Then this place value is 4 cubed.
So the value of this tree can write it now. These are the values I want us to compare. So this tree right now the value of this tree is 3 * 4 cubed.
That's the value of that three * 4 cub because it is in a 4 cub place value. It is in the 4 cub place. So the value of it is 3 * 4 cub. We're working in base 4.
We're doing a similar thing to this number that is in base seven. I didn't have to write out these. I'm just doing it to show you, right? Because if I wanted to convert it, it wanted to convert this entire number to base 10. It will be 1 * 4 to the 0 * 4 1 2 * 4 2 and 3 * 4 cube. And I add up those values to get the value base 10.
Then this value working in base 7. Now this place value is 7 to the 0. So this one means 1 * 7 to the 0. 7 to the 0 is 1.
This five is in the place value 7 ^ 1.
So you see you start with zero and you increase those powers up that way to get the place values.
So this five value is 5 * 7 to the 1.
This tree is in the seven squared position. 7 squared.
So the value of this tree is 3 * 7 squared. And if I wanted this place value is 7 cub.
So the value of this 6 will be 6 * 7 cube. We're not interested in that.
We're only interested in this value right here.
That value is really 3 * 7^ 2.
That's the value of that three and that's to base 10. We have worked out the values in our normal base 10 numbers. That's what that's what the value now in our normal base 10 system.
So this value right here work it out 3 * 4 cube. What would it be? I don't need to work it out. I can calculate different straight but anyway let's work it out just to show you to be 3 * 4 cub.
3 * 4 cube is 192. So the value of this is 192 to base 10 and the value of this one is 3 * 3 * um 7 squared is 7 squared which is which becomes which becomes 147 147 to base 10. So the difference is this minus this. That's the difference. You subtract them to get the difference.
So the difference is 192 from 192us 147.
The difference is 45. The answer is 45 to B. That's the answer for that one. 45 to B.
That's what that question is.
That's 45 to right.
Let's try the answer for this one here.
So I can shake this up. So the answer for this one is just 1 / 21.
1 / 21. The answer for this one = 1.
I'm just going to take this off.
and shift and shift the calculation this upwards that is understand that.
So the key thing with any number in base is to have the place values.
The place values we did a class with this already right maybe find the video and send it to you but go through it fully now. But the key thing to this question guys is to bring it to base 10 using place values and the place. So work it. If you have a number for example say 6 4 8 1 and this is base 5 for example.
Then this starts at 5 to the 0. This is 5 to the 1 5^ 2 5 cubed.
So the value of this 1 is 1 * this. The value of this 8 is 8 * 5 1 which is 5.
The value of this 4 is 4 * 5^ 2. The value of this 6 is 6 * 5 cub. You converting it to base 10. And I add up all those values.
All right? If you add add up all those values, if you have a if you have a number and you want to convert it to base five, for example, let's say I have let's say I have the number 381 and I want to convert that to base 5. This is base 10 and I want to convert to base 5.
I want to see how the base five place values add up to make this starting to from the largest possible value. So I have to write down the place val for five first which is 5 to the 0 5 to the 1 5^ 2 5 cubed 5 to the fourth power. Those will be the place values and you can only use these numbers up to four times.
When you're working in base five the digits you allowed is 0 to 5. If you're working in base four, the digit allowed is 0 to three. The digits you're allowed is always one less than the base. So since this is now base five, the digits allowed on these values is only 0 to four. You can use these values 0 to four times.
There's no five digit in base 5. Just as there's no 10 digit in base 10. 10 is a combination of one and zero.
broken in base 10. So the only digits that exist in our normal decimal system base 10 is the digits 0 to 9 0 to 9 are the only digits we see in the base 10 system.
One less than the base. All right.
So what to be these base values if we converted that I was doing this as an example. What are the actual values of these place values in place five? 5 to the 0 is 1. 5 to the 1 is five. 5^ 2 is 25.
25 * 5 again 5 g is 125. 5 to the 4 power is how much? Multiply by five again. 12 65 um 125 * 5 625.
You're trying to use these numbers to make up this 381.
That's in base 10. Remember I can use each of those place values only four times. 0 to four. You're going to see how many multiples of it you're using.
So this one out of it because the number that you're trying to make up is less than the 625. So you're not going to use the 625. So this place value is not going to be used at all. So the next place value that you can use is the 125.
How many 125s are in this? 300 381 two of it or three of it can check it out 3 * 125 it's 375 yes so three of this you can actually use three of this you can't use four of it so I'm going to put a four to say that we're going to be using this not four three of it to say three we're going to using three of the 125 so it's like you get it you take three of the 125 out of that so this number how much you have left to use you have 385 You have you have already used three of the 125 which is 375.
So I've used 375 out of it.
So you subtract to see how much you have left. So out of that we have six left. Only six left. Only six we have left. Maybe try use a different number. Let me create a number. Let me use a bigger number. Let me use 3 94. So big on mobile.
All right, let me just use 394.
So 394. So we still use 125. Sorry. 394 some chocolate cup.
Comes 394us 375 394 - 375.
I have 19 left that I can use. 90 left to use to make up this 394.
So I look now, can I use any of the 25s?
No, cuz 19 is less than 25. So I'm putting a zero here. I'm not going to use any of the 25.
So can I use the five? How many of the fives can I use? I can use three of the fives.
I can use three of the five out of the 19. I'm going to put three of the five for three years. So So I'm going 15 more. three, five or 15. So subtract the you have four left. So you want to know how many of the one they can use now out of to make up the four. Four, you're allowed to use four. So you put four right there. So So that's how you make up that number, right? So it means that this number to base 10 is this number to base 5. That's what it means. So So you can write the answer now. So you say 394 to base 10 is equal to 3034 to base 5.
That's it.
So you use the place values to go either way. You go from a number to base 10 use the place values from a base to a decimal number. I still use the place values.
We have a decimal you have a decimal number you're seeing how many how much of the place val you are using up to make up the total number and we have a base number now you just put them put them in the respective place and you're seeing like this one like this one here this is base 4 I wanted to know what this number was in this 10 I would say okay it is 1 * 4 to the 0 * 4 to the 1 2 * 4 2 and 3 * 4 cube if I wanted the entire number and add up all of those but in this example only wanted to know the value of the three the value of the three is 3 * 4 cub the three is in the place value of 4 cub by the base 4 so base 4 starts from 4 to the 0 then 4 to the 1 then 4^ 2 and so on those are your place values all right guys let's move on don't want to spend too much time on this one question.
All right. So I'm just showing you that this example down here. So a little revision of bases don't come on exam. In fact I think this is the second time in recent time I remember the exam paper second time.
All right. I don't exam paper but this one is a nice but it never get it over. again by yourself.
All right, good. Then we have this question here. Now, let me make this small and shift it up because make this a little small and shift it up. That small All right, come to this question now. So this was part two. Come to part B now.
Right here.
What does part B say? B says by writing each of the numbers in this fraction to one significant figure find an integer estimate of this value. So we need to write each of these numbers to one significant figure. So the one significant figure going to be the first nonzero value from left to right. All right. So this going to become equal can just put it right here. So this stay 600 the first significant figure I fix. So that's still 600 minus first significant figure is this eight and the eight is going to be increased by one because of that seven.
The size of the number cannot reduce this number is in the 10 because it's a 8 odd number. So it can become a 90 number. It can't become 8. 87 can't become eight. So we must make sense. So can't become 9. 87 cannot become 9.
There's no way estimated to it estimated to 90. So it make sense as well. So anybody put n that don't make no sense.
All right. The size of the number must be. So the number is in the t the estimation is in the t. The number is in the hundreds estimation is in the 100.
In the thousand, the estimation is in the thousand. Something like that. The number is in the ones estimation is still in the ones. can't have a number on estimated to be in the text like far in the text.
So this number so basically when you increase this right you're going to replace this this seven with a zero and so you're going to become a 90. So this number one significant figure is going to become a 90. But understand that and we're dividing that now this number on a year again we're going to change this decimal place that's going to become a 30 cuz the 9 is going to increase this two to a three and they replace that with a zero going to become a 30 that's what it becomes and you work that out now. So it becomes a fraction.
I have to show that working over tell you what to do. So it becomes 600 - 90 over 30. That works out to 27. So that's equal to 27.
That is what you okay with that right cross here. So that's what comes 27.
All right. So you see you're getting an integer for your answer. You can't put you can't rotate and put 27 nothing. You want a integer answer 27.
All right. Next question. Enter.
Remember inside a whole number positive or negative whole number. All right.
Next question says Alana and Bren Bren Brenn whatever that is shared this amount of money in the ratio 8 to 17.
What? What?
Huh?
17.
Oh, put something on the calculator. I put 900. My mistake. I put 900 in the calculator. So, it takes long to talk.
I put 900 in the calculator.
See, same thing. Have check. See, I put 900 in the calculator. That's why six. All right. Good. Get 17 bright 17. All right. Good.
All right. Let's go. This one says now they are sharing this amount of money in the ratio 8 to 7. 8 to 7. The 8 to 7 has 15 parts. What it 15 parts.
One getting eight out of the 15 and one getting seven out of the 15 parts. Show that Alana received this amount of money. Alana get the eight part don't it? So Alana receives Alana receives let's put it right here.
Um 8 out of the 15 let's put it right here. So times the amount of money of 9280 and that's supposed to not 928 17 something. Come on Mr. Focus.
17400.
And that's supposed to give it a figure, right? 8 over 15 times, 17,400. Good. I get the same together. I said, right, that is equal to 9280.
So shown, right? That is shown.
shown left off those lines of the paper.
Next part now says Alana invest her share of money into a business venture earning simple interest earning simple interest at a rate of 4% perom. She received $2,88 in interest. Determine the number of years she invested the money. So we're going to be using the formula I = P R T / 100. And in this question we're trying to find the time guys. We're trying to find the time.
So to find the time we can multiply 100 and divide the P R. So you will end up get 100 ID by the principle and the rate equal the time. So put the values in this and we're good to go. So based on this question, let's write them out.
The rate is 4.5. So the R is 4.5.
The interest is 2020 2088. The I is 2088.
And the principal was this amount of that she invested. The principal is the amount she invested the 9280 share. So those are the figure they put into this calculation. Let's do it over here.
So therefore the t becomes equal to t becomes equal to the 100 times the interest and interest is 2088 2088 divided by the principal which was 9280 times the rate which is 4.5 4.5 Five.
Put that in the calculator and we get the time. And we get the time. We put years on the word on it. We put the word years there on the time. It's a fraction 100* 2088 divided by 9280.
And under there is also the 4.5.
5 years. If 5 years believe the word years, they might lose a mark.
Five years. Five years. They said how long did they say how many years? Yeah, they said how many years? Number of years. This example put they did say number of years. Number of years is five. All right. That is good.
Very good. All right. So we finish that question.
Now come to this one down here. Now this one says we have to factoriize this expression. That's the difference of two squares right here. But this is the same as 1^ 2 minus t ^ 2. All right, everybody must remember 1 is a perfect square. The first square number is 1. So this actually becomes 1 - t * 1 + t. That's the answer. 1 - t * 1 + t because the 1 is the same as 1 2. This one is same as 1 2 that good. Then we have this thing. Now they want us to expand and simplify this expression. Expand and simplify that expression. See if we can fix it over here.
Expand and simplify that expression. All right, let me write it out.
So we have we have 5 R - 5 Q* 3 R + Q + 3 QR. All right.
Uh oh. Four. First one is four. Thanks.
All right. So, we're going to multiply like this. So, for this one, I'm going to multiply this times this and that times that. This time this and that times that, right? That's expand this bracket. So, each thing gets multiplied.
All right? So, careful now. So, the 4 R * the 3 R is 12 R 2. Right?
Then the 4 R * Q is 4 QR.
Alphabetical order. Put the letters in alphabetical order. So going to get 4 QR.
So the R is R * Q. You put QR alphabetical order. When you put RQ is still there, right? Then have the - 5 * the - 5 Q * 3 R going to become -5 QR.
Then the - 5 * Q* Q becomes -5 Q 2. And you add back this 3 R. 3 QR. going to add back this three QR right here. What good? They add up all the like terms now. So let's look at the like terms. So this is the only square term there. So that stays 12 R 2. Then the QR terms now. This one, this one, and that one are the QR terms. So that's -15 + 7. What's that? What's that? What's that?8 that's 8 QR and then this is the only square term there are now which is - 5 Q^ 2 and that is it 12 R 2 - 8 QR - 5 Q ^ 2 that's the answer for that one right that is make this on.
All right. Then we have this thing to solve. Now this thing that's an inequality is a simple linear inequality. The key thing to remember about this guys is that if there comes a point we have to multiply or divide with a negative number. We have to reverse the inequality sign. That's the key thing to remember about solving this.
But it's basically the same steps as if we were solving an equation like that.
The only thing with the inequality to remember is that if you're multiplying or dividing by a negative number, the inequality sign must be reversed. All right, so let's do it right here. So it's -4 P + 3 greater than or equal to 19 + 2 P.
I'm basically trying to make P the subject. So you can decide to group the piece on the right side or you can decide to group them on the left side.
Your choice. Let's say we're grouping them on the left side. So we're taking this and bring it to over here. Then it would be min - 4 P - 2 P cuz this comes over become a minus 2 P. This stays greater than because we did not multiply or divide. This stays 19.
Then this three goes over and subtracts and becomes -3.
This becomes -6 p greater than or equal to 16. Everybody say that you want to divide now by the -6 and reverse the inequality sign. Divide by -6 because we're dividing with a negative number. Now we have to reverse that greater than or equal to sign.
That's what have to remember to do.
So you divide by -6 it becomes P.
This sign is reversed to a less than or equal to. This becomes now -16 over 6 -6 / 6 N8 / 3 2 and 2/3. So good. It's P less than or equal to -2 and 2/3.
-2 and 2/3h.
Yeah, man.
-2 and 2/3.
All right. And just to show you, if we did it the other way, just to show you.
Let me make this smaller and see what will happen if we did it the other way.
All right. If you if you group on the right side instead, let's zoom up here.
So if we had the same 4 P -4 P + 3 greater than or equal to 19 + 2 P I decide to group the Ps on the right side and group the values the numbers on the left side. Then this would go over become a plus 4 P. This will stay over here as a three.
This 90 will come over become a 19.
the sign will stay greater than or equal to. You have this 2p on the right side.
Then this 4p will go over and become a positive 4p over here.
Then over here become a 6p and over here become a - 16.
Then again divide by six now. So now for this one we divide by six. That is six.
But now because we're dividing with a positive number, we're not reversing the sign now. So then we would get on this side the 2 and 2/3 again for the same 16 / 6. The sign would be now this way and you have a P.
So it be that this number now is greater than the P. But you can read it the other way. Reading it from right to left, it means that the P is no less than or equal to that. So you come back to the same thing. So say I. P that is if P is less than or equal to -2 and -2 and 2/3.
That is good with that. Good. So either way you're good to go. Next part now says determine the largest integer.
Integer is a whole number value of P that satisfies this inequality. Now the best way to do that is to show the solution on a number line and see what the first integer going to be within the range that they give you.
So if I to show this inequality on a number line right now, let me draw it down here. For to show this inequality on a number line, let me put a number line here and this is a P ais and we're working in the negative range.
So let's put zero here.
-1, -2, -3, -4.
The critical value is -2 and 2/3. That's where inequality to start -2 and 2/3.
The -2 here. So, -2 and 2/3 is close to the -3. We're not quite with about right here. So, -2 and 2/3.
That's where you have to put your circle and you have to shade it because it included in this one and then you'll be pointing to the left less than it.
But if it has to be an integer then the integers would be these values the -3 and the -4 then -5 then -6 and so on. If there were integers and so the largest one would be this, don't it? That be the largest integer.
All of them negative number. The largest of these negative numbers is the one closer to the right. All right. The one on the right. And that would be the largest one.
So the largest integer would be -3.
What is that? -3. Put -2. No. The best way to show it is to put the number line so you can actually see what is that good to be -3. So this answer will be -3 -3. And our solution for this was what again? P less than or equal to -2/3.
This answer is P less than or equal to -2 and 2/3.
And this up here was factor this one into what? Wipe it off already became this.
We can copy this right here. This paper kind of mess up.
answer for this one.
All right, I just kind of put them on the paper. All right, then we have this question down here. This one in this area.
It gives us this triangle. It is a right angle triangle. And once it is a right angle triangle, you know, you can use soal to It says for the right angle triangle we are to show this that that becomes the relationship between those sides sides are in terms of x that become the relationship it is the stere we're going to use to get this expression right so what do we know about this triangle we know that this side squar is equal to this side square plus this side square is theorem so that's how you start do it right that's what we know we know that 2x x cancel we know that 2x + 3 r² that's hypotenuse squ= the side square which is x^2 plus this side square which is 12 that's what we know from pi we're going to expand and simplify to get this expression that's what's going Right? Now remember when you expand this perfect square going to get three terms.
The first term squared is 4x^2.
The middle term is twice the product of the two product of two. You multiply these two things first. That's 2x * 3 get 6x and you d the 6x to 12x. So the middle term is 12 x.
Then the second term squared is 32 is 9.
That's what the left side becomes. x^2 + 12 2 and 12 square we can put 144 now 144 just take this and transfer over here.
So we take this x and bring over subtract from the 4x going to get 3x^2.
This is the only x term in it. So it stays 12x 144 and subtract from how much you get 9 - 144 subtract to work out to 135. So that become 135= 0. Understand that?
Huh?
Huh?
>> Yes, sir.
>> Tuesday.
Go through this much one million time already. When you expand a perfect square, you're going to get three term.
The last time now exam is Tuesday.
When expon is twice the product of a two. Pay attention and stop writing tell all the time. Get it in your head before you put it in the book. Go through this multiple times already. Exam is Tuesday.
When you expand a perfect square, it becomes the first term squared twice the product of the two. The middle term people always forget this one. How you get it? Twice the product of the two means that you multiply the two things in first.
You going to get 6x and you double it.
That's get the 12x. That's twice the product of the two.
See, generally speaking, look look a + b squared really means a + b * a + b.
You see, if you expand this out, you're going to become a 2 + 2 a + b 2. If you expand it by doing this, look expand by doing this. This time this this time that this time this this time that you get 2 A in A plus AB 2 A and get you're not seeing it.
Do this all you know like the same thing more than one time especially we spend a lot of time on it more than once. So if you expand out this step by step, you going to get a * a first a get a * b is a b. See going b * a is b a that's another a see two of them there two abs get b * b b 2.
So it becomes a squ + 2 abs + b 2. For some reason people love forget about that value right in the middle there. So twice the two numbers see there the two numbers in the bracket is a and b of them is ab and they double it to 2 a. If this was a minus inside your s then this will become a minus in the answer out here.
This will become a minus it's not complicated guys. You have to remember the way of perfect three terms something you care tell you if you make a mistake on something guys you have to remember it.
So if you expand two, if you expand a perfect square like this one day and you only write 4x^2 + 9 and you come and discover you have to do the three terms, oh you forget it again ever and ever.
Amen. You must remember that you made the mistake.
I say every time three terms you have to remember it's not just the first term squared and the second term. is wrong.
I don't expect people to be making that mistake at this stage. We go through this multiple times. So have to remember the key critical things especially when you make the mistake before with it redo the question.
Don't look back at correct it. Don't lodge in your brain that way. You have to literally redo it. do it again the right way. Keep going in your brain the right way cuz the wrong way is still in your head if you don't do it the right way. And that's keep repeating the same mistakes.
You have done this already multiple times. I think you have recently encountered the same situation.
I would do is nobody ask me that today. God know any exam everything good. So you know divide you by three right cuz this we're trying to get this and you can see we divide this by three cuz all the values of this can be divided by three.
So divide this by 3 becomes x^2. Divide this other one by 3 becomes 4xide 135 by 3 becomes 45 equals zero.
So that has been shown that is it. All right.
Okay. Good.
This paper clean. All right. Next question down here. Another question.
People have questions about you. This question about some rotational symmetry.
People asking me during the test for for rotational symmetry.
Uh the question says this is a regular octagon and the question like regular shapes, regular polygons. Yesterday we do a regular hexagon concept is the same. The question is asking us part one how many lines of symmetry does it have and and and what is the rotational um order right? What is the rotational order number? All right. So lines of symmet lines of symmetry right the line that divides half and half one half is the mirror image of the next right. So you have to basically just count them out. Just count them out. So the first lines of symmetry, you have one down there. So let's do the off sides first.
You have one across here. So let's count them this way first. One down here. So one up here. So one across here. So and one down here. So then there's another set of symmetry through the intersections. That's four so far. Right? So come across the B and the thing there. So that again is a line of symmetry. See it cut the half here and half your mirror image.
Right? And then you can cut another one like this.
Let's try better Mr. Sh.
See that? And you can cut another one like this.
And cut another one like this.
So four blue lines and four red lines.
Four blue lines of symmetry and four red lines of symmetry. So total lines of symmetry here is eight.
I know a lot of people put four know eight wrong get wrong.
Let's talk about rotational symmetry now. So I show you four lines of symmetry. Let's talk about rotational um symmetry. So just copy this thing. Oh no.
Copy this.
Show lines of symmetry is the pess.
All right. So those are the lines of symmetry. Right. Let me clear this now and show the rotational thing. Let me show the rotational thing by I copy the same object again.
So I put a red line up here as a reference line. I put a red line up here as a reference line. So I want to explain rotational symmetry now. Right?
So I'm putting a red line up here as a reference for you. Copy this thing and show you the thing I rotate. Copy this for you.
Um, I just want this stuff. I'm going to use I explain rotational symmetry to you. Pay attention everybody. You only need to see it explained one time.
So rotational symmetry are the order of a rotation order of rotational symmetry. It's basically like this. The number of positions where the shape is exactly the same as you rotate the object through 360° about its center point. So like taking this this octagon and you're rotating it about the center point here. So you're rotating it about the center point and you want to see how many positions it will look exactly the same as it is now.
So where we starting now that's called one position. Make sure put the red on top so you can see it as a reference.
And I'm going to rotate it so you can see how many positions it look the same.
And you know what? Let me put a line down here. So, so we can use this line as a reference line as well. So, I'm going to put a line down here on the base.
So, every time every time the flat side every time one of the one of the edges or one of the sides come flat, it going to look the same. All right. And the red is going to allow us to count how many positions we have so we can understand it. So, I'm now going to rotate this.
All right. So, that's wonderful. We start with one. You count the first position as one. You don't repeat any position. So the red is at the top.
That's position number one. That's order one so far. So every object have a rotational symmetry of at least order one. So there always one position look how it look right. Good. So watch it now. Rotate this portion look exactly the same as it did before. That's two.
Let's rot it again.
Rotate again.
That's a third position. Look exactly as it was before but not the same reach back where it was. Right. That's three.
Let's rot it again.
That's four. Don't it? Let's rot it again.
That's five. Let's rot it again.
That's six. Let's it again.
That's seven.
Let's show it again.
That's eight.
That's eight gone. You rotate one more time. It reach back where it was. So you can't count that. So the order of rotation symmetry of this object is eight.
Generally speaking, regular hexagons um the number of sides match the rotational um the order of rotational symmetry. So like for example an equilateral triangle equilateral triangle where all the sides are the same that's a regular polygon rotational symmetry of order three. I can rotate into three position and it will look exactly the same going be six. You get me? Just like this. So the regular the regular polygons same as the number of sides. All right.
So this rotation rotation now. So so whichever shape them give you can check it. So it give you give you any shape I give you. You can basically check it. All right. Which shape I can try. So try a square.
Rotational symmetry of a square is not muchh square of a square.
Just in a while regular polygon the same number of sides is four. Just in a while ago a square is a regular polygon with four sides. Listen to the details. So rotation symmetry of this one is four.
four different position it can rotate in exactly the same put the red side of again that is one next to red side going to be here that's two next red side going to be here that's three next red side going to be here that's four see that so regular one's easy all right let's let's tell me a circle rotation symmet Order of a circle is what?
Huh? Infinite. That infinite number of portion can put a circle rotate it and it will look exactly the same. So that's infinity. Um the shape we can try.
Somebody tell me a shape.
What kind of triangle?
So what kind of tri let's say we try?
Let's say we try an isosles triangle like this.
Let's make it better.
This one rotating rotate about here. For example, about the center.
If I take it and I rotate it, if I take it and I rotate it, guys, it never look the same again.
Right.
Remember not you're not flipping it. You only rotating it. You're rotating it. So have this and rotate it. Only time it to look exactly the same is is when it reach back the original position. So order one. You see it? See that? Let me set it like this. Make it more obvious.
Watch. Make it more obvious. Let's draw it like this.
Remember it's about rotation.
Think of as if you're rotating this thing. So you see order one right order one right now. So if you turn it, it will never look as original one. Hold on. Come again. Will never look like the original one. Rotate it. It will never look like the original triangle. Never.
never look like the original triangle until you reach back the original triangle. You can't count the same portion two times. You see that? So this have to order one as an isoses triangle.
But I get that. What about this one?
Give you another shape.
Suppose I give you what we call an isosles trapezium.
So like an isos trapezium look like trapezium. So that top side here you have a side come down. So side look exactly the same like that one up on the next like this and you have the line across here. So not so perfect but you get the picture. So what happen?
So we have this isoses trapezium. All right.
Rotate it.
So let me take it and rotate it like that. Right. Will it ever look like itself again?
No.
No. No. One.
See that other one equilateral triangle order three.
Take this and you rotate it. It look like it say this. So again, that's two.
That's three. Let me put one of the side in a color so you can see it. Put the side in a red so you can see it better.
I rotate it.
That's one.
That's two.
That's three.
three places exactly the same as it did before. That's order three. But basically that's rotation symmetry.
How many times if you rotate it 360° about it center point, it will look the same again. How many times it will look the same? All right, let's go. Then this one.
Let me tell you something. Whenever you get a polygon to work in a calculation, it is a lot a lot of the time it is easier to work with external angle or exterior angle because you know the total exterior angle is always 360°.
Total interior will change you know depend on the size depend on the number of sides you know but regardless of the number of sides the exterior angle will always exterior angles will always add up to 360° a lot of the time that calculation is easier to work with.
So to find an interior angle sometime better to find the exterior angle first and then subtract from 180 and the exterior and the interior must add up to 180. Let me show you again. We did this with the with the with the hexagon yesterday. All right. There are two ways to do it. You can work from the perspective of interior or exterior.
Exterior easier. So to find that angle G, what I'm going to do is find this exterior first. Find this angle out the first. Let's call the theta as an exterior angle. And is is a regular polygon regular um polygon. So mean that all the exterior angles are equal and all interior angles are equal.
And it have eight sides. It have eight of them exterior angle and eight of them interior angle. Eight of them.
The eight exterior angle. The eight of these exterior angle. So you have one more right. So you have one more right there. So you have one more right there.
So you have one more right there. So you have one more right there. So you have one more right. So eight of them. Those are the eight. Have eight sides. That's eight of everything. Eight sides. Eight angles, eight vertices, eight interior angles, eight exterior angles and we call it regular. All of them equal everything equal. All the sides are equal in length. All angles are equal.
So you have eight of these angles making up 360°.
But the exterior angles will always add up to 360° regardless of the number of sides. If you have 1 million sides, the total exterior angle is 360°.
If it has three sides of a triangle, total exterior angle is still 360°.
Yes, the total interior angle of a triangle is 180, but the total exterior angle of a triangle is 360.
So therefore this diagram using the concept of the exterior angle total exterior is 360° and there are eight of them. I'm calling each of them theta.
So three so 8 * theta or one of them then becomes 360 / 8 don't it? So theta then becomes 360° / 8 cuz there are eight of them making up 360. What is work out to 45°?
Therefore the G is simply 180°us 45° which is 135° easy like that. Working from exterior angle you can also work from interior angle. I know n minus two triangles a yeah that's what we say with triangles n - 2 * 18 can be divided into nus2 triangles right so this this polygon can be divided into six triangles six triangle look from here to here one here so that's one triangle Two triangles.
Three triangles.
Four triangles.
Well, actually six triangles. 1 2 3 4 5 6.
So the polygon because eight sides can be divided up into 8us 2 six triangles and each triangle has 180° in it. That's why the total interior angle is nus 2 * 180 nus 2 * 180 each of the triangle 180°. So if using interior angle to do it interior angle to do it you would say all right the becomes equal to total interior angle which is n minus 2 triangle / 8 all equal don't it? So it will be n - 2 * 180° as a total interior angle don't it divided by 8. So you could get it that way as well.
So G = um 8 - 2 * 180° and there are eight of them / 8 that going to give you the same 135° 6 * 180 / let's make sure 6 * 180 / 135 they could do it from the perspective of you have two choices your choice you decide what you want to I make use both of them.
This one small.
So for this one we saying total interior angle total total interior is n - 2 triangles n - 2 * 180° that's what we use all them equalide by 8 to get one of them that's it so this equals that right of 135° is by God creation.
God say spend a lot of time on that though.
G + 1 I don't know it says the diagram below shows two similar triangle on this similar triang are equal and the ratio of the square of the area is equal ratio of the square the ratio of the area is equal to the ratio of the square of corresponding sides. So the ratio of the sides corresponding sides are equal and the two similar triangles and the ratio of the areas of the two similar triangles equal the ratio of the square corresponding sides what they want us to do. So this this triangle now two similar triangles you know so this small one and this big one out here so similar see that small one or the big one and I said to you anytime you have a big triangle and there's a side in a line in it that is par to one of the outer sides we form similar triangle by doing that right similar triangles so have two similar triangles here um and it give us some dimensions So these two lines are parallel. Yes.
And give us some dimension. This side is five and this side is all right. I'll send the solution afterwards. This side is seven and this side is 21. So these two sides correspond to one another. See that those are corresponding sides, right? And the corresponding side horizontally is this piece here.
with the whole of this hotel. So those are the corresponding sides. So the reds are the corresponding heights and the purples are the corresponding bases. You see that? But you can't see that. All right. So they want us to target the length of yx. That is right here. So yx. Now watch this. Now you see this. You see this based on the big one.
Based on the big one out here.
is really five plus this length where the yx see that 5 + yx all right so I'm going to use ratio of corresponding sides are equal like this the ratio of corresponding sides are equal watch this now tell me to see it so I'm going to use want to use based on the small one based on the big one over base on the big one over height of the big one. Let's go again. Base over height on the big one equals base over height on the small one. You can see that. So base over height equal base over height. You can do it that way.
Ratio of corresponding size equal. So base over height on the small one base height on the big one equal base over height on the small one. You can set up ratio any way you want. So I'm going to use 5 + yx that's the base on the big one over the height of the big one which is 28 equal base over height right equal base over height on the small one. The base on the small one is five and height on the small one is seven. Everybody understand that? So we're using base over height as the ratio. See there base over height.
base over height.
So base over height on the big on the big equal base over height on the small but understand that you could have used height base if you want but I feel like I I like when the unknown that I'm trying to find on the top of my fraction that's started away cuz transposition is easier. So we're transposing to find the yx now. So going to take the 28 multiply it up here first and you're going to get going to get going to get 5 + yx = 5 / 7 * 28 and subtract it five and subract get yx = 5 / 7 * 28 and subtract five what out to this four 4 5 is 20 20 - 5 15. So 15.
So y is 15. That answer that simple as that.
Yeah. And then now I want us to calculate the term angle vx.
That's angleh.
that you say we could use any we can set up the ratio any way we want just remember the ratio of corresponding sides equal so you set up the ratio on one side you have set the ratio on the other side the same way so let's use base over height for this example so base over height on the big one equal base height on the small one simple all right so we want to find this angle now let's call it angle theta on the diagonal So to find that one now use the side to be 20 don't it? I just worked out that this is 15 or so just worked out that this side is 15 cm. So it's a 20 right and it's a 28. So going to use tan that's what we know in the big triangle. Now we know the opposite and the adjacent. So we're going to use tan.
That's going to use tan. So down here we're just going to say tan theta theta equal opposite side of 20 over the adjacent side of 28. So theta then is tan to the minus one of that tanus one of 20 over 28 and put that in the calculator and work it. So 10 to the minus one a fraction opposite over adjacent that's 35.5° remember said you must give answer to one decimal place angle in degrees 35.5° that is it okay with that all right beautiful let's go next question.
This question is a coordinate geometry but they don't give us any graph. to calculate. We have to remember our formula guys for coordinate geometry right give us the straight line and they say it pass through this point if it pass through that point mean that these coordinates can work in the equation that's what it means the point the coordinates of the point satisfy the equation and I want us to show that the value of c is 10 so in other words when I put this x value in there we can put these values in there put that one for the x and put the eight for the y.
That's what we're going to do and work it out to get the 10, right? Simple as that. So, it's going to become it's going to become 3 * -2 + 2 * 8 = c equals c. The point is on the line so it satisfies the equ of the line. You can substitute the coordinates of the point into the equation of the line and whatever you get will be true.
All right? So this works out to be 16 - 6, don't it? That's a 10.
C= 10. So C= 10.
Simple as that.
All right. Change the color of these two lines. Can separate. So put the put the -2 for the X and put the eight for the Y right there. That's what we did.
Substitute them. All right. So the point is on the line. So it satisfy. Then we want the gradient of the line. So to get the gradient, we're going to put it in the form y mx + c. Make y the subject.
That's what we're going to do. Put it in the form y= mx + c.
All right. So the equation now becomes um it becomes 3x + 2 y = 10. Cle this first. We get 2 y = - 3x + 10. Remember we're trying to get this know y = mx + c. We want equation to look like that. Not this c you know this c our normal c. Don't come don't don't confuse them. Not the same c. Then know to make y on the subject we need to divide by this two. We divide by two. We divide each value by two. So I divide through by two or say multiply through by a half whichever you want to look at it. So divide by two going y= divide this by 2 it becomes -3 * x because I want to be able to isolate the m and see the m separately from the x and then divide this 10 by 2 you're going to get 5. So the yaxis intercept is five. So the distance is 32 right there. So see there. So the gradient m is -3 / 2. What I go to that it says the line this that's another line intersect L at the point T.
Determine the coordinates of T. To do that we solve the two equations simultaneously.
To find the point of intersection we solve the two equations simultaneously.
That's all we have to do. So the equation we're solving simultaneously are 3x + 2 y = 10. That first one and this one now 3x + 4 y = 8 and 9 to give us it with two values the same. So we can subtract to eliminate the x. We're going to subtract them. So subtract these two equations.
Subtract So subtracting, let's subtract it this way. Let's subtract the bottom one minus the top one. Why am I doing that? A y value greater than a y. Let's subtract them that way. The bottom equation top equation. So the bottom equation the top equation. The 3x - 3x gone. Right? 4 y - 2 y is 2 y. I'm getting a 2 y over here.
= 8 - 10 is -2.
This means that the y=1 can say that see that divide by the two. Now that the y1, you can plug it back in any one of the two equations you want. Let's put the y =1 into sub into let's put in the in the top one 3x + 2 y = 10. Put it in there. Put the y= -2 there.
1 get 3x + 2 * -1 = 10. This 2 *1 -2 -2 come add to the 10 give me 12. So I'm going to get 3x = 12. Let me get x = 4. And therefore t is the point what t is the point this x value. Put x value in bracket first. Work on the y value first. put it in bracket second.
So the point t say the point t put it right up is a point four is the x value one is the y value that is it okay with that very good all right next part it says the diagram below shows a graph of line to give us the line this is the line they gave this line.
All right, that first line is see the yis intercept is five for true. It's really five, right? And they give us two other lines. X= X=1. That's this line. X=1. A vertical line and Y = 5. Y = 5. All right.
Question says on the graph draw the line y = xus one first draw this line first y= x - one so we can draw the line quick and fast we know say y1 don't it a gradient of one see the number in front there is not shown so it's a one right so for that line is c is equal -1 and the gradient is equal to one. Now one means one means rise one over run one don't it rise one over run one so we can rise and run from this we have to rise one and one we can make a bigger rise and a bigger run so we can maybe put like say what go up to this go up to about seven so we can go six over six so we can rise six or rise seven around seven is still one understand that so we can do one big rise and one big run and get another point on the graph. So we can draw the graph that way. Use the yaxis intercept and then use the gradient to rise and run to other points. So some people calculate coordinates you can put on the y intercept and then use the gradient to rise and run to get to other points.
Since the gradient is equal to one, one is the same as one. Meaning rise one run one. We do have to make that little small rise. We can instead rise seven run seven or rise five run five.
Remember also we're not rising and running on the grids. We're rising and running on the units. So we check the scale. So horizontally one grid is one unit. Good. And vertically one grid is one unit. So here we can write the grid.
The grid is the unit is the scale. So ready now. So ready now. So the yaxis intercept is this point right. Right.
That's one point on that line. I'm going to make a big rise of seven. So rise seven to six don't itative rise come to six up to here and then run six run six go to seven run seven so from z run seven come across to seven over here so 10 too thin you're not seeing it so rise six rise seven sorry I mean 7 over 7 so rise from one if you rise seven you come up to six and then from over here if you run seven come over to seven over here sir so a point on the graph rise six and six I keep saying six rise seven run seven so that's another point on the graph so all I need to do now is draw a straight line through those two points this yaxis intercept on that point gave me that Right.
So, I draw a straight line through that point now.
Like so.
That's it. That's my line y = x -1.
Simple as that. Some people might plot point and get it, but the gradient and a yaxis intercept is a nice quick way to draw one line. put on the yaxis intercept and then rise and run from it using the gradient because rise gradient is how you rise and run from one point to another point. So that's the line y = x - one nice and easy. Then I want us to now shade the region that satisfy these inequality. Look at let's look at them individual.
Let's put down what each one means. So this x= 1 means above count it above the line x=1 above a vertical line is to the right of the vertical line don't it? So means I want to the right of this vertical line.
Let's put arrow pointing to the right.
Next one this means below. Whatever is positive less than means below. So this one means this means below. We want to go below this line below the line y = 5. This inequality means that we want to go below the line y= 5.
So line y= 5 is this line. When I put an arrow pointing down to remember that I want to go below it. So right now I want to be to the right of this vertical line and below this one below this horizontal. says don't want so far some don't but let's narrow it down some more.
All right then this one now this line that we just drew a while ago and since y is positive it's greater than means above. So I want to go above this line now above the line y = xus one. That's what this inequality means. So I want to go above that blue line. The blue line is what we draw what we line that we drew. So I want to go above the sign and put an arrow pointing up above that put I want to go above this.
So the reason that I want somewhere near so far see below this one above this one and above the blue line so far I would want that triangle but I have one more inequality to put in. So let's go for it. So so far that's the region that I want.
All right, that's the reason that I want so far.
Let's go for the last one. So this one now the y over on the left side of inequality is positive. So this less than means below.
So this means below the line 3x + 2 y = 10. I need to go below this line. And that's the line that we had before. this black line here. So, we need to go below it. I'm going to put an arrow pointing down.
Let's zoom in on it now. So, this region, you know, because it is below this horizontal line, below this line right here. So, it's to the right of this vertical line right here. So, it is above this blue line. So, it's up here.
So, it's below this black line that gave us horizontally. So, it's below here. So, so the reason I want is inside the majority and purple for you is inside here. I want that region satisfies all the inequalities.
Lower the one that is supposed to be below the one that is supposed to be above. That region satisfies all the inequality. body can't see that that's the reason you can't put this part over it wrong that must not be in it and you can't put no part of the over here in it either wrong can't put no part of that in it cuz if you put any part of this in it going to be showing above that line not supposed to be above it it's supposed to be below it must be less than five less than equal to five all right so basically that is the region everybody go to that so that's the in purple that you would sh for this question.
Good.
Taking my time on this one. The last one that exam time and it's the most recent exam. So, let's go. Next question.
Pie chart.
This question says it says some grade two students were surveyed to determine the job sector in which they are most likely to seek employment after graduating. All of that fluff. It says their choices are represented in the pie chart below. So we see the pie chart and zooming in on the angles. So see angles here. So we have some angles in terms of P. Those three are in terms of P. This one is 90.
So they don't show the 90. That one is 90°. Cannot ignore it. That one is 90.
Let's put it there. This one is actually 90. A lot of people ignore it, you know.
But this is 90°. That's one of the angles for the healthcare. 90°.
All right. And then you have two known angles 67.5 and 40 40.5. That's for agriculture and ICT. So we know those three healthcare agriculture and ICT we know those three and the other are given in terms of P remember the total angular is 360 guys 360 total angler is 360 okay so all those angles must add up to 360 in the pie chart anyway that's what they want it says calculate the percentage of students who are likely to seek employment within the agriculture and ICT sector so combining the agriculture and ICT E sector but I want it as a percentage.
So we can do we can get it as a fraction of the 360 angles are and then express that as a percentage.
Right? So that calculation for this one is going to be the agriculture and ICT total which is 40.5 plus 67.5 as a fraction that is out of 360 but because I want as a percentage you multiply that by 100.
So this is a fraction as a percentage you multiply by 100. So to get a fraction to become a percent by 100. So basically that's it. So this will be the fraction of them of the total students but they want that fraction to be expressed as a percentage. So multiply by 100. All right. So let's do this calculation and you get they want to combine. So, it's 40.5 um Oh, over 4.5 + 67.5 right foolishness.
I'm writing foolish fraction 40.5 + 67.5 good over 360 that's a lot of fraction it's 30% then the the thing is.3 so it's 30% and say 30%.
All right done then I want to tell them the value of B the value of P. So to get the value of P add up all of those angles and total 360 right 360 total angle must be 360 let's do it right here what the P add up to add up the P you know it's 4 P 3 P and 2 P the P add up to 9 don't it add up to 9 so 9 P let's add up everything add up to 360 so we can say the 40.5 plus the 67.5 5 plus the 90. Those are the angles that we know. Plus the 9P we said, right?
137 + 2.9 + the 9 P that must equal 360 everything in degrees.
So therefore the P comes 360 minus the number divided by 9. Nobody say that.
So the P becomes the P becomes equal to 360° minus these three add up together which is 40.5 + 67.5 + 90 and you have you have divided by this 9 now by 9.
All right and put that in the calculator and see what you get. Let's see fraction 360 minus bracket 40.5 + 67.5 + 90 close bracket dividing that by 9 18. So P is 18° and we can put the degrees on it because they didn't put any degree on the P. So the P therefore is 18° that is it is 18° good 18° got 18° right very good next question says le has a set of red so this is probability now guys you see not setting they've been setting a lot of probability question these days already know so this one says Liz has a set Liz has a set of red, yellow, and white buttons in a sock. All right, she chose buttons at random.
You said the probability that she chooses a yellow is 0.3. Let's write them out. The probability for a yellow button is 0.3.
The probability for a white button, probability for a white button is 0.1.
It says determine the probability for a red button. Remember the total probability has to be one guys.
Probability cannot go between 0 and one.
So the total of all the possibilities in the space must be one. So since the probate of getting a yellow is.3 and the probability of getting a white is.1 that 4 g so far so therefore since the total probability must be one probability of getting a red must be 6 can do it this way understand that total probability must total one total of all the probabilities all the possible probability must one sample space which is the sample space of the bar the total probability of The red, white and blue, red, white and yellow can must come to one.
Right? So see the probability of the red must equal the total probability which is one minus the two we must add up to one. I can do it that way. Look look at it this way. Probability of the red plus the probability of the yellow plus the probability of the white must total one.
See that and you know these two already those two add up to 04. So the probability of the red then becomes 1 - 0.3 and also - 0.1 which is 0.6 0.6.
So a 60% chance is 0.6 probability of a red is 0.6 but the total probability is one. All right. Next part now says if there are 80 marbles 80 buttons in the sock originally determine the number of buttons that were red or yellow. Now generally speaking or means that you add the probabilities right and you can you you multiply the probability by the total to see what the actual are.
So the probability of a red or a yellow going to add those probabilities. So the red and the yellow combined see there red is 6 and the yellow is.3. So probability of the red or yellow.9 that so it's 0.9 of 80 marbles that there were 80 marbles in there right so let's set up like this remember now the probability of the red is 0.6 six and the probability of the yellow is 0.3.
Combine these. The probability of red or yellow. You're going to add them pro of the red plus the probability of the yellow which becomes 0.9.
So that's a chance of getting a red or a yellow. Right? But there are 80 marbles total. So therefore the number of marbles number of red number of red or yellow adding them combining it become 0.9 * 80 prob like a percent you know so you could say okay 90% so 90% of the 80 marbles is red yellow or red a percentage could have been expressed um probability can also be expressed as a somebody knocking the Yes, that's 72.
The compound number of yellow and red marbles is 72.
72 red and yellow marbles combined. You want to work the number of red marbles in the 80 and number of yellow marbles in the 80. And then add them either one either one.
Red oil.
Are they good?
All right. So, remember guys, J speaking in statistics. If a question asks for the probability of something or something else, you're going to add the probabilities.
You ask for the probability of something and something happening and you're going to multiply the probability. Now, the probability of getting a red and a blue from two draw. So prob of getting the red and the blue going to multip with probility of getting the blue. So remember this guys for probability at this level or you're going to add the probability or means that you add the probability and means that you multiply the probability. So for example, the probability of A or B becomes equal to the probability of the A plus the probability of the B.
Right? But the probability of A and the B, you want two of them to happen. Now becomes the probability of the A times the probability of the B.
So or you add and you multiply. So look out for that.
And a and situation is like something followed by something.
Something followed by something or consecutive events is also a and situation. This happen and then that back to back. That's a and situation as well. And all right. So remember and you or you and you multiply or you add probability of something or something add of something and something is happening both of them. This and that or this then that you multip you want this then that multip and that together you multiply the probability. All right that is it. All right, very good. Move on. All right, next question. The map and scale question. This question says a map is drawn on a scale of 1 to 20,000.
That means 1 cm on a map is 20,000 cm on the land. That's what that means. It says um them say on the map 1 cm represents n kilometers. And I want us to find what that value of n is.
We we need to convert this scale into a cm kilometer compar so we can set up like this.
So they give us 1 cm they give us 1 to 20,000 of the scale right it means that what does that mean?
It means that map to land or map to actual is 1 cm on the map because distance on the map will always be in cm guys.
Measure meters per me of paper in front of you. They're not going to measure me on measure in cm. Distances on the map going to be measured cm. So 1 cm on the map would now represent 20,000 cm on the land cm.
That's what it means. You want to know what the 20,000 cm is in kilome. So you convert the 20 cm to kilome. So you can know what 1 cm on the map represent in kilometers.
Now I like to carry the meters first.
You must know that 100 cm make a meter.
So divide by 100 first. So 1 cmide by 100 you get 200 m.
Right? Why did that? Because remember now right here from here to here you using the fact that 100 cm is equivalent to 1 m. So if you have cime and then now the other conversion now going to make kilome we're going to use we're going to use the fact that 1,000 m is equivalent to 1 kilome.
So if you have me get kilomeide by a,000.
So therefore the 1 cm on the map is going to be divide by divided by 1 2 3 the decimal point comes in front the two. So that's going to be 0.2 kilm and that becomes a scale for distance.
All right that's the scale of distance to the end. So we get 1 cm to 0.2 kilm.
So this end that they said is 0.2 2 say n kilm n k k k k k k k k k k k k k k k k k k k kilm n is 0.2 say that n is 0.2 say now um we must find the distance on the map in cm which corresponds to an actual distance of 4.8 km. So scale this now this is just scale or like a ratio we're doing a ratio question.
So what is now what do we know from this? We know that 0.2 2 kilm is corresponds to 1 cm. I'm reversing the scale. I'm just reversing this scale. I'm just putting the the actual side first to the map side and 0.2 km is that I want to know from this ratio what is 4.8.
So can divide by the 0.2 first and see what one is and then multiply by 4.8 to see what 4.8 is. So you can do a thing like this. Divide by the 0.2 first.
Divide by 0.2 first. I'm going to get 1 kilome to 1 / 0.2 cm. What is this 4.8?
So to get the 4.8 multiply by 4.8 know what one is multiply by 4.8. So to one first and then multiply back up to 4.8. So get 4.8 8 km then becomes 1 / 0.2 * 4.8 cm. Multiply this by 4.8. When you do this calculation, you're going to see what it is.
So it becomes becomes 1 / 0.2* 4. That's 24 24 cm.
So 4.8 km is 24 cm. So this is 24 cm.
All right. So use the same scale to do the conversion. That's not the only way to do it guys. But that is it. All right. I could build up this number until you reach 4.8. Just multiply it up until you reach 4.8. And that's I could say what number need to multiply this by for it to become 4.8.
And that will be 24. Get me?
All right. Anywhere you want to be. Next part now says Next says um the actual area actual area in square kilometers of a lake that is 12 cm square on the map. So remember now when you have a map when you have a distance scale I want to convert to a area scale you're going to square it.
You have to square the distance. Okay, let's do it right here. This over a little bit.
Let's do it right here.
Watch this now. So, we have we have the distance scale.
Distance scale is the distance scale is 1 cm to 0.2 kilometers to get to the error scale. So we can do error calculation. We square the distance scale. Just square the value.
Square and square. That's all you have to do guys. So the area scale becomes the scale that we're going to use for area.
It becomes 1 cm square one become one square one same way and you square the cm become cm then know we square a 0.2 there so it become 0.2 squared kilome squared 0.2 square is actually 0.04 so it mean 1 cm squared on the map is 0.04 04 kilome squared on the island or on the land.
And that's how you get from the distance scale to the area scale. Remember guys, just square it. Square this one. Square it.
Just square everything. Square the two sides of the scale. And to work in area now, we're using this. So if we don't want to know what 12 cm squared is, we just take the scale and multiply by 12.
Multiply this now by 12. The question want to know what what 12 cm squared on the map would be. So multiply the area scale by 12.
So multiplies by 12. You're going to see that 12 cm squared.
This times 12 is much 4.8.48.
Let's see. 0.04 * 12 0.48 kilome squared. This is the answer.
This is the answer for the question.
12 cm square on the mark is 0.48 km squared on the land right here.
0.48 48 kilome squared.
But understand that? All right. So remember if you if they give you the scale that they give you will always be distance. Scale that they will give you will always be distance.
And if you have the distance scale and want to get the area scale, you just square everything. Square the values.
Just remember that when you square the one, it stays one.
So we square the side, it becomes 1 cm square. And so you have to square this side as well to get this. And then this now becomes your area scale. Scale for area for calculating area. Area from map to area on land or air on land to this is the scale you know working with for the area scale. I will always give you the distance scale. The scale for distance. Get the scale for area. You just square the values of the distance.
All right, that is it.
This question stand.
So this question says the diagram below shows a running belt E F G H I J E that revolves around a running a running road on a treadmill a running board sorry on a treadmill in the diagram. So this like this like somebody going to be running on this treadmill, right? And that's the bench going around going around as the person run on the bench, right? So it's always good when you think of it realistically.
Just put it in real life. It becomes easier. All right. It says EJI that's over here and H H GF over here are those are um semicircles. So they have two semicircles at the end of the belt.
Um and the diameter is 0.07 to give us the diameter 0.7 not 0.0.7.
So it means that the radius radius over here. So the radius of this semicircle are 0.35 half of this right that's the diameter and that is the rad that half of 0.7 0.35 we like to work in the radius. All right. Then it says we must use pi = 22 7 calculate the length of the belt. So length of the belt is going to be these two long sides plus the two half circle don't it? Two half circles make a whole circle don't it? That's all. And the circumference of a circle is 2 pi r don't it? The length of the belt, the length of the belt is going to become 2 * this plus 2 pi r 2 * that plus 2 pi right here. So length of the belt comes equal to 2 * the long side which is 2.1 plus 2<unk>i r 2 *<unk> is 22 / 7 * r is 3.5.
So this calculation right that we did is 2 pi r 2 pi r two half circles we have make a whole circle don't it circumference right so put this in the calculator we're going to get the length of the belt let's put this all in the calculator so it becomes 2 * 2.1 plus + 2 * a fraction 22 / 7 * 3.5 26.2 26.2 What is that in meter? 26.2 m that's the length of the belt 26.2 m it says Glenda uses. That's what she said. Linda uses the equipment to exercise four times a week. Four times in a week she using it and she runs at 9 km per hour and should run for 20 minutes each time.
All right. Question says calculate the number of complete revolutions of running of the running belt um that the running belt makes during the week when Glenda is using it. All right. So we need to find the total distance run and divide it by the length of the belt. So we can know how many revolutions we need to work. Yes.
The radius is 3.5.
Huh?
God.
Oh sorry. Hey by mistake.
I have seven stuck in my head. 0.35 says 2.6 then no no my mistake my mistake I don't know why seven stuck in my head so I'm going to change this calculate there 0.35 say 6.4 before.
Oh, a good thing on the bright.
All right. My mistake. My mistake. My mistake. I have seven stuck in my head, guys. Right out said 3.5. All right. So, that is it. So, it's 6.4 m. All right.
Let's go again now. So, we need to find the total distance blender run for the week and divide by 6.4 to know the number of revolutions, right? Each revolution is 6.4. So we have to work out the total distance of the road. All right. So the total distance of run is first inside the speed and all right let's do a simple breakdown.
Let's do a simple break down. This 9 km/ hour really means it really means 9 km in 60 seconds don't it? at 1 hour 60c 9 km in 60 seconds was running 20 60 minutes and head gone to the boy 60 minutes 60 minutes 1 hour 60 minutes so it's 9 km in 60 minutes right but she only run for 20 minutes but 20 minutes is one/ird of the 60 minutes don't it so therefore she running one/ird of this distance.
You know, you can't say that.
So, she does. So, this 9 km per hour means she does 9 km in 60 minutes. You agree with that?
So, she does 9 km in 60 minutes. So, she does 9 km in 60 minutes. How much would she do in 20 minutes?
1/3.
Agreed. Because 20 minutes is 1/3 of 60 minutes. Why say that?
Good.
But can't say that, right?
Make it simple, right? So if you 9 km in 60 minutes, 60 minutes has three 20 minutes in it.
But each of those 20 minutes, you be doing three, right? So you like to divide this now by three to see how much you're going to do in 20. Divide by three to get this to come down to 20 minutes.
It means that she does 9 km in 20 minutes.
So it means that each that 9 that's three. Why gone to them? Oh lord.
Right. She does 3 kilometers in 20 minutes. Right. And she does 20 minutes every day and she run four times per week.
So the to the week is 4 * 3 km. So she does 12 km in the week.
All right, we're just trying to simplify.
So she does 20 km. She does this amount per day. So look look here now. This is the amount she does per day.
What she does four days in a week. Four days in a week. Therefore she does this per day. Therefore in one week she does 4 * 3 kilmters which is equal to 12 kilometers.
This is per week.
That's what she does per week. 12 kilometers per week. All right.
And and so watch this. Now the key thing now that this is in meters guys.
The belt is in meters. So we're going to have to convert this 12 kilometers into meters.
So we can divide by the length of the belt to know the number of revolution.
Understand that.
See that? So the total distance run is 12 kilm.
The length of the belt is 6.4 m. So one have to know how much 12 km is in meters. Divide that by the 6.4 cuz we cannot divide two different units.
Right? So therefore see here now the number of reps number of rest become the total distance which is 12 km divided by the length of the belt which is 6.4 m. Convert this to kilm. Now it becomes 12,000 m over 6.4 m and that's going to be the answer.
12,000 / 6.4.
So it's 12,000 over 6.4.
So she does 1875 reps.
1875 reps with us with it.
Understand anything understand.
So we have to work out the total distance that she did and we get the total distance from the speed right the total distance from the speed.
So she was doing 9 km per hour. So okay since I want the time in minutes we're going to break that 9 km I okay the 9 km is really done in 60 minutes.
So she does 9 km in 60 minutes and she clearly does 3 km in 20 minutes. So each time she runs, she runs 3 kilometers.
But she runs four times per week. 4 * 3 is 12. So she runs a total of 12 km in a week.
But the belt is 6.4 m long.
So you want to know how many revs it make. Every it makes 6.4 m. So convert 12 km to m and divide that by the length of the belt. And that's the number of times you bel it over again like this.
All right, let me focus. So I can't get focus good enough. Anyway, that finishes that question. Let's go. We come to number seven, the pattern question.
So this question gives us this pattern diagram.
more shapes. I don't want to complete the shape. So if you look at it, I am seeing look what I'm see I am seeing that they added this to this original sheet. That's what I'm see.
See, you take this and add it on to the bottom here to get this because this top part here stay the same.
This top part here so is the same as it was up here. So, so that this top part the same.
And I keep adding I'm going to say to keep adding this to the bottom. All right. So, the next shape is to take this take this and add it to the bottom of this thing down to form the next shape. body agree with that. That's what I'm looking at. Other people might see it differently, but no problem. Once the final answer, the same. So, well, firstly first have to draw back. Let me delete this now. I move this out here a little bit.
Firstly, I have to draw back this shape over here, sir. I have to draw that shape. So, what is missing?
What is missing? I'm going to complete this block right first.
And there's like this I have right now.
This what I have right now.
So I need this piece under here to complete that shape. So copy that down.
Draw it on there in blue and copy it down. So this shape is missing.
That shape is missing. I'm going to put back this shape to complete the original shape.
The fourth shape.
So, put this back down here to complete the fourth shape.
I'm now back at the fourth shape. It's a little bit bigger down here. So, no problem.
Right.
So, I'm now back at the fourth shape, guys. I just need to add back now one of these to get the fifth shape. Don't s this way. You going have to draw the exam. I don't have to draw it cuz I have the device.
So I can do it like that. But essentially that's what you need to draw.
That's what you need to draw.
So if you draw that, you're good. So you're supposed to have one, two, three, four, five of those lines and then piece on top.
If you have that, you're good to go.
However, however you set it up, but that's what you're supposed to have. So the blue is what I use. The blue is what I use to come to get back to shape four and then the red is what I put on to complete shape five. But understand back to shape four with the blue and then put on the red additional piece to make shape five. That's what that's what happen in this diagram right?
All right.
Oh here now complete this table. Right.
The same thing all the time. Complete this table. So the information up here is always the same. It tell you about the table related to the shapes and what have you and I want us complete the rows one two and three and then there's an additional question down here. So let's see now. So I like to complete the formula line first as usual especially in the time where adding the same number of val each time. So here you're adding two each time. So the formula starts with 2 n down here. 2 n and the first value has three is three.
2 * 1 is two. You have to add one to make the formula.
You're not sure. You can try the second on it. 2 * 2 is 2. 2 is 4 + 1 makes 5.
So you can try the values on it if you like. Then this one going up by five.
Right? It's going up by five. So the formula start with five n.
But the first figure has 10. So five times 5 * 1 is five. You have to add one another five to it. So it's 5 n + 5.
This one going up by this one going up by two as well, right? This one increasing by two. So the formula over here starts with a 2 n as well. But the first is 8. So add six to make it.
You're not sure. You can try the second value on it. 2 * 2 is 4.
4 + 6 is 10. So that formula is correct.
So now that you have the formula actually help on the next line. But this is the next shape. So they make it easy for you by giving the next shape. So you can continue the trend of adding two to this one to make 11.
Add five to this one to make 30.
And they have already added the two for that one. So we're good to go. Then to get the value for here, we're going to do is put this formula equals this this thing right here. Put the formula equals that over here. Let's do it over here.
So, so for this line, we're going to put the formula= 47.
So, we're going to put the 2 n + 1 = 47.
Subtract the one, we're going to get 2 n = 46. Divide by the 2, we're going to get n = 23. So the value right here is 23.
Good. And you can check it. 23 * this five give you what?
23 * 5 is how much? 115. 115 + 5 make 120. That is correct. Let's put the the 23 on this. Now 2 * 23 is how much? 46 + 6 is how much?
52. So the value over here is 52.
52.
All right. And to get that value, we just put the formula on it. So we did 2 * 23 + 6 just to show we got that value. And the table done, right? Table done. Next one. It says Mara says she can make one figure that has exactly 502 squares.
Explain why she is incorrect.
So these are the squares. What do you notice about the square?
The number of squares is what?
No, but what why is this number not possible for the number of squares?
You're saying it because the number of squares is always odd. The number of squares are always odd number and this is a even number.
See that the number of squares is odd number. Eh if what you have mean that they will accept it but answer my what did you say?
What did you say?
What you could do by calculation nor what you could do? What you could do is say, okay, put this equal 52 and you can't get a you can't get a integer number for n. You can't get a a whole number for the end.
That's it. And they accept that, right?
So the end has to be a whole number. So you can tell them that. But looking at it, what jumped out at me personally is that the number of squares is always odd because remember you have a odd one up here. See that? See it? So total number of squares is always odd. But I keep adding two squares each time I start out with odd number. See? So I'm going to say the number of squares is an odd number.
And 512 is not an odd number. 52 is an even number. Yes.
When we finish this paper, we're done. I did say we're going to 5:00.
I did say 5:00 now finish before actually finish. So anyway, for this one, I would say the number of squares must be odd.
All right.
52 is not an odd number.
That's why she's incorrect because two is not an odd number.
2 is an even number. But number of squares must be an odd number. You can't see it. But that's a formula. That's the expression for an odd number. 2 n + 1 or 2 n - 1. I gave you that.
And I gave you the possible formula that you must know. I gave you that one. 10 + 1 or 12 minus one is the notation for an odd number, right? So this the number is always odd. So there always I can see it on the shape too is a odd one out. I keep adding two. So there's always an odd number out. So the number of squares is always an odd number. 502 is an even number. That's the reason why. All right. The number of squares must be an odd number.
must be an odd number.
52 is not an odd number. It's not given up. That's why she's incorrect. She's giving and giving up. It's not possible.
All right, that question done. So, three more questions to finish before 5:00 unless we stop to talk about stuff. Uh this question says f and g are defined as this is f and that is g. So f is a quadratic.
G is a line. But this f is only for x greater than or equal to z. So from zero ones, right? It says must find g of1. So plugative in the g function. Now this is the same as this is the same as f of x = x^2 + 3.
And this is same as g of x = 2x + 2. Right? Remember that same as same thing.
So that's what we are familiar with.
That's what they give mostly. But yes, we can give it like that as well. I want g of minus one. So put minus one in that. So g of -1 becomes 2 *1 + 2 that becomes 0.
That's a -2 -2 + 2 is 0. Good with that.
Then I want fus one. That's inverse of this one. Fus. Let's do it here. So when I start by saying the f = x^2 + 3 want to inverse it and get x = f -1 2 + 3. And we're trying to make this fus one a subject. So take the tree and carry over here first. Let's continue here. So we get x - 3 = f -1 squared.
And then to get rid of the square root both sides. Square root both sides. I get the square root of x - 3 = x -1.
All right. And it's a positive square root because they did say that the x is greater than zero. So you form the inverse function. This has to be greater than zero. But let me not get into that.
That's for you're not concerned with that, right? All right. That's what's going to happen. So this means this means that x -1 of x can put it of x now is equal to roo<unk> of x - 3. That's what they're looking for.
Screw it of xus. Anybody got that?
Anybody got this for the answer?
Oh, okay.
You know get it.
All right. That's it. Then they want they want us to find g of f of x. That means f into g, right? That means f into g. Let's do it right here. f into g. So we have the f = x^2 + 3 and we have the g = 2x + 2. And so we say we want to g of x. So g of f of x it means the f into the g. So it means that we're taking all of this f.
I want to bring it and put it into the g right there where the x is. That's f into g.
All right, that's what we're constructing, right? That becomes equal to going to put back the two where that x is. We're going to put back x² + 3 there. Now, put all of the f there and then put back this plus two at the end.
Simplify this. Expand this out. We're going to get 2 x 2 get 2 * 3 is 6. 6 and 2 is 8. Plus this two make 8.
That's the answer right there. 2x^2 + 8.
So we can write it right here. So g of f of x is 2x^2 + 8. That is x now says f of g is equal to not g of f of g.
Notice the reverse f of g is equal to this. Um solve this equation.
Solve this equation. So we are solving we're solving f of g of x equals 2 g f of x - 15. All right, that's what we're solving. So the f of g is this one. So this one coming on the left side right here. So right that's f of g. So that is 4 x^2 + 8 x + 7. Let me shift this over so can have the equal sign here.
Then we're doing the 2 * the g of x. So doing 2 * this. That was the g of f.
Let's put it equ= 2 * g of f which is 2x^2 + 8 2 * g of subtracting 15 over here - 15 that I was solving that for x expand it out and solve it this 4x^2 + 8 x + 7 = expand this bracket it we're going to get 4x^2 + 2816 - 15.
So what happens now is that this x² term 4x² here and 4x2 here cancel each other right that and then now this seven goes over here.
Now firstly this 18 not 18 that 16 we're supposed to write 16 28 16 - 15 is 1. So this becomes a one then come over subtract from it get a -6 right 1 -6 we're going to get 8x 8 x = -6ide by the 8 get 8 = -6 / 8 which become 2 3 4 that's the answer x=3 That is it.
This is just this just some algebra guys. Let's set up an equation for you to solve. Solve. There's nothing to it.
This not even come like math.
Just create an equation. Substitute into it and solve for that's some algebra.
Right? That's it. The next part now says use the graph to plot the f function and the g function for x greater than or equal to zero.
All right. So plotting the graph like plotting the two curve plotting. We're plotting plotting the f and the g right.
So the f remember is x2 + 3 and the g remember is 2x + 2x + 2.
We can plot the line. The line is to draw the line straight and we can plot a couple points for the curve and see we only need three points. Well, four points 0 1 2 and three. 0 1 2 and three.
All right. That's already Let's do the line first. The line easy. This is the line. It passes one pass through two intercepted two. So it pass through here. So it has a gradient of two.
A gradient of two. So for this curve, for this line, the gradient is two.
is two guys and that grad of two is same as two over one right which means rise to run one so rise two unit run one right so from here when going to rise to run one remember I not rise in grid you rising units.
So this is one all the way to say one.
So if you're going to rise two from here, you're going to come up to four.
Run one, you're going to come all the way out to yourself. So that's a point on the graph right there.
Rise two again come to six. Run another one all the way over to yourself. That's another point on the graph. The line rise two again come to eight. run reach over to three over here. So those are the points on this line on the G the G function which is a straight line that cool. If you want to plot points and calculate coordinates fine but I like to just use the graph y intercept and the gradient you put on the yaxis intercept then use the gradient to rise and run to find other points easier. You don't need to be calculated.
All right you just do that rise and run.
So those points are on the line.
So a straight line to those points are good. So I'm just going to go like this.
That's my line. All right. That's my G line. That's my G.
Let's do the curve now. The F. Let's do the F now.
So for I'm just going to plot a couple points. So X= 0 F= 3. Everybody cancel that? No sir. Or the y value is three. We're doing the f.
We're doing the f over here. If x = 0.
Yes. Y = 3. Next one. x = 1. You put one here. So I get 1 2 1 + 1 is 1. 1 + 3 is 4. So I get f = 4. Same as y + 4. Put x = 2. 2 is 4. 4 and 3 makes 7. The fy = 7.
Next point 3 x = 3 3² is 9. 9 and 3 is 12.
So I get f = 12. So those are the points. Let's put them on. Let's put them on a different color. Perfect. So 03 is one point. 03 is going to be right here.
Next point 14 1 4 1 4 0. So same point as that one 14. Next point 27 27 is right here. See 2 up 7. 27 is this point right here. Remember that positive curve you know just half of the curve you have it curve. Next point 312 312 3 up to 12. That's this point up here. So 312.
And then you just draw a curve.
So um you draw a curve through these points like this.
That line might be a tangent. I don't know.
This might be a tangent.
That's that's what I'm getting.
Actually half of a positive curve guys.
Go like that. It's actually half of a positive curve. All right. So I'm going to leave that as the answer.
So like a tangent point right here as a point.
That's a tangent point.
Check it quick. What is it? x².
So dy by dx will be = 2x = 1.
dy by dx = 2. a tangent. It's a tangent to true.
It's a tangent. So the blue line is a tangent. So if they ask what's the relationship between the functions, the blue is a tangent to the f. This is the f function.
So this a tangent point that is it good?
So this plot some points. All right.
Number nine. Two questions to go.
Number nine says the diagram below shows a quadrilateral O. Quadrilaterals O M and N. Quad quadrilateral M and N are the images of O after it under goes two different transformations or two different transformations. So what is the original? The original one O right.
So O to M and over to N.
So O to N clearly is an enlargement, right? This is an enlargement and figure out that one.
All right, figure that one out. Let's see what they say. All right, it says describe fully the transformation which maps O to M. All right, so O to M clearly in canally see it. Um if you compare this length and this length are doubly so the scale factor is at two.
All right. Additionally they're pointing in the same direction. So for example this side up here so pointing so corresponding side are pointing the same way. So a positive two scale factor.
Next thing we need know is the center point. Where's the center of the enlargement? So the center of the enlargement is on the line join the corresponding point. So these two points are two nice points to use.
So they are on this line first.
All right. Because the scale factor is two. I can know that this is the point because the distance here is doubled to here.
See that? I can know that that is the point. But the best way to actually find it, the other way to find it is to join two other corresponding points like where like um that right like this point correspond I'm going this way the other way the point is below the other way. Let extend this down the enlargement in this way.
the point down. So the other way so the other way too. So join this to the other way somewhere down here.
So it looks like about here. No, not there here.
So this distance is double to there.
All right. So I can join two other points like for example here and here.
join them and extend it. Right? So, it's going to go this way.
Something like that. So, you see right here the center of the enlargement.
All right. So, that point is one one.
If I join two other points, you're going to get the same thing. If I join for example here and here, it's to go like this.
See that same center. So we know an enlargement of scale factor two about this point. That is it.
That is it.
So that is it.
So going to write that here. So this transformation right here an enlargement.
scale factor two about the point one or you could say with the center at 1.
Anybody got that?
All right, good. Next part. Now, what single transformation maps O to N?
O to N. That's this one to this one. All right, let's look at it.
That looks like if I join these two corresponding points like so, and I join these two corresponding points like so.
It looks like I have rotated 180° about the origin.
See that? Let's join two other points.
If I join these two corresponding points, these two like so pass through origin. If I join these other two and this like that.
So yes, that's what it is. a 180° rotation about the origin or a reflection to the origin. Same thing say that it's a 180° rotation about the origin or a reflection to the origin.
Same thing. So we can tell them that's what it is reflection to the origin or 180° rotation about. So this one this one is a 180° rotation. I don't have to give a direction. Clockwise, anticlockwise is the same.
There's a 180° rotation. If you want to put clockwise, I want put anticlockwise.
This time, it don't matter. 180° rotation clockwise and a 180° rotation anticlockwise is the same result. So, put whichever one you want, right? So, 180° rotation, clockwise or anticlockwise, your choice.
about the origin.
We could also say a reflection to the origin or you could say a reflection through the archive. Remember 180° rotation about a point same as reflecting through the point. That's what the second one is. All right. Then down here it says now on the diagram it says above but is actually below. Draw the image of O under after it under go the following transformation. And so we're going to do a five2.
So remember -52 means five to the left and two up.
That's what it means. Five to the left, two up. So moving this thing five to the left, two up.
Let's do it.
Five to the left, two up. So, we're going to do this point. Just move this point and then draw at the same shape around that point. Five to the left, two up. One, two, three, four, five to the left, two up. One, two to here.
That's five to the left and two up.
So, this point coming to here. And you're going to draw at the same shape of O around this point. We're going to put this little diagonal piece first.
Let's draw it in purple, which is here.
Going to put this other diagonal piece, which is across two boxes over to here.
I want to draw this diagonal piece up here, which is across two boxes vertically again up to here.
I want to put this piece now to connect them which is there like a sh that's it and must leave it what always tell you what to leave it must leave it lab it l so this one L that's what you're supposed to get everybody x1 must reflect o in the line y = -x. Now y= - x is a horizontal line.
That's perfect. This line is a horizontal line. y= - x. So y not - x y= -1. So y= -1 is a horizontal line down here.
So I'll put it out the way. This was what again? One.
this point. All right. So, y= minus one is going to be this line this line across here.
That's the mirror line y =1.
I reflected this in that line y=1.
All right. So, let's put the image in blue. So, firstly this point to the mirror line going one point on the other side to here. one point at a time.
And this point is two to the mirror line. It's going to go two on the other side to here.
And this point, this point is also two to the mirror line. This one is two to the mirror line. Going to go two on the other side to here. That's where it's going here.
That right.
Let's see at this point now is one 2 three four to the M line. It's going to go one 2 3 four on the other side here.
Now we join in these points. So join these two.
Join these two.
Join these two.
And join these two.
And that's it. It's like a reflection period. This reflect to this this line.
So you see do your transformation one point at a time.
All right. That is it. Leave this last one. Leave it P. So this is P.
So when you done guys, that's where diagram is supposed to look.
So the P is supposed to be there and the L is supposed to be there. That's where you're supposed to have.
You don't have that much mask for all of these. They have the mask here where you're supposed to have for each one.
All right, that's what you're supposed to have transformation one point at a time. Next question said, oh, so that is it. I finished that question. This question says a boy and a lighthouse L are 95 km apart. He said the boring of B from L is 230°.
So from L to B is 230°.
It means that from here we turn to the right 230° to get the direction down to B. So B some and that angle is 230°.
It says use a scale of 1 cm to 10 km.
So it means that the 95 km if you're using 1 cm to 10 kilome then the 95 km is going to be a 9.5 cm line. You see that 9.5 cm. So we need to construct a 230° and a line that is 9.5 cm. That's what you want. So when you done supposed to look like this supposed to look like oh the wrong thing we need a 9 cm line 230° so 230° clockwise. So our line is going to go somewhere down here. So we have to measure it. Use a protractor and measure it. But you can get 230° can piece by piece or you can do 230°.
230. It can be 40 below this line. 40° below this line to get it. 230. I can add it up little by little. Right. Can you can add it up little by little. So you have 180 there.
Um 230 can get ang this at about 50 50° in or you could say okay horizontal line 90 and you want um 40 below to get the 230 everything around is 270.
All right. So, if you want 230, you can subtract 40° from this line. You have to use the protractor to do it.
Won't have 230 on it. But that's what I'm trying to say. You want it at 230°.
So, you can mill it up. Okay.
90.
your ch anyway supposed line like this and you want to put on so the line is supposed to be 9.5 cm long that's going to draw a line that is 9.5 cm long and the angle is supposed to be 230° made that angle 230 you could also measure angle inside here what that going to be 360 - 230 is much 360 - 230.
What's that?
Huh? So 130° can measure 130°. You could measure angle. So that is 130° instead. Measure 130° over here. So measure this one.
Measure 130° here and put in the line. and they measure this line 9.5 cm long you represent your 95 km but the question is on a scale of 1 cm to 10 kilm so 9.5 cm would represent 95 km that is 1 to 10 right 1 to 10 all right so this will be equivalent to your 95 km so that is what you're supposed to have on your diagram something looking like that when you Measure it angle on this line like so.
That's a projector and you're reading up for 130° of your s reading up for 130° on your protractor of up.
But I understand that. That's what they put. Measure 130° this way based on this line.
I measure 130 up to here. 130 up to here. 130° up here the protractor. That's how you do it.
All right. So if you didn't do it before, go do it again. But follow CC.
So they didn't put they didn't put any straight bearings on this paper. They don't put any circular measure circle on it. Notice no bearings. No certification.
Um we didn't get any let me see what number 10 one question is right. So the the question kind of weird I wanted change it because this transformation question usually come on the last question kind of weird usually come on the last question and they didn't put any transformation in section one there's no transformation in section one so you see the people a little bit different guys all right so no matter where they put the question you have just do them where they put them but it kind of structure a little bit differently this time. That's what I'm about to say. Let's answer this question and we'll call it a so give us a vectors question here. So it says the diagram shows triangle OP r is the midpoint right here and n is the midpoint right here. Looks like a vector. Yes, it is. All right. It says O is U and O N is V. So they give us these two vectors.
Now can populate the diagram as we go from now since this is the midpoint on this line and the vector from O to M is U. Means that from up here is another U.
Same thing down here. This vector down here is V and this is the midpoint and this is the same length, same direction.
So the vector over here is another V. So you have V right here. So a U right there. So, so all of this at 2 u the whole at 2 V.
So those position vectors must get the vector MN.
So because we have the two position vectors, we can just use O* the last O the first. So this is O M O N O and the last is O N. The last is N. O NUS O M the vector vector.
So the O N is V minus the O M is U. We look at the diagram as well. To go from M to N, we go minus U going opposite direction to you plus E. Oh, by the way, they didn't put arrows on this diagram. Let's put them on. The arrows are this way and this way.
So they didn't put on arrows. You put them on. Right.
It's OM. Then I want NP. So last again out and the first we have the O N is V and the OP is 2 U the OP is 2 U. So we're good to go. So we can use last minus first again this becomes OP minus O N.
The OP is 2 U minus the O N is just V.
See there to go from N to P to go from N to P we go minus V see the minus V there and then plus 2 U plus 2 U the plus 2 U there that is it all right then it said now show that MN is paral to P R MN and P R we don't have P R yet we have MN already so we need to calculate P R now so we have MN already mN N is the vector V minus U. We now need P R.
So what is P R equal to? Can we get it from the diagram? E to R. Yeah, because we know we know O R is 2 V and we know O is 2 U. So again we can use O. What a easy question get or get O R minus O.
Why supposed to get 100% on this paper?
I know the O R is 2 V and the OP is 2 U that's what the P R is so we multiply this by two becomes equal to that right we multiply this by two we'll be able to equate because this now will be 2 MN = 2 V - 2 U and then these two things will be equal and so these two things now equal the P R is 2 * MN which means that they are parallel so when we equate we shall see that the big one P R is two times the shorter one M this means that they are parallel therefore So P R is parallel to the MN. The vectors are parallel. So the lines are parallel.
P R and MN. Let's see P R that's this line over here.
Par with this line over here. Those two lines par. Can see it on the diagram as well.
Done.
come to some maj like like this one people like this one kill it don't it yeah man this getone question says the matrix P written in terms of a don't think on this paper I would say is the is the is the base question it's not very frequently set and maybe the rotation of symmetry but you should understand rotation symmetry, right? And the base question just whatever base they give you, put the base values with base three, 3 to the 0, 3 to the 1, 3 2, 3 cub, those are the place values. I know what those values are. All right. Base val is the key for the base question. All right. All right.
So this matrix says it says A is real as follow. So this is a vector. This is the matrix P. Calculate the determinant.
Calculate the determinant of P in terms of A. Calculate the terminant. The terminant this product minus this product. Right? So the terminant of P equals this product which is 3 A * 2 A minus this product which is 6 * 4. Just work it out one time. Is 6 A 2 - 24.
That is it. 6 aus 2 something says determine the values say values meaning in plural two values more than one determine the values of a for which p is singular remember if p is singular it means it has no inverse they could have said that determin of a is for it p has no inverse and it has no inverse because the determinant is equal to 0. So we're going to put this equ= 0 because it's singular. So become 6 a^ 2 - 24 is = 0 / 2 by 6 you get a^ 2 - 4 = 0.
Transpose and get a 2 = 4. But when you calculate remember plus or minus as a key. So the a= plus or minus2 just prove what get plus or minus 2.
Another reason why it is plus or minus 2 I can show you right here from this part of the equation. So I said this side is a difference of two squares. The a minus 4 is the same as a - 2 * a + 2 = 0. If you factoriize the value that so this factor equal 0 then you're getting a = 2 and this factor equal 0 getting a= -2 so that's why plus or minus 2 right so this way as well that it says determine pus1 the inverse of p for which p is non singular so if it is non singular get in terms of a we're not actually get it.
and get in terms of a is already in the we don't know what we don't know what a is all we know is that if a is singular if the p is singular and a be one of these values right and it says now determined inverse of p for which it is non singular in other words you're saying it has an inverse we have to get it in terms of the AC so becomes one determinant where this is the determinant times the adjint you have to leave those expressions in the answer right I don't know what A so the P to the minus1 the P to the minus one becomes 1 / the determinant of the P times the adjint of the P the determinant of the P we just put the 6 the 6 A 2 6 A square - 24 this put 6 A square - 24 6 A 2 -4 * the adjint of the P. So to adj the 3 A and the 2 A, you switch them around. So put 2 A up here. So and 3 A down here. So and then these two other values on the second going to leave them where they are, change the sign. So this four becomes a -4. This six and becomes a -6.
So we're going to have -4 -6. I would leave that this time really dividing each of these by this. But I think if you leave that answer the answer all right I want to hear what going to get rush out.
Next one is a right on 2x2 matrix represent a counterclockwise 90° rotation about the origin. See this question some question give you some question expect you to know it just know them and done it's a anticlockwise 90° rotation matrix about the origin. This means that the negative one is at the top. See here the negative one is up here. So at the top one down here. So 1 0 0 call it x equals that. See it there? Easy to remember.
That's a anticlockwise 90° rotation about the origin.
Show that all the time. Next one. They want a counterclockwise rotation about the origin 90°. That's this matrix followed by followed by a reflection in the y ais. This is the base for reflection the y axis.
So what is this? Look what they want.
They want a X followed by a Q.
X followed by Q is what is X by Q?
QX.
It means that the T is equal to QX. The say it becomes equal to Q * X. That that's what we calculate q * x q is q * x component transformation. So the t = multiply the q * x - 1 0 0 1 * 0 - 1 1 0 that out and get the answer. So 0 * that is 0 and 0 * this is also zero. That's a zero right here. And the value is this time this is one and 0 * 0 is 0. This becomes a one out here. So one and don't know this row and this column going to get a one. 0 * 0 0 1 * 1 is one. So don't use a one.
And it's going to be a zero as well because it's 0 * 0 1 * 0.
That is it. It's supposed to be aative one is a one. Okay. And this matrix, which matrix is this? Which matrix is that?
Which matrix is that?
y = xction like y = x y = x. Reflection matrix. reflection in line. Y = X Y = That's what that is. Come down. Let me hear you get your packing up. H4.
All right.
By the way, class tomorrow evening everybody.
I think at I think at 4:00 you get what you get. Tell me what you get. Tell you get >> online student WhatsApp.
>> Send me a WhatsApp. How many you got, guys?
chance One time jump.
All the question you got wrong.
The question that you got wrong. Read the question you got wrong. Guys, redo the question you got wrong.
supposed to.
>> Yes, they are.
How much?
6.
That's a jump from the last time.
So still get me wrong.
Oh, all right. I'm going to I'm going to send the video.
All right. Everybody tomorrow we have a session tomorrow evening a bring your question session tomorrow bring your question to explain session tomorrow bring your question you want to explain by anybody online.
How much you get?
All right.
Bye everybody.
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