CAT Coaching – Understanding Algebra
First, we’ll understand the complete concept of a specific area. Then we’ll look at questions based on that and learning, then we’ll learn solutions. Ugh… maybe two, three different ways but we’ll finally decide the one with, to be used in an exam because there is a time constraint where you need to do it faster and after doing all these things we’ll start the from the other side. What are the questions which anyone can ever make? And that’s what we’ll do it in your pattern workshops. There are two, three levels over here. These concepts sessions are different. Not what you will see anywhere, not what you’ve seen before, not what you’ll see again.
So now, what is Algebra? If it’s about questions, leave it to me. It’s as easy as Algebra. I’ll consider it as flexible questions. Why I’m starting with Algebra? There’s a reason – huge number of questions. Like, on an average forty, forty five percent of the questions can be converted into an Algebra question, ‘can be converted’, which can be considered Algebra not in your conventional method, in our method. So, Algebra, there are two types of questions – flexible and fixed. So, what is a flexible question? Now, flexible question is where you want where you can use this word.
To classify further, you can say that Algebra is where variables are used, in one way or the other, ‘where variables are used’, right? Now, what are the question types in your objective exam? Questions can be in variables, options can be in numbers, Category 1. Category 2 you can say it’s the reverse. What is the reverse? Questions in numbers and options generalized. ‘Generalized’ is nothing but variables or it can be questions in variables and options in variables. Now, these three categories of questions, these are the flexible questions in Algebra, can be easily more than fifty percent in, of Algebra. These, these three questions why you should not prepare, it’s, I’ll just take one, one, one question from previous papers and will make you understand. You need to understand one question, after that, please don’t practice! Reason is, a variables to numbers question and a numbers to variable question and a variable to variable question. Which topic? Which topic? It can be any topic! Which level? It can be any level! Who can do this? Anyone can do this!
Let’s consider a, y, z, b are in geometric progression. You…you must be wondering, why do I need to solve questions like this to become a Manager? a, y, z, b are in geometric progression. a, x, b are in arithmetic progression. So, find (y3+z3)/ xyz. Now, this is not an easy question. This will be considered as moderate or difficult. Why? Because this is not any, this is not a form-lay based question. This is not a form-lay based question. These are the ones which we call ‘indirect’. ‘Indirect’- tricky questions, right? And so, when you’re solving this, now I’m writing the answer options here – one, two, three, four. That is why, what is the category? Questions are in variables and options are in numbers. This much is enough! The moment you realize this, it’s over! Now when you’re solving this, I’m telling you, you don’t need to prepare. Why? Because you can take any geometric progression. There is a pre-requisite. You should know what is a ‘geometric progression’.
What is a geometric progression? One to many of an arithmetic progression or you can say, you keep on multiplying by the same number. So if I start with 1 and I can write 1, 2, 4, 8. And 1 and 8 if I keep it constant, I should keep it constant, right? Because the same question and ‘a’ and ‘b’. What is the middle term in an arithmetic progression? That’s the average, right? We call it the Arithmetic Mean’. In common language we can call it ‘Average’. Arithmetic Mean and Average it’s one, one and the same. So, so, you get (1+2)/, (1+8)/2, 4.5. Now, this is what matters. You have y 2, z 4 and x=4.5 and you’ve, you can solve any question after this. You’re not preparing for this question. Now, after this, you can solve any question in x, y, z, right? You can solve any question in x, y, z and anyone can do that and that you don’t need to prepare. You need preparation for adding and multiplying? I don’t think so! Now it’s an addition question or a multiplication question. Down to an eighth standard level now.
I’ll rewrite the question now, in a different language, that is, here I’ll end up using numbers. If questions are in variables, we need to use numbers; if questions are in numbers, we can use variables and if questions are in words, convert them either into variables or into numbers. That’s the only place where you need some more time. Converting words to numbers and variables. So, now the answer is as easy as this, right? y is 2, so 23; z is 4, 43. x is 4.5, y is 2, z is 4. Don’t tell me that this is at your level. This is way below your level. Eighth standard, they’ll also give the answer as 2. So you can’t make a mistake, unless and until you don’t know how to do this and this, plus and minus and you need to learn how to add, multiply, subtract, divide, that’s it! So learn how to convert.
So what do you need to learn is, how to convert one question to the other format and this, it’s not that there are many, right? There are only four – words, variables, numbers and diagrams. You can even reduce it further, right? The best part is, you don’t need to take 1, 2, 4, 8 also. You can start with 1 and multiply by 1 and write 1, 1, 1, 1 and 1, 1, 1. Now, now, it’s down to second standard level because the question is (1+1)/1. Even if you want you can’t make a mistake.
Now see, the next part, numbers to variables is not even worth mentioning. Variables to numbers, I agree, you’ll still need some pre-requisites. Any question of this category if you’re preparing it’s a complete waste of time. That’s why I won’t spend more than two, three minutes here.
And this method, numbers to variables, some of you are already using this method. I’ll write a sample question here. If I write some two ‘n’ terms, this question is not at all important. It’s not at all important. Anyway, because any numbers to variable question you’ll end up solving if you just listen for the next one minute. So, I’m just writing two options here, you can write questions in numbers and options here are in variables, right? Options are in variables. So, this is why it’s much more faster because you can simply replace n = 1. What is the meaning of ‘n’? What is the meaning of n = 1 in the question, among the options and see where they are not matching, that’s it! The solution is over! So, n=1, what it means in the question? It means 6, it means 6. And among the options you just replace wherever you see ‘n’ you replace it by 1. So, this is not equal to 6. So this can never be the answer; that’s the common sense. That can never be the answer.
And this you can immediately mark the answer. So, previous one, here without knowing anything you can solve. Previous one, you still need to know a few things. You need to know what is an AP, what is a GP and all, for variables to numbers. Numbers to variable, you can simply replace your. Numbers to variables is very easy to solve where options are in variables you need to eliminate. Variables to numbers is just the reverse. And both works on conversion and the second one is, both are just single conversion. That is why both the categories you can consider as Round 1 questions because you will take only one step to mark the answer, whatever be the level of the question.
Now what is required is, variables to variable is the same thing. Variables to variables what you need to do is, I don’t even need to discuss the question. Convert the question into a number, convert options also into a number, eliminate wherever it’s not matching, over! Preparation is over. These are considered very, very time consuming. I won’t use the word difficult because nothing is difficult. Given time you can solve all thirty, in a section. But, in that time, solving it better and faster without making mistakes. You solve it faster and your accuracy is also, will also improve because you will always solve it in the easiest format. That’s why you need to learn concepts in multiple formats. Do that well; you don’t need to solve questions. How we’ll solve questions and you’ll soon understand that solving questions is actually at a lower level. So two are over, right? Variables to numbers is over once and forever.
Numbers to variable, it was some of you were already using it. Some numbers to variables some of you were already using it knowingly or unknowingly. I’m just extending it to all the other categories. Now, variables to variables, this is the approach. Convert this into a number, options also into a number, eliminate wherever it’s not matching. Maybe I can show you one… one… one question here also. So, x2-px+q=0. So, what is this? So, there is some pre-requisite. You should know what is this. It’s a quadratic function.
Quadratic equation, if I put a zero over there x2-px+q=0. As soon as you see this, now this is not your question. What I am telling is the kind of change which already happened is, anything after this can be solved. So without reading the next line you can be sure of marking the answer. You can be sure of marking the answer. You will read the next line before getting the answer but you will select this as a ‘sure shot answer’ question. So, selecting questions there are very very basic parameters like this. It’s not based on chapters and all.
What are the different chapters? Permutation, Combination and all is a chapter, you think so? That is complex because they make it complex in the books. You can teach Permutation, Combination to a seventh standard guy because we have programs in schools also and you can get ten out of ten IIT-JEE questions correct from most of those guys. It’s just that, what is Permutation, Combination? It’s not those ten formulae which we have memorized. Permutation, Combination is basic numbers; that too, very basic numbers. It’s about distributing numbers across some variables that is P&C. What I’m telling is anything after this can be solved by anyone and that’s a big difference, right? Anything after this can be solved by anyone without even reading the next line. That is why objective exams, objective type exams you need less preparation and a different kind of preparation and a better preparation.
So I’ll give you a sample question here, k and 2k2 are the two roots of an equation. x2-px+q=0. Now find q+4q2+6pq. Now this way you can’t prepare, right? You cannot keep on solving questions like this and, because every question will have a different method and you’re trying to solve it using variables. Th…that and options I’m writing here. Options I’m writing p3, q2, 2p3, etc…etc. Now, there is a pre-requisite here. You should know what is ‘p’ and what is ‘q’ which I am very sure all of you know this, if not now, all of you learn this later before you take the exam. What is ‘p’? Because it’s revision, nothing but revision. Even if you’ve not learnt quadratic equations before, to reach this level you need only fifteen minutes. But after the first fifteen minutes, you’ll go to a very, very high level which is not required for any of the exams in a, in a specific session on quadratic functions. I’m not doing that today.
Right now I am assuming that you know what is ‘p’ and ‘q’. But what is ‘p’? Some of the roots and ‘q’ is product of the roots. The moment you realize that it’s over. How is it over? You write 1 here and 2 here. So it’s over. I can’t write 1 and 1. That’s a violation. Put k=1, 1 and 2. So, even a question like this, this will be considered as a very, very difficult question or a time consuming one in your paper. This will come in the top five out of thirty in terms of level of difficulty, a question like this. But this is your easiest one, you’ll surely solve it. You…ugh…the only thing is now you replace p by 3 and q by 2.
So the logic here is, replace variables by numbers in as many questions as possible; as many, as possible. Easy ones you don’t need to, easy ones you don’t need to replace. So you’ll get ugh…substitute p 3 and q 2 you’ll get 54 here. If you think it’ll take some time; if you’re very slow in calculation keep it for Round 2 but it’s still a sure shot answer in Round 2 and 54, among the options, p3 is not 54. This is the logic, it’s a very important. So this can never be the answer. You need to eliminate the wrong ones before confirming the right one. q2 is not 54. So this can never be the answer. The other last option which came in the exam was 0. That anyway is not the answer. And 2pq without verifying you can mark it now because you need one answer option to be correct anyway. But if you verify you’ll get 54. So 2p3 is 54. That’s not the reason why this is the answer. Because the other options are not 54, you can say that this is the answer. There’s still one assumption. What is the assumption? You’re assuming that, question is correct. If the question is wrong then anyway, nothing will work. So, but still you won’t end up wasting time at all. You realize this very fast and you can move on.
Now, what is the change required in terms of your preparation? That’s very easy. I told you if you think you need any kind of preparation, it’s not like what I’ll do in the next two minutes. I’ll tell you what you’re not supposed to do because that’s what you’ll end up doing otherwise. Not after this session for sure and I’ll try doing it very fast because I don’t want to learn, I don’t you to learn this. So, see it should not be like this, you take quadratic equation or an arithmetic progression or a permutation, combination or topics like that and you can’t start like this, like what we’ve done before. What we used to do is, before we used to do it like this, right?
What is AP? AP is a+d, a+n-1 then which is nth term (unclear). Arithmetic Mean is nothing but normal ‘average’. What is geometric progression? (unclear). Geometric Mean is product and nth root AP, GP basic concepts are over. This is what we call basic concepts. Questions based on this you’ll call it direct questions. You’ll anyway solve them, right? That’s what you call fixed questions. Now, what are the flexible ones? The worst part is you can’t stop with this preparation because you won’t get mor…very few questions will be direct where there will be an arithmetic progression given and where they’ll ask you to find, let’s say, some two fifty terms or some two hundred terms.
So, you start your preparation like this and property, with properties. How many properties on arithmetic progression? Good books will give you twenty five and not so good books will give you fifteen. The biggest, the JEE preparation books you’ll have twenty, twenty five properties on arithmetic progression alone. Even though it’s one, two, three, four, five. Those properties are good, provided you’re understanding and learning, not memorizing. If you’re doing it like this like so pro…starting with properties.
I don’t know how to sing otherwise I can act you can actually sing all these things. Some of the first ‘p’ terms equal to some of the first ‘q’ terms and some of the first p+q terms is equal to zero in an AP. In an AP, if pth term if it’s ‘q’ and qth term if it’s ‘p’ then tth term will be zero in an AP. ‘m’ times the mth term and ‘n’ times the nth term then tth term is equal to zero in an AP . First plus last, second plus second last, two times the middle term they all are equal in a GP. (unclear). I can go on and on and on. So, it will never finish. Because your twenty five questions can be based on twenty five properties but the fact is when you are taking an objective exam you don’t need to even learn these properties. But, you will be in a position to extract all the properties, you’ll be able to find out new properties but you don’t need to because the fact is, that they all are flexible questions finally. Because, when do we call it as a property? If I’m telling that ‘m’ times the mth term is equal to ‘n’ times the nth term then (m+n)th term is equal to zero; that is, that means it will work in any, this is the key word. It has to work in any arithmetic progression. And if it is something which will work in any arithmetic progression, while solving you can take any arithmetic progression. Even 1, 2, 3, 4, 5. That is why you should not learn. So this is what you should not do. If you’ve already memorized, you don’t need if you have lot of memory space please don’t forget. You don’t need to forget it and all but you can stop using them. You can easily stop using them. Because all the property-based questions you can use this word and solve. This is the…this is the only thing.
If there is anything which I’ve taught you till now? I’ve just taught you how to use this word. And variables to numbers and numbers to variables and even variables to variables it’s about that only, right? Variables to variables k and 2k2 you can take any number and solve. So that’s again this word. You can easily learn, that’s the best part. It’s not because of practice; I don’t practice at all. You can also easily learn how to use these…these day, this is all logic nothing but common sense. So, three…three, Algebra is over! What I have not, I didn’t discuss is, fifty percent difficult part of Algebra is over. Easy part of Algebra is remaining for us to revise. That’s…that’s why I told you that this is actually a scoring section. It’s a boring section also that’s the scoring section. You might think that I’m a…I’m…I’m making it sound very easy. But the fact is, all topics are like this only, it’s…it’s all about these four languages – numbers, words, diagrams.