WATCH NOW 🔥 JAMB 2026 MATHEMATICS: 40 QUESTIONS THAT WILL REPEAT + FINAL REVISION

Added: 2026-05-06

[00:00:02]Welcome guys to my YouTube channel. In today's class, we are going to be taking the what the part two of this video lesson. That's why on the board we have question number 11. So I'm going to drop the part one of this video lesson where we solved question 1 to 10. Do you understand? Yes. Where we solve question 1 to 10. I'll put it in the first comment of this video. Now let's dive into what we have for today. The first question on the board says, if a quadilateral has the following angles 3 y + 10, 2 y + 30, y + 20, and 4 y.

[00:00:41]Don't forget that a quadilateral is a plane shape that has four side. An example of a collateral is a rectangle, a square. Do you understand? A rhombus, a parallelogram, a trapezix or a kite.

[00:00:56]They are referred to as quadilateral foursided figure. That's why they gave us four angles in this question. That mean in your jam exam they could give you five angles or six angles or they could give you angles more than six. Do you understand? So the question say find the value of what? Why? So what we do is they're going to give you a general formula that you can always use to solve for the sum of the what interior angles of polygon. Yes. So please take note of this. The formula is simple.

[00:01:30]Sum of angles sum is equals to n / 2 bracket 118. This is the general formula for polygon. Yes. And the general formula be that. So if the plane shape has five side, you simply put five here.

[00:01:46]It becomes 5 - 2 then multiply by 180 to know the sum of all the walls the angles. So as for this one now count the number of side it is 1 2 3 4. So it has four angles. So we are going to substitute four into this place. Do you understand? So since is a quadilateral for a quadilateral you have n = 4. So what do we simply do if n= to 4? I'm going to put four here.

[00:02:15]So you have 4 - 2 bracket 180. Please just do that without 4 minus 2 will give us 2. 2 * 180 will give us 360 what degree? That means the sum of all the angles put together will give us 360. Do you understand? So we are going to sum this this this this equals to 360. And I think we can cut it short from here.

[00:02:41]Yes, we can cut it short from here.

[00:02:43]Simply add the values of y together.

[00:02:45]Please check this question very well.

[00:02:47]This is 3 y. This is 2 y. This is y and this is 4 y. So instead of writing these steps down, we can just cut it short.

[00:02:55]Don't forget they appearing for jam exam. And for your mathematics questions, you have 40 questions in math for 30 minutes. Each of the papers is 30 30 minutes. So let's do that quickly. 3 y + 2 y will give us 5 y. 5 y + y will give us 6 y. 6 y + 4 y will give us 10 y. So what we have here is 10 y + let's have add the values together.

[00:03:22]10 + 30 will give us what? 40 + 20 will give us 60. So not that values again. 10 30 20 that's 60. So + 60 will give us a total of what? 360°.

[00:03:37]Like I said if the angles were more than four let's say 5, you would have done 5 - 2 to give you 3. 3 * 180 that will give you the sum of the angles. Then you add all the angles given to you equals to that sum. It is very easy. So what do we do now? Collect like terms together.

[00:03:55]So that in this place now I will take my 60 here. So that it becomes 10 y = 360 - 10. So 10 y will give us what? 360 minus oh sorry if you take my 60 over become 360 minus what? 60. So that will give us 300. Then we divide both side by what?

[00:04:19]By 10. This cancel this. Our value for y is equals to 30 what degree? Yeah, we have 30° as our value for y. y your y is equals to 30°.

[00:04:40]Equals to 30°. So that's the value for the angle. So the answer is not a b the answer is was c. So learn how to solve polygon by first of all look at the number of angles given to you. After that then you know this is your sum of your interior angle for a polygon. Sum of interior angle. Sum of interior angle.

[00:05:04]Yeah. Sum of interior angle is n minus 2 bracket 180. So you get it and you sum the angles equals to what? Now let's take the next question. The next question is gotten from the topic known as differentiation. Yes, this is differentation because they said if y is equals to this question is very common in your jam exam. Y is equals to x sin x find the value of dy over dx. Now first of all under trigonometric identity your trigonometric identity tells us that if y is equals to sin x then dy / dx will be equals to cos x. Yes we are told that if y is equals to sin x dy / dx will be equals to what? cos x. And if y is equals to cos x then dy / dx will be equals to - sin what sin x.

[00:06:15]So class we have if y is sin x then dy / dx will be cos x and then if y is cos x then dy / dx will be minus sin x.

[00:06:30]Finally if y is equals to tan x then dy / dx will be equals to sec² x. So these are derivative of the most three common trigonometric ratios. But when you look at this question class you will discover that the question says that y is equals to x sin what sin x. So you must understand that there are two different entities here. The first one here signifies for u while the second one here signifies for v. So in this particular condition where you have two different variables multiplying themselves, we simply apply what we call product rule. Yeah, we use product rule to differentiate these type of questions and we can simply use a shortcut like the steps I want to show you now. Simply do this dy / dx is equals to use this shortcut whenever you come across question like this and I'm going to leave a question for you. solve it and drop the answer for me in the comment section. But let's do this. First of all, if I differentiate x, I'm going to get one. Right? Because you have been told before now that if you differentiate 4x, you are going to get four. 8 x you are going to get 8. If you differentiate x, because x is 1 x, you are going to get what? 1. So if we differentiate x, we are going to get one. Then you multiply it with sin x. So you have 1 * sin x plus see what I just did I differentiated x I got 1 then I multiply with sin x so 1 sin x then let's differentiate sin x if you differentiate sin x they said the one dx for sin x will give us what cos x so if I differentiate sin x I'm going to get cos x and I multiply with this x is going to give me x cos x so my answer is 1 * sin x is what sin x plus x cos what x this become the answer to the question so you can actually get your answer in one line differentiate this one multiply with this plus differentiate this one multiply with this and that will be the answer so what's the answer to the question on the board it is x + x cos x so the answer is what b do you understand so what you do for me as your class work and then you drop your answer for me in the comment section is that if y is equals to s cos x. Please help me to do dy / dx. Do it. Drop your answer for me in the comment section to prove to me you actually understand this. Do you understand? So let's take the next question.

[00:09:11]Evaluate the integral value of pi / 2 then lower boundary zero of what? Sin x with respect to what? dx. Now what you must understand is that integration which is this symbol this symbol signifies integration. Do you understand? Integration is the opposite of differentiation. You must understand that. See in integration class in integration in integration look at this carefully.

[00:09:44]If you integrate sin x normally when you differentiate sin x you get cos x isn't it? But when you integrate sin x, you are going to get minus cos x. That's what you're going to get. And then when you integrate cos x, you are going to get sin x. Instead of having minus sin x as it is in this place, you have sin x here. So it's just the opposite. Then when you now integrate se² x, you are going to get tan x. So they will simply tell you that integration is the opposite of what? Differentiation. Do you understand? So if I am to integrate this sin x, what do you think will be my answer? It equals to - cos x. So let's quickly do that class. Let's quickly do that. If I want to integrate this is / 2 and here is 0 and I'm integrating sin x with respect to what? dx. So if I integrate sin x, do you know I'm going to get minus cos x? Yes, because sin x will always give us minus cos x. So sin x will give you minus cos x. But one thing you must know is that it has upper and lower limit. The upper limit is p<unk> / 2, lower limit is 0. So after integrating this to get minus cos x, we put our upper limit as what? Pi / 2. And our lower limit as what? Zero. Now one more thing to take note of is that whenever you see pi it is equals to 180 in degrees. Take note of that. So if my pi recall that pi is 180° so if pi is 180 if I put it here it means that this value will be 90° here will be 0°. Do you understand? So what we have now is minus cos x upper limit 90° lower limit 0°. Then what do we do from here? Simply substitute your values. The first the upper limit here is 90. You first of all substitute it for x. So you are going to get - cos what? 90°. You put it in bracket. You see now then minus the other value which is zero. You now substitute your zero. Now subute the upper limit first. Then subitute your lower limit. So you now have minus cos what 0°. Do you understand? So what's the value for cos 90? Cos 90 is equals to 0. So you are going to get 0 here that's - 0 minus bracket. What's the value for cos 0? It is equals to what?

[00:12:19]1. So you have -1 here. So what do you do? Minus minus will give you plus. So what we have is -0 + 1. Of course the answer will eventually be equals to what? + one. So the answer is not A. The answer is not B. The answer is what? C.

[00:12:36]So take note of that. All you need to do is to know the value for sin X and know the value for cos X. Know the value for se² X. Do you understand? Now one more trick I need to give you is this.

[00:12:49]Learn these simple tricks please.

[00:12:51]Whenever you are to integrate sin 3x for instance, let's want to integrate sin 3x. Take the reciprocal of this guy here. This guy three. That will give you 1 / 3. Then hopefully knowing fully well that sign s will always give us minus cos. So you can bring first of all - cos 3x. This is your answer then plus your k as usual which is your constant. So if you have sin 3x it will give you - 1 / 3 cos 3x. If you have cos 3x to give you 1 / 3 sin 3x. The reason why there's minus here because for every sign you have minus. Do you understand? So that means watch carefully. If I have a integration of cos 4x you simply take you simply take the reciprocal of four which is 1 / 4. Then cos as usual always give us sign. So it give you sin 4x plus your constant which is what? K. Do you understand? So this stuff is very easy.

[00:13:52]If you have sin 5x, if it is sin 5x, it give you - 1 5 cos 5x. If it is tan, if it is se² 3 x, do you understand? Like if it is se² 3x, it will give you 1 / 3 tan² 3x.

[00:14:16]Sorry, tan 3x plus what? K. Because for every sex square x you have tan x. So you first of all take the reciprocal the number which you want to be at the back then change your sex square to simply what tan and then you just put the value you get your answer. Do you understand?

[00:14:32]Let's take more questions.

[00:14:38]So class let's take question number 14.

[00:14:43]Question number 14. Question 14 says in a triangle P Q R Q is 8 cm R is C cm and cos P is 1 / 12. Okay, calculate the value for P in a triangle. Let's quickly draw the triangle to see what they are talking about. Now let's draw the triangle like this is the triangle class.

[00:15:06]This is a triangle.

[00:15:10]This is the triangle. Okay. Now if this our triangle, this is our P, this is our Q and this is what R. This is what said P Q R is a triangle. Okay, they now said that small letter Q is 8. The small letter signifies the side that face the big letter which is the angle. So if this one is Q, the small letter will be here.

[00:15:32]So you have 8 cm. Okay, that one by the side arrow is cm. So this is big letter arrow here will be the value for arrow.

[00:15:41]So C cm say we should find the value of what P. So there's a small figure here which is P. Then they gave us the angle here. Look at it. Cos of P is 1 / 12.

[00:15:53]Now whenever two side an angle between the side is given. The rule that should be applied is cosine rule. Yes is cosine rule that should be applied. Whenever you are given two angles and one side, two angles and sorry, whenever you are given two sides and one angle between them.

[00:16:22]Whenever you are given two side and one angle between them, you use what? Cosine rule. Take note of that. So that's number one. Number two condition to use cosiness when they give you three sides.

[00:16:37]Three sides are given. Yeah. If the three sides are given, it's cosine rule.

[00:16:44]If two side and the angle between the two side is given, it is also cosine rule. Do you understand? Yes. They gave us two side and the angle between the side is cosine rule. So we looking for the value of what? P which is a small letter the side here. So simply cosine tells us that P² is equ= to Q² + R² - 2 P Q co of the angle of P. This is the formula for P. If you're looking for P, it will be Q 2 + R² then - 2 PQ cos of the angle of P. So what is Q? Now Q is already 8. So you have 8 square plus R is what? 6. So you have 6 square - 2 brackets 8 * 6 * cos they gave us cos p already say cos p is 1 / 12 so * 1 / 12 this will give us 64 + 36 - let's do this quickly 2 * 8 is 16 * 6 will give us 16 * 6 will give us 96. So you have 96 / 12. So this plus this will eventually give us 100 - 96 / 12 will give us 8. So you have 8 and that will give us 92. So finally if I'm since p² is equals to 92 it means that p will give us the square root of 92 cm. That means the answer is what? B. So it's very easy whenever you are given a problem like this the square of this one is equal to the square of this plus square of this then minus 2 * this time is cost of the angle that fits the side you are looking for. So the next question on the board is from differentiation and this question is based on function of a function. You can watch that in my previous video. according to function of a function whenever you have multiple bracket like this you have to deploy function of a function instead of opening the bracket so I'll give you the shortcut on function of a function so if y is equals to 2x + 2 all to the^ 3 very simple we differentiate inside here differentiate inside differentiate inside if you differentiate 2x you are going to get two I told you before differentiate two you going to get zero so by time we differentiate everything this side we got what two then you multiply with this three that will give us what six I'll repeat myself again dy dx equals to let me repeat myself again differentiate inside differentiate inside differentiate differentiate inside what will that give you that will give you two because if you differentiate 2s you have two this one will give you zero so what do you have there is two you times it with three that will give you six then you Bring this bracket back again. 2x + 2. That's you bring the bracket back again. Then remove one from here. It becomes what? Two. That's the answer. You don't need to straight yourself at all. The answer is what? C.

[00:19:57]Very easy. Shortcut. The next one says is a question based on mean. Don't forget that the formula for mean is equals to the summation of x over what?

[00:20:09]N. Do you know that the formula for mean deviation?

[00:20:14]mean deviation.

[00:20:17]Formula for mean deviation is that mean deviation is the summation of x - xr / n. That's the formula for mean deviation. Then the formula for variance formula for variance is that variance is equals to x - xr² / n. That's variance. So that don't forget that this symbol signifies your mean.

[00:20:44]mean don't forget it signifies your mean.

[00:20:48]Then finally standard deviation is the square root of your variance and that settles everything. Do you understand?

[00:20:57]So in the question on the board they say we should find the mean. How do you get mean? Very easy. The sum of everything divided by the number of values. So the values here are what? What are the values here? You have t + 2 comma. You have 2 t - 4 comma. You have 3 t + 2 comma. And you have 2 t. So very simple.

[00:21:19]The mean is this. Let's just add all the t together. T + 2t will give us what? 3 t. 3t + 3t will give us 6 t + 2t will give us 8 t + then here you have 2. 2 - 4. What would that give us? 2 - 4 will give us what? - 2 - 2 + this 2 give us what? 0 over how many are they? 1 2 3 4 / 4. So the mean is going to give you 8 T / 4. And the mean is equals to what?

[00:21:53]2T. So my answer is equals to what? D.

[00:21:56]Let's take more questions.

[00:22:01]So class, let's take question number 17.

[00:22:05]Question number 17 says this is the age while this one is the number of people.

[00:22:10]The number of people also refer to as frequency. So they say we should calculate the median of the distribution. Now I'm going to solve for mean, median and mode because in this particular question now maybe I'm told to solve for the median but in your jam exam you may be told to solve for the mean or the mode. Do you understand? So it be important for us to do what? sub the mean median mode so that you can see how to sub mean median mode on a frequency table like this. Do you understand? So let's do that quickly.

[00:22:44]Now let's go.

[00:22:46]The question is what is the formula for mean for a distribution like this? The general formula for me is the summation of x / n. The one we did just now. But when we have a table like this, we do summation of fx / n. So number one is the mean. So what is the formula for the mean? It is the summation of fx over the summation of f. So understand this that f is the frequency. This one is referred to as the f. Why this one is referred to as the what? x. Do you understand? So we can solve it. So let's do it. FS me f * x. So let's do that quickly.

[00:23:28]3 * this will give us what? 60 + 5 * 25 will give us 125 + 1 * 30 will give us 30 + 1 * will give us 35 plus um 2 * 40 will give us what? 80 + 3 * 45 45 * 3 this will give you 15 5 carry 1 3 * 4 is 12 plus that one 13 so that will give you 135 all over the total frequency what's that frequency 3 + 5 is what 8 + 1 9 + 1 10 + 2 12 + 3 15 all over 15 so the mean the mean will be the summation of let's do that quickly We have 60 + 125 + 30 + 35 + 80 + 135 that will give us by 15.

[00:24:36]So the mean is equals to 31.

[00:24:40]That is the mean. Although we were not told to calculate for the mean but let us use as a revision guide. Then number two, let's solve for the median.

[00:24:50]What is the formula for the median? The median is the summation of f / 2 or the summation of f + 1 / 2. This one is when the summation of f is an even number.

[00:25:09]Yes, if the summation of f is an even number, we simply divide it by two. But if the summation of f is an odd number, we have to add what? One to it before we can divide by what? Two. So if I do all the frequencies together, I'm going to get 15. So since the summation of f, watch me carefully, is equals to 15. I'm going to use formula number two. So my median my median which is m e will be equals to 15 + 1 / 2 and that will give us 10 /2 to give us 8 position. It is not actually the media is not actually 8. It is the eth position. So we have solve for the media position. Now let's trace it. How do you trace it? You trace it by combining or accumulating the frequency together. So if I decide to create a small table here which I will call commulative frequency like adding all the frequencies together. The first frequency here is three. If I add the next one join to this three becomes what? Eight. If I add one becomes 9. If I add one becomes 10.

[00:26:18]If I add this two becomes 12. Add this three becomes what? 15. So cumulative frequency is when you adding this frequency as you go you add you start with the first one you add the second one to becomes eight add that's cumulative frequency that's where you trace your median your median so now that we have gotten eight if we trace 8 is found here then if eight is here we trace it to where the age is. So the median age median age is equals to what? 25 years or thereabout. Okay, do you understand? The median age is 25. So the next one I want to solve for is the mode.

[00:26:58]Number three is the mode. Although we not told to solve for the mode. The mode is the value with the highest frequency with the highest frequency.

[00:27:15]So which value have the highest frequency? Not this this one. The frequency here. Which value? Look at it.

[00:27:21]Five have the highest frequency. Trace it to this place. So find out that that the mode is also what? 25. The median is 25. The mean is 31. So you can find mean median mode for such a table like this.

[00:27:36]Don't forget that we are looking at possible likely jam questions and a strategic revision guide. So now that you have seen, make sure you master how you can trace your median, how you can solve for mean and how you can solve for your mode. Now let's look at this one.

[00:27:49]The next one, the question says that it say if the variance of 3 + x 6 4x 7 - x is what? Four, find the standard deviation. The question is very tricky but I think you shouldn't stress yourself because standard deviation is equals to the square root of your variance. So what is your variance?

[00:28:13]Four. It means that standard deviation will be the square root of four and answer is equals to what? Two. That's option D. Do you understand? So the next question question 19. It says that this is the score and this is the what?

[00:28:28]Frequency. They say we should calculate for the range. What is the formula for range class? Range is the highest minus the smallest.

[00:28:40]The highest minus the smallest. Highest minus smallest. So what's the highest values? Highest scores, not frequencies scores. The highest scores here is nine.

[00:28:52]And the smallest score, oh sorry, the highest score here is 10. And the smallest score here is three. So 10 minus 3 will give us what? Seven. That how to calculate for range. Very easy.

[00:29:04]No stress. Don't waste your time.

[00:29:06]Question number 20. In how many ways can a student select two subject from five subject? Understand this topic. This guy now under combination. Yes. When you hear the word combination, combination in mathematics it simply means to choose or to select. So whenever you hear select or choose know that that aspect of mathematics is under combination and the formula now n combination arrow is equals to n factorial over n minus arrow factorial arrow factorial. Now the formula be that the n there is the total number of subjects while the arrow there is the number of subject that is selected what you are choosing. Do you understand? So if my n is equals to 5 and r is equals to 2, I will simply put this value into the formula so that it becomes 5 combination 2 equals to 5 factorial over here becomes 5 - 2 factorial 2 factorial that will be 5 factorial over 3 factorial because 5 - 2 is 3 then 2 factorial this is the answer this is the final answer I beg whenever you combin ation it means to select or to choose. So that's why the answer is um a is the answer. You just only rearrange it. Put two factorial before 3 factorial is the same thing. That's why 2 * 3 is 6 and 3 * 2 is 6. So answer is what?

[00:30:39]A. The next one saying how many ways can three seats be occupied if five people are to seat.

[00:30:47]This one has to do with arrangement and this aspect of mathematics is known as permutation.

[00:30:54]Now permutation and combination are on a topic known as factorial mathematics. So in permutation please don't forget this in permutation it means arrangement.

[00:31:08]It means arrangement. And what is the formula? The formula is n permutation arrow equals to n factorius what arrow factorial very easy it means to arrange just to arrange nothing more.

[00:31:24]So the n is the total number of uh people while the three is the seat available which is arrow. So if I deploy it the bigger number is five the smaller number is what three that's how we do it. So that become 5 factorial over 5 - 3 factorial. That be 5 factorial over 2 factorial. Do you understand? So if I have 5 factorial 2 factorial, what do you think I should do? This is what we will simply do. Let's create a space here class. If I have 5 factorial over 2 factorial, I will expand my five factorial until it gets to this denominator. that is is going to be 5 * 4 * 3 * 2 factorial over 2 factorial. You stop at the denominator. 5 * 4. You keep going until you get to the denominator.

[00:32:19]So I will use 2 factorial 2 factorial.

[00:32:21]I'll be left with 5 * 4 * 3. 5 * 4 is 20. 20 * 3 is what? 60. So you have 60 ways here. Do you understand? So in this question number 21 the answer is what?

[00:32:36]60 very very easy just understand that combination means to select permutation means to do what to arrange. Do you understand? So let's take more questions.

[00:32:50]So class we take the next question. What is the probability that an integer 1 to 25 choosing at random is divisible by 2 and three? Yes. What is the probability that a number choosing at random from 1 to 25? Can we just list this number from 1 to 25? Let us go. You have 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 That's the number. What's the probability that is divisible by 2 and 3? Let's look for the ones that can divided by both two and three.

[00:33:46]Um this one one can divided by two and three. Um this one two can be divided by two and three. Then this one three then two and three can divide this uh four. Yes, there are four in numbers.

[00:34:07]Look at it. 1 2 3 4 can be divided by both 2 and 3. So what's the formula for probability? It is equals to probability is equals to required over total outcome. Required over total. So what they require from us is the one that can be divided by what? By three and two. So 1 2 3 four there are four numbers. The total number of um the total number here of values is 25. So you have 4 over 25. That's the answer. Very cheap. The answer is what?

[00:34:46]A. And I want to believe that your math exam will be very cheap like this. And I know it's very cheap. Now a basket contain nine apples, eight bananas, and seven oranges. A fruit is picked at random from the basket. Find the it is neither apple nor orange. Okay, let us do this quickly. Let's do this quickly, please. Number of apples. The number of apples is equals to 9. The number of bananas, the number of bananas is equals to 8.

[00:35:29]Then the number of orange is equals to how many orange do you have there? Seven orange. You have seven. So what becomes the total? The total is 9 + 8 + 7. And that will give us 9 + 8 is already 17. 17 + 4 + 7 give us 24. The total is equals to 24. It says you find the probability that it is neither apple nor orange. That means it is not apple, it is not orange. Therefore, it will be banana. Yes, it's as simple as that. If it's not apple or orange, it be banana.

[00:36:08]So, it becomes the probability of banana.

[00:36:13]Banana which is required number of banana over total. So, what's the number of banana there? Banana is what? Eight. The total is what? 24. Eight go here one eight go here three theory so it is one over what theory very easy I never knew math was this easy very very easy now let's take the next question number 24 evaluate this guy is a matrix question yes I decided to pick three matrix question so that you see it and I will teach you how to get inverse of a matrix by the time I'm done with this before we take further questions so look at this matrix Now 3 - 2 - 7 5 plus you are using two. Now if you watch it very well this guy is in bracket. So they are using two to multiply this one that is here. Do you understand? So let's do that quickly. First of all we are going to write our values. 3 - 7 what do you have here? - 2 then 5 + why not just let's use this two to times everything. So if I use 2 * -2, I'll get -4. 2 * this will give me what? 8. 2 * 3 will give me 6. 2 * - 1 give us -2. So let's add it up together to get our final answer. So how do we do it? 3 + -4 is -1. - 2 + 8. You just add them according to the way they arrange. 3 + - 4 is -1. - 2 + 8 is what? 6. - 7 + 6. - 7 + 6 is -1 5 + - 2 + - is - So that will give you what 3. This become the answer -1 6 -1 3 the answer is what? D very cheap. Take a look at the next question. Question number 25. Whenever you see this two straight line in metric, it signifies determinant of the matrix. Yes, that's what it signifies.

[00:38:18]And how do you get determinant of a matrix? You go multiply the product of these two minus sign with the product of these two. That's how you get determinant. It is a product of the principal diagonal minus the product of the ordinary diagonal. This is known as the principal diagonal. Diagonal.

[00:38:39]This is known as the ordinary diagonal. Do you understand? So let's get determinant quickly.

[00:38:49]Determinant. Let's do it quickly. I'll multiply this by this. What would that give us? Because they said they said if this is equals to -12, find the value of x. This is the determinant. So let's do it. 4 * - x give you -4x. Then -1 * 2 give you -12. So the determinant is equals to -2. Do you understand? Yes. So therefore you have - 4x - is + 12 = -2.

[00:39:19]So -4x is = watch carefully. Yes already -2. You bring this guy you have another -2. So what do you do? -4x now = -4.

[00:39:30]Then we divide both side by -4 - 4. This cancel this x = to what? 6. That's the value for what? x. Yeah. Take a look at this 26 question. You see these two straight line it always signifies what?

[00:39:45]Determinant like I said determinant. So how do you get determinant? Let's say you evaluate it means to get determinant. So how do you get it? Very simple. This matrix now is 3x3 matrix.

[00:39:58]So 3x3 matrix you first reduce them to 2x2. But before you reduce them you must assign your co-actor sign which are plus here minus plus. So this what we're going to do. The first one is plus zero.

[00:40:12]You pick it. That's zero. Then by the time you pick it, you cancel the column and the row that that element belongs.

[00:40:20]You cancel here. You cancel here. What will be left here? 7 8 5 4. Do you understand? Yes. So you pick the next number. I told you put plus minus plus.

[00:40:31]Now the rule that then pick the next number - 3. So when you write minus 3, you cancel here. You cancel here. What do you have?

[00:40:40]18 04. Then you go to the next one. You pick it again. + 2. You cancel the column and the row. It become 1705. 1 7 05. By the time you able to reduce it from 3x3 to 2x2. Then you are very close to your answer. So what do we do now?

[00:40:59]The normal way of determining of getting the matrix is to do what? So multiply the principal diagonal which is 28 minus ordinary diagonal which will be 40 here then minus 3 4 * 1 is what 4 this time this is what 0 + 2 bracket 5 * 1 is 5 then 0 * 7 is what 0. So what do you have here? 0 bracket 28 - 40 is -2 - 3 bracket 4 + 2 bracket 5. This will give you zero. But that will give us -2 + 10.

[00:41:33]The answer is equals to what? -2. -2 becomes the determinant of the what matrix. I said I was going to show you how you can get what?

[00:41:46]I said I'm going to show you how you can get the inverse. Let me just do that quickly. If I tell you that um please watch me carefully. They will give you inverse of 2x2 matrix. If I tell you that if a is equ= to 3 8 - 4 - 9 I said find a^ -1. This simply means to find the inverse of the matrix. Yes. A to the power minus one means the inverse. And here's the simple step, very easy step to find inverse. Take a look at it. Step number one, follow me carefully. Get the determinant.

[00:42:26]Get the what? Determinant.

[00:42:29]So, how do you get the determinant of a matrix? I've told you now. You simply multiply this time this minus 27. Then multiply this time this - 32. So, that will give you - 27 + 32. And that will give you 5. I've gotten my determinant.

[00:42:47]Do you understand? Then step number two, get the new matrix. Step two, get the new matrix. I will show you how to get the new matrix. How do you get the new matrix? Interchange this one.

[00:43:01]Just interchange the principal diagonal so that it becomes instead of having three and - 9, it become - 9 and 3. This one you don't interchange, you only change sign. If it's positive, change it to what? Negative. If it's negative, change to what? Positive. This become the new what? Matrix. Then the last step is to get the inverse. Step three. Step three. Get the inverse. How do you get the inverse? Very easy. It's equals to 1 / determinant time the new matrix. New matrix. So that will give you your answer. So what would that give us now? Let's do it here. uh 1 / 3 I mean 1 5 the new matrix is what - 9 - 8 4 3 this becomes the inverse of the matrix so you can multiply everything by 1 5 this give us - 9 5 - 8 5 - 4 5 - 3 5 you get it so let's take more questions that cut across other topics in mathematics So class let's take question number 27.

[00:44:16]According to the question they say we should simplify and you must understand that this guy is a conjugate sword. Yes a conjugate sword and sword that can only be rationalized or simplified using the conjugate of their denominator. For instance look at the denominator of this question. is <unk>2 + roo<unk> what? 3.

[00:44:36]The conjugate of this s is <unk>2 minus roo<unk>3. Do you understand? If it was <unk>2us roo<unk>3, the conjugate would have been <unk>2 + roo<unk>3. So the conjugate is just to change the sign that is in the denominator and you use it to multiply both the numerator and denominator. So let's do that quickly.

[00:44:57]Solution the s here is 2<unk>2 -<unk> 5 all over <unk>2 + <unk>3 so multiply by the conjugate which is <unk>2 - <unk>3 over <unk>2 - roo<unk>3 don't for what I said conjugate of the denominator so let's do that I'm going to use this to times this you use the numerator time numerator denominator* denominator do you understand. So let's multiply. Now we're using this one to multiply this one and this one to multiply this one. So let's do that quickly. First of all <unk>2 2<unk>2 * <unk>2 if you multiply you are going to get two brackets 2 because roo<unk>2<unk>2 is 1. 2<unk>2 * roo<unk>3 give you -2<unk> what? 6. Do you understand? So I've used this to multiply this and this. Now let's use to multiply this and this<unk>2<unk> 5 * <unk>2 will give us what -<unk> 10 roo<unk> 5 * -<unk>3 will give us what + roo<unk> what 15 all over let's multiply this whenever you are doing conjugate sort and you get to the denominator use this shortcut to do the denominator don't waste your time use this first one to times this one use this one to times this one it will have the same result as opening all the bracket. Do you understand? So <unk>2 * roo<unk>2 will give us what? 2 <unk>3 roo<unk>3 will give us what? 3. Do you understand? So that is very cheap. So 2 * 2 is what? 4 - 2<unk> 6 -<unk> 10 +<unk> what? 15 all over 2 - 3 is what? -1. So what do we simply do? We divide everything by minus one. So if I do that, I'm going to get divide this by -1, you have -4.

[00:46:50]- 1 you have - is plus so + 2<unk> 6 + what<unk> 10 - what<unk> 15 this become our final answer -4 + 2<unk> 6 + roo<unk> 10 + -5 you cannot combine them together because why the result are not the same so they are not comfortable for addition or subtraction so leave our answer like that do you understand so the next question is under inequality eval the inequality. Very simple.

[00:47:22]Whenever you have inequality involving fraction like this and you want to evaluate it, simply do this. Take the LCM of the denominator and use it to multiply everything.

[00:47:32]So do take the LCM the LCM of the denominator.

[00:47:41]Denominator. What is the LCM of the denominator? 2 4 6. The LCM is equals to what? um 12. The LCM is 12. You can confirm that. You write 2 4 6 12. LCM 2 go. 1 2 3 6 2 go. 1 1 1 3 3 go. So the LCM is 12. So I'm using the LCM to multiply everything. So multiply through by the LCM. multiply through by the LCM. So if I multiply everything by the LCM, which is 12. The LCM is 12.

[00:48:28]So let's do that. I'm multiplying this guy with 12. So it's going to give me 12 * x / what? 2 + 12 bracket 3 over what?

[00:48:39]4 less than equals to 12 bracket 5 x over what? 6. Then you have 12 bracket 7 / 12. Do you understand? So if you use to cancel 12, you have 6. Then this cancel this, you have 3. 3 * 3 is what?

[00:48:57]9. Less than equal to this. Cancel. This will give you what? 2. 2 * 5 will give you 10 x. This will give you 1 * give you - 7. So we collect like terms together. Do you see now? So let's collect like terms. You have 6 x. If this one comes here becomes - 10 x less than = -7. This one become - 9. 6 - 10 x - 4x less than equals to -6. So what do we do? We divide both side by the coefficient of of x which is -4. So if I divide both side by -4, divide both side by what? -4. If you divide by minus4, don't forget what will happen. the inequality sign that is less than or equals to will change to greater than.

[00:49:43]So whenever you divide with a negative number the inequality sign must change.

[00:49:47]So because it changes you going to get -4x -4 from less than equal to become greater than = -6 - 4. This cancel this x= to what? Four.

[00:50:01]This become the answer to the what question. Very easy. Very very easy.

[00:50:07]Let's take the next question. This question says we should find the value of x. The angles given to us are special angle and that's why it's a jam question. Do you understand? So how do we get the value for x? You use your sign rule not your cosine rule. Cosine rule is when you have three side given or you have two side and the angle between the two side given. I gave you the condition for cosine rule. Otherwise it's automatically waiting sine rule.

[00:50:31]And you know say for s rule the formula that you simply use take a look at the board. Look at the board carefully. Look at the board. You have a b. Then you have c. Don't forget that here is a, here is b, and here is what? C. So the sign rule says the side which is c.

[00:50:48]Okay, let's start with a. The side which is a over the angle which is a equals to the side which is b over the angle which is b equal to the side which is c over the angle which is c. So it is the side to the angle the side to the angle. So it cheap it easy to understand. So what do we do now? The side here is x and the angle is 60. So x x / sin 60 equals to then the side here is 10 there is 30. So 10 / sin 30. Do you understand? So it's very easy without skepticism or we just cross multiply like this. So x will give us 10 / sin what? 30 * sin what? 60. Do you understand? So x will give us 10 / sin 30 is 0.5 but sin 60 is <unk>3 / 2. So x will give us 10 / 0.5 is 20 * <unk>3 / 2. This cancel is x will give us 10 <unk>3 what cm. Very easy. The answer is what is b. So we'll take the next question. Question number 30. A cylindrical tank has a capacity of this is the volume of the cylinder. What is the depth of the tank? If the radius of the tank is 28 cm, they are basically looking for the height the height of the world cylinder. So what is the formula for the volume of a cylinder? Volume of a cylinder. The volume of a cylinder is V = R² H. That's the volume of a cylinder class. So what do we do now?

[00:52:32]What's the we make h formula? So h will give us volume over p<unk> r². So let's do that quickly.

[00:52:42]Let's do that quickly. So height is equals to volume. What's the volume?

[00:52:47]6160 over<unk> is what? 22 / 7. Then * r². R is what?

[00:52:58]28². So height to give us 61 C 0 / 22 / 7 * 28 * 28. So what do we simply do? We simply say this cancel this. You know 28 / 7 is going to give us what? 4. So that you are going to get 61 6 0 all over 22 * 4 * 28. So what would that give you now?

[00:53:31]What would I give you now? Let's do that quickly.

[00:53:37]Um 22 * 4 * 28 2 4 64. So you have 61 6 0 / 2 4 64 61 6 0 / our answer that's 2.5.

[00:53:58]So take note the height of the cylinder is 2.5 height equals to 2.5 m. That's the height of the cylinder. The next question which is question number 31 says solve for x in the equation. Let's do that quickly. This is simply um indices leading to a linear equation and it's quite very cheap. If you have 8 x ^ -2 = 2 / 25 what do you do? First of all the x the^ -2 if it's coming down the minus sign will disappear based on the laws of indices. Don't forget that laws of indices say a^ minus m is 1 / a to the^ of m. So if I have 3 a^ - m it is 3 a^ m. So if this a^us m is coming down the minus disappears. So that's what happen to this y becomes 8 / x² = 2 / 25. So what would that give us now? We cross multiply cross multiply. You do 2 * x² give us 2 x² = to 8 * this 8 * 25. So we divide both side by two. This cancel this x² will give us 2 will give you 4. 4 * 25 will give you 100. So s become the square root of 100 and that gives you what? 10. Do you understand? We have nine more questions to go that cut across other topics in mathematics. So let's take more questions.

[00:55:35]So class question number 32 says a cord of circle substain an angle of 120° at the center of a circle of diameter 4 <unk>3 cm.

[00:55:50]Calculate the area of the major sector.

[00:55:53]Now my question is simple. What is the formula for a sector that they will now be requesting for the area of a major sector? First of all, the formula for a sector is theta / 3 *<unk> r². Take note. Area of a sector. Area of a sector.

[00:56:14]I will give you this before the end of this class is theta / 360* r². This is the formula for the area of a sector. But whenever they say major sector, they refer to the bigger angle because why? and the angle at the center sum up to give you 360°. So if you already have an angle of 120 and they requesting for the major sector, it means 360 minus 120 and that will give us what? 240. So we are not going to use 120, we're going to use the bigger angle which is 240 because the request for the major sector. So we can simply say for the major sector for the major sector we have 240° over 360° *<unk> * radius they gave us diameter don't forget that diameter radius is diameter / 2 so our radius here will be 4<unk>3 / 2 and that will give us 2<unk>3 cm this is our radius. You should know that. That will give you the radius. So what do we do now? We put 2 <unk>3² which is our radius. 0 cancel 0. That one is off. 12 going to 24 will give you 2. 12 going to 36 will give you 3 *<unk> * can you please square this for me?

[00:57:39]What is the square of 2? The square of two will give you 4 * what is the square of the square root of 3? That will give you three. You must know that that if I have three square this cancel this so you be left with three. So that will give you three here. Do you understand?

[00:57:57]So I can use three to cancel this three once. Now my final answer 2 *<unk> is 2 pi * 4 8 pi. This is the answer. 8 pi is my answer. I'll give you the formula for area of shapes related to circle by the end of this video. So watch this video to the end.

[00:58:19]Do you understand? The reason why I have to do that because most of jam questions on measuration is centered on circle. So I'll give you the formula and I'll explain for you how to apply this formula by the end of this video. So stay to the end of this video. 323 say find the length of a chord which subain an angle of 90° at the center of a circle whose radius is what? 8 cm. How many of you know the formula for the length of a cord? In case you don't know the formula for the length of a chord length of a chord is equals to 2 r sin theta / 2. Yes. So um the angle here is 90° and the radius is 8 cm.

[00:59:11]So the length of a cord will be 2 * 8 * sin what? 90 / 2. 2 * 8 will give you 16. Sin 90 will give you what? S 90 / 2 give you 45. So what's the value for sin 45? You have 16 * <unk>2 /2. Don't forget that sin 45 is <unk>2 / 2. So this cancel this you have 8 * <unk>2. So that will give you 8 <unk>2. That is the answer. 8 <unk>2 is option what? Option D. So question number 34 is from coordinate geometry. Most times you see question on coordinate geometry from something like this. Something like this and something like this. So I picked three question from coordinate geometry and I want to show you how you can solve this question. Now let us go find the equation of a straight line through this point - 2a 3 and perpendicular to this equation. It's a very simple thing but there's a condition for perpendicular lines. And whenever you hear perpendicular lines in coordinate this is how to go about it. Let me show you.

[01:00:22]First of all you get the gradient from the equation. I'm going to get the gradient here from my equation. And how do I get the gradient of this equation?

[01:00:30]I will simply make y subject formula. So the equation says 4x + 3 y - 5 = 0. Please let us quickly make y subject formula. I will leave my 3 y here and take 4x become - 4x then + 5.

[01:00:53]How do I make y formula divide everything by 1? 3. So I'm going to divide here by 3. Divide by 3ide by 3.

[01:01:01]This cancel this y - 4 over 3x + 5 / 3.

[01:01:08]So how do I I've made y subject formula.

[01:01:12]So what I need to do now is to get my gradient. Now how do I get my gradient from an equation? You must understand that the equation of a straight line is given as y = mx + what? + c. So the n is the coefficient of x. So in this question now in this equation >> I can say that my gradient is equals to -4 over 3. But this is my gradient for m_sub_1. Do you understand? So if this gradient is perpendicular to a line, what will be the gradient of that line?

[01:01:46]I will show you the condition for perpendicularity. Now see for perpendicular lines perendicular lines for perpendicular lines understand this that m_sub_1 sorry m_sub_2 is equals to - m1 / m1 that's perpendicular lines so I've got my m1 to be -4 3 so my m2 now be -1 / -4 / three. Don't forget that minus cancel minus so minus don't go.

[01:02:22]Three will go up because normal if you get a / b / c like a bc this one will go up. That's what we have. So if three goes up it becomes m_sub_2 = 3 / 4.

[01:02:36]Don't forget that the minus has been cancelled out. So we are now going to use this guy to form an equation. Find the equation of a straight line through the point that is perpendicular to this guy. So we have gotten the gradient that's perpendicular this. Now let's use to form an equation. The equation formula is y - y y1 y is m x - x1. This is the equation formula.

[01:03:01]So y this place becomes your second gradient. So what is our y now? We don't know what y is yus. What is y1? Don't forget that this guy here is x1. Here is y1. Do you know that? So if this my y1 becomes 3 = to my gradient is 3 / 4. So 3 / 4 open bracket x - x1. What's my x1 - 2? So what do I do? y - 3 = 3 / 4 open bracket - give plus. So you have x + 2. So we have to write the equation now from that place. We have to write the equation. So what do I do? I think I need to use four. Cross multiply. If I cross multiply, I'm using four to multiply this. So I have four open bracket y - 3 equals to what? 3 bracket x + 2. Let's see it.

[01:04:02]So open your bracket. So you have 4 y - 12 = 3x + 6. So arrange it properly.

[01:04:11]You have 4 y - 3x okay - 12 - 6 = 0. So my final answer becomes 4 y - 3x - 18 = 0. This is now the equation. So if they give you this question for jam just follow this step.

[01:04:32]You understand? It's just one perpendicular line question that'll be given to you. Okay. So follow the steps.

[01:04:37]It's very easy. First of all make y subject formula get your find to that line by using the condition of lines m_sub_2 is equals to - n - 1 by n1 so we sub 3 4 they used to form the this equation this is the this formula for the equation of a line that's the formula you put your grad and you get your answer question number 25 say the gradient of the line joining x 4 1 2 is 1 / to find the value of what? X.

[01:05:11]Do you know the formula for gradient?

[01:05:13]What is the formula for gradient? Don't forget that grad also means slope. Yes.

[01:05:17]So gradient which is M is actually change in Y over change in what? In X.

[01:05:24]That is the formula for your gradient.

[01:05:26]So what is your change in Y? Actually it is equals to Y2 - Y1 / X2 - X1. Yes, that is the waiting formula. So class if they give you two points don't forget that they give us the point this and this. So what's our x1? This is our x1.

[01:05:45]This our y1 this our s2 this our y2 s1 y1 s2 y2. Do you understand? So I'm going to into this place to get my value for x. First of all my gra is 1 /2. So 1 / 2 = to y2. What y2? 2 - y1 4 all over x2 1 - what x from there I think I'll get the value for x because they say find the value of x so cross multiply 1 bracket 1 - x is 2 bracket I think uh 2 - 4 is -2 so 1 * 1 1 * s is s = -4 what do we do now collect like terms together so that you have take - 4 here you have 1 + 4 = x 1 + 4 is 5 = x so therefore x is equals to what five the answer is what a be careful the answer is a now question number 36 said you should find the midpoint you know what they call midpoint midpoint is the center or the middle of a line and what is the formula for midpoint just write this formula down you see this in your exam exam Now midpoint is equ= to x1 + x2 / 2 y1 + y2 over two. Yes. So that's the formula for midpoint. So like I said this is your s1 this is your y1 this is your s2. This is your y2. So let's the formula s1 is 5 + s2 is -3 all 2 comma y1 is 4 + y2 is - 2 all / 2. So let's do it. + - is minus. So this place now it give us minus. So 5 - 3 is 2 / 2 + - 2 - 2 4 - 2 is 2 / 2. So my answer is one comma one. This is known as the what midpoint. Don't forget that we have four more questions to go that cut across the other topics in mathematics. Hold on. Do you know that for perpendicular lines we say m_sub_2 is equals to -1 / m1. Do you know that for parallel lines, for lines that are parallel, hold on class, for lines that are parallel, if I have parallel lines, parallel lines, m_sub_1 is equals to m_sub_2 per pendicular lines. Follow me carefully.

[01:08:32]It means m_sub_2 is equals to something like this or m_sub_1 is equals to something like this. Any one of them is the same thing. Do you understand? So because they said it is perpendicular, that's why we put this law. Do you understand? So let's take the remaining four questions for today's class.

[01:08:51]Question number 37. Please watch this careful question carefully. 37 and 38 is from polomial. Yeah. And it is the major subtopic of that usually see questions in your jam exam. is called Rea's theorem because it's straight to point.

[01:09:10]Do you understand? Remember theorem and factor theorem are similar but they're not the same. This one is factor theorem. This is Rea's theorem. But let's see how we solve it. Use the same method to solve two of them. Look at the question. If x - 4 is a factor of x^ 2 - x - k then k is what? Do you know what they call a factor? A factor is a number that can divide another number without remainder. So meaning that if I use x - 4 to divide this guy, the remainder will be zero. That's why it is called a factor. So how do I prove that the remainder is zero? There's a step that used to prove for the what remainder.

[01:09:48]Please take note.

[01:09:50]Since I'm using x - 4 as a factor, it means that I can divide x² - x - k by x - 4 and that the remainder will be zero.

[01:09:58]But how do you prove that the remainder is zero? See the first thing to do is x - 4 you x - 4 = 0. Therefore if -4 crosses this equal to you have x = 184.

[01:10:13]Now how do I prove that x - 4 is a factor by making first of all find the value of x and then you put the value of x into the equation then you write equal to zero because when you sub the value of x to the equation you get the remainder. So since x - 4 is a factor so as substitute it we get equals to 0 because they said x - 4 is a what factor. So we already know that the remainder is zero. So the remainder is gotten by substitute the value of x from the d divis x - 4 something value for x into the equation. Since x= to 4 let's put x here. Then we put equal to z. Why did we put equals to 0? Because they told us that x - 4 is a factor. Otherwise we will not have put equals to z. Do you understand? So let's do it quickly. So instead of having a polinomial of x, this is the actual polinomial. It is what x² - x - k. So now that we have x= to 4, it becomes a polinomial of four.

[01:11:12]Take note of that. So what is now the polinomial of four? It becomes four.

[01:11:16]Anywhere you see x, you put what? Four.

[01:11:18]So become 4 to the power of what? 2 - 4 - k. Then we say the answer is equals to what? 0. because they said x - 4 is a factor that's why it's equals to 0. So whenever is whenever is a factor equal to 0 in pol. So now do 4² is what? 16 - 4 - k = 0. - 4 is what? 12 - k = So what do you have? You can take k here. So 12 = k. So therefore k= what? 12. That's the answer. Very easy. Look at the next one. The next one I say find the remainder. So this time it is not a factor. So we cannot say equals to zero.

[01:11:58]We just sub directly. Whatever we get becomes the remainder. Find the remainder when 6 pq - p² - 7 p + p / p - 3. So is divided by p minus 3. We start by doing p minus 3 = 0. Therefore p= 3.

[01:12:15]So any we see p we put three in this place now. So it becomes a polinomial of three. So any CP instead of writing 6^ 3 you have 6^ 3 - 3² - 47 bracket 3 plus what 50 then I'm not going to write equals to zero because it did not tell us is a factor whatever we get is the remainder this one remainder is zero so we can get the value for k because they say n - 4 is a factor so this will give us 6 bracket 27 - 9 - 47 bracket B + 50. So let's quickly multiply this.

[01:12:57]So you have you have 6 * 27 to give you 162 - 9 that's 3² - 47 * 3 141 + 50. So let's do it. 162 - 9 -41 + 50 that will give you 62. That's my answer. So the answer is equals to 62.

[01:13:30]How come we having 63? Let's check it out again. 6 * 27 will give us 162. 3² will give us 9. 47 * 3 give us 141. Then so we have 162 - 9 - 141 + 50 give us 62. So that is likely a mistake. So supposed to be 62. So the answer is equals to what? 62 which is a.

[01:14:04]The next one says this one is from the top we call binary operation. He said if the binary operation which is this is defined by x operated in y x + 2 y find the value of 2 asteric open bracket three asteric 4 very easy. How do you solve this question? This question should not bother you that much in your exam. First of all they said s operated in y is equals to x + 2 y. That's what they said and they say we should find this. If you look at what they say we should find we have this and we have bracket in mathematics you are told to attend to the bracket first. So you going to attend to three a four three a what? Four which is the one in the bracket. So anywhere you see yes you put three anywhere you see why you put four.

[01:14:54]So you have 3 + 2 bracket 4 3 + what? 8 that will give us what? 11. So when I solve this bracket I got 11. Did you see the way we did it? 2 * 8 2 * 4 is 8. 3 + 8 is what 11. So everything in this bracket is 11. So that means we now be left with two as what 11. Where did I get 11 from? I put the value of 3 and four as x and y. Do you understand? So we substitute it into the same equation.

[01:15:23]So we are going to say anywhere you see x now you put your two. Anywhere you see y you put your what? Your 11. See look at what I'm saying. Don't forget that they told us x they told us x asteric y is = x + 2 y but now that we have solve for this bracket and we got 11. So it's going to be 2 a 11. Do you understand?

[01:15:48]So it's going to be 2 a 11. So we do the same thing again become 2 + 2 * 11 2 + 22. So what become our answer? 24. The answer is what? C. very easy just do the rest of it's very easy this so the last question you say the fourth term of an AP is 13 and the 10th term of an AP is 31 find the 24th term it is very easy first of all what is the formula for the end term of an AP the end term of an AP is a + bracket n -1 close bracket d this is The formula for the n term of an AP.

[01:16:34]Do you understand? So the n is the number of terms actually. So if they tell you fourth term, it means you should put four as n. If they tell you eth term, you put a as n. Do you understand? So they say that the fourth term of an is 13. So let's put n as four into this place. Now see the formula here. n minus one. Let's put n as four.

[01:16:58]I'll put it here to become a + instead of writing this one become 4 - 1 do you understand while I look at it fourth term the fourth term is to put four into this formula 4 - 1 3 * d 3d now that is equals to what 13 that gives us equation wait 1 do you understand so they now say that the 10th 10 you now put 10 here 10 - 1 is what? 9. So it's 8 + 9 d. So you have a + 9 d = to you see the 10 is equals to what? 31. So this will give us a simultaneous what equation. Do you understand? So let's subtract this guy now. A c 9d will give us 6 d equals to 31 - 13. I think give us 18.

[01:17:56]Yeah. 18. So we divide both side by six to get the value for D. D is equals to what? 3. After you have gotten d, you have to get the value for a because this is a you need to simultaneous equation.

[01:18:09]So how do I get the value for a? Pick any of the equation. Substitute your value for d to get the value for a. So you have a + 3 d= 13. So you become a + bracket d is what? 3 bracket 3 equals to 13. So what do we do now? It becomes a + 9 = 13. a becomes 13 - 9. A becomes 4.

[01:18:33]So you must take note that your d is equals to 3 and that your a is equals to what? 4. Now the question says find the 24th term. So how do we solve for 24th term?

[01:18:49]Let me clear up the board. Let's get the 24th 10. How do we get the 24th term?

[01:18:54]Don't forget that the formula still remain the same. What's the formula? A plus bracket n minus one close bracket d. So for 24th term for 24th term remember that um anywhere you see n 24.

[01:19:10]So put 24 here you're going to get a + 23 what d? This is the 24 term. Now but a what's the value for a? a is equals to what? Four and d is three. A is 4 and D is equals to 3. So we are going to put it into this guy. So 24th term will be equals to 4 + 23 * what 3. Do you understand? So what will this give us? 4 + this will give us um 69 and that will give us 70 what? So the answer is what? C. Finally class we have come to the end of today's video lesson. Let me quickly give you important formula that you need to learn before your jam exam.

[01:19:58]Let me give you the formula now. Okay, let's do that quickly.

[01:20:03]Yeah. Hello class. So we are going to look at some important tips in mathematics. Number one tips you need to take note of before your jam exam. Tips in mathematics.

[01:20:13]before your jam exam is to know that number one you must understand area of plane shapes. Yes, area sorry measuration of plane shapes. Measuration of plane shapes. Are we together?

[01:20:30]Measuration of plane shapes. So but you have to focus on circle. Yeah. So we start with a area of a circle.

[01:20:39]Circle. We start with area of a circle.

[01:20:41]What is the area of a circle class?

[01:20:43]Everybody knows it. P<unk> R². What is now the perimeter of a circle? It is 2 pi arrow. It is also referred to as the circumference of a circle. Number two, you must know the length of an ark.

[01:20:57]What's the formula for the length of an arc? Length of an ark is theta over 360 * what? 2 pi. What? R.

[01:21:08]That's it. Number three, you must know the area of a sector. So basically, you must know stuff about sector. What's the area of the sector? It is theta / 360 *<unk> r². And don't forget that a sector is an arc banded by two radi.

[01:21:28]This is a sector. Yes, the angle at the center is here the length of an arc. So this area is sector. So what about the perimeter of the sector? A sector is made up of two radi which is 2 arrow plus the length of an arc which is theta over 360 * 2<unk> 1 r.

[01:21:50]Then the deep part of it that you must understand what we called a chord. Yes.

[01:21:55]So very important what is a chord? Look at it. If this is a circle, this line A to B is called a C. Do you understand?

[01:22:06]There are two ways to calculate for a C.

[01:22:08]Number one, let's look at it from this angle. This is the perpendicular bis sector of a cord. This is the height here is the radius. So if I label as M from A to M is going to give us half of the cord. Do you understand? Half of the cord. So the full cord is a to b. If I use pagoras theorem here I can get one set of the cord which is a n. So pyagoras theorem will tell me that a n is equ= to the square roo<unk> of r² - h². This will give us yes. So when you have a n m* 2. So the entire length of the cord which is a is now equals to 2 of a n which is now 2 of the square root of r² - a². So this will give us the length of a cord. Now there's one important formula for leng of a cord. It is equal to 2 sin theta / 2. The one we used to solve the previous question.

[01:23:07]You can use this directly if you have the angle given. Not half of the angle but the complete angle. If you have the complete angle, you can use this one to solve for the length of a cut. Then we go to the next guy E which is called a segment.

[01:23:22]Which is called what? A segment. Okay, if I have a circle like this now and a small circle inside such that both circles share the same center. Let's say from here to here now small radius while here is a big radius. Do you understand?

[01:23:39]Yes. A segment could be the region here that is what shaded that could be the segment. So how do I find the area of a segment? The area of this segment, the area of this segment is equals to area of the big circle big minus area of small. Take note of that. What's the big cycle? Pi R² is the one that have the the big letter R then - R². If I take out what is common, I have pi r² - r².

[01:24:13]So you can have a scenario like this in your jam exam. They will have the same center. They're looking for the area the shaded portion. This is just one part of a of a what?

[01:24:24]V. And this is D. This is E. So let's call this one I of E. Then is another section of a sector that looks like this.

[01:24:36]Yes. This that looks like this. The entire shape actually this entire shape is actually a what?

[01:24:45]a sector but they cut some part out of the sector. So how do I know the area of this segment? Now this shaded part it is area of a sector minus area of a triangle that's what will give us the area of the segment. So let's do it quickly. Area will give us area of sector minus area of triangle.

[01:25:11]So how do we do that? What's in a sector? We know that before the 360 time square minus area of a triangle this particular triangle is not have this time height. No if you are given a triangle like this where you have a and angle between them the area of this triangle is 1 / 2 a sin theta. Yes you must know that there are three cases of triangle. Look at the case of triangle class. Look at the case of triangle.

[01:25:39]Number one, if you have something like this base and height, the area can be half base time height. Yes, you can have a triangle like this. To calculate for the area of the triangle, it is half base time height. That's one. Number two, you can have a triangle where A and B is there and the angle between them.

[01:25:58]You can have B height. In this case, the area of a triangle is equals to 1 / 2 A sin what? Sin theta. Do you understand?

[01:26:07]Then finally you can have a scenario where three side of the triangle is given A, B and C. I told you in this one now you use the hero's formula.

[01:26:20]Hero's formula to find the area. The area is equals to the square root of x bracket x - a x - b x - c. Yes. But if you want to get the x, you go add a + b + c. You divide it by 2. Do you understand? Yes. Where x is a + b + c over two. So the area of a triangle is in three cases. This one, this one, this one. So but in this case now we using the case 2 which is 1 / 2 a now happens to be the same radius. So r² sin what?

[01:27:02]Sin theta. Do you understand? So take note of that class. It is very very expedent. Let me show you something.

[01:27:10]Watching my area of segment. I can also find perimeter of segment.

[01:27:18]Perimeter of segment. How do you get although this one is not common in jam?

[01:27:23]Because it's long. It's long. This one is common in jam. So perimeter of a segment. What makes up a segment is the length of a cord plus length of an arc.

[01:27:33]So we can say equals to length of a cord plus length of an ark. Yes. So length of a cord plus length of an ark. What's the length of a cord? It is 2 arrow sin theta / 2.

[01:27:53]Length of an ar theta / 360 * what? 2 pi arrow. So this is your what's common in your jam exam. Then you must also learn measuration of solid shapes. Not all solid shapes. So but some of them are we going to consider them right now. Number one you must learn is a sphere. Yes. The area of a sphere is 4 pi r². If you don't know this now run go and collect your book and your pen. Jot it down fast. The volume of a sphere is 4x3 par cube. You must understand this one.

[01:28:31]Very very important. Number two is called a hemisphere. Yes. So a hemisphere is half of a sphere. Actually a hemisphere is like this. Yes. This a hemisphere. So what is the total surface area of a hemisphere? Since it is actually half of a sphere that means it will be P 2² but because this heavy sphere has a circular cover it becomes 3² look at it this one only this place is 2 pi r² this one is p<unk> r² so everything will be 3 p<unk> r² then the c surface area will just be this part which is 2 pi r² then the volume is half of this which is 2 / per cube. You understand? You must take note of this. Number three, you must know the that of a cylinder. Very important. This how a cylinder looks class.

[01:29:33]Yes. So this is a cylinder. Here is P².

[01:29:37]Here is P². Here is 2 pi RH. You see now we have two type of cylinder. We have a closed cylinder. The one close at both end. And you have open cylinder. So I can tell you that the total surface area for a closed cylinder cylinder that is closed glass closed at both end is you combine these two together<unk> square will give us 2<unk> square + 2<unk> what h that will give you do you understand you then the if total surface area now of an open cylinder that's a cylinder open at both end sorry at one end if I command this place now I'll be left with P<unk> R² + 2<unk> H. So that will give me P<unk> R² + 2<unk> R H. Do you understand? Then we now have what we call the C surface area. Let's assume we open the top we open the bottom. We'll be left with this guy 2 pi arrow what?

[01:30:39]H. Do you understand? Then finally the volume of a cylinder. The volume is p<unk> r² what? H finally I'm going to give you cone that of a cone.

[01:30:52]Yes. For a cone, you know the way a cone looks like class. This how a cone looks like.

[01:30:59]Yeah. This is a cone.

[01:31:02]It has a right angle triangle base radius height and slant height. Yes. So what is the total surface area of a cone? This place only this place the bottom is r² while here is r. So it's r² + r l. Do you understand? Then what is the c surface area of a cone? Just this place is p<unk> r. Then what's the volume? 1 3² h. It's almost the same with a cylinder. Well, a cone look like a truncated cylinder. So that's why it's like this. Please you must take note of this stuff. Finally, you must know how to get your special angles without using calculator.

[01:31:48]Special angles. How do you get your special angles? Like the other time when we were solving question, I said sin 45 is <unk>2 /2. And some people were wondering how did I get it? So let me show you. Draw a square root like this.

[01:32:02]Then write over two here. Do you get it?

[01:32:04]This square divided into two. One will be for sin theta. One will be for cos theta. You understand? And then you put your zero 0° all your special angles.

[01:32:16]30° eh 45° H 60° 90°. These angles are called special angles and the angles that comes out in J. So you number it 1 sorry 0 1 2 3 4 for sign. For cos you do 4 3 2 1 0.

[01:32:37]Anything you are doing is over two. So if I ask you what's the value for sin 45 you just come here say what sin 45 sin 45° equals to the square root of two see it anything you do is over two do you understand what's cos 60 you come cos 60 square root of 1 / 2 root of 1 is one so you have 1 / 2 the same as 0.5 do you understand so if I ask you what is sin 30 now you trace 30 for 30 square roo<unk> of 1 / 2 and that will give you square root of 1 is what 1 / 2 that will give you 0.5. You understand?

[01:33:18]So any angle you want to you want to get you can always get it. If you're looking for tan let's say what's the value for cos 30° now what's cos 30 you come cos 30 cm square roo<unk> of 3 / 2. Do you understand? Finally, if you are now looking for tan 30, tan is not in this place, but tan is equals to sin 30 / cos what 30. So what be your sin 30? If you can remember how to draw this table, you don't get special both for mass and for the question. So what sin 30? We have done it before is 1 /2. You do 1 /2 / cos 30 <unk>3 / 2. So that will be 1 / 2 divide to time. So * 2 /<unk>3. So this cancel is 1 /<unk>3. So it should be noted that tan 30 is 1 / 3. And you can to get any of the angles of your choice.

[01:34:16]So if today's your first time coming to my YouTube channel, you are seeing me for the first time, please subscribe to this channel on your notification bell and share and give your friends preparing for jam and other related exam. Thank you very much. I'll see you in my next video.