NIOS 12th Mathematics Sample Paper Solution 2026–27 Score 95+ in Exam

[00:00:00]Hello and how are you class 12th And welcome back to your mathematics class.

[00:00:06]First of all, let me tell you this I would like this class to be The class is going to be held on your demand.

[00:00:11]Your demand was that Sir, please bring the sample paper which When has NS given that sample paper?

[00:00:15]We will have to solve it with some technique or the other.

[00:00:17][MUSIC] Tell me. Tell me the tax. So your Fulfilling the demand, brother, I have brought it.

[00:00:21]Today you have a maths sample paper friend.

[00:00:24][music] Now NS has already given this sample paper is authorized on its official website on the website. Right? You can get it in one click You can download it by going to the website Or a link to it in the description.

[00:00:36]You will get it. And also keep in mind one We must not miss our question. even one I will not leave any questions. whether anyone Why not a section? You do this once See. Right? So this is your paper.

[00:00:45]The entire portion in its starting It happens. It is written in section A and Section B. Now how do we get this entire paper?

[00:00:51]Want to tackle? So in the same way I I will solve the sample paper. suppose I I am a 12th class student and I I am going to give exam in NOS. Now I Went to give the exam. This is what is in front of me The exam paper, this exam The paper is in front of me. Now I this How will I tackle it? How do I solve this I believe that my marks are above 75%.

[00:01:09]Come on. So I'll pick that up first.

[00:01:10]Sections in which I am more confident.

[00:01:13]In which more average questions are being asked Are. Not even average, easy to average. Meaning Whose level is easy to average. So, That is the post-MCU questions. Attention Putting you first Never Before MCQs Have to pick up. Note this down. Why not Want to pick up? Dude will you pick up MCQs. Sometimes You will get stuck in some question. someone You will feel stuck in the question. Our You will take ego somewhere. You will tell me now brother I will show you by doing it. Something will happen. Now Brother, I will do it right now. Oh, he's just hanging on That question is for you, friend. MCQs Only he will be able to ask questions who The concepts will be clear. Crystal Clear Surely it will be. But if you assume it is simple You are an average child and you are good If you want to score in maths then you First take the questions that you know so that He can build your confidence.

[00:01:54]If your confidence builds then your paper will be It will automatically go well and you will score You will be able to do it. Ok? so quickly quickly Let's start without any delay.

[00:02:04][sound of clearing throat] So you look at this First of all, the section I start with I am here son, that is your after MCQ.

[00:02:11]As I told you about section A After MCQs which part hota hai 21 to 24 It has been said that the questions will be of two marks each.

[00:02:19]One question two markers means that You don't have to write much in it.

[00:02:23]You don't get more marks by writing.

[00:02:25]Accurate writing gives more marks Or if I get full marks, I will say okay.

[00:02:30]Come on, this is the first question brother.

[00:02:33]We have to match from here to here, okay?

[00:02:34]Let's do it, look, it said this matrix I have it, look at me, I don't I think it needs to be written about, but then See also a 2/3 matrix is ​​a 3/2 is the matrix of. First of all, both Can be multiplied. Check it because The first column is the second row.

[00:02:50]The numbers are equal. So a * b can be Brother. and if I do a * b then its The result will be a matrix of 2 / 2, right? 2 / Of 2. So what will be a * b? 2 / 2 of It will be a matrix. Similarly, what is b * a will be? When I b*a now I'll do these 2 And if I check 2 then it will become 3/3.

[00:03:09]Right? What? It will become 3/3. Hey look at it I took B first. Ok? Then I A taken. So now I'm going to look at this 2/2 This may or may not be multiplied.

[00:03:20]It may be multiplied. So what is the result will form? The result will be 3/3. Meaning the AB The result will be AB meaning I means BA which The result will be 3/3. It Means BA The order will be 3/3. became my Two questions. Isn't it? Do it like this if you want Show it to the examiner. Look, show me this If you please the examiner Examiner, there is no problem. Examiner is very happy. It's a simple thing. Come on brother Let's do the next one. The next one says it's simple.

[00:03:47]That y first point look at yours. It is written y = You have to derive ex + c. Derivatives means do it with respect to x. then e of ^ x is x itself and c is zero Is. So this is a derivative. it's over talk. Got number two. He said the degree How much is it? Degree means the most Most often, the derivative will be two The most frequent derivative is. The two most More often it is a derivative. In both of these What is the greatest power? Four. So only four You have your degree. Now here's a little definition.

[00:04:17]You will write. A relation of R on a set A Defined as A B belongs to relation. B A Belongs to relation. If brother A B relation I belong to. B A also relation belongs Does. means the opposite of that, so what does he Do you speak? It is called a symmetric relation.

[00:04:34]What do you say? called a symmetric relation Are. Right? What do you say? symmetric It is also written. So this statement is yours How are you son? It is true. A relation r is said to be an equivalent if it is Reflexive, symmetric butt knot Transitive. No to that relation brother will have to be reflexive, transitive It will have to be, it will have to be symmetric also. If Any relation has these three only then it It is called equivalence relation. here Said that not transitive. this this Your statement will be false. Because It has to be transitive. Equivalence To be called a relation. It's that simple. To you You should have known the definitions. just here Do you know? Is ours clear? Is it clear?

[00:05:12]Let's fill in the blanks we have to do.

[00:05:14]said a square matrix a is said to b if determinant of a = 0. square What could the Matrix be? When Its determinant should be zero. what could happen Is? That could be inconsistency. Right?

[00:05:30]That will not be an invertible matrix. So that will be a not invertible matrix. Then both Can write inconsistency and non-invertible You can also write. Ok? you write it down Brother. Please let me know by writing in the comments.

[00:05:40]A square metric can be expressed as A sum of symmetric and a dash matrix.

[00:05:45]We can express any square metric Can. Sum of Symmetric Ends Sciusymmetric. Right? Transpose of A of What will be the transpose? Meaning if you If you transpose the transpose also What comes? Matrix A itself comes.

[00:05:59]The number of okay what is it here isn't it four Think about that in marker question, two annas Needed How about four? Okay, here it said 25 28 is the four markers. Ok? Ok. one by one All of them are numbers. The Number of All Possible Matrices are matrices of order 2/2 with each Entry Zero and One Is What? So the number How many are possible? to remove it what needs to be done? There are a lot of possibilities Can. But how much is your entry? Only There are only two, brother. And how much is your order? 2 / 2 is the order. So you have four There will be places. Did you understand what I said? Ok?

[00:06:28]So brother, let's see its answer. Who me Will you answer in the comments? Ok? I get this Answer is required in the comments. Fill in the Blanks [nasal sound] still four Number questions means one by one number questions There will be all the questions. one by one all We do. Come on. First question. If y = How will this first result in dy/x? See It's simple. Don't do anything. I mean, do something.

[00:06:49]Let's do this first, the people inside.

[00:06:51]To. So 5 * 3 = 15x and x to the power of 4 And what happens to sec? sec is of sec Ok?

[00:07:01]Angle Sem * 10 Angle Sem. The angle should be the same simple. So 15x ^ 5 - 1 means 4 end angles same. Angle Sem sec of sec x is sec x * tan x exactly Isn't it? Now how will we do it here brother? donation What is the value of derivatives? See It's simple. Now what happens to cos here?

[00:07:22]Is? -sin And if it is square then what will happen to the square?

[00:07:28]2 cos power one term 2 - 1 now check this Whose formula became the formula? 2sin* What is the formula for cos? 2sin * cos It is sin 2 theta, right brother. 2sinθ * cos Whose flower is Theta? sin of 2x. then write Gave. Just because of the minus-- sin is 2x.

[00:07:47]The General Solution of the Differential Equation This is given by y equal to what? General We have asked him for the solution. So the general What will we do to provide a solution?

[00:07:57]Let's write it down here.

[00:08:02]So dy / dx = we can write 6x. write You can sir. I sent this dx over there.

[00:08:08]What did I send? I sent DX there.

[00:08:10]Gave. Now we integrate these two.

[00:08:13]Can. I can do it sir. So we both Let's integrate. So the variables are separable.

[00:08:18]By this method it easily comes into general solution.

[00:08:20]Will go. So here there is a forest which belongs to Brother y integration and it will become 6x² / 2 integration + c right? that means this to say To cut out. Right? So this becomes 3x² + c.

[00:08:34]This has me asking what will happen if y =?

[00:08:36]3x² + c will be Simple. Ok. sin inverse x What is the derivative of? do not know if I don't know son, take note of it. Watch Direct Questions come. Study the Figure Carefully Of this. If we look carefully, vectors I can see something. I a little bit Let me introduce you to vectors, brother.

[00:08:54]See this also. Brother, this is the y axis. These z Axis. This is the x axis. on this which xi is the cap. Actually we call it unit Also called vector. Right? which we denote by i Also denote. But it has said xi.

[00:09:05]Meaning x will be the distance. This is yj cap. Is No? This is ZK cap and you have an arbitrariness point which is your R denoting the cap Used to be. Ok? okay brother? Do study Taken for now. Now let's see further. what is The name of commonly given the vector like OP Vector. What is the name of the vector from O to P?

[00:09:26]Is it brother? What is a position vector? OP of vector? So OP belongs to R vector. position Vector. Yes and no. What is the name of the OP vector?

[00:09:33]Is it brother? R vector.

[00:09:36]What is the name given to vectors ij?

[00:09:39]What name have you given? Look, it's simple.

[00:09:41]What have you said to i? xi cap yj cap zk cap right? So xi cap yj cap zk cap What is or what are? The term or Terms used for collection of ij and k These are called unit vectors. What Do you call them that? Unit vectors. Attention Let us assume that ℓ is a unit vector. Simple C talk In given figure if r vector is 10 bhaiya bol Given that the magnitude of the r vector is 10 Is. Well, 10m. The value of x is 8m and y is The value of is 6m. So what is the value of z?

[00:10:20]Will it happen? r vector xi cap yj cap zk cap jo What did he say to you?

[00:10:30]10 8 6 so 10 means if we take the magnitude If we do it on both sides, then this is something like mine.

[00:10:39]Sits. Right? means the same magnitude Went. okay brother? So how much has he given?

[00:10:45]This is given as 8m, which means it will become 64m². and i The magnitude of the cap is one Is. Because that is a unit vector. Right?

[00:10:51]I also wrote unit vector, brother, unit They are all vectors. here it is y will become 6 means 36m² Right? And this will be z squared.

[00:11:02]k will become one. And this is 10. 10's Meaning 100 m², look this is also becoming 100. So Coming here, 100 became minus and then zero came. So Actually the magnitude of z is What is it brother? Find the value of z means This value means its magnitude Take it out. So what does magnitude mean, brother?

[00:11:17]Its magnitude is coming out to be zero. whether Square it or take the square root. Correct Is? It's simple.

[00:11:28]Come on brother, now he has said read the passage. Correct Is? Read the passage and answer questions one to six You have to answer in Roman numerals till.

[00:11:36]I said okay sir. Come on. in calculus Optimization Problems Involving Finding The maximum and minimum values ​​of a Function. typically with in a specified domain this problem these problems are Commonly encountered in various fields such As Economics, Physics and Engineering.

[00:11:54]One Fundamental Technique for Solving Optimization problems are of use to Derivatives to Find the Maximum and Minimum of a function fx okay na simple v First locate its critical points by setting fx = 0 ok brother fx = 0 say He has given Then Analyze the Behavior of the function around these points using the Second Derivative Test If f'(x)² is > 0 and end at a critical point and it Indicates a local minimum where as double The derivative is less than zero. Now look at this We have already told you everything Are. When we read the chapter maximum If you read the topic minimum then all these Things we learn that brother you are a single You can do the derivative test and get a double There is a derivative test. Right? Double What happens in a derivative test? When is less than zero, then we say It is maximum. And when it comes to greater than zero, Any value that comes positive is doubled.

[00:12:48]Of the derivative, we say it is minimum.

[00:12:50]And he said the same thing. Do you understand? So It's a simple thing. It Signifies a Local Maximum. So local maximum and local You have to tell me the minimum and nothing else.

[00:12:58]to do. Okay, right? I am the same thing Concept and questions below. So consider a Function 2x - this is this. first of all You don't have to do anything. of this function You have to find the derivative. Critical To take out points. So this becomes 6x² 6x - 12 that's zero. Right? six if you want Take the common. This is a question from the book. I Let me tell you. I have remembered. Now I I am putting this equal to 0. Right?

[00:13:24]So that I can find the critical point. So We can write x² - x - 2 = 0 which is We will do it by splitting method. What about this 2 * 1 = 2? It's simple.

[00:13:36]Let's do 2 * 1 = 2. Minus ka to me Who needs it. Right? So I would plus it Am. So this is x + 1 and this is also x + 1 and from here somewhere two multiples of x There are values. One is -1 and the other is +2.

[00:13:50]I don't have two values ​​of simple x anywhere.

[00:13:53]She is lying somewhere. So he said how Many critical points does the function have?

[00:13:57]There are two critical points. One then -1 to. Two I have critical points. Ok?

[00:14:02]Next he asked a question which statement About critical points on the function? Is Correct. All the critical points of function R positive. All the critical points are Negative. No, it is not like that. what is this Are you gone brother? Hey hey, give me the pointer To. Where from? There is one brother in my gym One is negative, one is positive. Isn't it? So, This one has said, some critical points of Function R positive. while others are negative and none of above as functions does not have any Critical point. No, it is a critical point.

[00:14:30]There are two critical points. One is negative, one is It is positive. I mean my option C It will be absolutely accurate and absolutely exact.

[00:14:37]Will sit. Ok? Ok? next question What about me? at positive value of What was my critical point of function?

[00:14:43]Critical point at positive value x = 2 One. then the function has local maxima, local minima, lower maximum and minimum, nor Minimum, none of above. Simply tell us this So your function was the 2x 3x² function.

[00:14:59]Let me tell you what it was, I was here 6x But what was it? It was 12.

[00:15:03]I don't know, I forgot + 8 + 8, so he He said what is the value on this, it is minimum.

[00:15:11]What is the local minimum, so how do you check it?

[00:15:13]Look, there is a simple way to check it.

[00:15:15]go do double derivatives they could also do It's a simple matter and the rest is up to you.

[00:15:19]Isn't the meaning of double derivative simple?

[00:15:21]Look here if I double derive this single derivative i know what will be? Isn't it? Test in Single Derivative I worked out that this is 6 x ² 6x - 12 right double derivative so this 12x right - 6 and 0 will come so obviously C The thing is, whatever you do, stay positive.

[00:15:41]Going to do. Isn't it? Put two here if you want.

[00:15:44]Look at it brother. So 12 * 2 = 24 - 6 means 18 It is coming. What does it's mean? Joe Double This is your derivative test, what is your Used to be? Greater than zero is coming. Now my What has been said to me in the passage?

[00:16:00]that if the double derivative is greater than zero et Critical point. If this is true then listen.

[00:16:07]If this is so then what do you say brother?

[00:16:08]Minimum. What is? Minimum. I told you Also kept. this is what happens. Isn't it? We I have read it in concept. When the double derivative becomes greater than zero, becomes positive Is. So we say that it is minimum. What Pay? At x = 2, this is the minimum value.

[00:16:22]What is this friend of yours, you too and when x = -1 What will happen if you keep it -1 I kept it here -2 -3 + 12 + 8 becomes 20 - 5 15 so this Your maximum value is So this is your maximum value and this is When was your minimum value, brother?

[00:16:45]When was it minimized, look, it was above Minimum is -12 simple So the minimum value is -12. And how much is it for Did you come? 15 So what is the maximum value?

[00:16:54]15 when x was my -1 and here when x was my There were two. Simple.

[00:17:00]okay brother? This was the entire question maxima of the minima. It was a good question. Excellent There was a question. Let us move ahead. So Find the equation of the hyperbola. Hyperbola What will be the equation brother? two types are hyperbolas. Here he paid x I have kept my focus. x pay means that its transverse axis You must call him Major. Ok? But You see, we don't say major or minor.

[00:17:24]inside the hyperbola. Major Minor in Ellipse Let's say. In this it is said that transfer and Conjugate. Ok? Meaning of Transfer Axis This is where the focus is becoming. So right now what The focus is being made, son, it is being made on x.

[00:17:36]Right? The focus that is being formed is on the x axis.

[00:17:38]Check it. One is 5 0. A -5 is 0. First It's about such a parabola, sorry, hyperbola What is equation? of such a hyperbola The equation is x²/a² - y² / b² = 1 This is the equation. Ahead He said what is vertex Brother? 3 0 What about Vertex, brother? 3 0 Okay, right? Listen to me carefully.

[00:18:03]Vertex means this thing here He is doing it. This is your 3 - 3 0 and this The thing is yours 3 0 this is a. Right?

[00:18:14]What is this brother? This is a. Right?

[00:18:15]This is your axis. is a. Transverse This is your axis. So if you look at a Value is somewhere same +- 3 you can say And c focus is the focus which is c. c The value of is somewhere around +-5. We know inside the hyperbola we know look Inside the hyperbola, a² + b² = c².

[00:18:36]If you don't remember this relation then right now Please write this relation in the comment section.

[00:18:40]You will remember it for life. So it means Keep the paper, I should remember it. a² means Square of 3. b² is unknown and c² is Meaning the square of 5. So obviously b² will come 16 which means b will come my + - 4 So this is my conjugate axis.

[00:18:56]Plus minus 4 I just meant b. from b Why did it mean? Because I need to understand the equation Have to take it out. So x² / a² y² / b² and = 1 this is the equation of Hyperbola. So did you write forest above?

[00:19:12]I have?

[00:19:13]This is the equation. Here it is a² 9, b² 16 this The equation of hyperbola is given above.

[00:19:21]Ok? Below it is said find the equation.

[00:19:23]It's a parabola, friend, you can do it yourself.

[00:19:25]yes. Ok? I will assume that It would be better if I get the answer in the comments.

[00:19:29]I would love to. Right? me more Would love to. I will like it very much.

[00:19:34]Right? So please answer it in the comment section.

[00:19:36]I will definitely give it to you.

[00:19:39]Find the value of determinant. yes yes Yes, that is a very bad question as it is called.

[00:19:44]I mean, not bad, I mean, what should I say?

[00:19:46]Such an easy question. Now I will tell you this too What? Let's pause the video, pause it right here.

[00:19:53]And tell me the answer in the comment section.

[00:19:55]What would be the answer to this? pause it Video here.

[00:20:00]I hope you have written the answer to this.

[00:20:02]In the comment section. Now let me tell you.

[00:20:05]So 6 * 2 becomes 12 and 8 * 7 becomes Brother 56.

[00:20:09]This will happen son plus this will come to 68 This is the value of its determinant if If yours is absolutely correct then brother will give a thumbs up.

[00:20:16]in the comment section in the section okay na Thumbs up marenge fast se find the miner of the elements of matrix A minor All the minors will tell me its meaning The previous question, what do you mean by -5 2 Your matrix A with -6 and 8 means I have to say minor in this These are the miners. How to remove Miners? It is simple. No bus sign change do. If we change the sign then it becomes cofactor Will you go? Don't do anything. its removal So all these elements are missing brother, eight Went. If you want to remove it, then all these Elements missing -6 arrived. It is to be removed So all these elements will go away. two come Went. If you want to remove it, then all these Elements Bhaiya disappeared. -5 has arrived. it Became a minor. And if you sign both of these You will change it because it is 2 1 and 2 1 2 So they become cofactors. Simple. next The question is also easy. It is said that a + b and Tell me a - b. So if you do a + b then see I I will tell you like this without writing. 7 3 10 be Went. 3 2 became 5. 4 2 became 6. 5 be Went. 5 and 1 become 6. and 3 and 9 are 12 Went. So this is a + b. will you do a - b So 3 - 7 means -4 2 - 3 means -1 4 - 2 Meaning 2 0 - 5 Meaning -5 - 1 Meaning 4 and 3 - 9 means -6, this a - b has come so easily I think this kid can do it I explained what miners are.

[00:21:45]How do they remove minors like once And let me tell you, like if I have to tell m1 then Here are the elements of the first row, first column.

[00:21:50]All the first rows and first columns have disappeared.

[00:21:53]This is your minor. This m11 is yours. See m11 That's it. Similarly, each element Take it out.

[00:22:01]The income is terribly poor. I am bad because I say it because look, it is right. Meaning This is a good question. Ok. It's fine now.

[00:22:09]Should I write the square? tell 2/3 Brother.

[00:22:14]Hey brother, we know this much that sec inver What is 1 / x? cos in1 is x and You can write what is cos inverse of 1 / x?

[00:22:26]sec in1 is x. Isn't it? reciprocal This comes through properties. Now we have this also know that sec of sec is inverse something What happens? Something happens. Something Should have come out. means to come out of it Shouldn't have gone. What should have come Brother? 2 / 3 2 / 3 So this is your proper answer will be.

[00:22:50]So this property will have to be written on the side.

[00:22:52]Son, this property will also have to be written down.

[00:22:54]On the side. These are his numbers only. number one You got one number, you got two total numbers.

[00:22:58]These are his numbers, brother. Let the limits come Have gone. If I go up to the top that's a good question because I'm I will do that first. So the limit x tends to π is written. and wrote here is sin x is written simply x. Take care. Good What do we know? We know that friend The sin that happens is first and second Absolutely equal in the quadrant, brother Equal means equal means that here But it is positive. First and Second Positive in the quadrant. If we are positive Speaking of quadrants, there is also a second.

[00:23:31]Talking about coming in seconds, out of π By subtracting something we come to the second quadrant.

[00:23:35]Can. This means if I do it like this I will write that sin π - x so what would you say that incense is its The answer should have been sin x Was Needed. Meaning I'm focusing on sin x instead of Listen from This means I replace sin x with Can I write this? Tell me brother.

[00:23:55]Tell me quickly. Can I replace sin x with this Can I write? You can write sin π - x, sir.

[00:24:02]There will be no impact. Now you know We also know that limit x tends to What is a sin x / x? There is a forest.

[00:24:12]Meaning, what is above should be below.

[00:24:16]Right? So what is this total value?

[00:24:17]It becomes equal to the forest. So this is the full value It will be equal to one. The entire boundary is of the forest.

[00:24:21]Equal. And when such a question comes, then What do I have to do? Look carefully. Now Seeing such a question. Brother, here you have a You must have read this form. Use of this form Do it on both the top and bottom sides. Of this form.

[00:24:37]Isn't it? What happens? n − 1 good limit It is also written that it is a different matter, isn't it?

[00:24:42]There is a limit to this, brother, you You could have written this as this is x 5 this is 2 5 Below this is the x. This is 2.

[00:24:52]Okay, the limit is running, the limit is running.

[00:24:55]is x tends to 2 okay is equal to limit x Tends to 2 is going on, what did I do?

[00:25:02]What did I do brother, I did this from x - 2 Divide it in the same way if when I did the above I divided it by x - 2, so I divided it by x - 2 Divided below also. Ok? So now What happened to me through Limit? This bun What happened to me? This whole part looks like this n - 1 then n a n - 1 becomes below as well. Below It's done brother. n a n - 1 in this manner. Correct It's a matter brother. n mine is three. Up n My 5 Mine has become simple. So x n - a to the power n It looks like a perfect hinto titto So how much is this, 5 2 4 2 is still 2 The balance will be higher, meaning 20/3 will come somewhere.

[00:25:49]Answer So you can do this too with great love.

[00:25:52]Right, this is also an easy question. Overview I have explained how to do it It's clear, okay, let's move on.

[00:26:00]So let's quickly show you the next one.

[00:26:04]y = december Have you given it to us brother? y has given x - 3x and into is written cot If you want to find the simple derivative Apply product rule Apply before rule What is the product brother 3x² - 3 and The second one as it is comes in the middle plus The former is a derivative of As It Is and Cot What will happen? cos x² This is the proper derivative.

[00:26:39]What will you do after this? after this Can't do anything. Then you just minus You can write minus. Such a minus plus It became minus. You can do this much and some more Can't do it.

[00:26:51]Divide e^x/1 + sin x by What will it take to do this brother? Tryant There will be rules to do this. So it is, right?

[00:26:59]Then do the one below. second question See.

[00:27:03]So here's y to the power of my e.

[00:27:05]x / 1 + sin x So the function below here is seconds Right then we will disinherit, look here, attention once from The one below will be written as it is like this In this way, the upper ones will make the hole square.

[00:27:23]derivative of the first one below such that such only Minus The derivative of the one below which only has cos It will be of cos only. sin will be cos and first The one is like this only. Now you will see brother e's Power x then came common. e to the power x common Went. remaining 1 + sin x - cos x and upon in 1 + sin x² There is no need to do much You. Now let it be just like this.

[00:27:52]Ok? If [nasal sound] If the area is πr² then if I divide this a by r I derive in respect to r How to do 2r right and r will be one Because we have done it in respect of r.

[00:28:08]Now he told me that if here If I replace r with two, then two here.

[00:28:13]But if you put it then it will become 2 * 2 = 4 brother 4π So this will be my answer. What happened brother?

[00:28:20]4π will be my simple 4π. Ok? Rate of It is a question of change. comes in the statement Is. He just picked it up and gave it to me. Take it brother Do it.

[00:28:30]Next, reduce the equation to the intercept.

[00:28:33]Form. There is an intercept form. You Would we have read it inside the plane? this is It was in line. It is of the plane. so much more You have to add it. So to make it like this Brother, can there be a way that you are equal?

[00:28:48]Make to 1. What to make equal to one have to do it?

[00:28:53]Nothing much, no hard work. 120 there Sent and divide everything by 120 Will have to.

[00:29:00]In this way, in this way, in this way. So The value of x came out to be 30. And this is 30 actual What's in it? This is also your x intercept.

[00:29:09]It happens.

[00:29:11]Well then your + y = will give 60 but I'll write it as -60 because there's a minus here.

[00:29:19]And what happens to you is this y intercept.

[00:29:21]It also happens Plus 6 * 2 = 12, so this is 20.

[00:29:27]This is the z intercept and is equal to 1 in Finished, he said tell me the intercept, so this x is 30, y is 60, and z is 20.

[00:29:40]That plane will cut on it. Okay, right? These is the intercept form. Right? that's it The intercept form is yours. Write this down and tell me No, first brother will understand the examiner. This much Why is he not there too? Find the Coordinates of the point which divides the Line segment. Hey friend, this is a 10th class question.

[00:29:58]How did you get into this? It feels like 10th class, right?

[00:30:01]But it is not so. Son, he is not in 10th class. 3D is in. Check this question. In 3D Is. It is said that one point is 2 243 and one is -45 -4 5 and -6 internally in the ratio 2 1 right Internally I divided it into 2 1 In. Now I don't know this point. from lats This point is x y z. Now we all know that My 10th grader will tell you this here.

[00:30:27]up to y. So isn't it so that sir, to find out x, we This is how it is done sir mx mx2 nx1 upon m + n is not the same age to extract y sir so As much as I am writing right now, this is our Bhaiya, a 10th class child, with great love I will tell you son, the one with Z will tell you because If he is watching this video on 10th Any child in the class would give him only this much Have to take z2 Have to take z1 Have to take x1 replace y1 and m + n end end this is Section formula is right so apply it Brother, the one with x is this, m is this, n is this Here is x1 y1 z1 Here is x2 y2 z2 Simple -4 * 2 - 8 right and that becomes 2 and 2 + 3 This is what happened -6 -6/3 how much would it be is - of 2 Let's see next what is the value of y It is coming brother, so 2 5 becomes 10 and 1 * 4 4 is done brother This is how much it comes to 40 3/3 see you next. So 2 * -6 becomes -12 z1 means +3 / 3 so what is z?

[00:31:47]-9/3 is coming, meaning -3 has come brother x y z is your coordinate which intersecting this line to the whole Internally 2 is dividing into 1 This is a simple section formula, friend.

[00:32:01]The coordinate a is its part Cartesian This is chapter 13 of the system.

[00:32:06]This chapter is part of it.

[00:32:12]Let's do this one first, now this one.

[00:32:14]Look at the question, it's a very nice question.

[00:32:15]Here it is said simply, simply, simply Is. Four numbers are being given to you for free Are. Number four here. How about number four?

[00:32:21]Will I get it for free? Just put one thing on you Have to prove it. So you pick it up from here Left hand side. Dude, what's the problem? And You can clearly see the formula here.

[00:32:30]Is. Every child will tell. who once If you have read ITF, you will tell me that tan is inverse Sir, the formula for x - tan inverse y will work.

[00:32:37]Here. And he will tell you about Phula also. that above You will get a minus, sir. And write plus below.

[00:32:43]Simple Phoola will tell you son. It's simple. Now Here you will see what it means What is? It means 27/1 and it means What is? Let's show it by keeping 3/5 Brother, the child is a child after all, examiner For us. So 27 / 11 - 3 / 5 / 1 + 27 * 3 / 11 * 5 Simple. So tan inverse Do it carefully. Now there's no mistake here.

[00:33:11]Needed So how much is this? 135 is - 33 / 55 55 is happening here also so 55 * 1 = 55 + 27 * 3 = 81 / 55 51 cancels out from 55 Got out. Right? What's left? what's left Brother? tan inverse This 102 is left.

[00:33:37]It seems to be 102 for me brother. The rest is fine Isn't it? 27 * 5 = that's all it is. Would So that's it, friend. So above save me attention Looks like 102 and below will remain brother 8 5 13 5 and 1 6 136 and both of these numbers Somebody comes to the table, these are 17 In the table. 17 * 6 = and 17 * 8 = right? And If we cut it then 3 * 2 = 6 3 * 4 = 12 Here tan inverse 3/4 means this much brother. This is mine is the LHS. But we have to prove it.

[00:34:11]In cos. So we have to convert tan inverse to cos Will have to convert. So what will we do?

[00:34:15]We'll write this as equal to. Late to late than this is theta. If this is theta, I I'll make this a function and it will be 3/4 of mine. That means this is my perpendicular And this is my base. So from here How to get rid of hypotension? By Pythagoras Triplets. we also know that 3 4 and What will be our hypotenuse? It will be five. So How hypotensive am I? 5 now Here we can easily write that the cost It will happen sir, if we do derivation here are cosθ of from the tanta to my If I get p then I should not get b And h I want here b means right now 4 h means 5 is now Find the value of theta How to calculate theta cos invar θ = cos invar 4 / 5 cos invar 4 / 5 4/5 come the value of theta and theta in actual you What did you believe? theta we assumed was tan What's the point of inverse 3/4? And this was the answer This was the answer to this, this complete LHS.

[00:35:18]This was my left hand side, and How much did this come, this came by deriving 4 / 5 with cos inverse and this cos inverse 4/5 How does this right hand feel to me?

[00:35:28]If it looks like the side is equal to the now hens Proved LHS = RHS over thing Four numbers free, did you get your formula?

[00:35:36]I will have to write it down. You will have to write the formula.

[00:35:38]Formulas have numbers. half number You will get the formula here.

[00:35:41]Understood? Understood?

[00:35:44]very good.

[00:35:46]Now a question that the child will say after seeing it Brother, I am skipping this brother. I not doing. are not I. But most Easy question, this is easier than this. No This is also easy. It's not like that. This is easy too Is. But this is easier than that. How? by I will show you. See We are given i = π / 2 sin x / sin x + cos x Don't do anything. Let us know a property Should have been 0 to a function. What Does it say property? This is the property That's us zero to if a is written as a + b which has the property a + b - x The alternative part is one that we can write We can write this as a - x dx.

[00:36:33]If I write it like this, then my What is made? i equal to listening attentively Just imagine, if I write anything, it will be less, right?

[00:36:38]Imagine if I had a - I mean π/2 - If I do x, what will sinπ/2 - x become?

[00:36:46]Think quickly and tell me what will happen brother? it You tell me. Because a is now my π/ Is 2. So what will π/2 - x become? that's it will go cos x Isn't it? Same edge down also cos x + sin x okay Isn't it true? So there's going to be a change down there.

[00:37:04]No, because both are different. this is it This will become sin, this will become cos, so I did this Wrote it in plus. Right? by reversing Simple. Now this part and this part Check it. This is the first equation from the let's.

[00:37:14]This is the second equation. So by adding first And the second What Will You Get? If both of What is the sum of i + i? 2i be It goes sir. Right? And when these two are added If you do this, the lower parts remain the same.

[00:37:25]Are. So there is no need to take LCM etc. It would have fallen. So both of these can be added easily.

[00:37:31]Right? So the first one above has sinx + cos x so now if you look carefully then this is complete If the part is cut, the forest remains inside and Every child knows about the integration of One What happens to the integration of Bhai One?

[00:37:46]What happens is x right and upper limit If you have to keep the minus lower limit here I put π / 2 - 0 upper minus lower two If you multiply this by π, what will be the result?

[00:37:59]/ 4 has arrived so this is my fine final accurate The answer is π / 4, so the denominator of such questions The exact answer is π/4. But here at What is the number for? Yes brother this is the number of the property which is being used for The child is demonstrating how to apply it.

[00:38:16]Right? Then integration, then cutting This is its number here. Did you understand what I said?

[00:38:22]Its number is here.

[00:38:25]Come on, what is the next question brother?

[00:38:26]Look, if I write this like this, is written dx / 2² - x² So how do you feel? Whose is this?

[00:38:37]Will integration happen? a² - x² whose Will integration happen? a² - x² whose Will integration happen? This will become sin inverse xy of a.

[00:38:45]Will it be made or not? Then do it and tell me that x Only x will come in place of. But what should I do instead of a?

[00:38:50]Will you come? You should look into that, brother. But here it is If zero and one are given then 1/2 will come. Meaning π / 4 will be the answer. Its answer is also π / 4 will be. The simple answer for both will be π/4. But Please try its answer once and see further.

[00:39:02]yourself. Okay, right? So let's go next brother Let's look at the question. The next question is Full six marker brother. number six There is a question. In this way, every question in the question paper The year comes and you cannot avoid it brother.

[00:39:12]Can do it. Don't skip at all.

[00:39:14]LPP question of this type of question Is. It can be seen clearly. Ok?

[00:39:18]How to do it? How do you not do it? His I will tell you the method. Look carefully. just before Read the question carefully. twice I have to read the question. First, your good Relax and then go back to your pace. first by half Get him to decode the question. As What he said earlier was that a manufacturer He makes two types of book cases.

[00:39:34]I said okay. The first type of case is Brother, he makes it. Right? So what is written?

[00:39:40]First Type Requires 3 Hours on Machine P And 2 hours on machine Q. I said okay Yes. Isn't it? The cases to complete Makes. Now how many does that make me?

[00:39:50]Don't know. Do you know? Me too Don't know. So this is the problem. Us I don't know how many such type one cases are being made.

[00:39:57]Are. Similarly, when you use the second type If you are making a case, you will have to work on the machine for 3 hours.

[00:40:03]I am going to have to spend the next 3 years brother You have to spend hours on the machine.

[00:40:08]Have been. Again I will say this, you type two How many cases have you made that you are making?

[00:40:12]had been? Do you know this thing? To you too Don't know. This means that first of all, this I will have to write that Sir, late number of cases Number of cases Which type of brother? Which type of off Type one which is the first type of cases Type One First Type Case I have given the value x Took. In this way brother number of again book Cases what is happening? g of which Type of? Obviously sir, type two him I assumed y. So now pay attention to my Listen to the talk. Listen to me carefully. To you When Type One cases had to be made, you used a machine Spent 3 hours at P. Isn't it? 3 on machine P hours and machine P next to B for 3 hours Spent. So you are spending it on the P&P machine.

[00:41:02]If I talk to you about machine P only. So If we want to talk only about machine P So he further said that the machine P is 18 It will take hours, brother. And the machine Q is the machine Q: That brother, it will run for 14 hours per day. Attention Keep this thing in mind. So here I am How have you differentiated the question?

[00:41:21]How are the categories divided? P machine and Q My category has been distributed through the machine That machine P will run for only 14 hours. It's 14, right?

[00:41:28]Brother? No sorry it will run for 18 hours and Q machine It will last for 14 hours. This means if I just P I get the work done by machine. Listen carefully.

[00:41:36]So machine P made X cases in 3 hours.

[00:41:39]I don't know how many cases were made. in the same way P machine regenerated y in 3 hours Cases. I don't know how many cases I made.

[00:41:48]Will be. But I know how much you You must have driven it for hours too. I don't mean But you did not drive for more than 18 hours.

[00:41:54]At most 18 hours, 18 hours is not much work.

[00:41:57]My machine P. It means more Should not work. means 18 hours of work either work less than that or work 18 Work longer hours or work less than that. Correct Is? Ok? Let's simplify a bit This. Friend, do this, take three mangoes. This way Common is not accepted in inkities.

[00:42:14]Dividing by three. Right? Dividing by In Three on Both Sides. Each element Divide and you will get less than equal To 6 * 3 = 18. Right? So let be the First equation. Simple simple simple. seconds The equation will be formed by your Q machine. Because You have given us the categories through P and Q machines.

[00:42:31]Have given. So how will Q increase with the machine? be careful See. It took 2 hours on machine Q. what to make In? In making the first type of cases. How many?

[00:42:40]Made? Don't know. Then how many within 2 hours Did you make it brother? This many would have been made on machine Q. And You have also processed the second type of case on machine Q.

[00:42:49]How many would have been made? How many Ys would have been made?

[00:42:51]How long has it gone, brother? Tell me. Machine Q for the This is not it brother. Where is it? 3 hours. 3 Hours on Machine Q. The second type is Again 3 hours is being taken by machine Q. Second Type 3Y ones. Now how much will these two together It should be an hour, brother, more time I shouldn't have only 14 hours Which machine will work brother, the Q type machine?

[00:43:12]I will not work more than 14 hours already.

[00:43:14]If I am telling you, 14 hours means 14 hours or less than 14 hours atomic most than less work will do right so let be the second If I can't do anything about it, Let us consider the second equation as It is clear here, it is clear so I have There are two equations. Now its graph from here It's a little awkward to make because inequality How to make the graph of till now sir? how not Make? The best way is that you Consider equations with equality. This Consider the equations. Ok? These are not inequalities These are not inferences but equations which are equal There is a sign because here both the signs There are also equals. 14 and 6 at the same time The equations will also be equal. So the first What are your equations brother? first equation Let x + y be the inequality From that equation is 6. Right. Well the children You must have read one thing in your life. Brother What have you read? Intercept form. Yes dear, Who all have read it? Intercept Form Who all have read it? We read x / a y / b = 1 where a and b are the intercepts on Respective Axis. means x under x is intercepted. below the y intercept of y Is. Is this clear? well what about this Can I convert like this? absolutely do I can brother. Look here carefully. I have Divided everyone from this. I started from six Divided it. So what does this mean Is yours gone? Now write it well. Isn't it? apart It would be better if you write it down than that brother, I have x The intercept is g. I have the y intercept Six. Did you understand what I said? Both of these are on me There are six each. Brother, the graph that will be made Six will also intersect y, six will also intersect x on the positive axis. Now here if we I want to check where its graph is brother.

[00:44:56]will form? Through whom its bid means away Is it from the origin or towards the origin?

[00:45:01]If nothing happens, we need to place x and y at the origin.

[00:45:04]If you want to check it then at 0 What you will get is 0 + 0 < = 18. 18 was six.

[00:45:14]Bhaiya it was six, sorry it was six, right, exactly six.

[00:45:17]tha bhaiya I am writing 18 again bhaiya so Here 0 comes and less <= 6 comes you Tell yourself by looking at this statement which is given below It is written that zero is not less than six.

[00:45:27]Is. 0 is exactly smaller than 6, sir. So this How is the statement brother? True. What does it mean The graph that goes towards the origin Will go. Ok? Let's make it and show it Graph. Come, just look here.

[00:45:42]Make it brother. Make it a little nice graph no. Something a bit like this. A big line.

[00:45:49]Ok?

[00:45:51]Here's my X axis.

[00:45:54]Here's my Y axis. Here is mine.

[00:45:57]Origin. Now friend, look at me, I have to go till six.

[00:46:00]It will have to be done. So 1 2 3 4 5 6 7 8 rough figure I am making it. Don't take anything seriously I am telling you first. Brother, I am a rough figure I am making it. Ok? So the rough figure has to be Don't take it seriously. Ok?

[00:46:24]Is this much clear? So I said no It will cut six at six. It means simply The thing is, here's six, here's six.

[00:46:31]Right? And here's six and here's six.

[00:46:34]We will combine these two like this.

[00:46:38]So this becomes your equation. This is still Whose equation is this? This is brother x + y = 6 Sign of equals. But what do we have to bring? lace Dayn. So to get the equation of less than What will we do? We will build some sheds here.

[00:46:52]In such a way that towards the origin it will go Shed. Because we had checked that brother The statement which is its statement origin It will satisfy you. Meaning right after keeping 0 Things will come equal to the hand side. Right What does hand side mean? This means 0 + 0 Less than equal to six when x and y are 0 May come. Equal to no less than to Will come. The younger one will come. It means whatever All the points below it on the back side All the points will be on this entire graph.

[00:47:19]Will satisfy. this line this line If kept in, the value will be less than six.

[00:47:23]Keep anything. Keep whatever you have I wish to. You can keep whatever you want, understood?

[00:47:29]Is it clear brother? Say.

[00:47:32]Ok sir. This much is clear to us. Now Let's draw another graph, which is 2x + 3y.

[00:47:38]Is. We will lie down here also. Come on. So let 2x + 3y = 14. From lats. Now I am in the same I will use the intersect form and divide it by 14 I divide it by.

[00:47:54]Ok? I divided both sides by 14.

[00:47:55]Do it brother. So I got this 7 Below But I did not find it here. Me Found here 14/3 = 1 Now if we look carefully then this 14 / 3 is it, friend, what does it mean?

[00:48:08]This means 4 * 3 = 12.66 Right? Same thing happens. So if we talk of the x intercept then the x intercept will be From. and the y intercept is 4.66 You Can Say 14/3 Right Now What Does This Mean That right here on y, this four is around 4 1/2 It will intersect at the top and at x it will intersect at seven No, let's make a graph of this, then x Where is it cutting? It is cutting on x, sir.

[00:48:38]Where is this seven brother, here is seven and 4.66 where is 4 5 a little over halfway here Write this point here. What is Point? Look here. what is the matter? these points Yours is 0 14/3.

[00:48:53]Okay, right? Is it okay brother? Now this point is Everyone knew it. This is easy. Come on brother Let's make this. This line will go something like Something like this. Come on brother. Hey, a little dirty brother It's done.

[00:49:09]Move a little bit up a little bit up.

[00:49:17]Will lower it a little.

[00:49:20]Ok? okay brother. accept here But do it rough. Okay, this equation Which line is this that has been made? Yes What does this line mean? The meaning of this line So 2x + 3y was equal to 14. Ok?

[00:49:33]But how will less than come? Let us see that We will have to find out where it is going?

[00:49:37]Towards the origin. Away from the Origin Is. Where it is? Let's check it once.

[00:49:40]Keep it zero. Keep it zero brother. Check zero.

[00:49:43]So if we keep 0 then 0 will come. If you keep 0, 0 will come.

[00:49:46]Right? End less <= 14 should come. What 0 Is it less than 14, brother? 0 is it small or short From 14? From? Absolutely. So Statement Wood B What's True? This statement must be true Needed If this statement is true. Its Meaning its shade will also go downwards, sir.

[00:50:01]What does going downwards mean, brother, this Kind. This is also going towards origin. This Check it like this. Check it brother. In this way.

[00:50:11]Now my question is, which one is this?

[00:50:14]Is it a region? Which shade is which part Where both matched and mixed. You You will say sir, there is no green here. Check do it and in this small one here It's not pink. Listen to me. You Get it eliminated. Pink in so many parts Not there. There is no green in this much part.

[00:50:31]So the part below this means from here Laker right?

[00:50:35]From here to this point, business can be made.

[00:50:38]Went.

[00:50:39]from here to the intersection point Then come down. So far only this much This is a region which is first somewhere.

[00:50:48]is in the quadrant. Right? Because profit First means positive will be first I am brother. This is such a reason somewhere somewhere. You can name it brother. Let's From A B C and these origins. This is what gives you A B CO, can you see this common region? In this First you need to find the intersecting point Will have to remove it. Let's find point B.

[00:51:08]and points B is made up of intersections. whose intersection What is this made of, sir? Of two lines. first line In which your 2x + 3y is equal to let's say 14 and one was x + y right? which is = 6 No? This was it. This was it. Now if I complete this If I multiply the equation by three, This becomes 3x 3y and 6 * 3 = 18 no problem then no. Let's eliminate so that we brothers can extract x. So this goes to -x = -4 where x = 4 If x is 4 then what should be the value of y wanted it? The value of y should have been two, sir.

[00:51:44]Because adding both of them gives six.

[00:51:45]Right? I made up my mind and quickly I did it with this. So this will come to my 4 2 Right. What is this point c point of mine? Check do it. C point is my 6 0 good A point What is mine? A is my point. wrote this It has happened to me. I will write it here then Too. 0 14 / 3 What should I write about this origin?

[00:52:04]Leave it. So this means somewhere three Points are such that they represent majorly common regions.

[00:52:08]There is also origin and common region, these three only There are points that are touching and these At any point on the three points, our The profit that is there can be maximum.

[00:52:16]Let's check the profit once. profit What was mine? Isn't it?

[00:52:27]Look, brother, now you'll see something here.

[00:52:30]What is profit? They see. wrote this There is a profit. oy hoy hoy there is a Profit of ₹30 on each book case of types.

[00:52:39]₹30 for every case you sell for the first book Available on every book. each book On cases. Meaning, it means the profit It is yours if I talk about profit.

[00:52:48]Actual profit as you would say brother What is our subject? Isn't it?

[00:52:54]What is our subject? So the subject Ours is z. Right? Right? and z equal to What? I sold x for 30*. I So only x has been sold, right? x sold. every one You will get ₹30 per book. I will get 30x Total money. Similarly, when I was reading the second book If I sell the cases, I will get ₹40 each.

[00:53:09]On the book case. So I will get ₹40.

[00:53:12]My profit is only after adding both of them.

[00:53:14]will form. So how many such x and y should I sell That my profit should be maximum. So I have Where did it go somewhere just like this?

[00:53:23]Brother? These are the only three points I have.

[00:53:25]There are only three points. These three I will look at the points. Look here. Isn't it?

[00:53:30]So let me think about these three points.

[00:53:33]I will have to decide on which one I can make maximum profit.

[00:53:36]I can earn. Right? So the first point Mine is late from A point that is 0 14 / 3 Right? The second point is my 4 2 and third is my c point 6 0 ok? Where there is profit, my obvious It's a matter of how much was it brother? writes here I am z = 30x + 40y So 30 * will be 0 brother and 40 * will be 14/3 How much will you get if you do it? 14 * 4 560 / 3 ok, you will see how much it will cost Give it to me brother, okay, but I know this much 180 something is going to happen right No, because 18 * 3 = 54, isn't it 18 * 3?

[00:54:22]= 54 then it will be 186 something, now four more Let's put two and see, come and put four 120 30 plus 2 then it becomes 80 Total 200 If I keep it, brother, it is coming to 180, friend, this is 180.

[00:54:38]If this 186 point comes then this and this small 200 from whom my maximum profit is What is my maximum profit brother?

[00:54:46]₹200 At what point where x is four and y my two Understand what I am saying, this will be my last answer.

[00:54:54]If the answer is clear then do it this way This question should be answered in a completely full flash manner.

[00:55:00]Your question. This is how it has to be done, son.

[00:55:02]Ok? What was the one thing here? Decode How we decode things.

[00:55:06]We had to make maximum profit. We The maximum profit taken is ₹200.

[00:55:10]How much are we making here x? four x are making cases and only two are making us There will be cases with y. Ok? then growing Yes brother, on to the next question. next question This is mine. Your question is number six.

[00:55:21]And it is a six mark question and in this what is written? Find the Cone of Maximum Volume. that can be einstein sphere of Radius r is a simple thing. here It has been said directly that whose brother is he?

[00:55:30]What we have to do is maximize the volume.

[00:55:33]Right? The volume has to be maximized. End He is inscribed in a sphere. end radius I have given it. Radius of sphere is given How much for me?

[00:55:44]radius of Sphere. He has given it to me. Ok.

[00:55:50]What have you given me? Son, this to me I have given the radius of the sphere, son. r ok Yes sir.

[00:55:56]ok sir. First of all, brother, see if I have the radius of the sphere and also the volume I should know, right? So I lay down First I will lay down the volume of cone v Take care. Look at what is in such a question.

[00:56:12]If you know three things? means no matter how much You know the thing. Three to four things At least three things. me in this question These are the three things I have come up with. Maximum Volume, Inscribe in a Circle and Given What did you give me? And one has said it, right?

[00:56:25]Volume Maximize Solve From There I'm starting to follow the volume, man.

[00:56:29]If we want to maximize volume I took it to be v. Now actually v's What is value? v is now 1/3 Look friend, you should also know about Kon.

[00:56:37]You need it, don't you. So you have to accept something or the other because joe lates se lates se lates joe height Who is of that, tell me what is there in the exam?

[00:56:46]So the height of cone I assumed is h, right? This So the radius of which is which is also And so is the radius of the angle So I accepted it, brother, why did I accept it?

[00:56:58]Because when I write what the volume is is π r²h 1 / 3π r²h then the examiner will think no brother this r and h where did you come from then you will have to tell him that Brother, look, I have accepted here that I I told you whose height is it and The radius is r. The second thing you need to know is if The radius of the sphere is given, so keep that in mind.

[00:57:19]We call this capital R as a constant.

[00:57:23]Will treat you. How? As a constant. Correct Is? Age a 1 minute spelling went wrong. Age a Constant will treat. means its The derivative can also be zero. Isn't it? Who The question has given you something too, right?

[00:57:37]Given means that you have given him as a Let's go by the constant value. Remember. Correct Is? Come on, now make it look like its figure.

[00:57:44]It must have happened. It is obvious. Circle K One inside like this brother, take this one now Who would you think that angle is. It seems Who is it? Who is it looking like? very good. Let us go Brother. Now here is a simple thing. This brother The base of the angle is the circle These. And here it is, the center of Sphere.

[00:58:07]Now whatever the length will be from here till here So much length here. This much length. Isn't it? Attention Look from. From here to here, This will be the length of your R sphere, brother.

[00:58:17]It's radius, brother. From here to here The length will be R, which will also be the radius of the sphere.

[00:58:22]Yes brother, it's okay, second thing, second thing The length will be from here to here.

[00:58:28]The smaller one will be the radius of the corner, r brother.

[00:58:31]And this will be the entire length Let us assume h as slant height right.

[00:58:39]Right, I did as much as I could.

[00:58:41]I have shown you my intentions, okay?

[00:58:44]I named it a b C O and let's I have also given you the names from M.

[00:58:52]It's a simple thing.

[00:58:56]You will see a small triangle in the middle.

[00:58:57]In. I can also see it here. These Look at this triangle. If this triangle If you see it being made, which one?

[00:59:02]Is it a triangle brother? You see meditation. in Triangle OMC Ok? Why am I doing this now? I Let me tell you it's cheap. So why do I do this Have I been? Look carefully. If we defer this Look, if we want to maximize, we need to double it.

[00:59:20]Derivatives will have to be done. Derive twice Have to do it. derivatives to our There is a way that both What's up? be in the same variable. Here is one There are only one variable here, two and this is also one.

[00:59:30]There is a variable somewhere. So these two There are variables. Putting these into a variable Will have to. That's why we are thinking like this. Understanding Are you listening to me? So we are thinking that Are. So I can't do it now Derivate. So we can draw some conclusions.

[00:59:41]Trying to get into a single variable We're doing that so we can derive With love. So OMC here is OMC I will take Pythagoras theorem. Big Sweet OM² perpendicular Just write it as CM brother. don't read anything else Take this. So what does OM actually mean?

[01:00:02]What is? That remove R from the entire H.

[01:00:05]Remove R from the entire H. the whole of When you remove the capital R from H, it means H - What does R² MC² mean brother? R² What do R² and OC mean? Large r² Ok? What is? Dude, what is happening?

[01:00:24]Large r² Now if you accept me from here as your younger self If we want to find the value of r², what will it be?

[01:00:30]She will become my r² - h - r². Oh yes and no.

[01:00:35]I took this out so I could keep it here.

[01:00:37]I can't. I can keep it here. Are you getting the point?

[01:00:40]Now it is getting into trouble? Now r² h has the form Check me. You will say Sir this r oh this Bhaiya is constant, right? Have you forgotten? What Did you tell me? The question has given it. These is constant. What is this brother? these variables Is.

[01:00:58]And I brought it into a single variable. Single Variable mean h. So the value of r² I put I can do it. You try putting. So What is my volume? 1/3 πr²h where where π and means 1/3π will remain the same. and r² What can I write? Write r² - h - r² I can. Sir, you have just told me. Correct Is? And I would leave h as it is.

[01:01:25]I am at the moment. Well, it's not that I I will differentiate it. Stop, stop.

[01:01:29]Let me simplify it a bit, r² and this becomes a - b² is becoming a - b². these checks do it.

[01:01:39]Ok? a - b whole square 1 / 3π such that Went. Minuses are taken inside. Right Right Right, right, right, something is cutting Brother. Something got cut. A lot has been cut.

[01:01:52]A lot was cut. And how much is h and h?

[01:01:57]It's done h How much is 2RH and * H? 2RH² This is it. Yes. It was V.

[01:02:08]Now you can designate it very cute Are. In respect of whom? respect for h In. So distinguish both sides with respect To H. Okay? I am also writing for you I am in short form. Disinfect Both Sides With respect to H. Okay? Come on brother. So What came down in dv up? dh has arrived. Now π / 3 will remain the same. It's a simple thing, isn't it?

[01:02:34]Ok. Since 2 * 2 = 4r and h this becomes 3h² Yes and no. Why are you keeping it equal to zero?

[01:02:44]Are? So that I can find the critical point.

[01:02:47]So that I can maximize the volume.

[01:02:49]Or I can also call it the double derivative from here.

[01:02:52]Can do it. But I'll tell you one additional step.

[01:02:55]I am giving you the idea that if by chance the question If you had asked me a little more, I would have told you that.

[01:02:59]I am demonstrating it. I will write additionally Am. This is an additional part. of the additional part This means that the answer we have to find now He has no relation with the one we are part of.

[01:03:10]Will take it out now. But by chance if in the paper Next year has come, this year has come, so what will you do?

[01:03:15]Do you want to do it? Look, you have to keep this zero.

[01:03:18]Right? This entire part became zero. This is complete The part became zero. So 4rh = 3H² will be equal to 0 because This will become plus here. one to one h will be cut brother. So 4r = 3h will come in And your h will now become 4r/3.

[01:03:35]So sometimes this also comes, keep this in mind It comes like if I ask in this question I would write that end Prove That Prove That its height Its height is 4r / 3 means Joe Kon's height is that is equal to 4r / 3. If I am so If you had written more lines, you would have written so many lines.

[01:04:04]After that you would have to do this work again. Understanding In? This would have been your proof here. But We don't have to do this work right now. Because Question had not spoken yet. But if If you had asked me the question, I would have done it for you.

[01:04:15]I showed it. Now what is there here? here We'll have to double derivative it.

[01:04:19]Why do I have to do it? so that we can Can tell the maximized volume. So the maximum To find the double derivative, we need to Have to go. what does double derivative mean Single derivative π / 3 π/3 is reached. First tell me the derivative.

[01:04:35]What is a single? 4rh - 3² How much is equal to? this is my dv/dh chal Used to be. I did double derivative of this.

[01:04:49]I did the double derivative. π / 3 fd like so Will remain. This will come to 4r - 3 * 2 6h Ok? Now you are not able to understand anything, right?

[01:05:01]Now you have to put the value of h. See Yes or no? You mean you have to do this work It was read, right? So you will have to do this work.

[01:05:07]This does not mean edition. in actual This will have to be done. So the value of h is 4r / 3 was because r is constant. So I told h Wanted to substitute in the form of r. h The value of is 4r / 3. g be 3 * 2 = 6 Went. 4 * 2 = 8. So 8 minus and How much is this +4? This is done brother - 4r So what will this entire part actually be?

[01:05:32]It will come negative. means the double derivative Right? h isn't it? That's your less than zero.

[01:05:39]and whenever the double derivative of a function If it is less than zero, what is the function?

[01:05:45]Is it brother? Function is maximum.

[01:05:51]Right? And here also we can write who now who has Who has Maximum Value So H = 4r / 3 because we used that It is here. If we do not use this h If we had to do it, we would not have done this work.

[01:06:14]But you will have to do this work. Ok? To you Here dv/dh will have to be kept zero. His Later this critical point will be revealed. It is clear So much? I know this question is very Time just goes over the head. Every time sir The question goes over it. But the method It has to be done. So try this method once Please repeat the entire video carefully.

[01:06:35]Right? Sometimes it is written in the question If you go, there is no need to be afraid. question same as will be. Right? I had not given it yet. But use So we had to do it then. It's simple. Let's go very Excellent. So your almost entire section should be Went. But you are having a PYQ session Is. This is PY speaking. MCQ section It is now yours. Ok? Let's MCQs I come to the sessions. Isn't it? not like that We will come to the MCQ section. Right? So straight We directly open MC. Stop One minute.

[01:07:30]We Look brother, this is your same paper.

[01:09:08]It is your right to download it.

[01:09:10]There's a link in the description, it's fine.

[01:09:12]No, we have 1 to 20 questions left.

[01:09:14]Which we call MCQs. I give you this I would prefer this and suggest that you Do it at the end in the paper also. Ok?

[01:09:20]Like we are doing now. Right? The first This is your question. Come let's solve it He has it.

[01:09:27]The coordinates of the midpoint of this end This is what? Isn't it? Isn't it brother? This is us Tell me what will be its mid point?

[01:09:35]Of coordinates? So brother, look, this is x1.

[01:09:38]This is y1. This is x2. This is y2. So mid To find the point, do x1 + x2 / 2 And make it y1 + y2 / 2. This way we are in the middle Let's take out the points. So 4 + 7 is 11/2 Is. 11/2 is in this also. It is there too. Now In this -1 and 2 -1 will come, meaning one will come.

[01:09:56]So 1/2 option number D. Ok? Now you too Keep answering in the comment section brother.

[01:10:02]You have to answer before me. The Intercept Made By This Line. Intercept What does it take to make it? Simple Yes sir. This -1, everyone is affected by this -1 Divide it. I got it divided here. Here But I still got it divided. So it comes to -4.

[01:10:20]Right? This is not what it means. It is -4. positive There is no four. And this will cut off minus minus.

[01:10:25]It will come to six. Meaning ne - ne 6 no A positive six will come. Option number B.

[01:10:30]There is a method to do MCQs. Same I am showing you how to do it here.

[01:10:33]Am. There is no need to write. He shows how to do it I am keeping you. The center and radius of Dec. Now this question will have to be done. Come Go. 4x² 4y² right? So let's do it here Brother. 4 x² 4y² what was it brother? 2x 3y I forgot to look ahead, friend.

[01:10:57]-If it is 6 then throw the six there first brother. Now Do we have to divide by four here?

[01:11:02]Because x² and y² are nothing. Meaning Quotient It should not happen. So what did I do?

[01:11:06]If you divide all of them by four, you get x/2.

[01:11:09]Right? 1/2 will come, right? I mean here too If you divide by four, you will get 3/4. And Here if you divide by four then you get 6/4 Will go. Look ahead. Now we know that we You will have to make a perfect square.

[01:11:23]So to make a perfect square we have y and also the part with x Separated them. Ok? Simple.

[01:11:31]Now halve it here. half of 1/2 How much will it be? 1/4 of 1/² plus one times Square it and then square it minus one Do it.

[01:11:41]Similarly, half of 3/4 will be 3/8.

[01:11:45]Ok? plus its square one times And minus the square of this once.

[01:11:50]So this perfect here becomes x - 1/4 Square.

[01:11:56]In the same way, it will be formed here in y plus y + 3/8 square And look at this in 6/4, it is 1/4.

[01:12:08]Bring the square which is in minus here, so this It's done minus, it will become plus 16 Minus will become +1 /16, this will become 9 / 64 64 minus 9 so what will be the next thing here Understand the plus, now we have to check it.

[01:12:24]Because brother, I will not do it completely.

[01:12:26]Yes. Come on, come. So one has 1/4 3/8 so 1/4 and - should be 3/8, right? Because The question which is our statement, The standard form is y - a Is. x - h, y - a so minus a is Needed in the middle. So this will come out to be -3/8 and 1/4.

[01:12:45]So where is 1/4 and -3/8 brother? Here it is In option number C. So I think that I need to get this out here now.

[01:12:54]Will have to.

[01:12:55]If you want to solve it then please solve it brother. Simple C thing. The angles between 2x + 3y = 4 3x - 2y = 7 Look whenever this happens when your x and y Whatever it is, the meaning may be same but different.

[01:13:09]be with coefficients. like here you are Look here, write like this. whenever you see it, if It can be written as ax by c1 now c can be anything bx or any plus Take the minus, anything can happen in between I am not talking about the sign, I am talking about the number.

[01:13:26]If I'm giving ay = c2 then these can be different Are. Signs may be different. Ok?

[01:13:34]There may be sign differences in between. matter does not. We need to see that a was here Then here came a. There were two here. Look, come here Went. There were three here, now three have come here. So In such cases the lines are always perpendicular.

[01:13:47]Are. makes π/2. Take care. In such What is the line always about brother? perpendicular Are there. Did you understand, tell me? Line Perpendicular Are there. Same brother angle means whatever identical if such crosses have at π/2 It will always be made.

[01:14:03]Meaning, the coordinates of the centroid have to be given.

[01:14:05]Brother. Add all three. x1, x2, x3, x1, x2, x3, y1, y2, y3 and down to three.

[01:14:12]What comes down? Three comes. So the option Number C is the formula of centroid.

[01:14:16]A 10th class child will tell you. has said a * b What will happen brother? Let's try it. Come on.

[01:14:27]Come on, let's get this done first. From this The multiplier will always be zero. Right? Zero.

[01:14:31]Both are zero. Now if we do this with Zero will come. 0 * 1 0 * 1 0 will come. This is zero If it is there then this will also come out to be zero. If this is 0 then this is also Zero will come. So everything will be zero. Here it is Option number C. This one 0 Ok? This was easy.

[01:14:48]What is the transpose of a + a? Asked He has. Brother a here it is. a What is mine? 1 2 3 4 will transpose it. Meaning Joe 1 2 It is written like this, we will write it like this. 3 4 such If it is written then we will write it like this. Ok?

[01:15:02]So 1 + 1 becomes 2. 2 + 3 becomes 5. 3 + 2 It's 5 now. 4 + 4 becomes 8. Option A is correct It is mine. Ok? Option A is fine for me.

[01:15:12]Check it. What an easy question. So MCQs also come easy. who what so Is? Anyone if you say that MCQs are easy They don't happen. Because she will think if you slap her twice.

[01:15:20]Here it is said that the value of q is two. and p + q How much is my value? six. Then p What will be the value? If q is two then p must be It has to be four because 4 + 2 is 6 will form. So the value of p should have been four.

[01:15:34]The sibling function is x², g(x) is 3. give the Value of f of g of x right? g of x is This means that in the function, instead of x, Listen carefully. Now in place of x in the function Pay I want to put g of x.

[01:15:51]And how much does g of x mean? Three. so me I have three subscripts in place of x in the function f.

[01:15:58]Have to keep it. So the square of 3 is 9.

[01:16:01]If my function is x + 3 then its f-1 x What will happen? Simple. So brother, it is simple. fx means my y, right? Meaning of fx I happen to be y. This is simple. These It happens to me. Ok? This is what happens Brother. Hey, tell me the truth, brother. fx means my y It doesn't happen. How many times have we read these things?

[01:16:22]Have been kept. I mean, I won't say it in the papers.

[01:16:25]I.

[01:16:41][nasal sound] [sound of clearing throat] Tell me quickly, if I have any of this here If I want to find the inverse, I need to find the invertible You have to do it and what not I have arrived no no no no invertible sorry Sorry sorry sorry what will be its invertible Buddy? Let me think a little. Its invertible We are taking it out, aren't we?

[01:17:06]This will be x - 3. x - 3 Ok. Ok. Ok.

[01:17:13]will be x - 3. Option A. Which of the following Function from z means integer. Two Itself are the explanation? There should also be one forest. Isn't it?

[01:17:24]I am on two and that too on integer. Attention Keep. So look friend, integer means 0 put it, put one, put two, put three gave, put -1, put -2, put -3 Gave. So on. So when I If I take it, I will not be able to maintain the route.

[01:17:45]So brother, this will not become one and only.

[01:17:49]Meaning this will not happen on two.

[01:17:52]And x + 1 should not be on either. Then x It can happen on +2.

[01:17:58]I will get it at x + 2, friend. This is at x + 2 I will get it. Keep anything and see, you will get everything Should go. If I keep one, then three miles Will go. I need something in the negative, so I I can keep some minus. accept me Should - of 3 then I can keep -5. Ok?

[01:18:13]So whatever comes will be related to my integer only.

[01:18:16]He will keep doing it. So my friend, the second function I am feeling fine. This is always an odd number I will get it, right? You will always get this odd number.

[01:18:23]Even number will never come. So I that Not Taken.

[01:18:27]Let if a binary operation star integer itself Is. Here it is, put in two and three first. This Kind of means put a two. Put B to three.

[01:18:40]So what is 6 - 3? It's three. Now Put this with four, put this with four. So 3 * 3 = 9 - 4 How much did you get brother? Five has arrived.

[01:18:49]Option number D. If x is n then its derivative What will happen? What will be n x n - 1? n x n - 1 will be. I am asking about the concept brother Nothing else is being asked in MCQ.

[01:19:01]Now look, this is from the question book and this We might have asked the question many times.

[01:19:05]You might have done the same with me. So its What is the answer? sin² a + y / sin a It happens. This is the answer to this. Quick Tell me quickly.

[01:19:16]This is what happens, friend. Isn't it? Now this We will solve it a little bit, not the whole thing.

[01:19:21]Hey friend, think for yourself. So think about it, if I here Let me write x equals sin y / sin a + y And if I apply the cosine rule here, then below Square will be made or not I have to take so much care and here One will come okay one will come up here Now whatever happens, a square will be formed here.

[01:19:45]So pick up this square and bring it over here.

[01:19:46]This will come up square and that will come here He would come down, so I thought.

[01:19:50]Sin per square will definitely be above. So That's all I could think of. After thinking so much I thought this answer.

[01:19:57]Which of the following is a vector quality Quality? There should have been quantity, son.

[01:20:05]quantity Look, there is no direction in mass. Time And there is no direction in length also.

[01:20:12]Force has direction. Newton.

[01:20:14]How much Newton force are you applying? Right?

[01:20:16]Isn't it? But in which direction are you applying it?

[01:20:17]I'm pushing someone. So which direction?

[01:20:19]I'm pushing? Right? The Distance You will be able to find out between the points, friend. How Do you calculate the distance? between three points Of. It's simple.

[01:20:30]x1 - x2² y1 - y2² z1 - z2 This is how it is taken out brother. It's simple. 9 - 3 6² how much? 36 has arrived.

[01:20:47]Ok? 2 - 5² 9 is reached.

[01:20:52]4 - 4 minus minus minus became +1 is 3 Will go. This also came to 39. This is also under route 18 This is it. 36 It's done. 36 18 * 3 = 54. 54 54 would mean There is 3 6, there is exactly 3 6 in the option.

[01:21:11]Option number D3 6, the rest is calculation You have to do it yourself, friend. Let's go down We do that. Come on. OC A is a vector. AB B is a vector. What will be the vector OA?

[01:21:23]Show me. OACB OACB Ok? He said what is my AB vector?

[01:21:31]Brother? Tell me quickly. AB is the vector my B Vector. This is my B vector. Right? Right?

[01:21:39]And OC is my A. From O to C This is my A vector.

[01:21:47]Ok? Right. Right. These have become diagonals And I don't know about OA.

[01:21:53]Where did OA stay? Here is the OA vector. Check do it. Here is the OA vector. I don't know this.

[01:22:01]Ok? So I have it simple. so much to me I know I have to get this out here.

[01:22:06]Buddy. Come now, right? Look, I know this much That it increases half by half, friend. Isn't it? here The intersection is the parallelogram half increased goes. Interests bisect Bisect diagonals. So if I talk I am just and only, only and only So this is going to be my b/2.

[01:22:24]And so this will be my a / 2. Yes and No brother, we That's correct.

[01:22:35]That's correct. Is it true or not? So y/2 / 2 should be b - a. b - a isn't it?

[01:22:43]It did not say which one was bigger, Which one is smaller. Isn't it? Let us make B larger Let's move ahead with respect. So its answer will be C.

[01:22:50]I will check the correct answer once.

[01:22:52]I.

[01:22:55]Which one is it? 17 We. Ok. C C. Ok. I did say It must be like that. Let's do the next one bro We are. If A B and A and B Why did you write it twice? it It should have been C. Then A vector, B vector and B vector r Coplanner. This should have been C. give the What is the value of P? But it should be C Was. This should have been C. somewhere or the other As far as I can tell.

[01:23:29]It should have been C. But no one. Let me I accept this as C. Let me write C I will give it to them. Ok? This is C, your brother.

[01:23:34]Ok? Let it be C. That is C. He is C.

[01:23:36]Ok? It's C brother.

[01:23:39]What does coplority mean? That You do this for all three. It's a simple thing.

[01:23:46]scalar stp scalar triple product which It should come out to be zero. Isn't it? You in Putting all three into one determinant do it. Ok? put in a determinant All three. Do this. Put in the determination Like this. and it will become zero because it is coplanar. So, 2 - 1 1 the first cry came.

[01:24:04]Ok? The second row reads 1 2 - 3 And the one with C is this C. Ok? 3P5 3 P5 What do you do to anyone? Now your Expand it. like I already Expand it, it becomes 10 + 3p, right? it If you go then it is 5-9. 9 5 is 14.

[01:24:29]Right? and this will be p − 6 = 0 g. So this is it 20 + 6p + p - 6 = -14 14 I sent it there. p and this p becomes 7p Done. 6 out of 20 are gone, 14 is coming.

[01:24:47]Right? So this 14 goes over there, so minus 14.

[01:24:49]Will be done. So 7p is equal to minus 28. So the value of p comes out to be -4. Then P The value came -4 means option number B option B Okay, that's what coplier means.

[01:25:01]that brother you are like A vector Here are the components of B, and Here is the third vector, you can keep it like this Take it, the shortened distance is zero Right in Copper, so shortened distance I am using right shorted You should use distance Meaning I want to say this a b c this It happens that your STp product is zero.

[01:25:24]Needed Now we can use it like this also. i.e First crossing and then dotting. Let us go Bhai if the vectors are perpendicular then Dot product r zero. Right? Yes and no.

[01:25:36]If two vectors are perpendicular, then the dot What is a product? It is zero. So 3 * 2 = 6 π is lambda, sorry, this is lambda end + 8 = 0 becomes 8 * 6 = 14. So this The answer is out, 14 plus 14 will come. a little Check it and see comfortably.

[01:25:55]Converse of the statement. Come on brother, last This is your question. Read this. what was written Is? If n is an odd number then n² is also An odd number. That's correct. This is good this So brother, this is true. This is the statement That is right. I have to convert it completely.

[01:26:11]Its converse means that if n² odds If n is an odd number then it means that n is an odd number.

[01:26:19]If I am a boy, I will get married.

[01:26:22]If I am getting married, I will definitely I am a boy. I am giving an example. So this I won't do it that way. Ok?

[01:26:28]I was just giving an example. So Converse Please tell me this statement first.

[01:26:33]Where from? If n² is not. The note is in all three This is the one. If n² is an odd number then It takes conjunction as a conversation to tell So what will be the sure n? It will be an odd number.

[01:26:45]Then n is also an odd number. So this one is B It will be fine.

[01:26:51]So I have prepared this entire sample paper for you.

[01:26:52]It has been solved. I hope you like this a lot You will get the benefit. Not today, but today's children No, not next year, the year after that.

[01:27:00]Even the children, the children of the next year Too. And whenever you any if baby this this If you are looking at the year then son, this year's paper also Math will emerge from this pattern. If the rest If we look closely, all these questions are nowhere somewhere that child could whose The concepts were clear. Isn't it? NOS has Even though this sample paper has been given and this Every question of the sample paper is the same child Can do. Those who have good concepts.

[01:27:22]The concepts should be clear. In concepts It should be deep somewhere. his concepts He should have deep knowledge. And all of that You get the knowledge of our batches Inside. Our batches like Lakshya Science It is ongoing now. It's going on. Isn't it? You You can enroll in this batch. Now What are you going to get in this brother?

[01:27:38]Live lectures will be available in this. Best Instant recording right through to the faculty You will get it. Class notes and PDFs will be available.

[01:27:43]You will find PYQ solutions, Question banks and revision tests will be available.

[01:27:47]There will also be question and doubt solving sessions.

[01:27:49]You can see below. Watch live class Which subjects are going to be examined?

[01:27:51]PCM is also there in it. Mathematics, Biology, Chemistry, English, New Syllabus, English right, data entry right?

[01:27:56]You will get all of these. Recorded Classes Will meet. Then you can download the app Manish Classes App Right Both Any iOS If you can do it on an Android phone, please let me know.

[01:28:04]Looks like the most popular batch is ours Which has brought the highest results every year It gives results, it gives output every year They have so many children [sound of clearing throat] output taken If so, you should try this one too.

[01:28:17]And I hope you understood today's whole You will get the rest of the lectures for the classes.

[01:28:22]andar batches andar and take care and Bye-bye