Transcript for:
Overview of Mock Test Paper 2

Hello everyone, I am Akash Agrawal your Maths Tutor. Let's discuss Mock Test Paper 2 for September 2024. Let's attack the questions. Before that, let's take an overview about my CA Foundation Google Drive Video Lectures One Stop Solution for your Maths Stats LR In this, we will cover things right from basics. So even if you are from non-maths background, don't worry. Everything will be covered in basics, all module exercises, additional question bank, past year questions from 2012 till the latest year, all chapter wise past year questions. You get test series chapter wise, section wise, full syllabus test papers, you get my color printed handwritten notes, very good, very nice and buttery and unlimited doubt solving support directly handled by my me and also a daily study plan. So if you and your friends and relatives are planning for Jan 2025 or June 2025 attempt Please drop a whatsapp My team will share all the details Let's attack So quantitative aptitude mock test paper 2 Which has come on 22nd August Let's attack First of all, there are math questions Then logical reasoning Then statistics Although your actual foundation exam All the questions will be in mixed order Let's attack What is the first question saying? A bag contains 23 number of coins in the form of 1 rupee, 2 rupee and 5 rupee coins. So total 23 coins are there. Denominations are 1 rupee, 2 rupee, 5 rupee. The total sum of the coins is 43 rupees. So the total value of the coins is 43 rupees. The ratio between 1 rupee and 2 rupee coin is 3 to 2. Then the number of 1 rupee coins. You can do this in two ways. One, let's do by reverse trick. What did the question give? The ratio between 1 rupee and 2 rupee coins is 3 is to 2. I said let the common multiple be x. So how many coins are there for 1 rupee? 3x. And how many coins are there for 2 rupee? 2x. Okay. What did the question ask? Total number of 1 rupee coins. So if the reverse trick wise goes, I said let's take a option. Let the answer be a option. So what is the total number of 3 rupee coins? For us, 3x. And as per the reverse trick, Total number of 1 rupee coins I have assumed is 12 Now let's check whether the rest of the story is correct or not Let's check So how much is x from here? 4 Common multiple is found So total Can I say this? Total coins of 1 rupee are 12 Total coins of 2 rupee are 2x and x is 4 So 2 x 4 is 8 Is it clear? Now Can you tell me how many coins of 5 rupee are there? Yes sir Total 23 coins 12 and 8, 20 coins So how many coins will be left for 5 rupees? 3 coins will be left Now, how is the answer to check this correct? We will find out the total value of these If the total value comes out to be 43 This option is correct Otherwise you will go to BCD Let's check How will we find out the value? 12 coins for 1 rupee, so what is the total value? 12 x 1, plus 8 coins for 2 rupees, so what is the total value? 8 x 2 And 3 coins worth 5 rupees, so 3 x 5, how much is that? 12 plus 16 plus 15. 31 plus 12, 43. Perfect. And the amount should have been this much. So, I guarantee this answer is correct. Is this clear? So, this is our first method to crack reverse trick. You said, sir, you have to understand reverse trick. You have to do it in a normal way. So, how did you do it? Did we just say, how many coins worth 1 rupee? So, this is also a detailed method. Please pay attention. Please note one more thing as I always say I am solving in detail I will be explaining using different methods It will be a little lengthy, you can do it faster Second method, how much coins are worth for Rs 1? 3x How much coins are worth for Rs 2? 2x How much coins are worth for Rs 5? I don't know sir So I said let the number of coins be 5 rupees Y So what did you say from here? Number of 1 rupee coins plus number of 2 rupee coins plus number of 5 rupee coins. How many are there? 23. Make an equation. 5x plus y is 23. Is this clear? How will the next equation be made? Sir, their total value is 43. What did you say? How many coins are there in 1 rupee? This much. Into its value. How many coins are there in 2 rupee? Into its value. How many coins are there in 5 rupee? Into its value. E is 43. So, what is the equation being made from here? 3x plus 4y plus... Sorry, 3x plus 4x plus 5y is 43. So, what did you say from here? 7x plus 5y is 43. So, now you have to solve these two equations. What is the best way? Sir, let's multiply this by 5. Look, write here. 5, 5 is 25x. y into 5, 5y. 23, 5 is, what else? 23 into 5. 115 let's subtract this subtract here 25 from this 17 goes and this is negative obviously here I have kept the big number below so this is minus 18 cut this will be subtracted this will be subtracted 115 minus 43 comes out to be 115 minus 43 comes out to be 72 and with negative sign so from here how much x comes out 72 upon 18 4 is coming, this minus minus cut So x is 4, this is the same that we got from the reverse trick But anyways x is 4, what did the question ask? Number of 1 rupee coins, how many 1 rupee coins are there? 3x So 3 x 4 which is 12 Hence option A is correct So both are normal, there is not much length You can do whatever you want Let's go next On simplification, we are solving this in detail We will do it quickly We have done the same question in the module This is a question of the model. We are doing this in a normal way. This 1 is visible here. Z raised to A minus B is A minus C. How will we write this 1? Z raised to A minus A. Z raised to 0 is 1. Now, we will write the difference like this. 1 upon, pay attention. Here, we will write Z raised to A minus A. Z raised to A into Z raised to minus A. Base same power added. So, that will come. How will we write this? Z raised to A. Z raised to minus B. Z raised to A. z raised to minus c. This is the first term. Next one is a little fast. Same way. We will write b minus b. Then plus z raised to b, z raised to minus c. z raised to b, z raised to minus a. Plus, this last term is visible. In the language of sub c. So z raised to c, z raised to minus c. z raised to c, z raised to minus a. z raised to c, z raised to minus b. Is this clear? Now, what will we do? What is common in all these terms? z raised to a is common. What is the result? z raised to minus a, z raised to minus b, z raised to minus c is saved. What is common in all these? z raised to b is common. What is the result? z raised to minus b, z raised to minus c, z raised to minus a. What is common in all these? z raised to c is saved. z raised to minus c, z raised to minus a, z raised to minus b. Is this clear? Now, is there something common in all these terms? This bracket is exactly same in all three. So I took that as common in all three. Z raised to minus a, z raised to minus b, z raised to minus c. It is common in all three. What is left? 1 upon a is common, so what is left in the first one? 1 upon z raised to a. In the second one, 1 upon z raised to b. In the third one, 1 upon z raised to c. And what is 1 upon z raised to a? It is z raised to... Oh, sorry. How can we write this? We can write z raise to minus a This z raise to minus b And this z raise to minus c Understood? I am taking the denominator up So z raise to minus a z raise to minus b z raise to minus c Now This is in the numerator This is in the denominator Same term cut Answer 1 Clear There are different ways of putting values Assuming values And solving Or cyclic All that It doesn't work in every sum, so you'll have to do it fast Or else it'll be solved quickly in a normal way Next 18 raised to 3.5 divided by 27 raised to 3.5 Into 6 raised to 3.5 is 2 raised to x Keep the calcib with it So how will we do it? 18 raised to 3.5 divided by 27 raised to 3.5 This multiplication is like this They didn't print it properly but it's okay. It'll be correct in the exam. Don't worry. Now what we have to do here is, whenever we get the game of indices, we always write the base in terms of prime numbers. What did you call 18? 2 x 9 is 18. And what is 9? 3 squared. So now, how did you write 18? 2 x 3 squared is 3.5. 27 is 3 cubed. 3 cubed raised to 3.5. How to write 6? 2 into 3 raised to 3.5 and this whole is equal to 2 raised to x. Let's solve. What happened from here? This is power 2. What happened? Remember, 2 into 3.5. This will be multiplied by law of indices. 2 into 3.5 is 7. So, what happened? 3 raised to 7. Let's go down. 3 into 3.5 will be? So, how much? 3.5 into 3 comes out to be 10.5. So, 3 raised to 10.5. Is this clear? 2 to the power of 3.5 into 3 to the power of 3.5. This all is equal to 2 raised to x. Now, pay attention. Take all the same powers together. 2 raised to 3.5, 2 raised to 3.5. Base same power add. So, 2 raised to 7 is done. 3 raised to 7, 3 raised to 3.5. Base same power add. 3 raised to 10.5. Below, 3 raised to 10.5. And this is 2 raised to x. This and this, death. What is left? 2 raised to 7 is 2 raised to x. So how much is x? Oh my god! 7. Super easy. Answer. Clear? Let's move on. What is it saying? The value of 243... Hey, this one, we have to note carefully. If you have done it, you will get it. The century of Maths from June 2023. I had taken a random question from outside. That century. This is the same question. Anyways, what is the question saying? 243 raised to 0.13, 243 raised to 0.07, 7, 49, 3, 43. Again. Indices game, base, in the language of prime factor 2, 43 goes to the table of 3 So that is the power of 3 If you don't understand this orally, then multiply 3 3 x 2, 3, 4, 5 So what is this? 3 power 5 raised to 0.13 Again, 3 power 5 raised to 0.07 Upon 7 power 0.25 49, 7 square power 0.075 And 343 is a cube of 7. You don't understand the story? 3 on top and 7 on the bottom. What is this? 5 into 0.13 5 into 0.13 is how much? 0.65 You can do it in a shorter way. I am just solving in detail. 5 into 0.07 That is 0.35 So you could have done it like this. 5 into 0.13 equal to 5 into 0.07 equal to gt So power 1 So what did you write? 3 to power 1 Let's do with base You can direct it Direct it directly Direct it directly This is 0.25 So 0.25 I will write it 0.25 I will write it once Then I will explain 2 into 0.075 is how much? 0.15 And 3 into 0.2 is? 0.6 Now if we add the power below So 0.25 and 0.15 is 0.4 0.4 and 0.6 is 1 So the power of 7 is 1 If you want to add this directly to the calcium Then you can use m plus m minus So 0.25 m plus 2 into 0.075 So 2 into 0.075 m plus 3 into 0.2 3 into 0.2 m plus If I recall correctly, 1 is coming 3 raised to 1 upon 7 raised to 1 means 3 by 7. Which option? 1. So, it's a game of indices. Next. The number of prime factors in. So, what did he ask for? How many prime factors are there? Again, what you have to do here is, the number in the base, we will write it in the form of prime numbers and simplify it. We will add all the powers. We will get the prime factors. Let's go. How did you write 6? 2 into 3 raised to 12. A little fast. 5 into 7, 35 is 15. 3 5 15 denominator 14 2 7 21 3 7 is this clear let's simplify 2 power 12 3 power 12 5 power 28 7 power 28 3 power 16 5 power 16 the whole upon 2 power 12 7 power 12 3 power 11 and 7 power 11 Is this clear? 2 is dead, can you see? Okay. Now, let's talk about 3. What is 3 doing? 12, 16. 12 and 16, 28. Out of 28, 11 is gone. 12 plus 16 minus 11 comes out to be 17. So, 3 power 17, can you see? Did you understand? This will go up and be negative. Let's see 5. 28 plus 16. 28 plus 16. How much is that? 44. There is nothing in the denominator, so 5 power is 44. Let's talk about 7. How much was this? 28. And this 12 and 11 is? 23. 28 minus 23 is how much? 5. Now, just add these powers. How much is that? 17 plus 44 plus 5 comes out to be 66. So, what will we get after adding the power? Total number of prime factors. So, how many are those? 66. So after writing in the form of prime numbers, we will add the powers. So we will get the number of prime factors. Next. Log of a square by bc minus log of ca by b square plus log of c square by ab. Don't take a load. What is visible? Log a minus log b. What is log of a upon b? So write that first. Log of a upon b. So, these two terms together will be log of a into b. So, now we are multiplying. We will multiply by c square upon ab. Is this clear? This will be our answer. He will flip. You could have written by flipping directly. If you understand. It's good. a square by bc. Flip. b square by ca. And who is here? c square upon ab. Come on. What is the simplifier? a, a, a square. Cut from a square. b into b, b square. Cut. c into c, c square. Cut from here. Everything inside is cancelled. Meaning 1 is left. Means who is inside? 1. And log of 1 is how much? 0. Easy. You will get log equations very easily. See, it should be written quickly. Next. 1 upon log of xyz to the base xy plus log of yz to the base xyz plus 1 upon log of xyz to the base xz. If we want to add this log, then we have to make the base the same. Can you see this? This log will go in the numerator. Base is swapped with the argument. This log goes to the numerator, base will be swapped. So this goes up. What will we write? Log of. Are you understanding this? This log goes up, so the argument base is swapped. This is already right. Very good. In the same way, what will be the third term? xz and in base xyz. Now what did you say? Log of. into xz with the base xyz. Is this clear? Now, what do you see? Square of x, square of y and square of z. Can you see? x into x, x square. y into y, y square. z into z, z square. So, this is how it happened. x square, y square, z square. Take the square common from all. xyz the whole square with the base xyz. Now, what did we say? After beating the argument power, we came. So, 2 came. Log of xyz with the base xyz. How much is this? 1, 2, 1 is a 2. Answer. Again, the question of lock is not difficult. We have done this question in June 24th's MTP. What is it saying? 4 tables, 3 chairs, together cost 2250. 3 tables and 4 chairs cost 1950. Find the cost of 2 chairs and 1 table. See, the question is made smartly. Table, chair. Table, chair. What did you ask? Chair, table. So, here you make some mistakes. I said, let's take table as x, chair as y. So, what is the first equation? 4x plus 3y, how much? 2250. Next equation, 3x plus 4y, 1950. Now, what is the one way? Add both, subtract both, solve those two i equations. Or, I said, let's multiply this by 4. Let's multiply this by 3. We'll make it a little faster. 4 x 4 is 16x. 3 x 4 is 12y. 2 x 2 is 50 x 4 is 9000. 3 x 3 is 9x. 4 x 3 is 12y. 1 x 9 is 50 x 3 is 5850. Let's subtract. 16 minus 9 is 7x. 12y minus 12y is dead. 9000 minus 5850. 3150. So, X is given. How much? 3150 divided by 7. How much is that? 450. Now, put this in the equation. Put it in A. Put it in A1st equation. 4 into 450. How much is it? 1800 plus 3Y is 2250. So, how much is 3Y? 2250 minus 1800. 450. Divide 450 by 3. So, what's y? 150. It's x, y. What did the question ask for? Cost of 2 chairs and 1 table. Who's chair is mine? It's y. So, whose value did it ask for? 2y plus x. Don't do 2x plus y. 2 into y means 2 into 150. 300. Add 450 to 300. 750 answer. Easy. Clear? So exact same sum from this MTP. If n is equal to m factorial, where m is a positive integer greater than 2, then the value of, again you can see one thing here, log log is there. Does everyone's argument look like n? If we take up log, then everyone's base n will be 1. So this went up. What did you write? Log of 2 to the base n. Are you understanding? Next, log of 3 with the base n. of 4 with the base n and so on last one log of m to the base n is this clear log a plus log b plus log c what happens everyone's base is same 2 into 3 into 4 into tuk tuk tuk till m. Is this clear? So, what happened here? Here, I add 1 by myself. Can I write like this? 1 into. Any number can be written into 1. I can write 10 into 1, right? So, you can write this entire multiplication into 1. You can write into 1 anytime. There is no problem. What do you understand from this? Sir, what is this that is being made? 1 into 2 into 3 into 4 into till m. Does he basically understand m factorial? How can you write n factorial? n, n minus 1, n minus 2 till 5, 4, 3, 2, 1. So, what is this basically? m factorial. Perfect. So, what did we say? Log of, who is this whole one? m factorial. And who is in the base? n. Is this super clear? It was a nice question. The question said m factorial. How much is it? n. So, we filled it. How much is m factorial? n, log of a with the same base a is always answer. Good question. Next. Aman walks a certain distance with certain speed. So, he traveled a certain distance with certain speed. If he walks half kilometer and hour faster, then what is this? Half kilometer and hour. Kilometer per hour. So, this is speed. If he is walking faster than half kmph, he takes one hour less. If he walks faster, it will take less time. But if he walks 1 kmph slower, if he reduces the speed by 1 kmph, then it will take more time. He takes 3 more hours. Find the distance covered by the man and his original rate of walking. What are the two options given in the question? This is distance and this is his speed. Is this clear? Sure? The reverse trick is amazing. Look carefully here. Let's do the reverse trick. Do you know this relation? Speed is equal to distance upon time. Let's assume A is the correct answer. So, what is the distance? 36. What is the speed? 4. So, can we subtract time from here? Yes. So, what will be the time? Distance upon speed. 36 upon 4 means 9 hours. Clear? Now, let's check the conditions. What did he say earlier? If he walks half kilometer faster How fast was he walking before? 4 kilometer Half kilometer fast means How much speed did you increase? 4.5 kilometer per hour Did you understand? Half kilometer fast Half means 0.5 He was walking from 4 kmph Half kilometer fast means 4.5 kmph speed He takes 1 hour less So how much time should he take? 8 hours Distance is same So let's check whether 8 hours are coming So we have taken out the time Speed, sorry Distance is 36 Speed is 4.5 If this is 8 Then the piece is looking good 36 divided by 4.5 8 hours have come Can you see that this is less than 1 hour Perfect Now one last condition If he walks 1 km an hour slower Means what is the speed now? Earlier it was running at the speed of 4 So now it will run at the speed of 3, what is the time? 36 by 3, how much is it? 12 hours. So what is the question saying? If it runs slow at the speed of 1 kmph, then how much time will it take? 3 more hours. Is 9 more than 3? Oh yes, this is 9 plus 3. Means guarantee A is the correct answer. Are you understanding? You said normal way, so just I am explaining. Although that is not that interesting, but we can do it. What is happening here? If you understand the story in the whole question, Distance is same in all cases. So, who is playing the game here? Speed and time. So, I said let the speed be x km per hour. And the time taken initially be y hours. So, what is the first case? So, how much distance will it travel if you understand? How much is the fixed distance? What is distance? Speed into time. Speed into time means x, y kilometers. Now, let's take case 1. What did we say in the first case? If he walks half kilometer an hour faster, so earlier the speed was x, now it has increased by half kilometer, so now what is the speed? x plus half. Understood? He takes one hour less. Earlier he was taking how many hours? Y. Now he is taking less than an hour. So how many hours is he taking? Y minus 1. Now what did we say from here? Distance is still the same. What is distance? Speed into time. New speed into the new time will still be the same distance he is travelling. This is our equation 1. Let's do the same for next. What did he say? Walks 1 km and are slower. So, the speed was x earlier, now it has become slow. How much? From 1. So, the new speed is x-1. Now, how much time is taking? It takes 3 hours more. Earlier, it was y. Now, it will travel the same distance for 3 hours more. New speed into new time. Multiplication is the same distance. When you solve equation 1 or 2, you will get xy. You can also get the distance from x into y. Although this is shandar. Next. If alpha and beta are the roots of this quadratic equation, the equation having the roots, very good, so how did we do this? x square minus sum of the roots. Now who are those roots? These two, pay attention. Alpha square upon beta plus beta square upon alpha into x plus product of the roots. So what is the product? Alpha square upon beta into beta square upon alpha is equal to 0. This is the required quadratic equation. Now let's simplify this. x square minus alpha square into alpha, alpha cube. Beta square into beta, beta cube upon alpha into beta. Here, one cut, one cut, what's left? Alpha beta which is equal to zero. So what do we want? We want the product of the roots and we want the value of alpha cube plus beta cube. So what did we say? What is alpha cube plus beta cube? Alpha plus beta the whole cube minus 3 alpha beta into alpha plus beta. Is this super clear? Now, let's take out all the values. From here, what is sum of the roots? Minus b by a. Who is b? Minus 4 by 3. So, how much is that? 4 by 3. Now, what is the product of the roots? c by a. 1 by 3. Is this clear? Fill everything in this. Fill everything in this. Answer. Okay, last one minute. What did you say? α plus β the whole cube means 4 by 3 cube minus 3 into 1 by 3 into 4 by 3. 4 cube is 64, 3 cube is 27. 3 4s are 12, 3 3s are 9. Let's make this 27 as well. 9 3s are. So finally what is coming? 64 by 27 minus, so in denominator 27 is commonly, 9 3s are 27. 12 3s are. 36. So how much is it? 64 minus 36. 28 by 27. Whose value is this? Take everything and fill it. The answer will come. x square minus 28 by 27 upon alpha into beta 1 by 3 x plus 1 by 3 is 0. Do you understand what I did? I filled its value in the numerator. Alpha into beta 1 by 3. Alpha into beta, 1 by 3. If you flip it, you can cut it. 3 ones are, 3 nines are. So, what is the answer? x square minus 28 by 9x plus 1 by 3 equal to 0. There is no fraction in options. So, what did you say? Multiplying by 9 on both the sides. Look, pay attention. 9 into x square, 9x square. When 9 is multiplied, it will cut. So, what will be left? 28x. If you multiply this by 9, then 3 ones are, 3 threes are. So, what is the answer? This is our required quadratic equation. 9x square minus 28x plus 3 equal to 0. Next. The shaded region represents, now see, do it a little smartly. Which area has been said in all three lines? Towards the origin. Can you see this? Everyone is saying the area below, towards the origin side. Let's go. Now let's check what the equation is. Less than 24, does this look like a problem in greater? If it was less, it would be origin side, so it's gone. Less, less, this greater is said, it's messed up. Less, less, less, and in first quadrant, yes, clear? How did you understand? What did all three of them say? Less means towards origin side, that's it, clear. We have done this in MTP, in MTP, PYQ. Come on, the solution of this inequality, can we solve it? How will we do it? Let's cross multiply this. Keep it like this. 5-2x upon 3 is less than or equal to 6.5 is 30. So, x-30 by 6. Is this clear? Now, it looks like a cancellation is happening here. 3 is 1, 2 is 2. 2 is here. 2, 5 is 10. 2, 2 is 4. Here, it is 1. So, x-30 is left. I keep x always positive. So, minus 30 is here. 10 plus 30 is 40. x plus 4x is 5x. Divide by 5, so what do I get from here? x greater than or equal to 40 by 5 is 8. So, we can say x is greater than or equal to 8. Sure. Which option? There is no option. None of this. If there is a misprint in the exam, then this will be x greater than or equal to 8. This is our guarantee. None of this answer. Clear? Next. The simple interest on a certain sum of money is 1 by 25 times of principal. So, what did the first line say? Simple interest is 1 by 25 times the principal. The rate of interest when Rate of interest and time are equal. So we have to find the rate of interest when r and t are equal. Let's use the simple interest formula. Simple interest is equal to principal into rate upon 100 into Time period is also r. P and P are cut. 100 comes up. 100 by 25. And from here, r into r, r square. So how much is this? 4. So what did you say? R square 4. So, R, how much? 2. Clear. Next. At what time a certain sum of money amounts to 400 at 10% simple interest and to 200 at 4% per annum simple interest? So, what is he saying? At what time? Means, how much time has he taken? When some principle amounts to this much. 10% on simple interest and that principle amounts to 200 at 4% rate of simple interest So, in which connection is this happening? Principle, accumulated value or amount So, we have to find simple interest So, I will use this formula a is equal to p into 1 plus i t We will use this twice, pay attention First, amount is 400 Principle is something, what is 1 plus i? 10% 10 by 100, 0.1 in some time This was the first equation. Second, it amounts to 200. Principal same, 1 plus. Rate is 4%, 4 by 100. 0.04 into P. Is this clear? What did he ask you to find from these two equations? At what time? So what will we do best? Divide them. What's happening? P and P are cutting. 200 ones are, 200 twos are. Let's do the calculation. How are you doing it? What's left from here? 2. From here, 1 plus 0.1 into t, 1 plus 0.04 into t. Let's cross multiply. Let's do it quickly. 2 ones are 2 plus 2 into 0.04, 0.08t. 1 plus 0.1t. 1, come here and subtract. 2 minus 1, 1. 0.1t minus 0.08t is how much? 0.02t. So, how much time has come? 1 upon 0.02. Let's divide. 1 divided by 0.02 comes out to be 50 years. Answer. Is this clear? This is your detailed, normal way of sorting. But if you want, you can use reverse trick. I'll just check the C option and tell you. What did we say? Let's assume it's taking 50 years. So we'll see the principle of this, whether it's the same or not. In the first case, what did we say? 400 is equal to principal into 1 plus 0.1 into 50 This will give us the principal And if the same principle is given in the second case Then our guess is correct So, how much is this? 0.1 into 50 plus 1 This is 6 We have to divide it by 400 So, divide equal to into 400 66.66 I am getting some principle Let's check this B case 200 is equal to principle into 1 plus 0.04 into 50 years If it also comes with same principle then it is correct 0.04 into 50 plus 1 3 is coming Now we have to divide it by 200 Divide equal to into 200 Brother it came same So what to understand from this This is correct Hence, C is the correct answer. So, you can check with A, B, S. Clear? Next. 6400 amounts to 7840 in 2 years at simple interest. How much will a sum of 84 invested at the same rate of simple interest amount to in 4 years? So, what does the question require? How much 84 will amount to? So, A is required. Does the principle 84 appear? I am talking about simple interest. a is equal to p into 1 plus it. So 1 plus i, we don't know. So you have to find the I from the first case. What did we say? We said 6400 is 7840 in 2 years. So how much was the simple interest? How much was the simple interest for 2 years? 7840 minus 6400 1440 Is this clear? Now for this use the simple interest formula Simple interest is equal to Principal into the rate that we don't know and how much is this? It is interest for 2 years so we will calculate What did you do? 64 into 2 will be divided so 64 into 2 how much is that? 128 we have to divide it by 1440 divide equal to into 1440 comes out to be 11.25 so rate is 11.25% So, what is I? 0.1125. Fill that in. The story is over. What did you do? 84 into 1 plus 0.1125 into 4. Answer. 0.1125 into 4 plus 1 into 84. 1 to 1.8. Answer. Is it easy? Next. A person gave a loan of Rs. 200 to Mr. X and recovered it at the rate of Rs. 35 for 8 months commencing from the end of first month. What is the effective rate of simple interest? Don't use annuity annuity. Brother, here clearly he is saying that this is not the case of simple interest. What happened here? A person gave a loan of Rs. 200. So, let's say that the principal is Rs. 200. Okay. And recovered it at the rate of Rs. 35 for 8 months. So, every month he paid Rs 35. 8 months he paid Rs 35 each month. So, how much money did he pay in total? So, the total amount that he paid, I can say, 35 x 8, 35 x 8 is Rs 280. Is this clear? The principal amount, the amount he paid, so, how much was the simple interest? Rs 280 minus Rs 200, which is Rs 80. Is this clear? So, we will use the simple interest formula from here. Simple interest is equal to principal into the rate that we don't know into the time period. Now, what is the time period? 8 months. 8 months is equal to 8 times 12. Today, your rate will be over. Do you want to solve it? It is happening here also. 2 times 1 is equal to 2 times 6. 6 times 80. Can you see this? 80 times 6 divided by 8 is the value of r. 8 times 1 is equal to 8 times 10 is equal to 10 times 6 is equal to 6. You can do it in Calcium, but this is being done orally. That was a good question. It was a good question so that you don't assume that it's a loan. I mean, the question of annuity is not present value. If the compound interest on a certain sum of money for 2 years at 4% be 510, then the compound interest is given as a value of 2 years, 510. It's simple interest of same time at same rate. So time 2 years, rate 4%, simple interest to be taken. So for that we need a principle, do you understand? How will we get simple interest? Principle into, we have to take the same rate, time period is 2 years, so we have to find the principle. That will come in compound interest. What did you say? Compound interest for 2 years, what did you say? Principle into 1 plus i square minus 1. 1 plus i the whole raised to n, 2 years. This is 510. We don't know the principle. 1 plus i is how much? 4 upon 100 square minus 1. We have to divide this from here. 4 divided by 100 plus 1 into is equal to so square is done. Minus 1. We have to divide this whole. So divide equal to into 510. How much is that? 6 to 5 is 0 approximately. Now put this here. So what did we do? 6 to 5 is 0 into 4 percent 4 percent into 2 comes out to be Next, on what sum will the difference between simple interest and compound interest for 3 years at 6% amount to 13.77? On what sum? Principle maanga hai. Simple interest and compound interest ka difference for 3 years diya hua hai. Bhai, apna favourite compound interest minus simple interest for 3 years. Pi square into i plus 3. Difference diya hai 13.77. Principle nahi pata. Rate is 6 upon 100 square and here If I put 0.06 plus 3 So how do we do it? We have taken out 3.06 First what we will do in Calcium? 6 divided by 100 into is equal to, because we want its square. Now, how do we multiply that? Into 3.06. So, this whole value is coming. It is being divided here. So, what did we say? Divide equal to into 13.77 comes out to be 1250. Answer. Next. Mr. X bought an electronic item for 1000. What would be the future value of the same item after 2 years if the value is compounded semi-annually? at the rate of 22% per annum. This is the latest MTP or PYQ. So, this 1000 is the present value. And what will be its value after 2 years? It's like a single cash flow. So, we'll use this formula here. Compound interest. Today's present value is 1000. 1 plus. How much is it? 22% compounded semi-annually. So, how will it divide? 200. And how much? 2 years. 1 year 2 half years, so 2 years 4 half years. Let's solve. How did it go? 22 divided by 200 plus 1 into 2 3 4 into 1000. How much is that? 1518.07. Or how could you have done the other way? What will you add to the principle? Rate. 22% is yearly. So how much is half yearly? 11%. And how many time periods? 4. You can do this. Clear? Next. Partners A and B together lent 3903 at 4% per annum interest compounded annually. After a span of 7 years, A gets the same amount as B gets after 9 years. The share of A in sum of 3903 would be... This is a PYQ. It came in June 2014. Okay? What are you saying? Partners A and B together lent 3903. So A lent some money, B lent some money. And how much money did both of them lend? 3903. So let's say, amount lent by A plus amount lent by B is 3903. Okay. What's the next case? At 4%, interest compounded annually. After a span of 7 years, A gets the same amount. Compound interest is happening. A got that much money after 7 years. How much B got after 9 years They get same amount Amount means accumulated value So we will use this formula What did you say? How much money did A get after 7 years? A's principal is PA That is the amount A is investing 1 plus rate is 4% So 4 upon 100 How many years after? 7 years This is the total amount that A received after 7 years. That is equal to the amount that B receives after 9 years. So, B's story. B invested this much money. Plus, the rate is the same. After how many years? After 9 years. This is his total amount. And what are these two amounts? Equal. If you understand, you can cut this. The power of 7 is cut from here. And 2 is left. So, what did you say from here? Amount invested by A. is equal to amount invested by B into, what is this? Square. So how did you do it? 4 divided by 100 plus 1 into is equal to how much is that? 1.0816. Is this clear? What did the question ask? Share of A. So what are you saying? From here, the amount of A upon B is how much? 1.0816 upon 1. How can you do this? Since we know the total share is 3903, you will say this. Share of A is, how much is A's value? 1.0816 upon, total how much? 1.0816 plus 1, so that is 2.0816 into the total value. Just like you do profit sharing ratio in accounts. Let's do it. 1.0816 into 3903 divided by 2.0816 comes out to be exact 2028. I hope you have cracked the exam. We have done the exact same sum in PYQs. X is given as ABCD. The elements of power set of P of X. Power set of any set. If we want to find out the power set, how many elements are there in that set? 2 raised to n. Now, how many elements are there in the set of X? 1, 2, 3, 4. So, how many elements are there in the power set? 2 raised to 4 which is 16. Check which set has 16 elements. It's already looking big. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. So, none of these have any answer. Clear? Null set will be there, A, B, C, D individually, 2 pairs of 2. 3 pairs of 3 are missing, and 4 pairs of 4 are missing. Are you understanding? Power set is collection of all the possible subsets of that given set. So, 2 raised to 14, nowhere else. Next. There is an X set, in which elements are X, Y, W, Z. Y is another set, in which 1, 2, 3, 4 are there. Order pairs are made in H. X1 y2, y3, z4, x4 h is a function, h is not a function, h is a relation So what did the question say? Is there a function or not? We have to figure out What did we see? In ordered pairs, if the first element is repeated, then the function does not remain y is being repeated Even if you see, x is also the first element which is repeated Means definitely h is not a function from x to y Is this clear? Mr. A borrows 5 lakhs to buy a house if he pays equal installments for 20 years. Equal installments. I talked about annuity. For 20 years. And 10% interest on outstanding balance. What will be the equal installment amount? Equal installment amount means annuity. I asked for annuity in the question. He borrowed 5 lakhs. So what happened? Sir, this present value is also like a loan. So, we will use the present value formula. What did you say? 5 lakhs is equal to annuity. And who will get all this? From GT. How will we do it? How did GT do it? It is 20 years. 10% rate of interest is being done annually. So, 10 divided by 100 plus 1 divided equal to 20 times. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. Then you press GT, this will divide, so what did you press? Divide equal to, into 5 lakhs. Comes out to be 58,729.81. Which option? It's 729 but it's 68. It's 58 but it's 239. None of this. Clear? This is not in the pass, there's a huge gap. So we'll do none of this. Next. Invest. Invest 10,000. There will be some man. XYZ invests Rs. 10,000 every year starting from today. If it is starting from today, it means it is an annuality deal. For next 10 years, suppose interest rate is 8% per annum compounded annually, calculate the future value. So, we have to use that formula properly. How much is A? 10,000. What will we write? 8 upon 100 into 1 plus i raised to n minus 1 into 1 plus i. Is this clear? This value is given. We will make it minus 1. So, subtract this from 1. So, what will come? 1.15892500 into What is this value? 1.08 1.08 into 10,000 into, if you flip it, 100 divided by 8. comes out to be 1,56,454.875. These two are also the same options. This doesn't happen in exams. The present value of an annuity which pays 200 at the end of each 3 months. End is ordinary. End of each 3 months for 10 years assuming money to be worth 5% compounded quarterly. What is it to be taken out? Present value of annuity. So, what is the formula? Present value is equal to... Annuity is 200. Now, where is this coming from? From GT. Do it smartly. Look, what is it saying? 5% compounded quarterly. So, what is the rate? 5 divided by 400. Is it clear? Now, what is N? End of 3 months for 10 years. So, what will be in a year? End of 3 months. Means, it is quarterly. Do you understand? Compounding quarterly and that 3 months means quarterly. So how many quarters in 1 year? 4. So what did you say? How many quarters in 10 years? 40. Is this clear? 4 quarters in 1 year, so 40 in 10 years. Now we have to do that here. 5 divided by 400 plus 1 divided. Equal to how many times? 40 times. 1, 2, 3, 4. Like this 40 times. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40. Now you pressed GT. into Kia 200, the value comes out to be value is 6265.38 so this is our present value which option? some weird options are given some weird option 6265.38 there are some errors in MTPs, I think there are more errors actually the value of present value of a sequence of payments the value of present value of sequence of payments of 80 rupees Made at the end of each 6 months and continued forever if money is 4% compounded semi-annually. Continued forever means perpetual annuity. Present value, what is the formula? A upon I. How much is the annuity? 80 rupees. How much is I? 4% semi-annually. So, 4 divided by 200. Flip it. 80 into 200 divided by 4. Super easy. Mr. X invests... We did this today only. This is the same. Just sold. We kept two options same there too. This option is different. Sorry. This option is correct. A company establishes a sinking fund to provide for the payment of Rs. 2 lakh debt maturing in 20 years. So, after 20 years, we need Rs. 2 lakh. Contributions to the fund are to be made at the end of every year. Find the amount of each annual deposit if the interest is 5%. Amount of each annual deposit means annuity is required. You have used the formula of future value. So, what is the future value? 2 lakhs. Annuity, we don't know. How much is the I? 5% into 1 plus I. How many years? 20 minus 1. Let's solve. 5 divided by 100 plus 1 into 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, minus 1 into Paltimara into 100 divided by 5. So, this is the value of the whole term. This is being divided. So, what did you say? Divide equal to into 2 lakhs. 6048. So, I am getting the answer. 6048.517. So this is very close to 6049. Next. A candidate is required to answer 6 out of 10 questions, which are divided in 2 groups, each containing 5 questions. So group 1 has 5 questions. Group 2 has 5 questions. And he is not permitted to attempt more than 4 from each group. In how many ways can he make up this choice? So what to do? You have to attempt questions from each group Not more than 4 So you can attempt maximum 4 out of each group And how many attempts you have to attempt in total? 6. So what happened in one case? You attempted 2 questions from the first group 4 from the second group You can't attempt 1 or 5 because maximum is 4 Or what happened in the second case? 3 questions in this group 3 questions in this group 6 questions in total 4 questions in the first group 2 questions in the second group Are there any other possible cases? 5 and 1 are not possible. So these 3 cases are possible. We just have to solve this. We need 2 questions. Out of how many? 5. So what did we say? 5C2 into 4 out of 5 5C4 Here. 3 out of 5 3 out of 5 5C3 into 5C3 Here. 4 out of 5 2 out of 5 5C4 into 5C2 Does the answer of case 1 and 3 look the same? How did we solve 5C2? 5x4 divided by 2 factorial. So, 10. 5c4 is 5c1. And how much is 5c1? 5. So, how much is this? 50. C option. C case. I mean, third case is also 50. Now, this. 5c3 means 5x4x3 upon 3 factorial means 6. So, how did you do? 5x4x3 divided by 6. How much is that? 10. And how much is this? 10. So, how much is this? 100. We will add all three. Q add. Either you can attempt two questions from first group and four questions from second group or three questions from first group three from second or 50 100 150 or 50 how much 200 is the answer Option may could be naive good, but hey is co-changed car they 200 Yeah, 150 co 200 up kiss Cody change career, but mistake a bias I examine eager sick. They printing mistakes both a MTP man next 1 by 9 factorial plus 1 by 10 factorial is x by 11 factorial. We have done a very similar question, right? This is very easy. How to make a square? 9 factorial is written in the language of 9 factorial. 1 upon 9 factorial. How to write 1 upon 10? 10 into 9 factorial. How to write 11? 11 into 10 into 9 factorial. Now what to say? 9 factorial will be deducted from all the places. If you want it here, then take common first. What is left? 1. Here 1 by 10. Here, x upon 11 into 10 into 9 factorial. Is it clear? cross multiply 10 ones are 10 plus 1 11 by 10 either but if this comes in exams none of this is the perfect printing mistake the value of n-1pr plus r into n-1pr-1 is npr direct our formula or important result we have done right sure now here catch one thing whenever in options both or both a and b something like this happens then check the options npr means n factorial upon n-r factorial means a and b both are correct perfect next If nC7 is nC5, then the value of n is? So what is one option? x equal to y. 7 equal to 5, not possible. Or else, what have we read? x plus y is n. So how much is x? 7. How much is y? 5. So how much is n? 7 plus 5, 12. Next. What are you saying? 2nC3 upon nC3 is equal to 11 by 1. Then the value of n is? Should we do the reverse trick? Let's put the best option A Instead of n, 6 2x6 is 12c3 Upon Instead of n, solve 6c3 How will 12c3 be? Pay attention If you understand orally 12 x 11 x 10 Upon 6 3 x 2 x 1 6c3 6 x 5 x 4 Upon 6 So both will be divided upon 6 So what will actually be left here? 12 x 11 x 10 Upon 6 into 5 into 4. Are you understanding this? Are you sure? The upon 6, 6 of these two is cut, right? 6 ones are, 6 twos are. 5 ones are, 5 twos are. 2 twos are, 4 and 4 are cut. How much is left? 11. We only need 11 is to 1. Your answer is straight up sour. Is it clear? You can do this on calcium also if you want. I have directed it. Okay? Shall we move on? Next. What are you saying? The end term of the series. whose sum to n terms is this? Brother, this is my favourite. Sum of n terms is given. 3n square plus 2n. From here, what did we say? What is the first term? a plus b. 3 plus 2 means 5. What is the common difference? 2 times a. 2 into 3. How much is that? 6. Question asked for nth term. So, what did you say? What will be tn? a plus n minus 1 into d. How much is a? 5 plus n-1 into 6, how much is that? 6n-6, so how much is that? 6n, 5-6 is minus 1. So what's the answer? 6n-1, option D. Next. What is the value of limit? y tends to 2, y2-4 upon y-2. You can do it in two ways. Either the L'Hôpital rule, or this factorization, do you see it? Put 2 instead of y. 2 square. 4, 4 minus 4 is 0. 2 minus 2 is 0. 0 upon 0. In determinant form, you can use L'Hopital rule. How to factorize? A square minus B square. A plus B into A minus B. What factor are you cutting? Now put a limit in this. 2 plus 2. How much is that? 4. How to use L'Hopital rule? Y square's differentiation is 2Y. 4's differentiation is 0. Y's differentiation is 1, 2's is 0, limit as y tends to 2, so limit as y tends to 2, who is left? 2y, 2 into 2, again answer 4, option B. Easy? Limits are easy. We have given x, y, dy by dx. We have done this in RTP, in September RTP. Come on, take dy by dt, the parametric function. 2t, the differentiation of 2 is 0. Take dx by dt, the differentiation of t is 1. So, only 2 ones are left, 5 is 0. We will divide these two. So, dy by dx is 2t by 2. How much is that? T. Sir. Easy. Next. F dash x diya hai. Equation of the curve y is equal to f of x passing through the point is given by. Reverse trick se kar sakta hai. dy by dx nikal ke aur woh point. Ya toh fir normal integration se. I guess, exact yaad nahi aara. MTP1 mein kiya ya RTP mein kiya hai. September ke. Check kariye ka please. And anyways. Normal tarikay se kare. F'x is given. So what did you say? What is f of x? Integration of f'x dx. So whose integration do we have to do? X minus 1. Integration of x, x square by 2. Integration 1, so x will be its integration. Plus c. And instead of f of x, I will fill y. Understood? Integration of x, x square by 2 minus integration of 1 dx. Its integration will be x over plus c. Now, Question says it passes through the point. What did you say? 1 for x and 0 for y. So 0 is equal to 1 square by 2 minus 1 plus c. So what is coming? Half minus 1 plus c. Half minus 1 is minus half. So how much is c? Half. If we put this in here, then how much is y? x square by 2 minus x plus half is the required equation of the curve. which is my option v x square by 2 minus x plus half oh yes next you can solve this normally you have to do integration u into v rule so you can solve normally now in these cases you can use shortcut in an easy way how to do it u into v rule is going to happen the answer that will come how to do it directly orally you know this how to do it orally First, in LAE, who is U and V? Algebraic and Exponential. So, this is our U and this is our V. So, first write U. So, what is written? X square. And we will do the integration of V. Integration of E raised to 3x. E raised to 3x upon 3. Is this clear? Now, we will always add minus sign. Now, do the differentiation of this U. So, what did you do? You did differentiation of this. The differentiation of x square is 2x. The integration of e raised to 3x by 3, note that the lower one is already 3. The integration of e raised to 3x is e raised to 3x upon 3. Are you understanding? Now this is minus, sin alternate, now add plus. Now you said, we will do the differentiation of this. The differentiation of 2x is 2. We will do the integration of this whole. Who is below? 1 by 9. The integration of e raised to 3x. e raised to 3x upon 3 plus c now let's check our answer x square e raised to 3x upon 3 minus 2x e raised to 3x upon 9 plus 2 e raised to 3x upon 27 plus c which is our option checker by 3 9 27 this is the option clear you can do normally this is a little shortcut method only works for u into v rule as it is next Integration of 2x upon 1 plus x square dx. Can you see which integration it is? Hey, can you see the differentiation of the lower one above? So, let's say integration of f dash x upon f of x dx. We will use this formula. What is the answer? Log of mod of f of x plus c. The differentiation of 1 plus x square is 2x. So, what is the answer? Log of mod of 1 plus x square. with the limits 1 to 2. Let's put 2 square 4, 4 plus 1, 5. So, log mod 5 is 5 minus log of 1 square 1, 1 plus 1, 2 mod of 2 is 2. So, what is the answer? Log of 5 by 2, which is my option A. Base in calculus is E. In calculus, like differentiation, integration, log's base will always be E. So, here we have We have covered all 40 questions of Maths. Let's now attack Logical Reasoning. A is 26, Sun is 27, so what is Cat? A is 26. A's position is 1 from the beginning but from the back it is 20. It is 6. So it is counting the position from the back and adding. Can you see the sun? What is the position of the sun from the back? 8. What is the position of the U from the back? 6. And what is the position of the N from the back? 13. So how much is this? 14 plus 13. 27. Perfect. Cat. Cat's position is 24. A is 26. And T is 7. 24 and 26. 50 plus 7, 57. Understood? Count the position from the back. In a certain code, language, premonition is written as something. Premonition. P-R-E-M-O-N-I-T-I-O-N So, there are as many letters as there are digits. So, there is one for each. How will the word monitor be written as? Monitor. M-O-N-I-T O R So first pick this up From fourth to M O N I T So this is M O N I T This is O And this is R 3049208 3049208 Easy This is a piece of cake We have done these questions in MTP and in PYQ as well. In a certain code language, 123 means bright little boy, 145 means tall little boy, 67 means beautiful little flower, which digit in that language means bright? Look carefully. 123 is there, 145 is there, so 1 is common. So which word is common in these? Bright little boy, tall big boy. So boy is common, 1 means boy, did you understand this much? So we cracked 1. Next, what else is being repeated? 1, 2, 3, 6, 3, 7. 3 is being repeated. So what is that word? Bright little boy, beautiful little flower. So what did you understand from here? The 3 is little. How do you understand how to crack? So what happened now? We got 1 and 3 from 1, 2, 3. 1 is boy, 3 is little. So who is 2? Bright. What did the question ask? Which digit in that language means bright? It is... 2 next find the next term of the series do this carefully take some time then watch the solution see here if you look carefully there are additions of 3 terms add these 3 1 plus 3 plus 4 8 take next 3 3 4 8 8 and 7 15 take next 3 4 8 15 12 and 15 27 so now which 3 will be next This 3 means 8 plus 15 plus 27 comes out to be 15. So, here it is a game of 3 3 terms addition. Come on, next. Find next term of the series. 1, 5, 14, 30, 55, 91. Again pause, take time, then watch the solution. Now, what is happening here? You take out the difference carefully. Here, what is the difference? 4. Here, what is the difference? 9. Here, what is the difference? 16. How much is 25 here? How much is here? 30 and 6, 36 So how much difference will be here? Square series is being made 2 square 3 square 4 square 5 square 6 square So this should be 7 square 40 sorry 91 plus 49 Comes out to be 140 Is this clear? Sure? So you have to figure out this by yourself Difference series Alternate Then we will see the addition series Here we have found the difference and pattern and the answer So we have to work out a little We have not come here with a very hard work But with practice it will improve What is blood relation? Q's mother is sister of P This is Q's mother Q's mother is sister of P So this is P's sister And daughter of M So this is M's daughter So these are siblings So there are two children of M, one is Q's mother and one is P's mother. Q's mother is sister of P and daughter of M. So M's daughter is P and these two are siblings. So this is M's child. S is daughter of P. P's daughter is S. S is daughter of P and sister of T. So S is P's daughter and sister is T's. So these two are siblings. How is M related to T? Who is M related to T? T is... P is father or mother of the mother. So, gender is unknown. But, the M is known as grandfather or grandmother. So grandfather or grandmother, we don't know the gender of M. E is the son of A. D is the son of B. E is married to C. So, this is a couple. C is B's daughter. C is B's daughter. So, B is B's daughter, C. And B has another son, D. So, these two are siblings. C is B's daughter. How is D related to E? Who is D of E? D. Who is E? Wife's brother. Wife's brother is brother-in-law. Easy? Next. There are six persons. A, B, C, D, E, F. C is sister of F. C is sister of F. C is sister of F. B is brother of E's husband. B is brother of E's husband. So E hai. E is the brother of E's husband. E's husband is his brother B. B is father of A. D is father of A. And grandfather of F. D is A's father. But D is grandfather of F. Means their child will be F. I don't understand. Question. There are two fathers, three brothers and a mother in the group. All of them belong to the same family. So what is he saying here? There is only one mother. So there is only one mother. So her husband should be A. Only then her child will be F. So this will be like this. Look. And this is also like this. C is sister of F. So I will remove this now. And this. Let's fill this whole here. E's husband should be A. Their father is D and they have two children. C female, gender F. We did this in the same MTP of 1st September. Anyways, what is he saying? There are two fathers. Do you see two fathers? D and A. We have two fathers. D and A. One mother. So we have one mother. Who? E. Three brothers. Now let's talk about brothers. So we will get everyone's gender. Look carefully. Who are the brothers? A and B are brothers? A is B brother One more brother should be F here Only then will we be 3 brothers, do you see? No one else will be a brother Oh, you don't understand Why did you take this male? They are brothers of each other Who else is a brother? Okay, you will also be his brother Only then will we get 3 brothers Understand? Next Let's crack it I hope it's easy Which of the following is group of brothers? Who are the brothers? A, B, F A, B, F It's not like that ABF is brother to each other Group of brothers means Brother to someone or someone How many male members are there in the group? Male, 1, 2, 3, 4 4, 4 How F is related to E? F is E's son It's F What's the difference? Definitely Next Again, I gave more data Read the following information and answer the following questions. A, B, C, D, E, F, G, H, J are 9 houses. So, there are 9 houses. C is 2 km east of B. So, C is 2 km east of B. C is 2 km east of B. A is 1 km north of B. A is 1 km north of B. And H is 2 km south of A. What does 2 km south of A mean? This is 1 km, another km. Here comes H. Did you understand? H is 2 km south. G is 1 km west of H. Who is in H's 1 km west? G. D is 3 km east of G. 3 km east of G. This is 1 km. If we go 2 km more, we see D here. D is 3 km east of G. F is 2 km north of G. 2 km north of G. So this is exactly the same. From here, one goes up and here, who comes? F. Next. I is situated between B and C. E is middle. H and D. So I is in between B and C. So who is in between here? I. 1 km, this 1 km. Similarly, E is between H and D. Between H and D. E. Is this all clear? Is the diagram super clear? What is the question asking? Distance between E and I. Between E and I, it looks like 1 km distance. So, what is the answer? 1. Distance between E and G. Between E and G. How much is the distance between A and F? 2 km Distance between A and F? How much is the distance between A and F? 2 km Is it clear? Easy questions. Just make the diagram correctly You will crack it 7 villages A, B, C, D, E, F, G are situated as follows This is also a game of distance E is 2 km west of B B's Who is 2 km west of B? E. E is 2 km west of B. F is 2 km north of A. F is 2 km north of A. C is 1 km to the west of A. A is 1 km west of C. F is 2 km north of A. C is 1 km west of A. D is 2 km south of G. D is 2 km... South of G G is 2 km east of C G is 2 km east of C Look at this 2 km east of C This one and this one Who is G here? Did you understand? G is 2 km east of C B is exactly between B and E Ok So first let's mix everything I'll keep this aside too. You'll get the full answer. Who? C, A, G, up here, F, this is 1, this is 1, this is 2. Look carefully. D is 2 km below G. How clear is it? And what does it mean? D is exactly between B and E. So, you can see this. D is exactly between B and E. And if the distance between these two is 2, then it will be 1. Understood? E is 2 km west of B Diagram ready What do you want? Which two villages are farthest from each other? F and E We'll discuss the story I'll blow all this up So it'll be easy for us to work out F and E What's the distance of F and E? F and E 2 to 4 So their distance is 4 km B and C D and C These two, these two, these look like hypotenuse. Do you understand? For D and C, these are hypotenuse. This is 2, this is 2. So for D and C, this is happening, if you understand. C, D, 2 km, 2 km. What is hypotenuse? Root of 2 square plus 2 square. So root of 2 square plus 2 square. 4 plus 4, 8. Root of 8 is how much? 2.82. So this can't be the answer. F. F and B, G and E. Can you see G and E? G and E. So, this G and E is like this. This 2, this 1. So, this will definitely be small. Let's check F and B. Let's check F and B. F and B. Oh, man. This distance. Should I change the color here? No, this is fine. F and B. This triangle is being made. How much is this? 2, 2, 4. And this is 2. So, if you talk about F and B, This is the distance, 4. Can you see? 2, this is 2 and this is 2. So what happened? Root of 4 square plus 2 square. 4 square is 16, 2 square is 4. 16 plus 4 is 20. 20's root comes out to be 4.47. So definitely, which one is the biggest? F and B are the farthest from each other. So you can say this is the answer. Very good question. It's a game of hypotenuse. If SE is called EAST, NW is called WEST, SW is called SOUTH then what will be called? Be careful This is our normal one North, East, West, South Now what is happening? SE is called EAST We will make all these with normal black This is SE North East, North West, South West. South East is called East. South-East is now East Means, I can say this 45 degrees anti-clockwise will remain South-East is now East So, 45 degrees anti-clockwise travelled Look ahead North-West is called West North-West is now West So, that is also visible 45 degrees anti-clockwise travelled So, in short, we have to rotate 45 degrees anti-clockwise in every direction South-West is called South South-West is now South Again, 45 degrees anti-clockwise What did the question ask? What will North be called? North B 45 degree anti-clockwise. What will it be? Northwest answer. Did you understand how I tackled it? A, B, C, D, E, F, G are sitting in a row facing North. North means facing this side. Come on, carefully. F is to the immediate left of G. E is fourth to the right of G. Fourth to the right of G. 1, 2, 3, 4. Who is on 4? E. I. C is neighbor of BD. C is the neighbor of BD. Like this or like this. It can be in any order. Person who is third to the left of D is at one of the ends. In the third left of D. Now, how is the story? Understand carefully. Let's take this diagram. If you, understand carefully, put these three here like this. Who? B, C, D. What did the question say? There is an end on the 3rd left of D. 3rd left of D. 1, 2, 3. This is not dead end. So, this arrangement is wrong. So, this is wrong. So, how will you put the guarantee? D, C, B. And count the 3rd left of D. 1, 2, 3. There is a person here who will be on the dead end. Clear? Now, who are left? A, B, C, D, E, F, G. A is not filled yet. B, C, D, E, F, G. Who is the guarantee here? A. So, what is the arrangement? A, F, G, D, C, B, E. First question. Who are the neighbours of B? Who are the neighbours of B? C and E. E perfect what is the position of A? A is at the left most position extreme left who are left of C? A, F, G, D, A, A, F, D and here is misprint this should be G then answer would be D. So again there are misprints here. Next. 8% E, F, G, H, I, J, K, L are seated around a square table. 2 on each side, there are 3 ladies who are not seated next to each other So first make a square table 2 on each side 2 on each side 2 on each side Like this J is between L and F So say J is between L and F L and F can be swapped too, so let's make one more diagram I said J is between L and F. So J is between L and F. This can also happen. G is between I and F. G is between I and F. So if F is here, I will be here. Then only G will be between. So same. F is here, I will be here. Then only G will be between. Next. H, a lady member, is second to the left of J. J's second left. 1, 2. Here is H, lady. In the inward phase. J's second left already has G. It won't go down. F a male member is seated opposite to E a lady member. F a male member is seated opposite to E a lady member. There is a lady member between F and I. There is a lady between F and I. So, there are three ladies and they are not sitting together. So, the diagram is ready. Let's see who are they. E, F, G, H, I, J, K, L. Kataam. Questions kya bol raha hai? Who among the following is immediate left of F? F, K, immediate left mein J hai. Toh ye wala answer J hai. How many persons are seated between K and F? K aur F. K aur F ke beech mein 3 log hai. Yeh side dekhe. K aur F ke beech mein 3 log hai. So, the option is 3. either up now LR cut about a let's now crack statistics for a moderate This is skewed distribution. Quartile deviation and standard deviation are related by 6QD is for HD. Option is given in the language of HD. 6 by 4 will be 3 by 2QD will be HD. On this option. HD is 3 by 2QD. Data are said to be dash if the investigator himself is responsible for collection of data. It's called primary data. Again, same question in MTP1 Which of the following is suitable for cumulative? O give The left part of the table Description of rows, who provides? Stubb Super easy, clear The following data relates to the marks of groups of students How many students got more than 30? Module question What is given here? Below 30 is 65 Below 50 means this is total N So what can you say? Out of 100, below 30 is 65 students. So, how much will be above 30? What did you say? 100 minus 65, which is how much? 35. The probability of a blue chip company is shown... It should not be probability. It should be profit. You asked about the model. The profit of a blue chip company will be shown by bell-shaped curve. If all the observations are increased by 6, Then the variance of the series will be If we increase it by 6 If we change the variance of the series by 6 It will remain unchanged Again super easy questions If the mode is 18 and mean is 24 Median Exact same MTP1 question What are we doing? We are using the normal relation Mean minus mode is 3 times Mean minus median This is 6 6 divided by 3 is 2 So, 24 minus median is left. So, what is the median? 24 minus 2 is 22. This was also in the same MTP1. If first quartile is 142, Semi-interquartile range is 18, value of median. What happened again and again? Q1 142. Semi-interquartile range. Semi-interquartile range means quartile deviation is given to you. You were asked for median. How will the median come? Q1 plus Q3 upon 2. We already know Q1. We need... Q3's value. Get it from here. 2 into 18, 36. So Q3 minus 142 is 36. So how much is Q3? 142 plus 36 is how much? 178. Now put it here. Add 2 in 178, 180, 180 and 140. 142 plus 178. 360 divided by 2. 160. Easy right? Stats toh fatta fatta hoga. Which measure of dispersion is not affected by the presence of extreme observations? Range, MD, HD, QD. Extreme observation se quartile deviation ko nahi farak padta. If standard deviation of x is 3, what is the variance of phi minus 2x? Toh sabse pehle di kya kar de? HD of x terko pata hai 3. Idhar se ye y le le, phi minus 2x. Toh aap kya bol sakte hai? Standard deviation of y kitna hua? X ka coefficient kaun? So mod of minus 2 into standard deviation of x. So that is how much? 2 into 3, 6. So what is variance of y? 6 ka square which is 36. Answer next. If x and y are related like this, standard deviation and arithmetic mean of x are known, then the coefficient of variation... Kitne baar same printing mistake yaar. Coefficient of variation of y maangaya usne. It will give clearly in the exam. This MTP is being copy-pasted, so I am assuming that it is not adding off y. What will be the CV of y? HD of y upon mean of y into 100 and you will get all this from that relation. We need y bar, so what did you say? You put it here. y bar is 2x bar plus 5 and how much is x bar? HD is 5, mean is 10. So 25 is here. Let's find out the standard deviation of y. What is the coefficient of x? 2. x is the standard deviation of x. So 2. The standard deviation of x is 5. 2 x 5 is 10. Let's fill it. 10 by 25 into 100. 1 x 4 is 14. 40. Next. If the relationship between two variables is given by like this. u bar is given. We need v bar. Put it here. 2 into... 10 plus v plus 7 is 0 20 plus 7 27 so v bar kitna minus 27 udhar chaayega x and y related like this mode of x mode of same change of origin change of scale mode of x dia 23 dal bhai 23 minus y minus 10 ya aisa likh deva y ka mode minus 10 is 0 23 minus 10 13 and minus y goes there so plus y, so what is the mode of y? Very easy statistics till now There are two boxes containing 5 white and 6 blue balls So first box contains 5 white and 6 blue balls Second box contains 3 white and 7 blue balls respectively If one of the boxes is selected So first you are selecting a box at random And a ball is drawn from it Then the probability that the ball is blue So what is happening now? What are you doing here? First one box select kar rahe hai, phir ush maise blue ball nikal rahe hai. To kitne boxes ke options hai? 2. Maine bola merko first box chahiye. To first box nikalne ka probability kitna? 1 by 2. Kaise? There are 2 boxes, but I want the first box. To pehle maine first box select kiya. Uski probability kitni? 1 by 2. And, now I am drawing a blue ball out of it. Total kitne balls? 11. Apun ko chahiye blue. To probability kya hogayi? First dabbe me se blue nikalne ki? 6 by 22 Or you draw the second box first So that's 1 by 2 again Into now draw blue ball from it Total balls 10 Blue 7 So what's the priority? 7 by 20 So what's the answer? 6 by 22 plus 7 by 20 Cross multiply it Do it like that Take it out in calcium Take it out in calcium Whatever Take it out in calcium 6 divided by 22 M plus 7 divided by 20 m plus memory recall kiya merko answer ara 0.62272 let's check 115 divided by 227 no 83 divided by 250 no 137 divided by 220 yes c option correct ya to cross multiply kar diya 620 120 22 into 7 154 22 twos are 440 So what is this? 154 plus 120 is 274 by 440. Cut from 2, 137 by 220. Again our option C. So you do in calculator or you solve the fraction. Whichever way you feel good. A box contains 5 white and 7 black balls. So 5 white and 7 black balls. Two successive draws of three balls are made with replacement without replacement. The probability that the first ball would produce white balls and second draw would produce black balls. What does it mean? You took out 3 balls with your hands, those 3 balls are white. Then when you took out the next 3 balls, those 3 balls are black. What did the first condition say? With replacement. What does it mean? Try to understand it carefully. How many balls are there in the box? 12. How many do you have to take out? 3. So you first drew 3 balls. Those 3 balls should be white. So what did you say? 5C3. So the probability of taking out the white ball is this. Now it said with replacement. So the 3 balls that came, those 3 balls are again put in the box. So how many balls are there in the box? 12 balls. Still we have to subtract 3 from 12. Now when you draw the second one, all the 3 should be black. So 7C3. Is this clear? This is the probability with replacement. If you understand what is 5C3? 5x4x3 upon 3 factorial. 12x11x10 upon 3 factorial. It will die. So what are we writing? 5x4x3 and this is 12x11x10. In the same way, 7 x 6 x 5 upon 12 x 11 x 10. Is this clear? You can do katakuti or you can multiply it. Multiply it below. Like this. 12 x 11 x 10. The answer is x equal to. Correct? Because it has a square. 12 x 11 x 10 x 12 x 11 x 10. So, x is equal to. Now, this is dividing. So, divide it equal to. Now, multiply all the numerators. 5 x 4 x 3 x 7 x 6 x 5. How much is that? 0.00723, check which option bro 6 divided by 321 6 divided by 321 No 1 divided by 20 No 35 divided by 144 No 7 divided by 968 Yes So this is actually 7 divided by 968 Let's do fraction wise, pay attention 4 3's are 12 5 ones are 5 twos are. So, what is left from the first fraction? 1 by 22. Okay. Here. 6 ones are 6 twos are. 5 ones are 5 twos are. What is left from here? 7 upon 2 to the 4 into 11 44 so now check 22 into 44 is how much 968 so easy answer but just for your reference let's do with without replacement With replacement, without replacement. Now what is happening? The first one is the same. Still, there are 12 balls. 3 out of them are needed. The first draw should be of white ball. Now, what do you mean by without replacement? The 3 white balls that you have taken out, You threw it to the side Now how many balls are left in the box? There are 9 left, you want 3 Understood? 3 white balls are not replaced You threw it to the side Now there are 9 balls in the box, you want 3 Now what is the probability you have to get? Black ball Black is 7C3. To solve this, your answer will be 5 by 264. Sure. Clear. Good question. Next. Priority of A, priority of B, priority of A intersection. B is given. We need its probability. We have done so many questions based on this. What? How did you write this? Probability of A dash intersection B dash upon probability of B dash. Then how did you write this? Probability of A union B the whole dash. Oh, yes, this is not it. Sure. Now what will we do? Complement law. Here we used D-Morgan's, now complement. What did we say? 1 minus probability of A union B upon 1 minus probability of B. What do we know? We know B. So, 1, you can see this. What will be the denominator? 1 minus 1 by 3, which is 2 by 3. Let's solve this. How will the probability of A union B come? Probability of A plus probability of B minus probability of A intersection B. Let's do that in CalciDirect. 1 divided by 2, M plus. 1 divided by 3, M plus. 1 divided by 4, M minus. Memory recall. This is coming out to be 0. Same values are always coming. Same question. So put this here. Doing this negative. Did plus 1. Now this turned. So into 3 divided by 2 comes out to be 0.625. Which option? 1 by 2 is not possible. 7 by 8 is also not possible. 5 by 8 is possible. Yes. Next. Or else you can play with fractions. What is the question? X and Y stand in a line with 6 people. What is the probability that there are 3 persons between them? Good question. Although we have done this in MTP 1 of June 2022. Or RTP 1 of September 2024. We have not done this twice. Let's crack it. MTP 1. What is the question? X and Y stand in a line with 6 people. What is the probability that there are exactly 3 persons between them? I am teaching you from 2 methods. Pay attention. How many people are there with x, y? 6 people. So without condition, how many people are there in total? 8 people. 8 people can be arranged in 8 places in 8 factorial ways. So this is clear. 8 factorial is in sample space. Now let's talk about our probability. Whose? We will take out n of a. Okay. n of a means 3 people. Between x and y, exact 3 persons. But how are they doing it? Now understand the whole story. Please. What is happening first? 3 people are needed between X and Y So I will say 3 people are between X and Y So these 5 people are a unit We will do it later 3 people are needed between X and Y Now who can be these 3 people? How many people are left? 6 Any of the 6 can be 3 So what did you say? Who are these 3 people coming in between? So what did you say? 6C3 Out of 6 people You have chosen 3 people who will come in between Now, if we are arranging in a line, then the permutation, the arrangement will matter. The 3 people out of 6 that you have taken, those 3 people can be arranged in 3 factorial ways in these 3 places. So, how can you say? 6C3 into 3 factorial. Or what did you directly say? 6P3 cube. Out of the 6 people left, arrange 3 people in these 3 places. Is this clear? So, 6P3 means the 3 people which are between X and Y can be arranged in this many ways. Oh! Question is, what did you say? X and Y stand in the line. In between them, there are three people. Did you say order? First X will come, then Y or YX. No. So what does this mean? X and Y can be arranged among themselves in two factorial ways. So what happened in two factorial ways? X and Y are arranged among themselves. Now the story is not over. We just talked about these five people. Now there are three more people left. Do you see? This five and remaining three more. So now what we have to do? We have to arrange this 5 along with the other 3. So what did you say earlier? Sir, make them one unit. You tied them together. Why? There should be 3 people between X and Y. So this should be like this. Now this one unit. So this is one unit. With whom do I have to arrange this? Now there are 3 left. This 3, 2, 5. How many remaining? 3 more people are remaining. So there are 4 units. This is 1 unit and 3 people are left. So 3 plus 1 is 4. 4 units can be arranged in 4 places in 4 factorial ways. So what is the story now? What can you say? What is n of a? 3 people have to be arranged in 6p3 ways. x and y can be arranged in 2 factorial ways. And, we have to arrange this unit with the remaining ones in 4 factual ways. We don't have to do anything by opening a gap. I have already done that. The 3 arrangements in between and x and y are the same. Why did we take this unit? Because we have fixed 3 people between x and y. Now, those 5 people will be arranged with these 3 people. This is the total number of ways of our event. Now, what is the probability? So what is the probability? n of a, what is that? 6p3 into 2 factorial into 4 factorial upon 8 factorial. What is 6p3? 6 into 5 into 4 into 2 into 4 factorial and what is 8 factorial? 8 into 7 into 6 into 5 into 4 factorial. Clear? Cut it, cut it. 6, 6, 5, 5, 4, 2, 8 and 8 cut. See what is left? 4 by 7. Clear? This is one of the way of solving. There is another way which is very good if you click. Look, understand this carefully. If you are arranging x, y and 6 people in a line, then what possibility can there be? How many people are there between x and y? Look, x and y are stuck together and 1, 2, 3, 4, 5, 6. So there are 0 people between x and y. Are you understanding? Or what can happen? There is 1 person between x and y and 5. Can you see? So, there are 0 people in X, Y, 1 person in X, Y, 2 people in X, Y, 3 people in X, Y, 4 people in X, Y, 5 people in X, Y, and 6 people in X, Y. There can be more people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X, Y, and 6 people in X Or you may ask why did you take xy? Let's take yx So it will be like this, 0 in between yx and 1 in between yx So how many of those? 7 more So this 7 and yx's 7 Total 14 We are interested in 3 in between xy So this 3 or y... Sorry This 1 or 1 in between yx So 2 is 2 by 14 Again 1 by 7 which is our answer Clear? I guess this is much better if you can understand or else That is the detailed one. Next. For two events A and B, probability of A union B is probability of A plus probability of B only. Means, who is missing here? Probability of A intersection B is 0. Can you see? When does that happen? A and B are mutually exclusive. That is when the probability of intersection is 0. If X is a binomial variable with parameters N and P, then X can assume Any value between 0 and n? Okay. Any value between 0 and n? Both are inclusive. Yes, 0 can be and n can be. The value of x. Let's find out. Probability of x is 0. If this coin is faked 5 times, you can say that the probability of x is 5. So it is inclusive. C option, any whole number? This is a much better answer. Any number between 0 and infinity? No. This was very correct. B option is correct but C is even better. So C is the correct answer. Next, the PDF of a normal variable x is given by, so PDF of normal variable x, what happens? 1 upon sigma under root of 2 pi, e raised to minus x minus mu the whole square upon 2 sigma square, sure, which option? See, here sigma is missing, it cannot happen, here also sigma is missing, it cannot happen, 1 upon sigma under root of 2 pi, this is absolutely perfect, what is the problem? This should be, between minus infinity to infinity. So, either it will be B or none of this. If this is a printing mistake, then B is the answer. If this is not a printing mistake, then I will click none of this Q. The range of x is infinity from minus infinity. Continuous variable. Next. What is the first quartile of x having the following PDF? There is a problem here too. It should be infinity from minus infinity. What is the formula of the first quartile? mu minus 0.675 sigma. So you have to find mu and sigma. You can find mu very easily. x minus mu looks like this. This e on top of that. x minus mu the whole square upon 2 sigma square. So, what is mu? 10. Now, what do you have to find? Sigma. How will sigma come? See, who is this 72? 2 sigma square, understood? 2 sigma square 72. So, sigma square is 72 upon 2, 36. So, sigma is square root of 36, 6. Just solve it. 0.675 into 6. Whatever the answer is negative, add 10 to it. 5.95, I am getting tired. Next. The average weekly food expenditure of a group of families has a normal distribution. So, the question is about the normal distribution. Mean is 1800. Standard deviation is 300. What is the probability that out of 5 families belonging to this group, at least one family has a weekly food expenditure in excess of 2100. So, what is the probability of this? Excess of 2100. This is a little unique. This is a little unique. This is a good question. Pay attention. Let's keep the red marked one aside for a while. What do we know? We know mean is 1800. We know standard deviation is 300. So, what can we say? What is Z? X minus mu by sigma which is X minus 1800 by 300. Is this clear? Question me, kiska priority maanga? Excess of 2100. Toh aapne kya bola? Probability of X greater than 2100. Is this clear? How you can say that? Probability of Z greater than X ke jaga 2100. 2100 minus 1800, 300. 300 by 300 is 1. So probability of Z greater than 1 means 1 minus probability of Z less than equal to 1. And does this question have value? This is F means this is phi of 1. So what did you say? 1 minus 0.84 which is 0.16. Is this clear? So what did you understand from here? Probability that the food expenditure is is excess of 2100 has a probability of 0.16. But what did the question ask? What is the probability that out of 5 families belonging to this group At least one will have this So now what is it saying? Take n, phi And give me the probability of this At least one family has So x is Or we will take a new variable x goes there So here we say y y greater than equal to 1 What probability do we have to get? More than 1 Which this probability is following So now what is needed here? This is called probability of success. Now I am using binomial distribution. Why? The question asked whose probability? If there are 5 families, if at least one of them has an excess of 2100, then find out their probability. Till now, whose probability have we found? If our expenditure is greater than 2100, then the probability of its expenditure is 0.16. But whose priority is to be taken out? If there are 5 families, at least one of them, at least one of them, has expenditure greater than 2100. Clear? So that will come from the binomial. So how did we do this? P is 0.16. So Q is 0.84. Let's put here. How will you do greater than equal to 1? 1 minus priority of Y less than 1. Less than 1 means who will work? We have changed the variable x to y. What will be 1 minus 5c0 into p raised to 0 into q raised to n minus 0. How much is this? 1 minus 5c0 is 1. This is also 1. 0.84 power 5. 0.84 into 2, 3, 4, 5. Subtracting from 1, we get 0.4182 something. Subtract this from 1 0.5817 is the answer 0.5817 which is option B. So it's a good question. Normal meh binomial. But it's a good question. Next. What is he saying? X is a binomial variable such that 2 times probability of X equal to 2 is probability of X equal to 3. Mean of X is known to be 10 by 3. Mean means the value of NP is 10 by 3. Next, what would be the probability that x assumes at the most the value 2? So, what is the probability? At most 2 means 2 or less than 2. At the most means more than 2 or less than that. Now, first of all, we will need NPQ. It will come from this relation. Pay attention. Pay attention. 2 times, probability of x equal to 2 means? nCx p raised to x q raised to n minus x. Is this clear? Here. Priority of x equal to 3. nCx p raised to x. x is 3. q raised to n minus x. Now, what all is being cut? Pay attention. If p is cut square, then here 1p is left. Now, pay attention. Instead of n, let's assume value 5. For this piece. 5 minus 2 is how much? 3. 5-3 is how much? 2. So, here 1 power is more. So, this cut with this, Q is left. Are you understanding this? You can understand this by looking. N-2, N-3 has 1 difference. This is big, this is small. If you cut, then 1 will be left here, Q. Now, here 2 is left. Now, Nc2. Look, if I said 10c2, then how do you write? In shortcut, you reduce 10 twice and below 2 factorial. How do you write Nc2? N twice. Then once less, n-1 So write 2 terms, below 2's factorial would be 2 How will you write nc3? There's a little gap 3 terms, n, then n-1, then n-2 And below 3's factorial, 6 Is this super clear? 2 and 2 cut, from here n, from here n cut n-1, n-1 cut Now what's left? Pay attention 6 goes to q 6q is equal to What's left there? n-2 into p bracket open n into p, np 2 into p, 2p. Now, np, do you know how much? 10 by 3. Now, what did we say? What is q? 1 minus p. So, 6 into 1 minus p. Let's solve. 6 ones are 6. 6 into p, 10 by 3 minus 2p. Let's take 10 by 3 here. So, 6 minus 10 by 3. 6p minus 2p, 4p. 3, 6, 18. 18 minus 10. 8 by 3 is 4p. 4 ones, 4 twos, so P is 2 by 3, so Q is 1 by 3. We need N. How do we get N from here? Pay attention. Put N here. N into P. P is 2 by 3. E is 10 by 3. Are you understanding this? NP is 10 by 3. 3, 3 cut. 2 ones, 2 fives. So N is 5. Now whose priority is asked? At the most, 2. Good question. What did you say? Probability when x is 0, probability when x is 1, plus probability when x is... How did you do it? Phi c is 0. What is your P? 2 by 3 raised to 0, 1 by 3 raised to 5. Probability when x is 1. Phi c is 1, 2 by 3, into 1 by 3 raised to 5 minus 1, 4. Phi c is 2. 2 by 3 square and 1 by 3 cube. Let's go. Phi C0 is 1. This is also 1. Here, we need 1 by 3 power 5. Phi C1 is 5. 2 by 3. Here, we need 1 by 3 power 4. Phi C2 is 5 into 4 divided by 2. So, 20 divided by 2 is 10. Into 2 by 3 square and 1 by 3 cube. We have to solve the normal fraction. Do the fraction. Do it in the calculator. 1 by 3 power 5, not into 1. I did 1 divided by 3 into power 5. Into 2, 3, 4, 5. M plus. Let's do this. 1 by 3 power 4. 1 divided by 3 into 2, 3, 4. Into 2 into 5. Into 2 into 5 divided by 3. Clear? Now, carefully. This is also M plus. This. 1 divided by 3 into 2,3 cube is here, now carefully we cannot do the square of 2 so what we did, into 2 into 2 3 x 10 divided by 3 square 9 equal to m plus. Memory recall. I got the answer. 0.2098. Let's check. 16 divided by 81. No. 17 divided by 81. 2098. Exact same. You can solve fractions too. For a standard normal distribution, the points of inflection are given as standard normal distribution means mu is 0, standard deviation is 1, so points of inflection is minus 1 and if two random variables are related like this, standard deviation of y is easy. Coefficient is minus 3, so what did you say? Standard deviation of y is what? Mod of minus 3 into standard deviation of x. So how much is that? 3 times standard deviation of x which is our option b. Yes, if you don't know mod, this is the question for you. Probability of getting a head when two unbiased coins are tossed simultaneously. So, you have thrown two coins simultaneously. So, what will happen in sample space? HH, HD, TH and TT. What are you interested in? Getting a head. If this is one head, then what is the probability? 2 by 4, which is half, which is 0.5. Next. If x and y are random variables having expected value as 4.5 and 2.5 respectively, expected value is x minus y. You know what addition is? e of x plus y is e of x plus e of y. Similarly, subtraction is e of x minus e of y. 4.5 minus 2.5 which is 2. Easy. The interval This M is Mu and S is Sigma. Mu-3 Sigma Mu plus 3 Sigma How much does it cover? 99.73% Or what did we say? All but 0.27 So, except for the whole 0.27, we'll modulate. What is a spurious correlation? Bad relation, very low correlation, correlation between two variables having no casual relationship. Perfect. Direct theory based. Two lines of regression become identical when r is equal to plus minus 1. Can you see? a and b. Two lines of regression are given. If the variance of x is 25, standard deviation of y is required. Good question. Variance of x is 25. So, how much standard deviation will be obtained from this? 5. What did you require? Standard deviation of y. Which game is this question from? Look at this. What is byx? r into sigma y upon sigma x. You will get these two from the regression line. Sigma x is known, sigma y will come. Let's go. Case 1, case 2. First, if I assume this as y on x, then it is said 10y is minus 8x plus 25. So, y is what? Minus 8 by 10x plus 25 by 10. So, for now, let's take byx. So, that will be our x on y. 16x is minus 5y plus 12. So, x is what? Minus 5 by 16. Byx, bxy. But how will you understand this case correctly? Byx into bxy's product should be less than 1. Let's check. Minus 8 by 10 into minus 5 by 16. 8 ones are 8 twos are 5 ones are 5 twos are. How much was it? 1 by 4 less than 1. Means this assumption is definitely correct. Who would this be? R square would be. Byx into bxy, 1 by 4, which is r squared. So what is r? Square root, but since both are negative, r will also be negative. Fill it here. Byx, how much? Minus 8 by 10. r, how much? Minus half. Sigma y, we don't know. Sigma x, phi. Pay attention. Minus, minus, cut. Two phis are 10, and this 10 cut. Understood? Phi twos are 10, and this 10 cut. So how much is sigma y? 8, answer. Clear? Good question. If the regression line of y on x and x on y both are given, arithmetic means of x and y. We will do reverse trick. Put this option. 1 instead of x, 1 instead of y. You can do in calc, but I guess this is easy. 2 ones plus 3 into minus 1. So how much is this? 2 minus 3 minus 1. Oh, matched. Let's check with the second one. 5 into 1, 5. plus 6 into minus 1 is minus 6, so 5 minus 6 minus 1 matched answer. Samjhe na? If y is equal to 3x plus 4 is the... Regression line of y on x and the arithmetic mean of x is? This is minus 1. So, I think x bar is minus 1. What is the arithmetic mean of y? We will put it in this. Why? How do we solve regression line? If we solve both simultaneously, we get x bar and y bar. If we get x bar, if we put it in any equation, we get y bar. So, we did that. So, how will y be? 3 into x bar is how much? Minus 1 plus 4. So, minus 3 plus 4 is how much? 1. Clear? If it is plus 1, then the answer is 7. But I guess it is minus 1 mostly. When the product of price index number and the quantity index number is equal to corresponding value index number, who says this? Factor reversal test. Clear? The formula for conversion to current value. What is deflated value? Current value upon price index of the current year don't know be current current Perfect next if the index number of prices at a place is place in 2008 is 280 2008 as base here then the prices have increased by 2008 make it not a base here at 100 2018 make it now. Yeah 280. So kitten is a bird. I got one eighty percent say buddy So many questions we have done in additional question bank, PYQ, same type of questions. In 1980, the net monthly income of employee was 800 per month. The consumer price index was 116 in 1980. It rises to 200 in 1984. If he has to be rightly compensated, the additional dearness allowance to be paid to the employee is... So how did we do this? 1980-1984. Here is 160 CLI. or CPI and here 200. Here it is getting 800, here how much? So we will cross multiply. How? 200 into 800 upon 160. This comes out to be 1000 rupees. So how much additional do we have to give? 200 rupees. Next. Consumer price index number goes from 110 to 200. From 110 to 200. The salary of a worker is also raised from 325 to 500. Therefore, in real terms, to maintain his previous standard of living, he should get an additional amount of. So, he is getting 500. But how much should he get more than 500 so that same standard of living remains? Last question. What did you say? How will the x come? 325 into 200 upon 110. Look, pay attention. 325 into 200 divided by 110 comes out to be 590.90. So, in real terms, how much extra should we get? 90.909. So, 91. Clear? This is 9090. So, 90.91. Is this super clear? Just like the last question. The prices of a commodity in the year 1975 and 1980 were 25 and 30 respectively. Taking 1980 as the base year, the price relative is. So, what is price relative? Price of the current year upon the base year. Here, base year is 1980. So, current year is 1975. So, 25 upon 30 into 100. It's important. Because base year is 1980. Again, addition question. 2500 divided by 80. Sorry. 2500 divided by 30. 83.33. Clear. Here, our whole mock test paper ends. I hope you enjoyed it. Many printing mistakes but some questions were unique. I hope you cracked it. Very less time. Please prepare very well. Rone ga nai, phorne ga. Perfect. All the best. Jai Hind. Intro Music