Hello dear children, welcome to another amazing session of cheat sheet series on binomial theorem. In binomial theorem, we will know the latest trends, i.e. what kind of questions are coming, and what are the things that J.E. Maynes has removed from this topic, or such topics which you do not need to read anymore. And along with that, we will see a lot of things that will teach you. So, let's go to today's special question.
Let's go on this beautiful journey. First of all, let's talk about the syllabus. If we look at the syllabus, J.E.M.A.S. 2023 Binomial theorem and if you look at this, you will see here properties of binomial coefficients.
This topic is now removed. That is, the questions related to the properties of binomial coefficients which used to come in 2023 till now, they will not come now. Let me tell you, binomial theorem is one of the most important chapters for J.E.
Maynes as far as past five years are concerned. I mean, in the last five years, there have been a lot of questions from the binomial theorem. Let's see what is going to happen this time.
Because a maximum of 30% of the problems that used to come from the binomial theorem, used to come from this topic. Almost 30% of the problems used to come from here. And now this topic has been excluded.
This is why... Now, we have to see how many questions will come from binomial theorem in 2024. So, in your shift, you can expect one question. Because in some shifts, there have been two questions from binomial theorem. But now, we cannot say anything about the past trend.
The past trend will continue. But still, we can expect that you will get one question in your shift. Here, let me show you what I was telling you till now. The problem concerned with binomial expression or binomial theorem, they constituted 69% of the total problems that came in JEE. If 100 questions came from binomial theorem, then 69 problems came from these topics.
In which, the power of x, term independent of x, multinomial theorem. KS term from the end, middle term, all these problems are full. And remember this data is from last 5 years, from 2019 to 2023. And here binomial coefficient and their sum, as I told you, about 1 third of the paper, 1 third of the question came from this topic, which is almost zero now.
So this is now gone in a way, there all the focus will go to K-means, so there you will see more problems. So the 69 will increase, definitely it will increase by 100%. What should be the general term? General term containing n terms, containing x power r. You will remember, general term in x plus y to the power n.
You will remember what is the general term? Let's write pr plus 1. How much is it? Remember it quickly and tell me. ncr is x power, n minus r and y power is r. So that is the general term. Sir, middle term?
K term from the end divisibility problem or multinomial theorem. In this I would like to mention a special mention divisibility problems or reciprocal problems sorry remainder problems and divisibility problems. This is the question that is definitely asked. In 2023, many many questions have been asked from this.
In fact, I would say a lot of problems have occurred from this topic. So, read this topic very well. Now you must be having a question in your mind that sir, from where to cover all these things, especially divisibility problems?
You can follow the Manzal series, there is a detailed lecture on this topic. There all topics of binomial theorem are covered. By watching that lecture, you can make your binomial theorem strong and correct.
If you have time, you can read the topics of numeric greatest term, integral and fractional part also. Because in 2023, only one question was asked on this topic. It can be done in one shift. Because now properties of binomial coefficients have been reduced, so its percentage should increase a little.
There is a very mechanical way to solve such problems, that is, there is a set pattern on which these questions are asked and there is a set pattern to solve them. So if you have covered this topic in a way, then let's assume that 100% there is hope that you will definitely solve this problem. What to leave? You can now leave the properties of binomial coefficients.
Let's talk about the latest trends. Problems containing x power r are always in the trend in binomial theorem. Problems are always asked on this. For example, the coefficient of x power 7 in this expansion and the coefficient of x power minus 7 in this expansion are equal. So, you have to tell the relation between a and b.
Here, If the constant term is alpha, then alpha's greatest integer value is to be told. So, the problems on the constant term x power r are definitely asked. Then multinomial theorem. Multinomial theorem is also an important part. In binomial, you have only two terms here.
Here there are more than two terms and binomial theorem is applied on it. You can apply multinomial theorem on it. For example, the coefficient of x power 7 is asked in this expansion. Many problems have come on it.
Reminder and divisibility problems, I have already told you that it is one of the most important topics, you must read it. And try here, the problems we have given you till now, we have given you almost 6 problems, not 6, we have given you 5 problems here. The answer to these 5 problems, do write it down and tell me, how many marks you have secured out of these 5, i.e. 20, counting with a negative. marks also.
If it does not match with the answer, it is the first time. Then today's special question, if the constant term in this expansion, this question again, because here it is not binomial, but it has 200 other terms, i.e. it is multinomial. So, here, if the constant term in this expansion is 2k into i, where i is an integer, then find, is an odd integer, then find the value of k.
So, if I write the general term of this expansion, Then the general term of this expansion shall be 10 factorial upon r factorial s factorial into p factorial or into 3x cube k power r minus 2x square k power s or 5 upon x k power 5 k power p. So, the general term comes out to be 10 factorial upon r factorial. s factorial, p factorial, 3 to the power r, minus 2 to the power s, and 5 to the power p, into x to the power, x to the power 3r, plus 2s, minus 5p, minus 5p.
Remember, r plus s plus p will be 10. and R, S and P will be whole numbers. Now, what we need? We need constant term. That means, for constant term, which is the power of x, that means, x should not be there. So, for constant term, we should have what?
3R plus 2S minus 5P is equal to 0. That means, 3R plus 2S is equal to 5p. So, and the second equation we have is, r plus s is equal to 10 minus p. If I multiply the first equation by 2, multiply the second by 2, and subtract this, then I will get r is equal to 7p minus 20. This is the value of r. And if I subtract this equation, I will subtract the second equation by multiplying it by 3. Then I will get minus S is equal to 9P minus 30. Let's see once.
9P will be equal to 8P. 8P minus 30. That means S is equal to 30 minus 8P. So, We have taken the values of r and s in terms of p. r is 7p-20 and s is 30-8p. Obviously, we can understand that the value of r and p should be greater than 3. So, if I take p as 3, then s is 6 and r is 1. Apart from this, if I take P as 4, then this will work, but if S becomes negative, then we will not get any other value.
This is the only possible combination of R and SP for which we get a constant term. Therefore, the constant term will be 10 factorial upon R, 1 factorial into S into P into 2 power. Let us see. 3 to the power r minus 2 to the power s. 3 to the power 1 minus 2 to the power.
S into 5 to the power p, 5 to the power p, once you see, yes, it is coming, so that comes out to be 10 into 9 into 8 into 7 into 6 factorial, 6 factorial into 6, 3 factorial value is 6, and this is 3 into 2 to the power 6, 5 to the power 3, now you can see very well that 6 factorial. and 6 factorial gets cancelled out and here also if you see 3 2 is 6 3 3 is 9 2 4 is 8 so this 3 and this 4 gets cancelled out so that is write this 2 into 5 into 3 into 4 2 square into 7 into 3 2 to the power 6 into 5 to the power 3 so if I 5 count to 5 power 4 and 3 count to 3 square and 7 and 2 power 6 and 3 9. This is odd integer. So, here the k is coming power because we wrote constant term i is equal to i into 2 power. So, the correct answer in this case comes out to be IIT Delhi, one of our dream colleges. So, that is IIT Delhi is the correct answer.
You will get the same type of multinomial question in the latest end. Do let me know whether you got it or not. So, let us meet in the next chapter analysis. Till then.
Jai Shri Krishna, Radhe Radhe. Thank you and bye-bye.