Engineering Maths-01 (M1) Endsem Paper Solution Full | #engineering #sppu

Added: 2026-06-03

[00:00:09]Students, in this video, we will see the detailed solution of the M1 paper that was held in June 2025.

[00:00:17]Friends, we have a time limit of 30 to 40 minutes, so this will give an outline for each question without going into details so that you will get an idea of how to solve the question. Yes, and after that, I will not give you a line-by-line but an outline. It will give you an idea. Everything will be clear to you.

[00:00:34]Once you understand how to solve the question, then first watch the video properly and understand it. And then this PDF is also given below. Then solve the PDF with the help of the video. It is necessary to understand it first. Okay, I will try to teach in Marathi in simple language.

[00:00:49]I will skip some questions that are very easy, but let's concentrate on what is important because our paper should be scored and if you like our work, then definitely suggest your friends about your subscription. June 2025 is a 24 pattern paper. So, this question paper is the first one in this question paper. Question paper 25 is a small MCQ, I have already given you some MCQs, so you can see them, there is nothing to say, there are MCQs, etc. Let's go to the top now, okay, otherwise I will tell you quickly, first of all, what to do in this is, take the derivative of A, take one with one and then equate both zeros, so that x and v come out, your point is very easy, as Mk is correct, then what is the derivative of this function and a, then one plus x is zero, put it in X, and similarly, how do you calculate it, x is common, x is common, one by one, cut, x and v are zeros, and the formula of Y is the value of one, put in your use formula, in this way, similarly, how do you calculate it, how do you do it, we went to the top, okay, R2 is 13, so it is zero, then R3 is zero, how much is one two, Leave it, then this is the standard solution and here is the answer, what is the meaning of your diagonal element m, it is very easy, look at your questions, first, the LMT, second, the Fourier, third, Taylor's, see what is the first function in LMT, is it continuous or differentiable, we give it, then we put this point which we consider as A and B here and F comes out right, then we take the derivative of that function, so we get A x x and put C in it, and then this is our LMT formula, put all the values ​​in it and finally we take the value of C, 1.71, now we want to check what is the point, what is our one and one, how much is it, usually it is written here, see, it is 2.71, it will come before and one between the two, 1.71, 71, so what happened, verification is very easy, what is to be done, first, continuous, then differentiable, then derivative, x in place Enter C, enter the formula, calculate the value of C, check it, that's all in LT.

[00:03:23]Similarly, look at the question of Four.

[00:03:26]What is the function given to us in Four? We have to solve it in the interval x- pi to pi. The diagram is not necessary. Okay, look, this is our regular formula limit. Our minus pi pi will now be converted into the formula. It will come to zero to pi. Aof x means that x is the same as the series. We have to solve it by taking the derivative. We will solve the integration k. X is the same as the integration of x. Here it remains. Integration minus co a. Next, the derivative of minus xv is the same as the integration of a and c. It is correct.

[00:04:05]And then we put pi to x.

[00:04:07]So this came to minus pi co a. Pi a is the same as the zero. Then we put the zero. Then we put the zero.

[00:04:16]See the limit. Low limit. Zero here. Zero here. Zero here. That's the answer.

[00:04:22]Answer all yours and this by the same answer all minus co pa a and then your answer came bev met b met b3 b fourser that's all your answer this diagram etc. no need to do anything just remark this is given for you to understand it is very easy main is this integration you can draw this answer will come your final answer next write down till here o this is your standard courier format sodhus final answer no need now tellus theorem tellus theorem is very easy x minus given means x minus a to compare with h how much came three right this f of x is taking derivative till we reach zero and put the value of in each value put the value of in three see the value came and all the values ​​came in our Taylor formula just this Taylor's formula put all the values ​​in the formula The solution is very easy. What is the derivative of this given in Taylor? Take H from this.

[00:05:34]Put H in the derivative and put the resulting value in the formula. That's the final answer. Similarly, another problem of Taylor. I have just given it as practice. The same problem is there. Look at it. Next, now these three questions are one microliter question.

[00:05:50]Element derivation. IMP derivation.

[00:05:54]Look here. Sa is the inverse. So is it possible to get Eves or Lag or L. These three can still come.

[00:05:59]Perfect the four derivatives. See what is the important thing. Underv.

[00:06:04]Plus Sa X and Plus Sa X. What is the formula? Sa X is the base plus Co. The square root of the square is canceled. That's all that remains.

[00:06:12]Now the series is Sa F and Co. The series is just the difference. What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do? What should we do?

[00:06:24]Let's see x ba so this series just simplified and you will get this final answer yes so what is the main here you only need to know 1 p sa formula sin x ba p co x bal su andto cancel this then sa and co ch series x place only x ba simplify it o it is a very easy paper see this derivation is important sa eves ok so what to do same first f of sa eves x limit to b first continuous differentiable as then we did a math of lt same f of a b to take derivative to take x put c in place of x put our regular formula and just take this standard condition now see here here c scr one minus under root what what c scr one minus undert as there will be three changes first see this standard formula scr sr as squared c self Minus one minus, see mass s one minus, see under root, under root, up to here, up to here, now there is 1 a, see under root, one a, one a, sign will change again, sign will change twice, after minus one, after adding one, up to here, now this term has value, we have put this here and multiplied it here and here, our answer is ok, I did n't ask you anything, but just for extra practice, sometimes they ask you to prove it, write as much as you think, just to top it, look at this for extra practice, the question of four was explained in the same way, now here children make mistakes in derivatives, see how we can calculate the derivative of y, We have the same equation as 1. Now we have x So it is a problem of derivative of the derivative in this way.

[00:09:45]This problem is simple. Next, how will we solve our regular years problem?

[00:09:52]Take the x s common and the x s cut, so that v is one. Now our formula of the years f is sa. Its derivative co is right. We call it the second formula. What is the formula of mass? What is its s? What is the Similarly, we took the second second, first we took x and y, now we have to do the same for x and y and we have divided both of them. Here we will take this box, the LS but the same answer will come and so both will be equal so look at this simple question I will not go into details now another important question is what will come from here x and by from here x ray and bath will come in common under the root and then it will come out see xv ba x and bath common came yes next this under root went out this happened ba x equal what do we write for this we aofwa ba x went up and bass and baw ba si equal and under root what rev baw bat how much bat m came after you got regular then do the same thing like a saw did a your cosec its derivative minus cose cose cose cava a co means this minus came out so this came you came second formula mass one its date s minus one same way s ch value and plus s simplified then your answer comes and goes 12 common take 12 144 very easy Question three steps first properly draw a from above x batu xv ba from below xv bath take common a draw first first formula a will meet from it and cha second formula mass simplify se slav pson s put solve 12 common draw 12 144 it came your honor s ya similarly composite magashi question is very easy question as told u m jiachment xva means m first x is right and which three values ​​are given to us three equations to consider them l so first how will it come da baba da baba x baba baba x see right next die baba a baba x came three equations now these three values ​​should be drawn so lcha x with lcha x with date how much x mass z mass z this second date always draw in composite we a x with here x is not zero minus zero more term zero x with z mass came minus x here will remain baba this minus comes out all-sunx Baba, what did you do when you first met, Y M X, next, Y M Z, you have to remove Y M Z, remove these three, see y with u L M N Y with Y M Z, do the division of the three, everything will be automatically canceled and your answer will be zero, so remember this simple trick, if there are three equations in the composite, then 0% answer does not come, I understood that I have to adjust it a little and write it down, I do n't have to think about it, now the next functional dependence independence simple question is this, it does n't mix it up much, just Y B D is removed, after looking at it, important questions have come up, after that, the same thing, but mathematics is also easy, this is our regular implicit function's R B J, the rest is easy, how does R B do it, the left box is here, now look at the first R, the lower box, then the T was adjusted, it was not here, the upper one came, the X came here, this is The term is minus, so now the box f and a left x box f and a da all similarly f and f and look at f2 determinant is f and given derivative, simplify and simplify, solve the above date determinant, solve below, automatically you will get the final answer, given that you will get x x ba final answer, it is very easy, yes question solve maximum minimum same type question is very simple question, take the derivative of maximum minimum, differentiate it, compare with the given zero, x or will meet, again you want to do double derivative, if you will meet the value of r st, put it at that point where x or will meet, simplify, the value of maximum minimum will meet you, a minimum a maximum answer, this is the final simple question, so why do I do it? Does not explain maximum minimum the easiest part Jacobian same way as before chain example our look what is asked in the question box aba now we have function kush ch what xva with how our chain r is formed xva na so box aba xva then xva with who r with da xva abata this our chain r fits and then same dad look left y ba left dad x left here but same hell take the derivative of given and see see how much x with x yachay with minus x how much will come with xv all similarly x with r what will come here co x with r minus sa right take the derivative in this way cross multiply generally we solve determinant then cross multiply sa scr cos sr theta what happens one and x s plus s what happens in r scr so final answer It will come to you, it is a very simple question, functional dependence, independent maximum, minimum and implicit function are very simple topics, error function is very simple, simple topics, one tells you that error function is important mathematics, the rest is a very simple example, see it, you are direct, it is very easy, now what is given here, E A S A As your mathematics is given, see the value of A, what did you say first, where did the energy change from here, 49 49.5 and velocity 16 150, so you will know that from here, it has changed, it has become 49 original, so 00, right now, the change that has happened, we call it derivative, right, so how much has changed, 49 495 has changed, so how much has come, derivative means change and 1600 to 1590 means it has decreased by 10, so this velocity is the derivative of velocity minus 10, look and then use your regular log.

[00:19:04]What will come here lag 1 ba m sorry lag-yan lag 1 bavarati gela-su zala-yan lag direct write lag pl lag ba will work because its date zero is coming date and bati's zero a md addi came after derit comes what do we need just ask what we want we asked approximate change in t we want the value of t i.e. change of dt i.e. we said here derivative dt so t will go here and just put all these values what is the value of t that m s a means 1 ba m² dm look here f he put m's value 49 value of 49 and 16 equal value of f 4d i.e. minus solved it it came our change sus and if we want to remove the error how will we remove the error this math will be repeated here we will multiply it by hne hne right and put all these values ​​given just all these values ​​it comes our Error is important math is changing a little bit math is very easy this math is a little tough tough means there is a little adjustment here because till now for error we had to have direct value here let's say f% 10% change here a little slide deflection give here kids get confused so I don't explain otherwise math is probably not explaining very easy similarly here is a simple example here is a small example to take derivative compare value will come x wasz simple math look at it we don't have that much time what is important it tells system of linear equation matrix is ​​easy chapter but in it they ask value of k and math this look this is math what is value of k here ok fu mass thz thw k your regular what do we have to do to get rank what do we do for rq now f r2 f r2 minus aw yes see cross multiply What will be the value of f2 minus 15 minus 31 minus 15 and 15 will be zero and zero.

[00:21:25]Now let's see here. We will do this operation. This value will come up with the same operation.

[00:21:37]So f k p si came. Then R3 p R2 is zero. This is 14 s 20 zero. Now look below. We have found out what happened here. What is the mathematics? In mathematics, non-trial solution is called non-trial solution. What is the condition in non-trial solution?

[00:21:56]R is equal to R.

[00:22:01]So what will we have to do? When will this trial happen?

[00:22:02]When will this zero happen? When will this zero happen? Rank will come. When will rank come? What will happen when rank comes?

[00:22:07]How much will our R be? How much will our R be?

[00:22:11]How much will our two be?

[00:22:14]And that is, unknown. How much will our X be? So, when two are equal to three, then only will our X be? It can be a non- trivial solution but now what happened rank also came three and one two three look here value means rank also is three and that means unknown but is three so non-trivial does not happen what will be required for non-trivial it should also be zero so to make this zero take zero this came here how much came minus fo when k put minus fo here this minus plus zero this row will become zero and when rank comes two this condition will be satisfied so I know what to do for non-trivial solution which k should be equal to zero k to take away k if put here then it will become zero it will become raku and unknown is three lesson then it will be satisfied simple math is telling me once again what did you do non-trivial math or finding the value of independent depend is very easy do your regular row operation this zero in zero ass zero after zero comes in zero equation To make zero equal to k, to remove it, it is very easy, do regular operations, which will give the value of k in the equation, compare the zero, and then the next linear dependent independent is very easy. What do we have to do here? We need to bring the zero to the zero, so we can process it regularly. I will not explain much here. Because the same R and M are interchanged, one zero has gone up.

[00:24:12]Now what I did is to multiply R by A and M and then the zero will be zero. After doing all the operations, see this complete zero comes and goes. And then we say, "C2 C3 zero z One C3 C3 what is it that comes here minus C2 C3 but what came minus C and T also came with zero also came with zero means what happened linear depend then to find the relation our regular equation has found the values ​​of C2 C3 in it minus discard them solve it any way suppose I put minus here two minus went here minus common subtracted minus common subtracted from here it cut x and x2 plus x3 this all relation is so easy so in linear independent just do regular row operation below this zero zero should come zero Given is its transpose how to transpose row what to do column row column row column we will multiply both row column same answer is same if it is not one z write it down ortho means show and its transpose same answer always write ortho and inverse means what property is orthogonal inverse means transpose just write this row column this row column row column it became your inverse o how easy is it similar question not consistent inconsistent same depend independent similar to do this zero operation regular operation this zero zero came now if you look we this matrix a is ch see how much is one two three this is zero means two r and this a s b s b its rank is same so what is that matrix become consistent yes and consistent what does it do after separating x as mentioned earlier Put the value of the wave in it, X will be found, X will be removed, it is finished, consist, consistent, it is very easy, linear depend, independent, I saw the same thing, now it is not repeated, I say, in the same way, do this zero, zero, X3, put the value, C2, C1 will be found, draw the linear depend, independent relation, it is simple, I explained it, current network mathematics It is in the same way, but look at it carefully.

[00:27:06]If possible, do not solve the current network. Solve its questions or do the questions in the options.

[00:27:10]In the current network, some corrections are made by the children. Therefore, it will be easier to solve the regular other questions than solving the network questions.

[00:27:19]Therefore, I will not explain it. Now look at question 10. Similarly, to find the eigenvalue, this is a minus Lada. Subtract S from the diagonal.

[00:27:29]How to get the dish of diagonal S?

[00:27:33]These two, these four sides, these two come to the determinant.

[00:27:39]I have taught trick after trick. See in your video. I have put the correct value in the equation. On the calculus, minus one to one. Then what will you put in Lada? We will put minus one in this.

[00:27:52]Then we will put lambda to Lada. We will put one minus one. The equation comes to come. We have to solve it.

[00:27:59]X is wise. We have met them and always come to come. How to take the rule of minus one? Let's take the rule of minus one.

[00:29:02]What to do in eigenvalues ​​Just remember the steps First of all, A minus L is the determinant A1 A2 and the determinant is to be removed from it.

[00:29:11]From the equation, the eigenvalues ​​will be found. The eigenvalues ​​will be put into all these determinants and solve using Crum's rule. Solve three eigenvalues ​​three times, solve two eigenvalues twice. The example is over. The colonial form is the same. The example is the same.

[00:29:28]Here I have given a photo of the book I gave. Yes, but perhaps eigenvalue mathematics, Cali Hamilton mathematics and modal matrix mathematics are easier than the colonial form mathematics. If you do this, then the modal matrix and the eigenvalue are the same.

[00:29:43]Solving the eigenvalues only increases one extra step in the modal matrix. The last step is to remove the p and d, which means that the modal matrix p and the diagonal matrix only need to be removed, which means that there is only one step. Increasing eigenvalues ​​Eigenvalues ​​Eigenvalues Eigenvectors and modal matrices are both the same except that the last step in the modal matrix increases. Therefore, the modal matrix Cayley Hamilton theorem and eigenvalues ​​are only three examples, such as the colonial form, etc. Rather than explaining the difficult ones, I will explain what is important, but I will explain them in the beginning. Out of 70 or 60 out of 60, out of 70 marks are obtained. The given pattern question number is the same as the modal matrix. Now, we have explained the eigenvalues.

[00:30:24]First, understand this matrix just by looking at it. This is a negative number. We will draw the diagonal direction S2. We will draw the determinant. The equation equation eigenvalues ​​are obtained. The eigenvalues ​​that have been obtained go into the eigenvalue matrix. So, the three values ​​of the eigenvalues ​​are obtained. We will use Commerce Ru three times. Right up to this point, we have solved the eigenvalues. Xv X X3.

[00:30:46]Now what happened extra? The model only increases by this step. Look, I said that the eigenvalue is the same as the modal matrix.

[00:30:51]Only one last step needs to be added to the modal matrix. The eigenvalue is the only thing that would have been mathematically solved. The math would have ended here. x3 increases by one step here. p = X123. This minus sign that came is sorry. 3-and X2. How to write the number and how to write the number. I wrote that much. Look, I only wrote two lines. The modal matrix has increased by one mark. And the seventh step is diagonalization. The diagonal means λ3. That is, all the three values ​​that came from this value λ.

[00:31:21]Write them down. Look, the diagonal matrix is ​​known. The eigenvalue is the math. The math would have ended here. The modal matrix has only increased by one and one. When three values ​​are found, write them together in P and in D.

[00:31:40]Write the value of the diagonal, how easy is it to pass the diagonal, perfect what I tell you, besides that, how to take each pattern out of the mark, how to pass in two days, according to your requirements, the pattern video, but the IMP pattern etc. folder that I have given in the course, watch it in your course, the idea will become clear.

[00:32:00]Salimilton, last portion, same way as in Calimitan, but first two steps, first, calculate the mass matrix, calculate A2 and calculate the determinant. This equation is done with regular expressions, just as in eigenvalues, the same method was used in Cali Hamilton. Here, I calculated the values ​​of lambda, that is, I calculated the eigenvalues, but there is no need to calculate them. If I had calculated them, it would have worked because in Calimitan, we do not use values. Do not calculate them. Now, I do not remember, so I calculated them, but there is no need to calculate them.

[00:32:32]Write directly up to here. I am done.

[00:32:35]Take out the scr, why is the matrix a twice, now the row column is not multiplied, how to solve this on the calculus trick, we learned in the video lecture of Already, I taught all the tricks, just take it out in that way, what is the same as when it comes to a s, write the value of a s, write the value of a and get the direct answer on the calculus, what to do now, we had this equation, what to put in it, just put a, put a minus, what will happen to it, a cube s, sn, which is a constant, I have done the dot, in tally Milton, what to take for the constant that is a, put all of these, see the value of s, here I have taken out a s, here I have taken out the value, see on the calculus, I have put s, minus s, a s, put in the question itself, what does s mean, our identity matrix is ​​​​and does not solve, sit down, write down all the zeros, zeros, all the caly Hamilton, reified, that answer Zero will come, see how much sinv has happened plus si -1 p si how much a -yin f 11 a means six minus s left zero a see zero all zero will come solve write directly first step cleared matrix call and calimilton pr now what do we have to do re fo when will it happen m four when multiplying this by a cube how much will this three become four then fo will happen multiply everywhere see everyone has multiplied soti f this 2 + 1 3 a * a a² i ai a means a and just all the values ​​we have to remove a 4 so take all these here minus has become plus plus has become minus minus has become plus now si k then write it down by trick 11 s also write it down by trick si given in question it does not solve sit fo but value direct calcs trick i taught that much write it down on direct calcs now I want to calculate s while calculating but sem is the main equation this is the main equation I want to multiply s by s see while calculating fo and I want to multiply s by s while calculating s I multiply everyone s is correct and two and x becomes s becomes s is correct now I want to calculate s send it here the signs will change and multiply s see six multiply that div now what to do next now let's see by s a s already calculated it wrote down s is in the question but we know and it doesn't solve it sit directly I taught on calculus write down the direct answer is over math is clear so it is so easy remember what is done in cali hamilton once I tell you in short once again first time first process sem a mass matrix a1 a2 find determinant equation will become come up to this point no need to calculate equation s Take it out on the calculator, then put the equation of this case and show it zero by zero. After Milton's correction, the equation has been multiplied by the equation, it will become a fo, send all the terms here, take the fo on the calculator, take all the answers, take the main equation, multiply it, put the rest there, take the answers on the calculator and write them directly, it's done, friends, it's so easy, in this way, this was our entire June 25 paper, which I have explained here, along with this, I have explained all our M. Ch papers in detail, the outline in this way, now you may have got an idea, how to solve the MO paper, if you are still struggling, watch the video once or twice, now it will work even if you watch it in fast forward, the speed will increase, now once you understand the video, then now I will give you this PDF below, solve it, see, you will get M. Ch out of mark in a very easy way, I have taught you on the calculus trick in Marathi, take advantage of it, if you liked teaching Suggest it to your friends too and I wish you the best of luck because after watching this, you will definitely get out of mark because I have solved all the papers and explained them to you on the platform, but there is no benefit anywhere else.

[00:36:56]Best of luck.