This video is part of a CSEC Mathematics exam preparation marathon, covering key topics including direct and inverse variation (y = kx and y = k/x), linear functions (y = mx + c), gradient calculations, parallel and perpendicular lines, function identification using vertical and horizontal line tests, transformations (translation, reflection, rotation, enlargement), similar triangles and area ratios, trigonometric ratios (sin, cos, tan), and vector operations. The instructor demonstrates problem-solving techniques using past exam papers from 2024-2026, emphasizing practical application of mathematical concepts to exam questions.
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CSEC Maths MC Marathon 2026 - Day 4 of 4Added:
Guys, internet looking a little bit shaky.
I'm trying to see if I can just sort that out.
is going to give the best signal over four different Wi-Fi sources.
Trying to see which one.
All right, we are at day four.
We are at day four of this thing.
Exam is day after tomorrow should be ready.
So, we're just going to do this last little bit.
You see, it's not coming up on the delay.
And the students at home, can you hear me and see the screen of a delay?
Let me let me um break the break the sh All right, let's work.
All right, so objective today is to get through these two last topics relations functions. I think I want to do some vectors and trigonometry 4 hours. So we start about half an hour late about 45 minutes late. Still have to do have to do so 8:00. So it be so be it. All right. 9:00 to go tonight.
So what we have to do.
All right. So let's get kicking. We know what it's about about doing this and getting ourselves familiar with them.
You should be familiar with all these questions. But just in case of an issue with how to work out a question, I'm just going through them with you. If there's something that you don't understand, you let me know. Students online, same thing goes for you. All right. So we're going to go through these functions. How many pages? 1 2 3 4 5 6 7 8 9 10. And not all the pages are full. So we should be able to get through these fairly quickly. Then we look at the 2026 paper, 2025 paper, 2024 paper break, and then we come back and do that. I might just run into the vectors after this one. All right, because we have not yet done vectors. All right, let's go. So within functions, relations and graph relations, function and graph given that y varies directly at x is a direct variation and y = 19 when x= 5 152 which of the following equation represents the relationship.
So what it is is that we're going to say y directly directly as x. It means that y is equal to a constant * x and what it is you're going to use the sphere of values to get an expression for the constant. So based on this if you divide by the x the constant is going to be y / the x that's what the constant is going to be using these values the constant becomes um the y 19 over 152.
And so this equation now this relationship becomes y = 19 over 152 times the x.
All right. And the question is which one of these is it 192 1 8 or is it 1 to 1?
No.
All right. So basically that becomes relationship 192 and it's eight times. So 19 to this goes one time 19 to this goes eight time. So it's this one y = 1 8 x.
This is it.
Right? Simple as that. So it was inverse variation. If they change this word to inversely, you have to put y = k /x and do the same thing. Use these values to find the k and then write back the equation. Right? That is it. Let's see what the next question is.
This one says the distance L that an elastic string is stretched by hanging an object of mass m varies directly as the mass of the object.
That's a direct variation. So the L is directly proportional to the mass M. Where's the mass? The mass M. That's what they mean. So the relationship is that the the length is K time the mass again. Just like that one before. Those are the two variables.
Then it says if 5 kg that is the mass when the mass is five of an object the string is 31 cm the length becomes 31.
Let's read it carefully. The distance L that an elastic string is stretched by is stretched by hanging an object of mass M varies directly as the mass of that object. If 5 kg this is the mass if a 5 kg object stretches the string 31 that is the length.
How far will 29 do um stretches the same thing? So when so so basically how it is how it is is like this. You can do it by finding the equation again with the K by find the K plug these values in here and get the K or you can do it like a like a proportion because what happen by the L you see that the K is equal to the length over the mass. So length over the mass is a constant. The length over the mass is a constant. Right? So it over here. So the length over the mass is a constant. We put this up here.
So this is a constant. Now length over the mass is a constant at all times.
Right? This is the information we're going to use right here. So the length over the mass is a constant. Then the length over the mass for this pair equal the length over the mass for the pair that you're trying to find with that 29. Right? So we can set up like this. The length the length of 31 over the sorry the length of 31. Sorry I mistake my mistake. The length of 31 the length of 31 over the mass of five is equal the length that you want. No, from here sir let's call that L over the mass of 29.
That's what they kind of work with.
That's what you're working.
All right. And you can literally estimate because these values are wide enough so you can estimate it. So you can estimate this maybe as almost 30.
This is approximately 30 this one.
So it's almost like six times almost like six times this right because if if you if you want to find multiply the 29 up here now right so the L that you want would be 31 over 5 * the 29 you multiply the 29 up here look at this now in can't work it out this is almost 30 right that's almost 30 30 / 5 is 6. So the value that you want is approximately 6 * 31 but actually less. This is the closest value to 6 * 31.
6 * the 30 180 181. But remember this value was less. So you're looking for a value close to that. You don't have to actually work it out. But I could work it like this. This value is approximately 30. This one approximately 30. So say the L is approximately equal to 31 over 5 * changes to 30.
But we know it's going to actually be less than this. We increase that to 30 just to make the calculation a little bit easier. And then going to say, okay, 5 into this go one time. 5 into that go six times. 6 * 31 would be 186 right but it's actually 6 * 29 you do so it's a value less than 186 close to 180 this is the one that it has to be right that is it so I would know that that is the answer right here this value is almost 30 right so like that you go So to be um 31 * 29 / 5. Yes, that is the correct answer.
All right, that is the correct answer.
179.
All right, that is the correct answer right there. This ear inverse variation.
This one says given that y varies inversely as x this means that y is equal to k /x inversely multiply the x up here now you get the x y is a constant what that mean the product of x and y is the same all the time a constant right the product of x and y is the same value for any period that you have the product of x and y is the same it says when y = 6 y = 16 x = Half means that the K is 8, right? It means that half * 16 is equal to the K. So the K is equal to 8. The K is equal to 8. So which of the following equation represents the relationship? The K is equal to 8. So from that now this equation comes Y = 8X.
She don't y that is it go so you set up the relationship use they give you to find the key some question you don't need to find the key as I did up here did not actually just know the key will always be this if down here they gave us some valate I would use the product of the numbers the same all the time is a constant so the product of one pair equal to another pair of X and Y. That's what that's what you will do. All right, that finishes that one. Let's look up here.
This one says the point where we are linear functions crosses the horizontal axis. I don't know if this is function or geometry. The horizontal axis is the x-axis, don't you?
So that is the x-axis intercept. So this are really really behind by far. This is the X-axis intercept.
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>> Yes sir.
What are you seeing on the screen now?
>> Question 26.
>> Huh?
>> Question 26.
>> Seeing question 26, but I'm I'm talking about something else, right?
So, the video is lagging.
You're not see you're not see you're not seeing me on question 45 right now.
Yeah, this thing is bad. This thing is bad. Let me let me try a different internet is bad. Let me try. It's lagging badly.
It's laggingly badly badly. Let me try my data as the this thing is very poor.
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Internet is poor guys with the lime.
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Flow. Yeah, it's very poor. It's very very poor. All right, let's see if we're making any It's just lacking badly.
It is lagging badly badly badly.
All right, let's go. It's a little bit better than before. All right, so the point where a function crosses the x-axis is x ais intercept. That's what it is, the x intercept. All right, the point where a function crosses the vertical axis. Another question this time is a vertical y axis.
That now is a yaxis intercept or y intercept this one.
So you see they can ask these question you have to careful right all you have changes the word horizontal to vertical in this question. So we have to be careful.
All right down here this one says the equation of a line which passes through the 05 and has a gradient of four. Easy question.
It's y = mx + c. I'm going to give it the point 05. It means that the c is equal to five right here and this is the gradient n.
So putting them in this equation.
So the equation becomes y = the gradient is 4 x and c value is 5. That is the answer 4x + 5.
All right, that is that. Next question.
It's like some um you get to the function part square geometry. A line L is paral to this. So line paral to it is the same gradient. Par lines have the same gradient. They could have changed this word to perpendicular on the same question but they say parallel they have the same gradient find we need to make y the subject of this equation right so this equation going to write it firstly I'm going to transpose the I can transpose the 3x over here and I know I'm going to get 7 y - 7 y - 7 y = 3x take the 9 and car over as well I get positive 9 then divide by -7 you dividing through by -7 -7 I divide through by -7 this becomes y this becomes -3 over -7 x this becomes 9 / -7 this right here know is the gradient that they want this thing is the gradient that I want the m -3 -7 the negives cancel to give you 3 over 7 so the gradient is 3 over 7 right here all right could have done it another way but it is key thing you have to get the line in the form y = mx + c y alone must be the subject y = mx + c when it it in that form the y alone as a subject then this becomes your gradient that's what you want the question instead ask for the yaxis intercept it will be 9 -7 this one orative 9 this will be the yaxis intercept this one if the question ask for the y ais intercept instead which they can ask you all right here we go again this line is parallel to this one Now line L is par to that again have the same gradient. What's the same gradient par lines of the same gradient? So again I'm going to write this equation in the form y = mx + c y = mx + c. Let's do it another way. I'm going to take the 5 y and car over on the right side instead.
So this equation I'm going to write it as 2x - 8 = 5 y over there.
5 and make it positive. Then divide two by this five to get y by itself. So I'm going to divide by 5ide by five. Now this becomes 2 5x - 8 over 5 equals the y by itself. This can be spun around. It is the same as i.e. This is the same as let's put it here. y = 2 5x 5x - 8 over 5. Again, this here is your m, your gradient. This is your gradient m.
All right. And so it is 2 five.
Next again this time our line is perpendicular. Now see change it a line is perpendicular to this line. So perpendicular lines have negative reciprocal gradients.
So what is the gradient of line L?
Negative reciprocal of this. If this gradient is m or m1, this gradient is m_sub_1 = 2.
Then m_sub_2 is -1 / m1 or ml then line of the line l is minus1 of line one.
This calculation can be done quickly by flipping this one and changing the sign.
Flip the two fifths. It's going to become 5 / two. And then change the sign to a negative. This is positive right now. Change it to negative. So become 5 / 2.
5 / 2. See right here? So 5 / 2. Flip the gradient that you have and change the sign and you get the perpendicular gradient. All right.
So this line they give this line a different question. Look different question same same they ask the same thing but they keep this equation the same y = 37x what is that + 9 you're going to become 73 so the answer will be this one -7 3 this one here this is -7 over three for this one the per if it was this equation the perial gradient would be this It was the same question and a different question this equation instead and this is the answer down here right flipping this and changing the sign 3. All right. So that is that. Next question up here. This one says what is the G of this straight line? You won't put -3. You know have to get rid of this two first.
I have to have one alone by the subject.
So you have to divide by the two first.
So you take this equation and you're going to divide by two to get rid of this two beside the y. And when you divide by two y = this by 2 -3 it becomes -3 / 2 - 3 Hello.
No, I don't want go.
What's up? What's up?
Huh?
28. What?
28.
You need it right. Right now, I'm in class.
Oh, all right. Send me a WhatsApp number and everything.
Send me a WhatsApp.
Want to do something computer.
All right. This one says what is same line you know.
So because of the same equation it is still works out to be y = -3 / 2x - 4. So the y intercept becomes this 4 right here.
That is your c value. Over here the isus 2 right here.
So you see on a different exam paper they ask something different for the same line.
All right.
Next question.
Here this one says give us that diagram of that straight line. A straight the straight line cuts the y axis. That's right here.
That point is 0 -2. 0 is the x value -2 is the y value. 0 -2 right here.
Next value there is zero 0 -2.
Right. Next one. This one says give us that straight line again. And it says what's the line? The gra of a on the graph is what? So we just to count the rise and count the run from here to here. Count the rise and count the run.
Make sure that each grid is one unit which it is vertically. Each grid is one unit horizontally.
So counting the rise and counting the run to go from A to B. So the rise is this to here.
It's a rise of two.
And then you count the run. One, two, three, four. One, two, three, four. The run is four units.
Run four.
So rise to run four.
So the gradient is rise three run four. 2 4 is a half one / two. So the answer is this one.
All right.
Next question. Which of the following pairs of lines are perpendicular? So remember the perpendicular lines are the ones with the product of grad one. When the product of the gradients equal -1 they are perpendicular.
So we need to look at the of each of the lines. So for this one if you two getus 32 look at it this line if you transpose it and make y the subject you going get y = - 3 / 2x - 2 get four.
So this is the grid of this one.
For this one subject you can take the two bring it over here first and divide it under the tree. So for this one you will get y = 3 / 2x and this will come over and become a + 8 and by two as well become a + 4. So these two gradients when you when you multiply them you don't get one. So those are not perpendicular right right here. So the two gradients are the same.
So they are parallel. These two actually parallel them have the same gradient, right?
That should have par. Let's see. What about this one? For this one, the gradient is -4. See the gradient is -4.
To get the gradient of this one, we need to divide by this four. So, we divide by that 4. This one will become y = 1 / 4x + 1 / 4.
So this gradient for this second one let's call it m1 for this one.
Second gradient will be 1 / 4. If I multiply these two values they become two gradients. So for this one, for this one right here, m1 * m2 is -1. -4 *1.
So these are the perpendicular.
Those two are perpendicular. So you're looking for the one where the product of the gradient is one. Those are the perpendicular lines. But the product of the gradient is one.
All right. down here. Which of the following is equation of a straight line? A straight line equation in the form y = mx + c.
So in this now you don't have no products or quotient or power of the x and the y. So this is not a line because it is a product of x and y. This is not a line because it has a product of x and x to the x².
This is not a line because again it has a x square term in it. So the line is this one can be in the form y = mx + c it becomes y = 2x - 3. All right. So you're looking for the one that does not have any product or quotient of the x and y values. So it must not have any x y must not have any x² must not have any y^2 must not have any x y like that right to be a line x and y val to be separate and linear equation which of the following represents the equation of a line again again it can't be this one look at it if I take the x and divide it up here I'm going to get x y = 4 that's not class.
This one has x in it. No, write this one. This one can become x y = 4. The product of x and y. You don't want that. This one has x². You don't want that. This one x square. None of these are lines. So, this is the line again in the form y mx + c. Obviously, right? Obviously, y mx + c. Again looking for y = mx plus c line in that form that all right that one done functions functions now ready functions now this question gives us this diagram and it say the relation that best describe the mapping diagram is what this one is that one many to one don't you have two values for example here mapping to the same one value up here.
See it that's many over here mapping to one over here.
So that is many to one and many to one relation is not a function. Yeah, it can be a function.
So one to many that can be a function, right? Once you map to one, it can be a function. So it is many to one.
So one to one or many to one can functions, right? Can be functions. This one down here gives us this h of x= that and asking us for h of -6. Just take -6 and plug it in for the x and work it out. So it to become 3 * -6 - 2 / 5 that's -8 and -2 that's -20 / 5 -20id 5 is -4.
So the answer is this one up here -4.
That is it. All right. Then have another one here. Now same function they give us but this time I want h of four. Again, I'm going to take the four, plug it in there. So, it's going to become 3 * 4 - 2 / 5. 3 * 4 is 12. 12 - 2 is 10. So, it becomes 10 / 5. 10 5 is now equal to 2.
So, this value now therefore becomes 2.
Right? Nice and easy stuff. Functions are easy part should enjoy functions on the multiple choice. Same thing here.
Again, you want just -6. Put the -6 in there. So, it's going to become 7 * -6 - 3 / 5. 7 * -6 is -42 - 3 / 5 - 45 / 5 which become 9. 45 is 9.9 right here. and so all even stuff so far down here they give us that mapping diagram and say which of the following best describe the function and actually the first one the y value the y value is three more than the x value every time can see it see this value is three more than that three more than that so you're adding three to the x value to get the y value see so the y value is the x value plus three so Answer for this one the first one relatively easy you got to put that as the first one but I that straightforward all right this one says the value of what is the value of fus of one if given that f of x is equal to for this one you can actually try val down you can actually try these values in this and see which giveative one give me the Watch this. They want fus1.
Pay attention.
So this is the function that they give us and this function is f = 5 - 2x. Right? Write it like that. And now we're trying to find trying to find fus one. What I'm going to do here to do it quickly, I'm going to put some number, right?
some number. Therefore, one is equal to f of that number.
So if I'm in the exam, I can try these values in here. I say which one equal one. Don't put it to put it.
I put in one in this, I get 5 - 2, which is three. That won't work. Put two in this, I get 5 - 4, which is one. So the answer is this one right here.
So whichever one of these give me one put in the function is the answer.
That is it in the do that. But otherwise you can finish this by putting 1 = f of n. So you put n in this.
So you get 5 - 2 n.
So you find the end that makes this equal one.
like this 2 n and the 1 get 2 n = 5 - 1 5 - 1 = 4 2n = 4 which therefore means that the n is equal to two which is the same answer that I have here but for okay which is valid pro it in this function the result is one and by the second right That's an easy which of the following which of the following best illustrates a function. All right, somebody talk to me. This one a function. Stop one. Why?
Why that? Why is this one not a function? This one. First one.
the domain the doain. So all these values must map and when they map one time so there are two values here causing that this not to be a function. This one don't map that makes it not a function and this one map twice having two coming off it. So those are the two values that causing this one to be to to not be a function. Remember it is the domain that decides a function on the domain.
All right. What about this one? Is this one a function?
Huh? Why this value mapping twice? This value is mapping twice, right? So that makes it not a function.
What about this one?
Well, all three of them call it to be a fun. That one not mapping. This one mapping three times and this one not mapping. So all three values causing that this not to be a function. For it to be a function each value in the doain must every single one of them must map and when they map one time this one you see it this is mapping one time which is good. This is mapping and it's mapping one time which is good.
This is mapping and it's mapping one time which is good. Doesn't matter how the mapping end up over there. You have a function. So this one is the function.
Okay with that good.
This one says the set of ordered pairs below represent a relation that is a function. Which of the following points when added to the set would form a relation that is not a function?
Remember it's not a function. If it becomes a if it becomes a one mapping to many becomes one to many is no longer a function. So each of the x value must only one one y value for it to be a function. If it is a one to many it is not a function.
So any value you add to it right it can't be that you end up where you have a doain value mapping to more than one value you get me so look at this now add this one to it would it still be a function if this up here and this join this join the set of values up here look at it this way if this value joins the set of value I extend it 28 would that be a function? No.
Would that be a function?
If I add this 28 to this set of values, would everything know still with all of them still be a function? Which of the following when adding to the set would result in a relation that is not a function? Look at the first one. If I add the 28 to it, would all of these four now form the values in a function?
You're looking for this. You're looking for this thing. You're looking for one to see it, man. Looking for the doain values.
You know, see the two up to seven and the two up to eight.
That is what will happen if you add that to it. You have the tomb of the seven and you also have the tomb of the eight.
See that if you add this to it. This is the one that you cannot add to it.
Because if I add this to it, you're going to have two map to seven. Yes. So, and also going to have the two mapping to eight.
You can't have that. That's why that would make it not a function. If you put 28 in it, it no longer function.
If you put one in, good one in the first place, right? If you put in zero, in other words, the point that you're putting in, you can't have any repeat of the doain.
You have the doain value again. The domain must only map to one value alone.
Huh?
is do that the size of function or not.
Look what the size function or not these values you do mean you don't want to be repeating this any of these values in the set up here. So you don't want no point that those values are repeating the doain val you get me because you cannot have a case where doain value is mapping to two different values that will be a one to many for it to be a function you cannot have a one value in the doain mapping to more than one value in the so you cannot have one to many this way you must not have you must not have one too many so any point we put in add to You can't have the same x value as one that is already there. So since we already have a two x value, I cannot go and add this point to it with a two years as well. I already have the two mapping to a value already.
So we can't make the two map to another value.
Understand that? Look at it carefully guys.
You cannot have a case where a value in the domain and x value.
Your x values cannot map to more than one y value.
If I add 28 to it to this set, if I take 28, I take this 28 pair and add it on here.
So see it right then you have two mapping to seven and have two mapping to eight. You cannot have that. Twos are doain values. No matter what valot doain can have otherwise it's not a function. So that is the play that I cannot add to it because I cannot set it up so that have two doain values have the same domain value mapping to two different doain. So this is the one that I cannot add.
If I add that to it, it is no longer a function.
If I add that to it, it is no longer a function because it becomes a one to many. You should understand this.
Let's continue.
Same question here. We're looking for again. Which of the following points when added to the set would form a relation that is not a function? Look at that.
See you can't add a point already have a doain you don't want any doain already come back you understand that that is the point you can't add C beautiful so you cannot add this one because you will have -2 to 7 you have -2 mapping to 8 which would not be a function that would be a one to n which is not a function.
All right, good. Let's continue. Which of the following which of the following points lies on this line? So to find a point that lies on that line, it's clear the last one, don't it?
Like the last one is a yaxis intercept.
See there? This is already in the form y mx plus c you know.
So we see this value right see this value c= -3 is that the pass to the point 03 and they give it right here. But if they give if they didn't give 03 how you could find out which point which point is on the line cuz when you plug in the x value in the equation you're supposed to get the y value. See that?
That's it. If a point if a point is on this line, it means that the coordinates satisfy the equation of the line. When you plug the coordinates in in the equation of the line, the equation stays true. So in other words, because this already have y as a subject, it mean that I plug in two here. So for example, I get two 2 is four and 4 - 3 is one.
But this y value is not one. So say this point is not on it. This point is not on it because it does not satisfy the equation of the line.
Get me guys. So this is the last one.
That's actually the yaxis.
Take the zero. I'm pl3 which is what it is here.
That's actually the yaxis intercept.
Next question. That function and I want f of minus3.
So going to plug the minus3 into this.
So we're going to get we're going to get 2 * -3^ 2 x - 3^ 2 - 1 now - 1. Now the - 3 2 is a positive 9. So this becomes 2 * 9 - 1 which then becomes 18 - 1 is 17. So the answer for this one is just seven.
All right, let's continue. This one says item 45 refer to the following arrow diagram mapping diagram above describes a relation. What relation?
Let's look at it. All right. So these are bigger than these which are a process of elimination.
It says y is less than x. That is not true, don't it? is y values are actually greater don't it? So this one is not it a process of elimination. It says y is a factor of x are these y factors of these. No actually multiples don't it bigger. So it's not true. The y values are not factors. They are actually multiples. Next one. x is a factor of y.
x is a factor of y. That looks more true. Let's look at it. Who is a factor of the eight and the 10? Which is true.
Three is a factor of the n alone which is true. Four is a factor of the eight alone which is true. Yes. So the x value are factors of the y. So a question like this says go down is a process of elimination. This is this is the answer.
This one says x is a multiple of y.
Again that is not true. Three is not a multiple of n when mean a whole number multiple. It's not right. So this answer also is not right. the last one and by a process of elimination we come out with our answer being the third one.
All right. This down here a mapping diagram. Same thing again. You notice the same but look now switch around these val and put them as the output values. See slash and these output values these y values have now become the input x values. So the same values.
So you see again kind of process of elimination to see which one is the correct answer.
All right. So this one says y is more than x not true. These are actually less than the right. So this one is not true.
It says y is a multiple of x. Why a multiple of x? No factor.
Right? Small are not multiples. Right?
Smaller than the x. So no this is not right. Said x is a factor of y. x is a factor of y. That's not true either. X is not a factor of y. factors are usually small, right?
Like one and and and so on. So those are not false. So this is also not true. X is a multiple of Y. Yes, these are multiple of these.
So yes, this is the one that is correct.
The last one. So it's a similar question on a different paper. Same two sets just switch them around.
That is it.
All right. There are some number lines.
Which of the following lines represent this? Look at it carefully. So you're not including -2. So you put a open circle on the -2 and you're going up to four. Including the four equal on the four. So for unshaded circle on the -2.
So right away you know that these two out of it, right?
And they're looking for a shaded circle on the four equality. They're looking for a shaded circle on the four. So a shaded circle on the four and a shaded circle on the four. But this is the one.
All right. So I'm just exiting off the one that I don't want. So I want a shield. So I don't want this.
I don't want this. I already don't want this one. So that's must be the one that I didn't off anything. See this is the last one. Yes. Home circle on the negative two shaded circle on the four and you want interval there.
So sometime I think I explain too much.
This one says um show the number line. It says the graph of the inequality is what? So you notice it's an unshaded circle on the one don't it?
So you notice a less than you're going to the left the values over here are less. So it's a less than this is a less than question less than x is less than square value is one. So it has to be this one right.
They can change it and shade the circle guys.
Then answer will be this one.
cuz you know now including the one. All right. You just shade the circle becomes that.
All right. But that is good over here.
Give us this graph.
Check this graph a lot. They ask us for the maximum point. Maximum point is right up here. Coordinates of this point up here. Look at it. It is 24.
That's the maximum point. You must familiar with this question. The maximum point has coordinates 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4.
This one said the equation of ax of symmetry to be the same graph and one axis of symmetry. Ax of axis of symmetry is this blue line passing through the turning point coming down here and pass through two as well. That's axis of symmetry.
So a vertical line passing through two says x = 2.
Right? x= 2. So this is the x = 2.
That is it. Next one. This one said the values of x at the point where the curve cross intersects y = 0. Now we have to realize about y= 0 guys that y= 0 is actually the x-axis. Right? So that's the key thing with this one that the y= 0 is actually the x axis.
So they're asking for where the graph crosses the x-axis which is now at this point at the origin at this point at four. See that right? x= 0 and x = 4 x ais and four question on it the value of x at the point of the graph the x ais same thing x-axis again right x = 0 x = 4 again for this one this one which are the following represent That's just a number. This one right here straight line equation.
So questions repeated straight line straight.
Uh up here it says which of the following sets is represented by the function f ofx = x^2 + 3. So this really is f(x) = x^2 + 3.
So you can look at this. You plug in these values in this function x2 + 3.
Let's look at them. So the zero will become 0 2 + 3 which is three, don't it?
So 0 into three. That's why you have z to three in all of those. Then the one if you take the one and plug it here. So it becomes 1 square is 1 1 + 3 is 4. So this one is mapping to four. So one map to four. So you know say this answer. So you say this out of it and this out of it. So that this answer or this answer.
So we can look what the two does now. So the two we take the two and put it here. Then become 2^ 2 is 4. 4 and 3 is 7. So the two is mapping to seven which now then is this one two seven the last one you get one map to four and two seven this one that is it extra give + x cub for x cub the -2 up to -2 cub -2 is -8 the only answer don't this one so this must be the answer8 in the range so that is the range -2 maps to8 and this is the only one with a8 that has to be the All right. See you know - x². You have to careful with this one now guys cuz this minus not squaring you know you're only squaring the x². So all your values will end up negative. So for example for example this -2 -2 this negative not squaring you only square negative on the two. So this becomes a four. So this become4 in the end.
So these three4 based on the fact that these three values are4 the second value which is what you get with the one with the one right don't it right? We're getting a zero. So you plug in the zero in this it is going to become zero. So zero maps to zero.
So the last one zero maps to zero. This one maps to net. No, that's not it.
That's not it. You know that's not it.
So you have to repeat because what happen guys this will always come out negative. Be careful because you only squaring this negative this square. So all the values are going to come out negative or zero.
See what I'm talking about. So maybe the safe thing to do is actually work out all of them. So see here this one. If you plug it in there, it's going to become -1.
You're going to get -1 squared, which is still -1 cuz 1 square is a positive 1.
If you plug the zero in there, you're going to get zero. So this one maps to zero.
When you plug the one in there now, you going to get 1^ 2 not on this negative. It's only on the one. So become1.
So this one also maps to1 and this two also maps to -4.
So the domain value you go to -4 are repeated and the negative ones are repeated. So this4 repeated it can all appear in the set once.
So you know4 this is this this or this in there but these out of it because you cannot have any positive value none of the answers are positive. So those out of it so the answer is only B as in boy to that.
All right, remember this one guys. Put a negative out here. Not square, not negative.
That's not the square. Then we put xr² that difference. I would make any difference xr same as positive xr² negative anyway. So it don't matter.
All right.
Anyway, let's continue.
This one. Which of the following is a graph of a linear function?
A linear function.
You know one of these two, don't it two lines, right? So we try to think this one or this one. A linear function.
Try a linear function. That one or that one.
All right. The first thing you need to make sure which first you need to know which one is a function and it is linear. What how do you know what a function is on a graph?
Vertical line test. Remember this thing.
Vertical line test tells you a function or not.
Same question and the solution for this on multiple papers the same solution.
This is not a function because you put a vertical line on it. It will intersect it at multiple points.
So it will fail the vertical line test.
A function can't have a vertical section on it.
So that's not a function because if you put the vertical line on it, it will fail the vertical line. So this is linear is a line. But it's not a function, not a function.
Because if you do the vertical line test like this, a vertical line cross it several times.
So it's not a function. That's a Vline test.
It fails the V line test. Fails.
Fails.
Fail the V line.
That's what this one pass line is at one point intersect it. Draw vertical line is one point. So that that's a that is a that's a function cuz any vertical line I draw it can only intersect the graph of one point. So it's a function and it's also a line.
That's why a linear function linear function. You see it? So this is the answer.
This is a vertical. It's a a linear function. It's a line.
It's a function. What about this one?
What kind of function is this? Is that a function or not?
Is that a function or not? Yes, because any vertical line I draw can only intersect at one point. See there?
That's how you know it's a function. Is a one to one function or a minute to one function. How you how you know how you know to you know you decide if a graph of a minute to one or one to one function.
You do the horizontal line test. So the vertical line test is to tell you it's a function or not. The horizontal line test is to tell you what kind of function it is. Right? So this is a function because it pass vertical line test. The vertical line test pass the V test.
So you know a function. The vertical line test tells you yes a function. What kind of function is a one to one function or many to one function? Then you do the horizontal test.
And in horizontal I cross at only one point mean it's a one to one function.
So this is the H test or the H line test horizontal line test at only one point and therefore that's a one to one function with a curve a nonlinear one to one function. That's what this is not a limit non linear one one function that's what this is what about this one is that a function or not function or noth First quad function guys function function first draw a vertical line test remember always do the V test first the vertical line before the horizontal line look any vertical line I draw can only intersect this graph at one point you see that any vertical line I draw intersect the graph at only one point why did you say It's not a guessing thing guys. You want a function or not a vertical line.
If any vertical line can only graph looking like this for example like so y ais x ais do a vertical line right here it's going to cross that two points. You know what that mean? It mean that the same one x value for example three is mapping to two different y values like one and five that's what that mean 31 planet I have a 35 planet same whenever I want that's three map into one and three map into five that will be a one to many which is not a function that's why we do the vertical line test the vertical line must cross the graph cross the graph at only one point.
Come on man exam is for tomorrow doing let me leave this on the screen. So this is not a function. This one would not be a function. This one down here not a function. Not a function.
Come on man repeat all.
So this is a function. What kind of function is it? This one. What kind of function is it?
Many to one horizontal line test. Now it crossed at two points. Why many to one?
Two different x values. X value over here. So x value over here. So two of them give it to the same y value. So that two different x values give you the same y value. So like for example, you have three right here. and one and the two of them have to for example value of five right so that many to one that's a many to one function to do the horizontal line that intersect at more than one point makes it a many to one function so horizontal line test tells you what kind of function it is after size of function it is a many part function using the horizontal line test. So this is the H line test intercept at two points. So we know it's a to one.
So what?
All right.
What else?
Which of these are function? Which of these are functions?
Which of these are function?
Huh? They say B.
Anybody else?
You can't be my boss. Look two points right there.
All right.
Those points right there.
Look no right. See there? Draw a vertical line here. It's going to cross at two points and here same way the points here cannot have any vertical section on a graph guys otherwise not a function draw vertical line here again pass through these three points fail the vertical line test this one again not a function Because of all these points down here, I put a vertical line through those points to intersect more than once.
Those are not functions. Look at this one. Any vertical line can only at one point.
So that is a function. What kind of function is it?
Oneh draw horizontal line one point draw horizontal line one point anywhere I draw horizontal line can only inter one point so this one will be a one to one function this one is a one to one function What is it?
One to one function. This one says what is the value of the point of intersection of the system of equations?
This and that. So when f of x= 3 is the same as y = 3. This one the same as y = 3 g of x = that same as y = 5 / 2x + 1.
You're solving that right?
It says what is the x value you want? X value alone you want. So you can eliminate the y value. So you basically need to put this equal to three y is the same. So, so you're going to put 5 2x + 1 = 3. Subtract that, you're going to get 5 / 2x = 2. Multiply by 2, you're going to get 5x = 4.
5 x = 4 5 which is 8 / 10, which is 0.8.
0.8. A that's simple is equations really.
All right.
And then we come down here after this. I do think bring up the screen with the next 20. I give you a couple minutes to them while I do something for work.
Right. This one said a parking lot charges $1 plus 25 cents. This is 25 cents for every hour h or part thereof in which a vehicle is parked. Function C of H that represents the parking cost is withdrawal. So the fixed $1 first fix $1. So this are this and you multiply the hours by the 25 cents. If you multiply the hour by 25 cent so to get the total cost. So, this basically is the answer right here.
All right. All right. So, that finish with that. Let me bring up uh let me bring up a couple papers.
of 2026.
I was um all right here 2020. 26 up a second one transformation.
All right, this page a page of functions all these questions take a couple minutes up another side. Let's put the screen of another table.
Oh, get it.
25 26 2025 some repeats.
I see them here.
You guys can see them right for me. Just do this thing for quickly.
All right.
Sabine was awesome.
Sorry.
August.
Yes. Teaching dance and y I don't know why I can't guys youngsters.
Thank you.
I said finish. Yeah.
Ready, ready, ready. Let's do this question.
Huh?
Huh? I'm going to do the same question repeat which is a straight line this one point where line crosses horizontal axis is the x-axis intercept again same question um which which is the mapping diagram many to one many to one many to one many to one this 1.
What's the line? We need to get rid of this 2. So you have to write this line as y = -3 2x - 4. So the gradient is - 3 2 right here.
This one they want 8. Put four right there. So you get 3 * 4 - 2 5 that's 10 which is 2.
Then fus one of one fus of one everybody should know that this just equal one guys f of f to the minus one of any number by the same number like map do reverse mapping so the answer is going to just come back to one reverse go fus one of f of any number equal the same number. The inverse the function inverse basically net out anything like drive forward and drive come back.
All right.
The function are reverse of each other. All right.
That finish that right there.
This one ask the maximum point. The same point appear again. Same question. We did this one already. Point of 24.
According to the graph solution between this and y= 0 y= 0 x ais x ais we're talking about these three points on xaxis y. So 0 and 4 0 and 4 0 and 4 0 and 0 is the answer.
All right, next one. So see the same question try change it up a little bit. 0 and four the two x values 0 0 is the y on the x axis. Y is always on the on the on the x axis.
This one h to the -/ h of -/ in here becomes 4 * -/ squared assume that's a square - 6 assume that this is a square.
So -/ square becomes a/4. So it becomes 4 * a4 - 6 1 - 6 which is5.
All right.
It will be 4 * - cube. Let's see if this will work. No - 6 that answer for the should be - 4 / 8 which is - so that has to be a square lines per lines one. So this line is going to be two. I take this one and bring it over here. I get y = 2x 2 x - 8 is two. So the purple of 2 isus a half and change the signus a half the line two.
All right. Next one. Same question we do earlier. So the following function is this. You plug in these values in there.
Plug that value there first. Get the 03 and plug the one going to get four with this. Sorry, right two with this one.
52.
All right, that's that. Let's look at uh 204.
Let's look at 2024.
2024. January 2024. right here.
After this, we take a short break and I'll come up solution is that solution solution.
2024.
Let's see if there's anything else on this that I'm not done. After 36 here, what is the y intercept of this line? Y intercept going to be -4. Remember you have to divide by the two get it in the form y = the form y = -3 2x - 4x -4 do this one already this four and this one already this one of one do that one already as well right like I said f ofx = 5x - 2x But this equation that you're trying to solve, we got f -1= some number. So 1= f of that number. So you can try which one of these val you plug in this you're going to get the one. Going to get the one. It's going to be the two. You put the two and plug it in here get 5 - 4 which is one. So it's a two answer. This is the answer.
All right. So we did that one already.
Then do this as well of x at which point intercept the um x-axis you put this= 0.
So on xaxis y = 0. So put this = 0. So 4x - x^2 = 0 - x = 0. So that this equal 0 and this = 0 which is x = 0 or 4 - x = 0 4 - x = 0. The x = 4. So 0 and four are the two x right here. 0 and four.
All right. that and that finishes that.
This one says which line perpendicus 52 right this oneus 52 perceptical let's look at June now let me take a June June 2020 June 2025 24 you see a different June 2025 and 2024 if it's in a different so come back here which a linear function this one see a lot of paper I don't know people stop and guessing on that one come almost every That made a sense guys for the maximum point again which is right up here. So 24 make sure it's 24 change it make sure then I give this function that's what I said for this one it's going to look different which of the following function is undefined at a point x= -2 remember it's undefined when the dime right undefined when the d= 0 you're going to get 2 - x = 0 you're going to get x = 2 that's Not the one that they want. You get x = 2. That's not the one that they want.
Huh?
So it is the last one. See that? The last one. Last one. You have x + 2 = 0.
So you have x= -2. So this last one is undefined when x= -2. So the x on the last one. All right. Undefined.
All right, that one look different. G of G of minus one. If I put G of one minus one first and then G of again, right? So it say G of the G of minus one putus one in this. So -1^ 2 + 1 - 1 square is a one. One is two. So doing G of two now.
Then G of two put two in this now. So it's going to become 2^ 2 + 1. So that's five.
This one I don't recall seen this one before.
All right. But it's not difficult. We do this one already. Parking lot for $1.
$1 is a fixed cost 25 $25 25 cents per hour. That's exact one that one already.
This one up here that's relationship is three more than so. So the y value is x + 3. That one as well. All right.
Lockdown 2024 June June 2024 June 2024 let's see June 2024 here we go function Same question.
It is this one right?
This one right here. The function see it.
Same question we did earlier.
Which say crosses the horizontal axis question already ask the vertical axis asking for the horizontal axis on this one now. So this one is now talking about this point 30 now. So another question to be 02 that's read the diagram and just think the same question and we set the same diagram and ask a whole different things. All right.
To do that equal -2. So that one is this one.
Rise to run three.
All right. See rise to run three to go from here to here.
That is it. So is two or three.
All right. So that is that here again ask for the maximum point which is again up here through four.
Keep setting this same question. We create a box of symmetry again the line x = 2 x = x turning point vertical line down here is a symmetry x = 2 and that's the maximum point and then here now they want gus -2 they want so I would say something like this g2 put -2= of some number and any of them number here now plug this equal -2 is the answer see that g of some number -2 so I can test them or you can solve it by putting this as a function so you have -2 = 7 n - 3 when you find you find the answer two so I get - 10 = 7 n - 3 I add it I get 7 = 7 n if I seven I get -1 so the n =1 so the answer is1 that is it see that good all right and this one is parking that question again really careful make sure the same question so the fixed charges $1 and then 25 cents per hour right so this is That is it. I think we're done. 52 range of function.
Oh, so range of function on this one again. The last one, right? This question early. Same exact question.
That is it. All right. So, let's take a short break.
Put 15 minutes. take a sharp and come back and go into the measurements the tricks the rest of the questions we really reach 50 something so we're looking at the rest of the questions to finish the paper we like we're like 50 52 right now so we finish the question on the paper right but we do the bank first you have to finish the bank come back on this paper the last set and we try so we can get in vectors before we finish cuz we did not know vectors we can get some vectors. All right.
I put up a clock on the January and I put up a clock.
Let me put up the clock.
What's up?
What's the progress?
You get back.
the set the glass jump.
The glass jump.
Sorry about that. Sorry about that.
Let's go again.
All right. Sorry about that, guys. So, let's do this again.
Yes, it's okay.
All right. So, I'm saying how the translation works that you have the point and the point is moved by these amounts. This is the X amount that it is moved. That's the Y amount that it is moved. So, you have the point 35.
I apply a translation to it of two -3.
This X value is going to increase by two to five. And this x value is going to decrease by three mean decreasing it by three. So -5 subtract three become 8.
And that is why this is the answer is here.
This is the same kind of question. So this x value is going to be adjusted by -2. So it's going to go to -7. This is the only answer with a -7 x value. So this is the answer right here. Again can see -53 under the translation of -23.
It's going to go to -5 and -2 makes -7 and 3 and 3 makes 6. This is actually a three right there.
So that's the answer right there. Next one. This point again under the same translation. And I'm going to move this by three. So you have -2 + 3 becomes 1.
The only value one at x value is this one. So again -2 3 under the translation of 3 4 -2 + the three going to one 3 and the four is going to 7.
So 1 7 is the answer for that. All these questions basically the same right with different values. Another one here again this one is asking for the translation.
So you have it says the image of this point and translation is this. So you can set up this way. So like you have the point one two car translation let's call translation A and it goes to the point54.
So what happens is that this one has been reduced to 1. You can say 1 and a 1 + a equal -5.
So a care make it negative. So the a= -6.
So this is the only answer= -6. So that have to be the answer right there. This one.
Something like that. All right. Or from one to go down to5 you have subtract six. Anyway you want to look at it. This question says if you diagram below what transformation maps PQR onto P prime R prime. Now that's a reflection. To confirm it's a reflection you can check orientation. So P to Q to R on this one is clockwise.
But P to Q again. Oh, I did it the wrong way. P to Q to R is the other way. Minus K. E to Q to R is this way anticlockwise. E to Q to R. This one is the opposite way clockwise. Orientation is reverse. So you know our flection in a so transformation is our flection but the orientation is reversed when you're a line orientation is reversed.
All right. So when P to Q to R that's this way and P to Q to R is this way. So one is clockwise other one is anticlockwise.
Same thing happening over here. P to Q to R and this one is this way. That's anticlockwise. P to Q to R and this one is this way. That's clockwise.
Orientation again is reverse. Same answer reflection. Two different exam papers. I set them like that.
All right. Down here this again says line CD is the image of AB. So under what CB is the image of AB looks like a reflection in this linection this are the same distance same distance and this one is same distance over here same distance a reflection in this line so reflection in the y ais here reflection in the y axis which obvious y= x. What is the image of y= x when it's rotated anticlockwise about the origin through 90° so that you have this line and you're going to rotate it going to rotate it 90° is going to go like this not working out But anyway, so you have this line and you want to rotate it 90° this way anticlockwise, it's going to come to here.
If you rotate this line anticlockwise 90° this about the origin such that this angle right here 90° that clearly is the line y= x y =x y =x that's what we're going to get right here if we rotate y asking a similar question. It says what is the image of y=x when it is rotated again anticlockwise about the origin through 90° you get back yex this rotate it down there again it's coming to there 90° so you're going to end up getting this Right?
If you rotate this 90° this way about origin 90°, you get the line y = x. That's it. Y = x. So you see same question exam paper.
All right. Oh, that's that. Here's a popular question asked all the time.
This one a lot on past papers. They said this line is mapped onto that line and asking us what transformation it is.
But it telling us for the translation vector that maps a AB on a prime bime. So to count to the right upwards.
So this point goes up to there, right?
So horizontally you go from two over to 7 that's a plus two and then you go up from 7 to 8 that's a + one.
So the translation is two to the right one up to one two to the right one up.
If it's from this line to this line and change it it's going to be two to the left and one down. So it be -21 if I change it the direction right I reverse the mapping. So it's 21 we're looking for right here one nice and easy this one it says in the diagram the translation of ABC prime B prime C prime is what? So you see this is what happened to one point for example C to C prime all the units how much it move horizontal and how much vertically so horizontally move to the left that's a negative movement horizontally 1 2 3 4 5 to the left as a five to the left is5 and then it go one 2 3 upward wards that's a positive three so the translation is -5 353 cannot double but I think this is the 53 this is the five three I think that one is it this is 53 So -5 -3 this is 53.
So this must be one of these two and I'm not sure but answer 53. Whichever answer 53 is the answer. Let's put that's 53.
So if that is the three this should be the answer5 3.
All right. Use that score the unit horizontally go to the left negative. If it if it went down vertical negative as well.
This one this one give us this is map onto that one and they tell us it's an enlargement and it asks us for the scale factor very even just to compare one side. So this side was one unit.
It has grown to two units.
That's a multiplier of two.
And so the scale factor is therefore two. A positive scale factor cuz the image of the object point in the same direction. For example, CA pointing up on the object. CA is still pointing up on the image. So that makes it a positive scale factor. So scale factor is positive2 right here.
All right. Positive scale factor image of the object point in the same direction. Another thing is that both of them are on the right side of the center. Both of them are on the same side of a center. That's another way you know the scale factor is negative.
Positive. When scale factor negative center between them scale factor negative scale the center between the image and the object of enlargement.
Here's another popular question right here which says the length of M because this appear of similar triangles similar triangle ratio of corresponding size equal. So this side to this side is a multiply of a half.
Then from this to this which are the corresponding side you have apply the same half you're going to half this if half you're going to get 3.5 3.5 so this side is going to become 3.5 the mo it's going to be 3.5 m 3.5 of 7 the ratio of corresponding sides are equal look like an enlargement similar to enlarge All right, next one. Same question here.
So we have similar terms again. Now this one is about area. So remember when it comes to area now the ratio of the area is the ratio of the square of corresponding sides. So you know the area of the big one that's the area of the big one. Let's call that area big over the area of the small one. error small is equal to a big side squared which is 6 squ over the corresponding small side square which is 3 squ ratio of the area is the ratio of the square of corresponding side what question it says if the area of the big triangle is 20 this can do another way let's do it two ways to give some dimensions to actually find some sides.
So this is one way to do it. And they're asking us for the area of the small one.
So we can take reciprocal of this and flip it. Let's flip it. So instead of doing it that way, we can flip this side and say the air of the small one over the air of the big one equal a small side squared over corresponding big side squared. So we can do it that way instead since the small one we want. We put what we're trying to find on top of the fraction to transpose and they tell us that the area the big one is actually 20 right. So we can put 20 on here and multiply it up there. So the area of the small one becomes 9 / 36 that becomes 9 3 2 is 9 6 2 is 36 * 20 right the 9 / 36 is 1 / 4 90 that goes one time 93 go four time 20ide by 4 is five so the small one is five.
So the answer here is five right here.
But another way we could do it since they tell us the area of the big one, we could work out what the height is and then use about the same height over here half it and calc.
So we find the big call this big H and work it out and call this small H. Small H is half of the big H because the same scale factor, right? Same um similar triangle, same factor, they reduce. So we could say we could say since the area is 20, we could say half 6 * big H is equal to 20. So therefore big H is equal to 20 / 3. Say half of 6 is 3. So big H is 20 / 3. So big H is 20 over 3. Then small h is half of the big h. Half of the big h which means that the small h is half of 20 / 3 which therefore is 10 / 3. So therefore the area of the small one is half of three * this half base times height. So you have the small one. You have the small one will be half the base. The base is three times the height which is h and h is 10 / 3.
10 / 3. So you see you get five same three count as three and then half of the 10. So it's five. So I could have done that as well. This is five. So two ways it could be done.
All right come to this diagram here says refer to diagram which shows the rectang xyz and it's image wp prime x prime y prime z prime and wp prime double prime after it under go a composite double transformation. So let's see what this one. So basically go from you go from this one to this one first and then from that one to this one.
See now pay attention not learn nothing.
Can't tell people act like serious act like serious but serious.
All right. All right. So from this to this look like a reflection to me. How I know because this distance from here to here the same same thing to Z to Z prime and then this one. Now the distance is halfway from this line that looks like a reflection in the Y ais first first transformation. Then from this to this is clearly an enlargement.
From this to this is an enlargement guys. But skill factor is a half. One is a half. But this side was two units and it's gone down to one unit. So skill factor is a half.
Hey class, you know one of you and then to find the center of the enlargement guys. Join the corresponding points like this to this and look like the origin here and you can draw maybe Y prime Y prime extended down here the meet at the center right so center of that enlargement is the origin and the scale factor of the enlargement is a half so that's a reflection in the y- axis and then an enlargement of scale factor half about origin.
Let's see what the questions are. It says sequence of transformation which maps this one to this one. We know the reflection y axis first. Let's look for that translation that wrong. A reflection on the y-axis followed by an enlargement center origin scale factor half. That is it. This is it.
That is the one that we just describ.
All right. This one here, it said the type of ah remember I told you guys about the positive and the negative rotation. Remember I keep telling you clockwise rotation is actually negative.
Anticlockwise rotation is positive. You can't remember just remember the x and y axis. Yes. So the positive quadrant.
So if you want to move your x-axis up into the positive quadrant in other words move the x-axis the positive quadrant you have to go anticlockwise. That's why anticlockwise rotation is positive rotation. Clockwise rotation is negative rotation.
See that? I see a question here that give you that and give you the negative and the positive. So you have to know say that one year is clockwise and a one year is anticlockwise.
That's what you have to know. 90 means clockwise 90° and positive 90. That's what 90 means.
90 means clockwise 90.
90° means clockwise 90°. This positive 90° at positive 90° means anticlockwise.
98 tell everything come on tell multiple times clockwise anticlockwise drill serious about retaining information you guys don't do things to retain information but say it's special because you They see it go through one person get it wrong.
So which transformation is this? It is from this one to this one. So you can look at it. This line that was horizontal O to Q has gone vertical.
So the rotation has been up this way.
See, and it's show 90° between here and here 90°. So that's clearly an anticlockwise 90° rotation about origin.
Right? If you look at this one, this line that was here has now gone to there.
Again rotating it this way about the circular origin.
So it's again anticlockwise.
Anticlockwise rotation is positive rotation. So the positive one. This one is the positive one.
Likewise rotation is negative. So the answer is the positive one. The positive rotation. See them two are clearly out of it. You know, right? So one of these two. How you going to know which one it is? You have to know what positive rotation is and what negative rotation is.
Right? You have to know.
So see the answer is this one. The anticlockwise one.
Anticlockwise rotation is positive rotation.
All right. Antic clockwise rotation is positive rotation.
I think that's basically almost it. The last one says in each of the diagrams Ara is image of A. Which of diagram shows a reflection in the X-axis?
A reflection X ais can't be this. This reflective Xaxis coming out to here. So it can't be that. Same thing here. This reflecting X coming out to here. Can't be that. This reflecting the X axis coming out to here. Can't be that.
This is must be it then. This reflecting the X axis coming out to here. So it must be this one.
I know can't teach this guy. This just straight obvious and straightforward.
Same question. Same question.
See say each of the following O is image of O A. Which of the diagrams are question the y ais they change it y axis up here they said x ais same question by the same question I say y ais now so if you go there might be the wrong reflecting the y axis so the yaxis this supposed to come over here so which it didn't this come over here so which it does but that's what supposed this looks like it that is it look at This supposed to come which didn't that's wrong across here again which is wrong. So clearly you have seed right here.
All right come to some trigger number.
All right. From tricks.
This is so the theory angle center is twice angle circumference don't it? So this is 100 right?
100 angle center is twice angle circumerence.
This one this one says what's the size of angle alpha. Now remember now these two sides are the same. So that's isosles. So if this angle is alpha then this angle says alpha don't it and the diameter 90° so right 90° so these two order must add to 90° don't it?
So 2 * alpha = 90° so alpha equals 45° don't it? So alpha is 45°.
All right and easy. X1 relation between two these lines Z angles go along here come down here go down there these two angles are alternate angles equal right alternate angles are equal right here that's it next one same thing alternate angle again say P and Q this time again equal right X and Y P and Q same thing equal angles are equal do do this.
Take this off from here and put the two over instead.
What's the relationship between them?
Huh?
Add up to 18.
they become co interior angles.
So this line that line parallel you are drawn by this like you have angles in the U angles. So in that case the P + Q will be 180°.
Similarly them two up to 182 in that case looking this way this way.
So you like this now. So these two angles this is x this is y x x + y as well. All right. So just giving you that I could make a simple change on this question and ask those things one here plus = 180 that tells me that you can put the Q and ask you that same question and then this will be the answer I just show you see right answer show That's strong to 60. So the rest of it take you up to 180 must 120 don't it? So 120 along with 180. Next one angle is X and one angle is 2X. What is the third angle? 180us of these two. So 180us 3x 180us 3x easy same thing here is x one angle is x what is angle c same question 180 minus 3x same question all right and q down there so them say a b and c it's same question really all right over here same thing I One angle is kx and that's what this one says that they're trying to change something. It says if angle P is formed by two equal if angle P formed by two equal sides of an isos triangle PQR let's look at it and P is X. So this is angle P.
anything you're not sure you can just make a sketch.
So this is P and angle P. Angle P is formed by two equal sides. These two equal sides form angle P. Angle P is equal to X up here. So that's what they basically say. What is the angle Q or R?
So this is P Q R. They asking us for these two angles. Now let's call them theta.
equal right angles.
So equally 180us x and then by two don't it don't it theta is going to be 180°us the x / 2 which therefore is this answer right here.
Typical is triangle question down here say in a polygon angle A is X and angle A and angle B are supplementary supplementary mean supplementary mean I don't know supplementary mean I used one million time I don't know supplementary which class I can't buy class supplementary mean that they add up to8 80° complimentary mean that they add up with 90°.
Come on man Thursday September.
Oh Jesus.
Right. So what is the size of anger be?
This one add up to 180.
See that supplementary add up to 180. So one is 180 the next one.
Oh boy.
So much time we use so much time opposite angles of a psychic supplementary opposite anglementary up to 180 time write to 180° understand it for any do my job and do my job very very well right don't take the learnings and the teachings and apply we know me exposing to everything over and over again so blank on the word supplementary so much time theory Anyway, angle of depression is X from Z to X, right? Angle of depression is 30° here. Um, this 30° here, 35° here. Remember the angle of depression from the top up here equal angle of elevation from the bottom down here.
You're basically forming Z angles.
So if out is 35° then in 35° as well. Normally write it inside the triangle. You want it inside triangle to help you 35° as well.
So you can work in this right angle triangle. Question say what is the height y over here sir. So that's the opposite side and we know the adjacent opposite and adjacent connected with tan ratio. So tan of 35° equal opposite side that you're trying to find the adjacent side of 12. Take the 12 and multiply it up here. You get the opposite side that you want. So the opposite side becomes 12 * tan of 35° which is right here.
That is it. Opposite side is the y that I want that triangle. Same question down here. So this time I want the x y is 12. So let's try to make changes on the question but the first thing is the same Z angles equal the pressure from the top equal angle elevation from the bottom. So inside it says 35° know that you're trying to find the adjacent adjacent is what you're trying to find out the side the adjacent side right XY the adjacent side you know the opposite this is now the opposite so you don't know the opposite the same time same thing kind of 35° equal opposite side of 12 over the adjacent side that you're trying to find. We find the adjacent side multiple I* 4.
So the adjacent side that you want comes 12 / the tan of 25° right? So don't go sweating. All right.
So you have made a change of huh yeah find other angle the 55 this answer but this answer not there so it means that they using this angle here let's put on the red so it's clearly using this angle which is 55° in that case now the tan of 55° This equals the opposite sides the adjacent over the opposite over sorry to that red angle. Look to that red angle that's now on the adjacent is now the opposite. So to the red angle this is now the opposite and this is now the adjacent.
Understand that to the red angle. So stand 55. We got the opposite. We want from the red angle. We want the opposite now from the red angle which is the x y over the adjacent which is 12.
So take the 12 m up here and you're going to get the opposite side of your becomes 12 * the tan of 55.
That's why that is another possible answer that they gave us right here.
But this one is missing. So don't get panic. Don't panic. From that up there.
If you don't find that one, don't panic.
All right, that is it. That work from 55. I can see the 55 in the answer. So that is it. Um.
All right, that is it. The same question again, but they put 10 now. And they wanted to find this side. So the same calculation. Okay, so it's 30. So inside here, so it's 30 as well.
That is the key asking you for the opposite side and you know the adjacent side. So same calculation of 30° equal the opposite side that you want the adjacent side of 10. Then I multiply up here that opposite side that you want comes equal to 10 * the tan of 30° 10 * of 30° is the side that you want.
That is it right there.
All these questions the same with minor little changes. This question a lot says which trig ratio is equal to four 8. So there are two possibilities can be working from the y working from the y this is always hypotenuse and this becomes the adjacent working from the y.
All right.
But if you're working from the X, this is now the this is now the opposite. If you're working from the X and this let me set up let me set up this way.
Let me set up this way. So this is hypotenuse regardless. Right? So this is the adjacent to X. This is adjacent to X but opposite to Y. Good. And this side is the adjacent to Y but opposite to X.
Depending on which angle you're using, it can be opposite or adjacent. Those two sides. Let's work it. So if we're working the S the X, then this becomes the opposite of Y. So that is sign, right?
So it means that the s of x is the opposite which is four over the hypotenuse which is 8. Working from the y and the cos of y is the adjacent to y adj.
So the sign of x or the cos of y is the answer. See that s of x or the cos of y is the answer for that.
Which one is there? Sign of X is there.
That is it.
All right.
We have this triangle down here.
And this is the hypotenuse of 40. That is 30°. What do I want us to find? Want us to find BC? BC is the opposite side, don't it? Hypotenuse here.
opposite hypotenuse is sign ratio. So I want to say the sign of 30°= opposite over 40. The 40 I'm going to write up here the opposite side= 40 * s of 30° 40 sin 30 this one.
All right. BC1 the opposite to the angle 30° up here. Now this give us this diagram.
This angle says 68° cross. We said the angle of 58 don't it vertically opposite angles. Right? And then now this angle and this angle up to 180.
Right? So right here so becomes 180°us 58 as a total straight line.
All right. And then now when you get this angle, this angle here is the same as that.
All right. Let's see what I want us to do. Measure angle a b e a b e that's this angle right here.
That's 180us don't it? That angle is equal to 180° - 58° which is 42 42° 142° 142°.
All right. Total straight line is 180.
Next angle. So which of the following angles are equal? Let's look are equal.
The vertical opposite angles are equal.
Right.
Vertical angles. Vertically opposite angles.
Huh.
Huh.
122. I missed the 122.
122.
122.
Yeah. 12. That's all right. Um, so what is the which are the four are equal? Let's look at the vertical opposite. So ABC and CB.
Let me take this out.
So let me say D B C D B C that's up here. So and C CB E right that's wrong. They're not equal. Right. So the first one wrong.
Next one. CBE CB E and A B E That's No, they're not they're not equal. Looking for the vertical opposite angles. Next one. A B D A B D which is the 58 and C B D C to B to D. That's this one.
No, they're not equal. So that's must be the last one there. A B D A B D which is inside here and C B E inside here. Those two angles are vertically opposite. They are equal and that is it.
Right. Nice and easy.
All right.
Now next this one says the plane flying a direction of 045° changes course in a clockwise direction to 135°. The angle through which it turns is what? So it's like you're going this way. See an outline here. So and it is going at 45°. going south at 0 45° right now when it reach right here. But still again it is now going on a bearing of 135° going down where now this angle is 135°.
So what angle did it turn through? It was going this way.
That's where it was going, you know. And it turned and is now going this way.
So direction has turned through 90°. You see that? What you say that? That is it.
It has turned through 90°.
That is it.
All right.
So this was the original direction and now this is the it is direction. So it has turned through 90°. We go from a 040 to 135. Maybe subtracted.
Anyway, that is that. Anything else?
Yeah, we have this last page down here.
This one says a ship sailed 8 km due east from A to B and then sail 6 km due north. All right, which are the diagram represent information. All right. So say 8 km to east to east means towards the east. So go 8 km suffers from A to B.
All right.
And then you go 6 km to north. That's towards the north. 6 km to C. So that's the diagram we're looking for. Right.
Eight across and six up. Eight AC. Eight to the right. Six up. Eight. Put it there. Eight to the right. So that's it.
That's the answer right based on what said this question like to set this question a lot as well.
Let's see the answer the safety. So a surveyor sets up an instrument 12 m of the foot of a building and record of elevation above the building. An estimate of the height of the building is obtained by calculating what? So what it is that it length is 12.
The center is also 12.
This height is 1.6.
So this height of the building part up to here is also 1.6 m.
And to get overall height of the building, we would need this additional height up here.
Let's call that H.
And so working in this triangle working in this triangle now in that triangle right triangle we could say tan of 40= opposite over adjacent.
So we could say tan 40° equal opposite of 8 over the adjacent of 12. Then the 12 multiply up here that h in the diagram becomes 12 of 40. But that's not the total building to either be the h + 1.6. So if I add them, so the h + 1.6 or 1.6 + h. This one, this one become 1.6 + h. So this is the answer right here.
Very good.
And that finishes that.
Um, time is it now? We have half an hour left.
Some vectors on this document, right?
Maybe tomorrow touch again 2020. I I'll let you do the three papers tomorrow. No, the five papers. You can print them out. I come with them tomorrow. I tell you so many pages and plus I don't know many people coming at the next lot of documents one day come and people not turn up and ask show of hands you come in people a lot of people put up in the group I do bas Number print your own papers said not printing any I'm not printing any I you know my pull up my waist already no printing Not printing not one. Not printing any print paper. Not printing any note for them.
Yeah. But I ask a specific question. Who come to the thing? A lot of hands went up. I come in, I come, I come in. I'm print.
Anyway, let's go. We finish all the vectors. This question says P and Q are those two vectors. I don't want minus P and minus 2. Be careful. They set this thing all the time and they keep changing this combination of the P and the Q.
All right. So if you want minus P, you want minus this. You want - 32 - 2 * this one -4. So you have those negatives. This one first becomes -3 -2.
Then the minus two times this becomes you can put minus and transpose in the one alone. So put minus transpose in the one the two sorry 2 * -1 is -2 2 * 4 is 8.
So you leave this minus out there. You use minus you change it to a plus.
They subtracting these now -3 and minus -2 becomes -3 + 2.
This is the only value with a negative one in it as the first value. And then -2 - 8 there - 2 - 8.
So that becomes that vector right there.
So with this one careful negatives, right? Pretty straightforward. Another one here. Now just want P minus Q.
All right. So this minus this - becomes a plus 3 -2 7 - 5. That's what you're doing. This becomes a five up here. So it's 52.
52 is right here.
So ask the same same vector but ask different P minus 2 Q. Yes. P min - Q is P - 2 Q. So you keep changing what the setting guy careful. So the P is 3 7 - 2 * this 2 * the cube. 2 * -2 is -4.
2 * 5 is 10. So you're subtracting -4 10 becomes a plus. So it becomes 3 + 4 is 7. 7 - 10 is -3.
So you're looking for 7 -3 that's not there. No seven. Oh see 73.
All right. So now same thing I get. See those two P minus 2 again. So the P is 3 2 - 2 * this. So subtracting - 28 2 * this. So again theus minus becomes a plus 3 + 2 2 - 8 2 - 8 that works out to be 5 -6 5 -6 right here. So these questions are pretty straightforward guys just have to careful with the signs got this one give us this par opposite sides are equal vectors. All right so the vector here and the vector is the same. The vector here and the vector here is the same. Let's see what they said.
They said based on the information above Z R is equal to what? Z R.
And I want to know which one of these is Z R. So look at this diagram to go from Z to R. I can go Z to Y and then Y to R.
If you notice that I given these answers in terms of the wx and the wz right. So we have to realize on this is that wz is same as xy.
All right.
Um or y to r is a half of wz. See that wz and xy the same y. is actually minus half of that because the whole of this up here would be wz same way and this is half of wz so this is half of wz that's what this is y r and then now this z y is the same as wx this is same as wx so This dy is the same as wx.
So that's what you're looking for in this question. Wx -/ wz wx -/ wz wx -/ wz. That's it.
All right. So the opposite vectors opposite sides are equal vectors here on here. Similar question.
So it gives us that diagram it says this vector is V down here. So so you automatically know that this whole vector is V as well but this is the midpoint of this side. So the whole thing is V. Then from here to here coming down is half of V and from here to here coming down is another half of V. And if this vector is U, then the vector across here didn't this way is also U.
The question I want to know the vector W R W. So WR clearly becomes U -/ V. See that U -/ V. So it's clearly U across here. And then why is it minus/ V? If I go in the opposite direction to the half V. Say u - half it becomes u - half of me.
First one.
Next question. Another similar question.
You again. So this vector is k means that right here. So is also k going that way. The vector coming down here. So is M. The vector coming down here is also M. Opposite sides are equal vectors guys in a parallelogram. What is Eg.
So obviously it is K + M or M + K. Don't it K + M?
Same thing over here.
OQ is A and QR in B. They want P L. L is the midpoint of right here. They want PL.
They want PL. PL is half of P R, don't it?
So you're going to write P L is half of P R.
What is P R? from P to R. P to R is - A + B. Can you see that? Why is it minus A? Going opposite direction to A, don't it? So it becomes half of - A + B - A + B same as B minus A, right? So it becomes half of B - A half of B minus A.
B - A / 2. B - A / 2. That is it.
Very good.
All right.
Next one down here.
This question gives us the diagram again information of PS. So it tells us that O R is 3 A minus 2. It says um R is the midpoint of O right here and O R is this vector. O R is this vector right here.
So they give us O R. So it mean that O is 2 * this right?
O O L is 2 * this because L is the midpoint and it also give us the vector O M out here in terms of A and B. All right. I don't remember seeing this question.
Question want us to find LM.
L M. So to go from L to M we can go we can just use Ous on the first. So LM is O MUS O L right watch this now O R is this.
So O L is 2 * O R don't it O is 2 * O R.
So, O becomes 2 * this which is 6 A - 4 B don't it?
O M. Now the OM is what they gave us right here. 2 A + B. So it says 2 A + 3 B not B 2 A + 3 B minus this. Have careful the bracket now. Put in a bracket. 6 A - 4 B. Put in a bracket.
Expand the bracket on this. Now you're going to get 2 a + 3 b minus the 6 a + 4 b now + 4 b consolidate now the a valid first this become - 4 a - 4 a and the b value become 7 b + 7 b so it's 7 b minus 4 it can be spun around can be written as 7 B - 4 E which is this answer right here.
All right. Nice question. That one that part similar question here let's say O R is A O T is B and R as the midpoint of this side and T is the midpoint of that side. So clearly then if O if O R is A and this are the midpoint means that okay another A don't it?
And if this is the midpoint to here and that is B then this is another right.
So this is 2B and that is 2 A.
The question wants us to find RM.
R M.
So clearly RM is OM minus O R O the last O* the first the O in this 2 B O J 2 B and the O R is just A right here so it's 2 B minus A first answer that is it similar question now same diagram but now they want LM want LM ln M is just OM minus O L.
The OM is the same 2B OS. So, and the O is 2 A. So, it's 2 B minus 2 A. 2 B - 2 A 2 B - 2 A. This factoriize it. Put it like this. Factoriize the 2 into B minus A.
So it's 2 into b minus e that we're done.
And we are done.
Huh.
All right, we have 20 more minutes.
Let's look at them. We have 20 more minutes.
Let's use the 20 minutes that we have.
So, let's look at the January 2026.
Huh?
Why did my try on serious discussion? Right.
Just did the spot ground, right?
Sure.
Change it to find EG Egrices.
>> Yes.
Huh?
>> It's a major seed. I think we went through this already. All that's going to happen, guys, is that this P equal three, right?
And Q + 2 = 1. That's all.
And then this S -1 = 2. So S= 3. Get the val matrix here that equal matrix the corresponding val equal.
So p = 3 q + 2 = 1. So therefore q is1 s -1 = 2. So s = 3. That's all for that one.
I don't see any more vectors than this.
But there are measurements question.
So regard some of these questions already.
This question down here says based on the diagram the bearing of the bearing of E from F. E from F. E from F is this bearing. You see this right? All right.
Z put the zero in front of it.
that we did already a while ago.
That one you see the questions repeat a lot guys a lot. What about this question?
What about this question? this question and I send it in the group. I did not send this paper in the group. So I'm going to send this solution for this paper in the group just in caseh.
So the question for this one says must read which are these statements based on the graph which following statement about the cyclist is true. So this is speed up this is the speed graph speed time graph of a cyclist on a journey all the information is there for this one is D it says from rest start from rest let's look at the first one and see why it's wrong cycles accelerator of 0.1 m/ second for 50 seconds that's the first part 50 seconds we check the acceleration if it's correct acceleration is good Then um the rest for one night. No, it never rest. So this is this is what makes it wrong cuz it did not rest. Even if the acceleration is right, it did not rest up here. So you're traveling at a constant speed. Rest means that you're not moving. Rest is when you don't when the speed is zero. These are the two places that that rest. Not moving. Speed is zero. So this makes it wrong up here.
This up here makes it wrong. Did not rest.
So this makes it wrong. Did not rest.
That makes it wrong. Second cycle start at a constant speed. No, start at zero.
So this makes it wrong. Never start at a constant speed. So this makes it wrong.
From rest the cycle is decent. No, the cycle didn't decelerate. It accelerates.
So this statement right here makes it wrong.
The statement makes it wrong. So that's the last one. From rest. Yeah. from rest. The cyclist accelerate at a constant rate. Yes, constant rate because a straight line we have on the graph. So it's one constant graded to a speed of 6 m/s after 50 seconds. See there? Yes. True. You reach a speed of 6 m/s after see there down there. So all right which is true.
It says after maintain the speed for 190 seconds. So that's 190 is 190 because it's 240 - 50. So that's 190 and it maintains that speed up holding that constant speed of six for that length of time.
What else it says? Then it says then decelerates to rest in 30 seconds. Yes, from here to here this is 217 from 240 to 27 30 seconds and here it will decelerate it speed is cutting. So this is the last one that is the correct. So watch this now give the star now and I ask questions like what's the acceleration going to be this gradient here.
which will be the rise of six over the run of 50.
That will be this acceleration gradient over gradient which is r.
The answer for the deceleration it will be the fall on this side.
Fall of six over the run of 30. That will be the deceleration. Fall of six over 30.
That gives you the deceleration.
The question should ask for the deceleration. If they ask for distance travel is the air under the graph. You don't want it want the distance when it accelerating right here.
You don't want to hear this when you travel a constant speed of this rectangle inside.
Remember distance equals area. You don't want the distance to decelerating is the area inside this triangle which is decelerated down here.
All right. Stuff like that. You don't want the average speed for the whole journey then to want the total distance total everything sides time the distance between the side which is six. So you're going to have these two side this is 190 and this is 270. So the distance between them is six. So the distance will be half the 190 plus the 270* the 6 between them that the distance travel if they wanted that and they want the average speed I put this distance now over the total time of 270.
All right, I think the solution I'm going to send the solution and I added some I put some additional stuff on it that I'm talking to you about now um on the question. I'm going to send it in the group so you can see it. All right, let me see if I can find the solution and show you.
So see this is it.
So I gave you all those things a while ago.
Oh, I'm tired. Mr. Sh tired coming to the end of my journey this class.
So much distinction I get.
See see bless me with a distinction.
See bless me with a distinction January 2026 solutions here bless straight in got this straight A guys. So I did some work.
Yeah, I did some work on it and sent to the group. So I worked out some additional stuff on it just in case.
All right, just to show you the average speed and the distance and all them down there. So, some additional stuff down here. You just should ask those additional stuff.
So, I'm going to send this in the group so you can see you know how you work out.
All right, that is it. So tomorrow last lap going to do five papers tomorrow. No I'm not printing any paper very serious paper bring so forget it. I'm not going to I'm not even going to be.
All right. All right. All right, guys. I want to focus on doing the most recent five exam papers at least 2026 January two papers from 2025 two papers from 2024 every single question on those papers then solutions are in the group right in the group I sent them in the group a lot of them here today over the last four days so tomorrow come last bring your question whether they want to understand from it is true.
All right. That's the plan for tomorrow.
Tomorrow. All right. Okay.
Everybody.
Oh no.
Right.
Okay. My boss.
Hello.
Hello.
Hey, That's not what's stopping.
one surface.
more down the road. More down the stretch.
What's up?
Hello.
Get I know.
Yeah.
Yeah.
One one Let's put them right there.
So tomorrow My dad.
Huh?
Are you are you ready?
Good luck.
Let's keep The one show was set.
med.
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