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2nd PUC Maths Fixed Questions & Solutions | ಇದನ್ನು ಓದಿ 75+ Marks Pakka in Midterm Exam 😎 — Transcript

by PW Kannada · 17,859 words · 2,910 segments · language en · Watch on YouTube

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0:01Hello Makla. Namaskar.
0:03Good evening all. Good evening all my
0:06dear students. Namaskar. Good evening
0:08all. Good evening all my dear students.
0:13Preparations.
0:18Are you all consistent?
0:27preparations.
0:33So preparations.
0:45Yes. Hello. Hello. Hello. Good evening.
0:48Good evening. Ganda. Good evening.
0:51Good evening. Good evening. Yes. Hello
0:54extra motivator. Arya. Arian. Arian.
0:58Hello Kan. Hello Lakshmi. Hello Muzam.
1:03Hello SA. Hello Asha. Better sir. Okay.
1:07Many more. Happy birthday. Aishu tak.
1:13Okay. Fine. So SA good evening. Yes.
1:16Hello Sharat. Good.
1:19Okay.
1:21Fine. Makla.
1:26Second
1:41engineering
1:43board exams.
1:47Okay.
2:00Yes. So students, hello. Hello. Yeah.
2:03Thank you. Thank you. Thank you.
2:09Thank you for your wishes students.
2:11Happy teachers day. Wishes. Thank you.
2:13Yes. Good evening. Good evening. Tak
2:16stop chatting.
2:26At least
2:33model already
2:43exams
2:46And
2:49application of derivatives.
2:52Application of derivatives.
2:55Very good. Very good. It's great.
2:59Superb.
3:04Okay.
3:06100% great. So differentation is
3:09difficult.
3:17Yes.
3:20So now
3:22Okay. Fine. So
3:25students Hello
3:28students attend
3:32students.
3:37So
3:44Yeah.
4:03So,
4:06yes, I know you said already. Yes.
4:10students
4:16see don't worry don't worry about it so
4:18already ma'am we are okay
4:24important questions
4:27fine fine don't worry don't worry
4:30so
4:38topics
4:43What I would like to suggest you is yes
4:45what I like to
4:49not
4:51so class
4:54so
5:11You can get it in the PW
5:24WhatsApp
5:27WhatsApp.
5:30WhatsApp.
5:38So integral lesson when will crash
5:41course start crash course.
5:45So KC road.
5:49Okay. So
5:51fine. So
5:54derivatives
5:58okay
6:00properties of properties of integral I
6:02don't think we can do here okay so
6:06exactly second order derivatives so five
6:09questions
6:11second order derivatives five questions
6:14should I solve the five math questions
6:16from second order derivatives
6:18so up to 11:00
6:2010:30 11:00
6:23Okay. So yeahvember
6:32or December somewhere in that time we
6:34will solve all model questioners don't
6:36worry
6:40so
6:45okay fine
6:48linear programming problem.
6:52Okay. So, properties of exams
6:57district
7:01in the blanks
7:07fill in the blanks. Go and watch it. So,
7:09important question
7:134 minutes you can get the video there.
7:17Okay. So, exam syllabus up to 7.6. Okay.
7:23like integration by fine.
7:28Okay.
7:45Fine.
7:47Okay. So,
7:55okay. Fine. So, coming to this
8:01five mark questions. Let me come to the
8:03five mark questions.
8:05Questions.
8:10Solve solve by matrix method. Solve by
8:12matrix method.
8:15Solve by matrix method.
8:18I will suggest you one thing. I will
8:19suggest you one thing students. First
8:24okay. So complete four marks problem.
8:26Complete four marks problems. So four
8:28marks. Complete four marks problems. So
8:32four marks.
8:34So complete four marks problems.
8:37You need to do this first. You need to
8:39do this students.
8:42Okay.
8:47So,
8:50complete this first
8:56question.
9:03So 10 questions
9:05and next complete five marks problems
9:09problems.
9:11So five marks
9:17go stage by stage.
9:20I believe this and I also ask my
9:22students to follow this
9:25stage by
9:28confidence
9:30and sense of safety.
9:38So first try to get that 30 marks 30
9:46I want to take out of
9:5030.
9:53If you want to get 70 or 75 first you
9:55need to cross the stage of 30. So I
9:57suggest you to go with marks wise rather
10:00than chapter wise
10:06but exams preparations go marks wise go
10:11marks wise don't go chapter wise chapter
10:13wise
10:14you will feel that you need to complete
10:16lot of things
10:21you need to be very smart enough you
10:23need to go only marks wise so first four
10:25marks complete go with five marks
10:28question students five marks problems
10:32relations and functions yes relations
10:34and functions
10:39Okay, next one.
10:42So I will get in matrices students and
10:44next determinants and next to
10:46determinance.
10:48Okay. Determinance and then students
10:49finally continuity and differentiability
10:51that is 5.7 5.7
10:57final exam
11:01right
11:16like half of the
11:18Sorry, final exam.
11:25I hope you getting clarity. Okay. So,
11:28next
11:33so perfect. So, perfect.
11:41See
11:46you will try preparing for 70 75 marks.
11:48You will try preparing for 70 75 marks.
11:50You will end up with 20 or 25 marks or
11:5330 marks.
11:5730 marks.
12:04Prepare for 50 marks. Prepare for 50
12:06marks and try to get 45 and above.
12:13confidence level
12:16you will feel like you can win the whole
12:18world. Okay. Instead of that
12:2325
12:32prepare this okay
12:37already 24
12:41then what you need to do next
12:45selected two marks three marks questions
12:48selected two marks and three marks
12:50questions. Perfect. Selected and
12:53selected two marks and three marks
12:54questions. Selected two marks and three
12:57marks questions.
13:00Yes.
13:01Okay. Selected two marks and three
13:03marks. Okay. Determinance. Determinance.
13:08Determin
13:19Okay. First one. Second one. Matrices.
13:22Matrices. Sum of symmetric and sky
13:24symmetric matrices. Okay. Sum of
13:26symmetric and sku symmetric matrices.
13:29Sum of symmetric and sku symmetric
13:32matrices. symmetric and sku symmetric
13:35matrices.
13:39Sum of symmetric and symmetric matrices
13:46right
13:49next differentiation okay so next
13:52differentiation
13:54so differentiation students
13:55differentiation complete 5.2 5.3 okay
13:595.4 okay all up to 5.6 5.4 5.5 and 5.6
14:05important important 5.6 5.3 these two
14:09are very very important.3.6
14:2434 marks you got now 34. So
14:36okay we'll get two questions
14:40simple question and okay three marks
14:44question
14:49show that a whole inverse equals to b
14:51inverse into a inverse.
14:54So inverse inverse
14:57inverse inverse
15:05show that show that a whole inverse show
15:07that a whole inverse equals to b inverse
15:09into a inverse e proof important. Okay.
15:13Next one. If a is a square matrix. Okay.
15:16If a is a square matrix. If a is a
15:19square matrix.
15:22If a is a square matrix then if a is a
15:25square matrix students then a plus a
15:27dashes then a plus a transpose is
15:30symmetric. Okay a plus then show that
15:33then show that okay then show that a + a
15:36transpose is symmetric a + a transpose
15:39is symmetric and a minus a transpose
15:41equals to sky symmetric and a minus a
15:44transpose equals to sk symmetric.
15:48Important questions all these are
15:49important questions.
15:52Okay.
16:0350 marks
16:0550.
16:19Main
16:25you will be passing.
16:37So believe my word students. Every year
16:39I just train my students like this
16:45and each year they're excelling very
16:46well. So they are getting students they
16:48are getting more than like Makla they
16:51are getting like 17 last year 170
16:54members got out of our students.
16:58Okay so just give me a moment
17:20Okay.
17:23Yeah. Thank you. Thank you. Madaga.
17:24Thank you. I think you are a previous
17:26Okay. This year student
17:29you are this year student.
17:32What is your name? So question came for
17:35mid. Yes.
17:38questions.
17:43Okay. So,
17:49sorry question. So, how to become
17:51perfect sir? So,
17:59perfect.
18:03So,
18:05question
18:07Five questions.
18:09Okay. So, which topics now do you want
18:11students? So, yeah. So, what should I
18:15reply?
18:21Okay.
18:27So, different
18:29but don't worry. See
18:34first
18:3915 days.
18:44So
18:49perfect.
19:08Yes. Yeah. Okay. Fine. So
19:12let us come to this. Okay. So let us
19:14come to this students. I think five
19:17questions. Okay. So we we can start with
19:20five questions.
19:23So proof.
19:36Yeah. So coming to this
19:41Okay.
19:44So coming to the students five problems.
19:48These are five problems.
19:53Watch here. So problem solve.
19:57So investometry two and three mark fix
20:00questions.
20:01Yes. So investometry model. So should I
20:06start withometry
20:07then? Okay.
20:11You want inverseometric functions.
20:19Okay. Let me start with inverse
20:20trigonometric functions. Okay.
20:33Fine. So
20:41important.
20:42Yeah. Two marks
20:44problem important. Prove that. So prove
20:47that. Prove that students. First one. 2
20:49sin inverse of x is equal to. Okay. Sin
20:52inverse of sin inverse of 2x into
20:54roo<unk> of 1 - x².
20:57Okay. Then second one prove that equals
20:59to prove that 2 cos inverse of x is
21:01equal to sin inverse of 2x into roo<unk>
21:04of 1 - x².
21:06So proves important these three proofs
21:09sorry these four proofs are very
21:11important 3 sin inverse of x is equal to
21:13it is sin inverse of it is 3x - 4 x cq
21:17and then students and then it is prove
21:21that prove that students 3 cos inverse
21:24of x is equal to it is cos inverse of it
21:27is 4xq - 3x in proof important proof
21:33do you know these four proofs in proof
21:35proves. Do you all know these four proof
21:38students? Yeah. Thank you, Mega. Thank
21:41you.
21:51Okay. So,
21:57should I should I solve Sorry. Should I
21:59prove all these four?
22:04should I prove this for or
22:08yeah
22:10proof
22:19no need proofs.
22:27So we want Romesh already I have done
22:30MCQs Romesh morning I had MCQ session go
22:33and check it morning I have done MCQ
22:35session go and check it so prove it okay
22:38so okay first one prove that yeah prove
22:42that students 2 sin inverse of x is
22:44equal to it is sin inverse of 2x into
22:47roo<unk> of 1 - x²
22:50very simple the proof please watch so
22:53very simple students proof so inverse of
22:57sin inverse of x. So take this sin
23:00inverse of x= t. Put sin inverse of x=
23:03t. Therefore sin x= sorry x= sin t.
23:07Therefore xalin
23:09t.
23:10I hope this is clear. And then here and
23:13then here students roo<unk> of 1 - x².
23:16Okay. Root of 1 - x² is nothing but
23:19roo<unk> of 1 - sin square t.
23:27So roo<unk> of 1 - x square nothing
23:30roo<unk> of cos square t okay therefore
23:32we got here it is cos t okay so roo<unk>
23:35of 1 - x² roo<unk> of 1 - x² equ= to cos
23:40we get here roo<unk> of 1 - x²= to cos t
23:43so therefore we get therefore students
23:45we know that
23:47we know that
23:54we need to take the same things. Okay.
24:05Observe this.
24:10We know that. Okay. Sin 2 theta is equal
24:13to it is 2 sin theta into cos theta. We
24:15know that sin 2 theta is equal to 2 sin
24:18theta into cos theta. So what is okay 2
24:21sin t sorry. So let me write your t. So
24:25sin 2t equals 2 sin t into cos tin.
24:33What is sin t students? Sin t.
24:37What is sin t here?
24:45Okay, what is sin t students here? Tell
24:47me first.
24:50It is 2 into x into roo<unk> of 1 - x².
24:54I hope this is clear. And sin 2t.
24:58Okay. So now here we get 2t equals to it
25:02is sin inverse of it is 2x into roo<unk>
25:05of 1 - x².
25:08I hope this is clear. What is t
25:09students?
25:11What is t?
25:13What is t students?
25:16Sin inverse of x. So therefore we got
25:18here it is 2 sin inverse of x is equal
25:20to. So therefore 2 sin inverse of x is
25:23equal to it is sin inverse of it is 2x
25:27into roo<unk> of 1 - x².
25:30I hope this is clear.
25:43I hope this is clear.
25:49Keep okay. So
25:53fine.
25:55Okay. Fine.
25:58I hope this is clear.
26:00Makla comment.
26:11So toxic boy.
26:14So
26:29but this proof.
26:32So second one students 2 cos inverse of
26:34x is equal to it is co sorry it is sin
26:38inverse of prove that. Okay. So second
26:41one students prove that. The second one
26:44it is prove that 2 cos inverse of x= sin
26:46inverse of sin inverse of 2x into root
26:49of 1 - x².
26:52Okay. So solution student solution
26:56watch everyone.
26:58So
27:00inverse of x
27:06inverse
27:09inverse
27:14inverse
27:16inverse
27:18cos inverse of x=
27:21cos inverse of xal tal
27:26and please Watch please watch students
27:28roo<unk> of 1 - x² is nothing but
27:31roo<unk> of 1 - cos square t. So what is
27:34roo<unk> of 1 - cos square t? It is
27:36nothing but roo<unk> of sin square t.
27:38Okay. So therefore we got your student
27:39sin t. So roo<unk> of 1 - x²= to sin t.
27:44So now
27:542 sin theta into cos theta 2 sin theta
27:58sin t into cos t. So
28:012 into 2 into sin t into cos t. So what
28:05is this? This is nothing but sin 2t.
28:08This is nothing but sin 2t.
28:19There is there are ins only 2x into
28:23roo<unk> 1 - x²
28:26into sin t into cos it is nothing but
28:28sin 2t. So write that formula formula.
28:32We know that we know that sin 2t equals
28:36to We know that sin 2t equals to it is 2
28:39sin t into cos t same thing same thing
28:46what we have got here we have got 2 into
28:48roo<unk> of 1 - x² into x
28:52right and here we have got sin 2t so
28:55therefore 2t equals to it is sin inverse
28:58of sin inverse of 2 into x into roo<unk>
29:01of 1 - x² square right and
29:05what do we get now? Now what we get? So
29:07therefore students what is tak
29:11what is this tiny is nothing but cos
29:14inverse of x. So we get here it is 2
29:16into cos inverse of x equals to it is
29:19sin inverse of it is 2x into roo<unk> of
29:231 - x². I hope this is clear.
29:29CL.
29:49I hope this is clear.
29:53Any doubt students?
29:56Hey Kan.
29:58Hey,
30:00stop chatting there and go out of the
30:01class.
30:05Please stop commenting there and move
30:07out of the class. Don't
30:18Don't spam here.
30:24So fixed questions.
30:28So already it will be uploaded.
30:42Yeah. Thank you, SAS. Thank you.
30:55Fine.
31:00So, thank you. Thank you. Thank you.
31:05Okay. Fine.
31:08Yes. Yes. Fine. So, important question.
31:14You will get you will you will get
31:16important questions of everything.
31:19So why because I'll be having classes in
31:21paid.
31:28Okay. Fine. Yeah. Fine. Fine.
31:35So important questions.
31:40Okay. Fine. So
31:44Okay.
31:46So let me give the proof of the third
31:48one. Prove that. Yeah. Prove that
31:50students 3 sin inverse of x is equal to
31:53it is sin inverse of it is 3x - 4x cube.
31:58So same thing
32:01inverse.
32:03So we'll start from this.
32:05So proof coming to the proof students.
32:09What we'll write here? Put put students
32:11sin inverse of x= t
32:14put sin inverse of x= t. Therefore x=
32:17sin t.
32:19Okay. And then students here.
32:25What do you need to observe? You need to
32:26observe this 3x - 4x
32:28we know that 3 sin t. Okay. So 3 sin t -
32:334 sin t. What is this 3 sin t - 4 sin t
32:37and what is that? This is nothing but
32:39sin 3t. This is nothing but sin 3t.
32:43Okay. Sin 3t. So what we can write? So
32:47we can start like this students. We know
32:49that sin 3t is equal to we know that sin
32:523t equals to students it is 3 sin t
32:55minus 4 sin cub t.
32:58Okay. So therefore what is sin t
33:01students? Sin t it is 3 into x. This is
33:05nothing but x minus minus students 4
33:08into 4 into what is this student? This
33:12is nothing but x cube. Okay. So we got
33:16here it is 3x - 4 xq and here it is sin
33:193t. So
33:22what we can write? So 3t equals to it is
33:24sin inverse of it is 3x - 4 x cube. I
33:29hope this is clear. And what is t
33:31students? T and makali. What is this t?
33:34tally
33:36what is this t students t is nothing but
33:37sin inverse of x so we get here it is 3
33:40into sin inverse of x 3 into sin inverse
33:43of x equals to it is sin inverse of 3x -
33:464x
33:48I hope this is clear
34:03Yes.
34:09Okay. So, yes. Yes. Mesh may concentrate
34:14on this.
34:16So, fourth one. Fourth one students
34:19prove that 3 cos inverse of x is equal
34:22to it is cos inverse of 4xqub - 3x.
34:27So same thing students 3 cos inverse of
34:30x cos inverse we'll be utilizing this
34:33we'll be utilizing this students
34:36put cos inverse of x= t therefore x= cos
34:40t x= cos t so
34:463x so
34:49we know that we know that 4 cos theta 4
34:53cos t okay - 3 cos t equals to about mlu
34:58this is nothing but cos 3t
35:00this is nothing but cos 3t so utilize
35:03the same thing students utilize the same
35:04thing here so we know that we know that
35:07students cos 3t equals to cos 3t is
35:10equal to 4 cosqt
35:124 cosqt - 3 into cos t okay so therefore
35:16we get
35:18it is 4 into what is cos cube it is x
35:21cube okay - 3 into what is cos t it is x
35:25okay so Therefore we got 4x cq - 3x and
35:29here it is cos 3t. So therefore students
35:323t equals to it is cos inverse of it is
35:364 xq - 3x.
35:39Okay. And now students here. So what is
35:42this t? t
35:44what is this t? t is nothing but cos
35:47inverse. So 3 into cos inverse of x
35:50equals to it is cos inverse of it is 4xq
35:53- 3x.
35:57I hope this is clear.
36:02So this is a practice class. This is a
36:04practice classifi
36:24particular topic.
37:45Fine. Important questions.
37:51Important questions.
37:56Okay. Yes. So differentation chapter
38:01integration important questions
38:04partial fractions partial fraction.
38:11Yeah.
38:14Okay. Fine. How the part A part B? It
38:19will come in part B.
38:24Okay. So coming to this
38:28question
38:30you all know how to solve this question
38:39part A part B.
38:43Okay mass integration questions
38:46six proofs
38:481x² + a square 1x a square + x² so I
38:52better suggest not to study that right
38:54now.
38:55So five questions.
39:08So listen to our words.
39:22But exam be smart enough. Be smart
39:26enough. Listen to our words. So first
39:31finish these four chapters matrices
39:33determinants continity and
39:35differentiability 5.7
39:38relations and functions.
39:43Okay.
39:47I'm not going to prove to anyone here
39:51exam
39:53prep.
39:55Are you getting my words?
40:01See
40:05okay but exam point of view
40:09change your mindset. This is not the
40:11right mindset which you're having.
40:14So
40:18try this.
40:20So simplest for y = inverse of it is 1x
40:24roo<unk> x² - 1
40:27x²
40:29x²
40:31this is similar to similar to what
40:35similar to yeah similar to square
40:40this is nothing but put yeah put x=
40:45okay because we know that square minus 1
40:47is nothing but yeah seeant square theta
40:49minus - 1 is nothing but it is tan
40:51square theta.
40:56Yes. So what we get? So put students.
40:59Yeah. Put here put x is equal to secant
41:02theta. Put x is equal to secant theta.
41:05Therefore theta is equal to secant
41:07inverse of x. Okay. So therefore we get
41:10here y equals to it is c inverse of c
41:14inverse of students 1 divided by square
41:16root of secant square theta. Okay. the
41:18secant square theta minus 1. So what do
41:20we get?
41:22We get here y = c inverse of it is 1 /
41:26root of tan theta. Yeah. Square root of
41:28tan square theta. Okay. Square root of
41:31tan square theta. Therefore we got here
41:33it is y = to c inverse of y = to c
41:36inverse of 1 / tan theta.
41:40Okay. So therefore we get now here we
41:43get it is y equals to yeah y equals to
41:46students cot inverse of cot inverse of
41:49cot theta c inverse of cot theta
41:51therefore y equals to theta y= to theta
41:54therefore we get y equals to yeah what
41:56is theta students theta is nothing but
41:58seeant inverse of x so what is the
42:00answer students seeant inverse of x is
42:01the right answer
42:08yeah yeah yeah so next not coming to
42:10This
42:13how do you convert the students
42:16question? This is a three marks
42:17question. This is a three mark question.
42:21How do you solve this students?
42:24Observe your tan is there.
42:28Observe here. You got tan.
42:33Okay. You got tan. So
42:40okay
42:43it is of the form tan of a + b. So tan
42:47of a + b is nothing but tan a + tan bid
42:511 - tan into tan ber
43:05which are doing wrong which are doing
43:07wrong students
43:13that is not the way you need to go. So
43:16okay. So watch here.
43:21Okay. So yeah let students please watch
43:24please watch your students. Let sin
43:27inverse of 3x 5 equals to a. So
43:30pythogoras theorem.
43:32So you get here it is 3 and then five
43:34and here it is four. So we got here tan
43:37a equals to tan a equals to tan a equals
43:41to this 3x4 and then and then students
43:44let let here let here c inverse of 3x2 c
43:48inverse of 3x2 equals to b. So using
43:52using the Pythogas theorem here. Okay.
43:54So here we get three and then two sorry.
43:58Yeah. So three. Okay. So directly
44:02directly
44:03we can write it as we can write it as
44:05tan inverse of sorry. Huh? Tan inverse
44:08of tan inverse of 2x3. This is nothing
44:11but tan inverse of 3 2x3. Tan inverse of
44:142x3 equals to b. Okay. Therefore tan b
44:17equals to 2x3. tan b= to 2x3. I hope
44:21this is clear. So therefore therefore
44:25therefore students tan of tan of tan of
44:28sin inverse of so sin inverse of 3x 5
44:31plus c inverse of 3x2 c inverse of 3x2
44:36is written as
44:39this is written as yeah this is written
44:41as this is written as students tan of a
44:43+ b this is written as tan of a plus b.
44:47Okay, because this is a and this is b.
44:49So what do we get? We get here students
44:52it is tan a plus tan b. Okay, here we
44:55get students tan a + tan b divided by
44:58tan a + tan b divided by it is 1 - tan a
45:01into tan b it is 1 - tan a into tan b
45:05okay so here we get here we get students
45:08tan a is nothing but 3x4 plus tan b is
45:11nothing but 2x 3 / it is 1 - 3x 4 into
45:152x3 I hope this is clearly
45:23did everyone Understood this. Reply me
45:25fast.
45:40Did everyone understood this? Reply me
45:42first.
45:53What? What doubt you got here?
46:02So I have got some other batches.
46:07So I'm tightly packed with the schedule.
46:26What do you mean by no sir?
46:35Okay fine. So LCM
46:39take the LCM here. LCM what is LCM
46:42students? 12. Okay. And here also 12 is
46:45the LCM.
46:47Okay. So 12 LCM
46:4912 LCM. So we get here it is 9 + 8 / 12
46:55- 6. So what we get what we get here?
46:59Yeah. What we get? We get here it is
47:02nothing but it is 17 / 12 divided by it
47:05is 6 / 12. So what is the answer? 17x 6
47:09is the right answer. 17x 6 is the right
47:12answer.
47:14Formula
47:23don't use it.
47:25mention
47:31don't do that.
47:34Don't do that. You will lose marks in
47:36the final exam. Tan inverse of x plus
47:38tan inverse of y formula in a textbook.
47:41So since they are not given that formula
47:42you cannot use it.
47:45So if you bored you can move out of the
47:46class.
47:48So formula
47:5211.
48:18How will you solve it? How will you do
48:20the problems in exam?
48:34Do you know
48:39exam?
48:50Okay. So starting.
49:05Okay. Learn this.
49:15Okay. So coming to the next one.
49:35important
49:39questions. Miscellaneous questions.
49:41These are miscellaneous questions. Okay.
49:46Miscellaneous questions. important.
49:48These miscellaneous questions are very
49:49important students.
49:54Important. So solve your method.
50:01So
50:041.0.
50:07Okay. So next
50:17okay fine
50:19prove that. So watch here. Prove that
50:22sin inverse of 8 by 17. Okay. Plus sin
50:26inverse of 3x 5. Okay. Equals to tan
50:30inverse of
50:33I will teach you this.
50:37Watch
50:46in between plus. Okay. First is nothing
50:49but in between plus in between positive
50:52sign. Okay.
50:55Tan inverse. RS. In RS there is tan
50:58inverse. In RS. Okay. There is tan
51:02inverse.
51:04there is tan inverse. Okay. So from this
51:07we need to decide
51:13we need to use formula. So we need to
51:14use formula. We need to use we need to
51:18use formula
51:21for tan of a b formula.
51:26So
51:29we need
51:38from this we need to decide it is tan of
51:39a bone.
51:48Okay.
51:49Okay. Prove the following.
51:53Watch here. Observation.
51:56Observation. In between plus
52:00one. First thing is in between positive
52:02sign. In between positive sign. Okay. In
52:06between positive sign. In between
52:08positive sign. And in RS. In RS there is
52:12cos inverse. In RS there is cos inverse.
52:16Okay.
52:20From this we need to use which formula?
52:23We need to use which formula?
52:27We need to use which formula
52:35which formula will use here. Tell me
52:43it is only for two marks and three
52:44marks.
52:46answer.
52:51We need we need to use which formula
52:57that I agree.
53:01Exactly. Exactly. Super. Super. We need
53:04to use cos of A formula.
53:09We need to use cos of A plus B formula.
53:15Yeah. So
53:20tan
53:22tan students tan of a plus b and it is
53:26nothing but tan a + tan b this is
53:28nothing but tan a + tan b divided by 1 -
53:30tan a into tan b
53:38right so
53:40so proof watch your student solution
53:43solution please watch your let Let
53:45students sin inverse of 8 x 17. Let sin
53:48inverse of 8 x 17 equals to a okay and
53:52cos inverse of sorry sin inverse of sin
53:55inverse of 3x 5= to b. Okay. So pythog
54:00apply the pythog.
54:02Okay. Apply the Pythagoras theorem
54:04students 87 and then it is nothing but
54:09yeah this is nothing but student square
54:11root of 289 - 64 which is nothing but
54:14square root of 225 so which is equal to
54:1715 okay so what we get here so we get
54:20here it is tan inverse so tan inverse of
54:24okay so tan inverse of or yeah so tan
54:26inverse or tan a equals to so tan a= is
54:30nothing but 8 by 15 I hope this is Clear
54:33tan= to 8 by 15 watch here.
54:38Okay. Here we have got three and then
54:41five. So we get square root of 25. So
54:45square root of it is 25 - 9 which is
54:47equal to square root of 60 which is
54:48equal to 4. So we get here tan b. So tan
54:53b equals to is 3x4. I hope this is
54:55clear.
54:57Tan a tan tan a meth tan b. Everybody
54:59understood this. Everybody understood
55:01this tan and tan.
55:15Okay.
55:17I hope this is clear.
55:22Okay. Fine. So what do you need to use?
55:24As I just mentioned, as I just
55:26mentioned, we need to use tan of a plus
55:28b formula. So therefore what is tan of a
55:30plus b formula? Therefore, so we know
55:32that we know that tan of a + b equals to
55:35we know that tan of a + b equals to we
55:38know that tan of a + b equals to
55:42tan a plus tan b yeah yeah tan a + tan b
55:45okay divided by 1 - tan a into tan b 1 -
55:49tan a into tan b what is this tan a plus
55:51tan b students n what is this tan a plus
55:54tan b tan a is nothing but what is tan a
55:57students tan a is 8 by 15 okay tan a is
56:008 x 15 plus it is 3x 4 divided by 1 - 8
56:05x 15 and into 3x4
56:08okay so what we get we get here LCM is
56:1260
56:14okay LCM is 60
56:17so LCM is 60 okay so we get
56:23it is 32 32 + 45 and here it is 60 minus
56:27yeah 60 - 24 I hope this is
56:33So this and this get cancel this 60 and
56:36this 60 get cancel. So we got tan of a +
56:39b is nothing but therefore tan of a + b
56:42is nothing but it is 77. Okay 77 divided
56:46by it is 36.
56:55Reply me first. So now watch your
56:57students stop wring. Stop writing and
57:00watch. Stop writing and watch students.
57:03So therefore a plus b equals to
57:04therefore a plus b equals to it is tan
57:06inverse of it is 77 divided by 36. What
57:10is a student? A what is a a what is a
57:14students? Ara it is nothing but sin
57:17inverse of 8 x 17.
57:20What is a a sin inverse of 8x 17. So we
57:23got here it is sin inverse of 8 by 17.
57:27Okay. Plus what is b students?
57:30B sin inverse of 3x 5. Okay. Sin inverse
57:33of 3x 5. Okay. Equals to it is tan
57:36inverse of tan inverse of it is 77 by
57:4036.
57:41Hence proved. Hence proved.
57:45Is this difficult students? Tell me
57:47fast. Is this difficult?
57:55Is this difficult? Tell me fast.
58:07Is this easy or difficult? Students tell
58:09me first.
58:24pure observation.
58:31Okay.
58:44So we need to use cos of a formula.
58:46Okay. So coming to this let us come to
58:49the solution students. Let us come to
58:51the solution here.
58:53Okay, let us start with the solution.
58:55Let us start with the solution students.
58:57Let
58:58Yeah, let students let cos inverse of 4x
59:015 equals to a. Okay, so and your use of
59:05pythogas theorem. So either four and
59:07then five and we get here roo<unk> of 25
59:12- 16= to square root of 9 which is equal
59:15to 3 therefore therefore students
59:17therefore so here uh sin a equals to sin
59:21a equals to
59:24it is 3x 5 and cos a equals to it is 4x
59:275. Okay. So now students here yeah
59:32coming to this so let let students let
59:35cos inverse of 12 x3 equals to b cos
59:40inverse of 12x3 equals to b. So now here
59:44apply the pythogoras theorem. So
59:46pythogoras theorem apply mimla here we
59:48get 12 13 and now here we get it is
59:52square root of 169 minus 144. So which
59:55is nothing but square root of 25 equals
59:57to 5. So therefore therefore students
59:59here we get okay so here we get students
1:00:02sin b okay sin b equals to 5 by3 and cos
1:00:06b= 12 x 13
1:00:10I hope this is clear
1:00:12everyone understood this reply me fast
1:00:22only during exams I'll be taking here
1:00:24that's all is this clear students
1:00:28Reply. Did everyone understood this?
1:00:30Reply me fast.
1:00:34Did everyone understood this? Reply me
1:00:36fast. Reply.
1:00:58So question will be solved in the month
1:01:00of November. So So we know that cos of a
1:01:03plus b is equal to we know that cos of a
1:01:06+ b equals to it is cos a cos b. Yeah,
1:01:09it is cos a cos b minus sin a sin b. It
1:01:12is cos a cos b minus sin a sin b. What
1:01:15is this students? cos a cos b minus sin
1:01:18a sin b. So what is cos a cos a cos b
1:01:22cos a stop what stop writing and watch
1:01:24here stop writing and watch here
1:01:26students cos a cos b any 4x 5 12 x 13 so
1:01:32here we get 4x 5 and then 12x3
1:01:36okay so minus sin a sin b what is sin a
1:01:38sin b 3x 5 and 5x3 so here we get 3x 5
1:01:43and then 5x3
1:01:46okay so here we get yeah here we get
1:01:48students it is 65 and it is 48 - 65 and
1:01:54then 15. So we got 48 - 15 divided by 65
1:01:59therefore it is 33x 65 therefore it is
1:02:0233x 65 therefore it is cos of a + b.
1:02:07Okay. So therefore we get Yeah.
1:02:09Therefore we get here
1:02:13Yeah. Therefore students we get here a +
1:02:16b equals to a + b equals to this cos
1:02:18inverse of it is 33 by 65. Okay. What is
1:02:23a? a a
1:02:25is nothing but a is nothing but
1:02:27students. Yeah. A is nothing but it is
1:02:30yeah cos inverse of 4x 5. A and cos
1:02:33inverse of 4x 5. This is cos inverse of
1:02:364x 5 plus what is B? B. What is B
1:02:40students? B is nothing but cos inverse
1:02:41of 12 by 13. It is cos inverse of 12
1:02:44by3. So therefore it is cos inverse of
1:02:4733 by 65. I hope this is clear. So
1:02:50therefore it is hence proved. Therefore
1:02:53it is hence proved.
1:02:58So most important
1:03:02check in the live section. Check in the
1:03:04live section. I have done it.
1:03:31So easy.
1:03:38Are you feeling difficulty or easy with
1:03:40maths?
1:03:42Math. Math.
1:03:44But
1:03:51this problem difficult problem
1:03:58in which batch are you?
1:04:01So previous
1:04:07important questions
1:04:11important question.
1:04:21Don't worry.
1:04:28Yeah, math is very interesting always.
1:04:30Exactly fine. So
1:04:39can you tell me students what we use?
1:04:51What you can use
1:05:09WhatsApp
1:05:122.0. Okay.
1:05:16Fine. Fine.
1:05:27Exactly.
1:05:28Exactly.
1:05:33In between positive sign. First thing is
1:05:36in between positive sign. Okay. In
1:05:39between positive sign and RS in RS. In
1:05:43RHS there is there is in RS there is sin
1:05:46inverse. So formula therefore use so use
1:05:51student sin sin of x + y formula. Use
1:05:54sin of x + y formula.
1:06:00Okay fine.
1:06:03S of a s of x of a
1:06:07sin of a b formula.
1:06:12You will try this as right. So
1:06:26yeah.
1:06:31So
1:06:34and already you are listening to what
1:06:36I'm teaching
1:06:38fine so next topic do you want which
1:06:42topic you want next
1:06:50question two mark questions important
1:06:52two mark questions simplest form two
1:06:55questions these are all two mark
1:06:56questions Contra.
1:07:02These are all two mark questions.
1:07:08See?
1:07:17Yeah. So, differentiation. Fine. Two
1:07:20more questions. We shall start with
1:07:22differentiation.
1:07:48Okay. So coming to these problems. Yeah.
1:07:50Coming to these problems students.
1:07:56Do you know all do you know chain rule
1:07:58students? Yes
1:08:03sir. batches
1:08:06see if you're preparing for KCT 1.0 2.0
1:08:09into any batch is fine. So exams
1:08:15in I only teaching.
1:08:20So important question.
1:08:31So
1:08:34don't worry.
1:08:40So
1:08:44sorry.
1:08:51So integrations.
1:09:01Okay. You want those problems?
1:09:10Oh very good Ashoda very good.
1:09:14Okay. So is asking for derivatives.
1:09:17Sorry, application of derivatives. Mana
1:09:21problem.
1:09:23An I think you are from
1:09:26you. You are from B right.
1:09:29Are you from batch
1:09:32problems
1:09:34x + y= 15=
1:09:38x² + y square
1:09:43okay fine
1:09:46most important chapter
1:09:50okay
1:09:54yes fine Listen
1:10:04to me. I'm going to give you a sgest.
1:10:06Watch here. Important questions.
1:10:11Important questions. Go with this.
1:10:13Sorry. Type like this students. Type
1:10:15like this. Batch. Okay. So, batch LP
1:10:22LP.
1:10:24So type like that
1:10:36you will understand the complete LP
1:10:38properly.
1:10:40Okay. So LP problem.
1:10:46So already
1:10:48so you need to type what batch and type
1:10:51the chapter name batch
1:10:54you will get everything
1:11:01fixed questions question
1:11:05okay what is the answer for this what is
1:11:09the answer for this students 2x + 3 y
1:11:12okay so 2x + 3 y = to sin yh makla If I
1:11:18you need to write students with respect
1:11:20to
1:11:22different respect
1:11:25must and should you need to write this
1:11:26and 2 into 1 + 3 into dy by dx okay
1:11:31equals to it is cos y into dy by dx. I
1:11:35hope this is clear.
1:11:39Did everyone understood this?
1:11:41Okay fine.
1:11:53Okay fine. So what we will get here? So
1:11:57dy by dx common. So dy by dx common
1:12:00students 3 - cos y equals to -2.
1:12:03Therefore dy by dx equals to it is -2 /
1:12:083 - cos y.
1:12:11answer. Did everyone get this? Reply me
1:12:14fast. Yeah. Thank you. Thank you for
1:12:15your wishes, students.
1:12:20Yeah.
1:12:22So, Shhat, you are from my Laksha batch.
1:12:35Yeah. You from my luxury batch?
1:12:40Shhat you are from Laksha batch.
1:12:45Okay fine. So coming to the next problem
1:12:47students answer
1:12:49what is it they are given solution.
1:12:51Okay. So x into y + y² equals to
1:12:55students tan x + y
1:12:59differentiate with respect to x. So
1:13:02differentate with respect to x.
1:13:05Differentiate with respect to x. So,
1:13:08so
1:13:10rule
1:13:13x into dy by dx
1:13:16plus y into 1. Okay.
1:13:20Product rule. So, x into y x into x into
1:13:24dy by dx.
1:13:26Okay. X into y. Y differentiation is dy
1:13:28by dx + y into differentiation of x is 1
1:13:31+ students y square y square. What is
1:13:35the differentiation of y square? Y
1:13:36square differentiation is nothing but 2
1:13:38Y into dy by dx
1:13:45equals to it is
1:13:48tan x tan x differentation see square x
1:13:51plus it is dy by dx I hope this is clear
1:13:54okay so dy by dx common yeah dy by dx
1:13:59okay so x into dy by dx minus dy by dx
1:14:03equals to it is seeant square x
1:14:06Yeah. + 2 y into dy by dx + 2 y into dy
1:14:10by dx - x sorry minus dy by dx equals to
1:14:18okay so equals to students it is secant
1:14:20square x - y so therefore we get now
1:14:24okay so now students we get dy by dx
1:14:27common so dy by dx common so it is yeah
1:14:31it is x + 2 y so -1 = to x + 2 y - 1
1:14:36equals to it is sec square x - y.
1:14:40So therefore we get therefore we get
1:14:42students dy by dx = to it is secant
1:14:46square x - y
1:14:48okay divided by divided x + 2 y - 1.
1:15:00I hope this is clear.
1:15:13I hope this is clear.
1:15:29properties of determinist
1:15:47problems already
1:15:50problems already. Go and check this
1:15:54problem.
1:15:56Please do this. Okay.
1:16:09Batch LP
1:16:10type this batch LP and you'll be getting
1:16:13this videos check you will get these
1:16:16videos go and check it in the YouTube
1:16:19okay so type this and every chapters you
1:16:21will get
1:16:24okay
1:16:26every chapters are here
1:16:29okay fine
1:16:34okay okay so coming to the next oneh
1:16:37get me the answer for This solution
1:16:40answer X into Y X² + X into Y + Y² = 100
1:16:46answer this answer
1:16:55exam
1:17:04depend.
1:17:13Okay. So
1:17:17exam syllabus depends on your district.
1:17:28Okay.
1:17:30LP. So it depends on your district
1:17:35but first chapters
1:17:39all the six chapters are there.
1:17:42So
1:17:43even
1:17:47delete. So now students coming to this
1:17:50different with respect to x. Yeah
1:17:52differentiate with respect to x.
1:17:58Differentiate with respect to x.
1:18:00It is nothing but 2x plus it is x into
1:18:03dy by dx plus y into 1
1:18:10x into y x into y x into differentation
1:18:14of y is nothing but dy by dx plus y into
1:18:17differentation of x is 1 plus 2 y + 2 y
1:18:21into dy by dx equals to zero because
1:18:25differentiation of 100 differentation of
1:18:27100 zero
1:18:29So we got here it is 2x okay plus dy by
1:18:33dx common. So we got here it is sorry uh
1:18:36x + 2 y x + 2 y + y =0. So we get here
1:18:42it is dy by dx yeah dy by dx into x + 2
1:18:45y equals to this - 2x - y. Okay. So
1:18:49therefore we get dy by dx = to therefore
1:18:53dy by dx = to it is - 2x - y okay
1:18:58divided by x + 2 y.
1:19:01I hope this is clearly
1:19:10did everyone understood this?
1:19:12Okay so coming to the next one. Okay. So
1:19:15solution students solution
1:19:18sin square y. Okay. Sin square y + cos
1:19:21xy = to k.
1:19:26Okay.
1:19:28xy.
1:19:39So
1:19:42doubt
1:19:52first of all
1:20:08discipline.
1:20:19First of all,
1:20:24mobile
1:20:27practice first sitting for at a at a
1:20:29place for 10 minutes, 15 minutes, 20
1:20:32minutes.
1:20:37Just sit quietly for 10, 15 minutes or
1:20:3920 minutes. Okay?
1:20:4330 minutes, 45 minutes mobile.
1:20:54Okay. So you will develop some patience
1:20:57level.
1:21:07Okay. Automatically third, fourth day we
1:21:10start studying
1:21:13mobile.
1:21:28Sorry.
1:21:35password.
1:21:39So
1:21:43do you think really you can study
1:21:45first try to be genuine? First try to be
1:21:48genuine. Try to be good. Try to develop
1:21:50some discipline. Try to develop some
1:21:52patience level.
1:22:01Okay. So coming to this differentiate
1:22:04with respect to x.
1:22:07Differentiate with respect to x
1:22:08students.
1:22:10What is this? This is nothing but 2 sin
1:22:11y 2 sin y into cos y into dy by dx plus
1:22:17it is it is minus sin xy. Yeah. It is
1:22:20minus sin xy. Okay. into into x into dy
1:22:24by dx plus x into dy by dx + y into 1.
1:22:29Okay. Equals to z.
1:22:32As everybody understood this. As
1:22:35everybody understood this. Reply me
1:22:36first.
1:22:41Okay. So we got here it is.
1:22:45Okay. This whole thing students this
1:22:46whole thing is nothing but sin 2 y.
1:22:49Okay. This whole thing is nothing but
1:22:50sin 2y into dy by dx into dy by dx.
1:22:55Okay. So then minus then students minus
1:22:59x into sin xy into dy by dx. Okay. Minus
1:23:04y into sin xy equals to z equals to
1:23:09zero. So now we get
1:23:13yeah now we get students here dy by dx
1:23:15common yeah dy by dx common. So we got
1:23:18here it is sin 2 y sin 2 y - x into sin
1:23:22xy okay equals to it is y into sin xy
1:23:31okay so therefore we got dy by dx equals
1:23:34to it is y into sin xy
1:23:37divided by it is sin 2 y sin 2 y - x
1:23:41into sin xy
1:24:10What is your second PC or first PC? It
1:24:12is second PC. So these are all important
1:24:15questions.
1:24:17These are all important questions.
1:24:20Yes. Yes. Sean how discipline is equal
1:24:23to success. Discipline plus consistency
1:24:26equal to success.
1:24:29Okay. So
1:24:32let me answer for this
1:24:35bas
1:24:39you know chain rule.
1:24:42You're asking this particular step.
1:24:44You're asking this particular step or
1:24:45this particular step. This is a
1:24:47question. The question
1:24:49first step question.
1:24:52Okay.
1:24:54Question.
1:24:59So fine. So
1:25:02sin square x. Yeah. Sin square x + cos
1:25:05square y equals to 1. So differentiate
1:25:08with respect to x. Yeah. differentiate
1:25:12with respect to x. So in what we get? We
1:25:16get here it is 2 sin x it is 2 sin x
1:25:19into cos x + 2 cos y into minus sin y
1:25:24into dy by dx equals to 0. So now we get
1:25:28here it is yeah. So 2 sin x into cos x 2
1:25:32sin x into cos x minus 2 sin y into cos
1:25:35y
1:25:37okay into dy by dx okay equals to 0. So
1:25:41we got here it is nothing but it is sin
1:25:432x. So this is sin 2x minus sin 2y okay.
1:25:48So into dy by dx. So therefore we get
1:25:52equals to zero. Therefore we get
1:25:53students here. Yeah. So sin 2y.
1:25:59Yeah. Sin 2y
1:26:02into dy by dx equals to it as equals to
1:26:06sin 2x. So therefore dy by dx equals to
1:26:09it is sin 2x / sin 2 y sin 2x / sin 2 y.
1:26:14I hope this is clear.
1:26:23Okay. So
1:26:28basics
1:26:43let me explain the basics students.
1:26:47Let me explain the basics. Please watch
1:26:48here. Suppose if you given students
1:26:51suppose if you given sin square please
1:26:53watch sin square
1:26:58y= x.
1:27:05So this is chain rule
1:27:11you can't even solve anything
1:27:16different
1:27:18people lag with chain rule chain rule
1:27:3110 Anybody
1:27:36wants chain rule here?
1:27:41Do you want me to teach the chain rule
1:27:43students? Chain rule.
1:27:48Do you want me to teach the chain rule
1:27:50students? Reply me first.
1:27:55Yes.
1:27:57Okay. So, if you want
1:28:04Okay, fine.
1:28:34Okay.
1:28:38get your minus sign. So
1:28:42we have written this minus sign
1:28:52cos square y differentation 2 cos y and
1:28:54then minus sin y. I hope this is clear.
1:28:58So
1:29:00okay bring this minus sign to this side
1:29:02and you get this.
1:29:10Okay fine. So let me teach you the chain
1:29:13rule students. Watch your students.
1:29:16Feose
1:29:20if you compare this with sin of log x.
1:29:24Okay. So s of log x if you compare this
1:29:27students if you compare this here. So
1:29:29here student this implies fal
1:29:32okay and g of x= to log x
1:29:36do you all agree with this? Do you all
1:29:38agree with this
1:29:41f of g of x? Okay f of g of x suppose I
1:29:47take here okay suppose I take your
1:29:49student sin x or yeah sin x power sin x^
1:29:533. So first
1:30:00which function is inside which function
1:30:05here students log x is inside here
1:30:08students log x is inside
1:30:11log x is inside what is inside s log x
1:30:15is inside
1:30:17log x
1:30:22which function is Inside which function?
1:30:25Which function is inside which function?
1:30:28Function function. So you can observe
1:30:31here. You can observe here students. You
1:30:33can observe here. So here here students
1:30:36here sin x here sin x is inside. Here
1:30:40sin x is inside what? Yeah. Here sin x
1:30:43is inside what students this cube. I
1:30:45hope this is clear.
1:30:50Okay. So what we get here? So f okay f
1:30:54is nothing but cube okay and g of x is
1:30:57nothing but it is it is what sin x
1:31:03did everyone understood this
1:31:06I hope this is clear
1:31:12okay suppose suppose students now I
1:31:14write here f of okay f of g of x
1:31:18okay f of g of x watch
1:31:23Sin x²
1:31:26sin x²
1:31:28What is f and what is g of x? What is f
1:31:31and what is g of x students? Can tell me
1:31:32first f g of x.
1:31:36No. Okay. So
1:31:40tell me what is f? What is g of x?
1:31:48What is fn? What is g of x?
1:31:51Fn g of x. Tell me fast.
1:31:55What is f and what is g of x? Students
1:31:57tell me first.
1:32:01What is f and what is g of x? Yes. No. F
1:32:04is nothing but sine and g of x is
1:32:06nothing but students x². So
1:32:15yeah exactly.
1:32:18problem. Let me start with
1:32:19differentation. See f of g of x
1:32:24log of yeah log of students tan x log of
1:32:28tan x
1:32:30what we get here f what is f and what is
1:32:33g of x students here f is nothing but
1:32:36log function f is nothing but log
1:32:38function and g of x is nothing but tan
1:32:41x. So
1:32:45which is inside which function?
1:32:53Which function is inside which function?
1:32:56Let me start doing this problems. Let me
1:32:58start doing this integration
1:33:00differentiation. So first one students
1:33:02first one y equals to so find dy by dx
1:33:04not find dy by dx. If y equals to it is
1:33:09sin of log x.
1:33:18Okay, please watch
1:33:21g of x.
1:33:23What is f and what is g of x? F is
1:33:25nothing but s and g of x is nothing but
1:33:27students log x of something.
1:33:31F is nothing but sin of something.
1:33:37Okay. So what we get here we get please
1:33:40watch
1:33:43fdash g dash of x fd g dash of x what is
1:33:47this? This is nothing but cos of
1:33:49something and here students 1x. I hope
1:33:52this is clear. I hope this is clear.
1:33:56So what we get solution? Solution
1:33:58students dy by dx dy by dx is nothing
1:34:02but it is cos of something into 1x
1:34:05something. What you will get here? What
1:34:09you will get here students? cos of
1:34:10something whatever this log x.
1:34:15So answer is what? Answer is what
1:34:17student? It is cos of log x into sorry
1:34:19cos of log x / x is the answer.
1:34:26Reply me first.
1:34:28I hope this is clear.
1:34:31Okay. Let me come to the next problem
1:34:33students. Let me come to the next
1:34:34problem.
1:34:35Yeah.
1:34:38So if y equals to if y = to sin qx find
1:34:42Yeah. Find dy by dxal.
1:34:46What you will get here? If y= x find dy
1:34:49by dx. So what do you get?
1:34:52So
1:34:54okay y= sinx we can write it as sin x
1:35:00right. So f g of x f g of x f is nothing
1:35:06but whole cube and this is nothing but
1:35:08sin x. Okay. So what we get fdash okay
1:35:13fdash we get here fdash here and here we
1:35:17get here gd dash of x so what is fdash
1:35:20fdash is nothing but it is 3 into power
1:35:232 2
1:35:25x formula
1:35:28x what is x^ n into xus
1:35:34cos x
1:35:36okay so therefore what is dy by dx what
1:35:39is dy by dx students. It is 3 into power
1:35:422 into cos xl
1:35:453 into 3 into something 3 into something
1:35:48power 2 into cos x. What is this
1:35:50something? Something we get here it is
1:35:53sin x we get here sin x.
1:35:57Is this difficult?
1:36:02Is this difficult students?
1:36:07Reply.
1:36:09Is this difficult students? Reply me
1:36:11fast.
1:36:20Okay. So let me come to the next one.
1:36:22So next one sin x². Okay. So y = to. So
1:36:28third one students if y = to if y = to
1:36:31sin x². Okay. Find the dy by dx. Find dy
1:36:35by dx. Okay. So f g of x. What is f and
1:36:40what is g of x here? F g of x. What is f
1:36:44and what is g of x students? F is
1:36:46nothing but cos of something. Sorry,
1:36:48sine of something. F is nothing but sin
1:36:50of something. And this is something is
1:36:52nothing but x². Okay. So what is fdash?
1:36:55Fash gdash of x.
1:36:58fdash incl Fdash is nothing but cos of
1:37:01something and G dash of X is nothing but
1:37:032X
1:37:05fdash is nothing but cos of something
1:37:07and G dash of X is nothing but 2X so
1:37:10what do we get solution
1:37:13S solution students dy by dx equals to
1:37:16yeah it is dy by dx
1:37:19equals to students it is cos of
1:37:20something okay so cos of something into
1:37:24it is 2x so something x² Right? This is
1:37:29the answer. I hope this is clear. So
1:37:32what do we get? We get here 2x into cos
1:37:34x².
1:37:41I hope this is clear.
1:37:46So partial differentation partial
1:37:50partial
1:37:54I hope this is clear. Any doubt? Any
1:38:02doubt here? Reply me first.
1:38:05Reply.
1:38:08Have you got any doubt students? Reply
1:38:09me first.
1:38:18Okay. So fourth one. Fourth one
1:38:21students. If y =
1:38:26log of tan x find dy by dx. Get me the
1:38:29answer for this. Find dy by dx. Get me
1:38:32the answer for this.
1:38:34What is f and what is g of x students?
1:38:37Get me the answer for this.
1:38:40It is log of something and this is tan
1:38:43x. Okay. So therefore fdash therefore
1:38:47students fdash and g dash of x. Okay.
1:38:50Fdash and gdash of x. It is nothing but
1:38:52this is 1 by something and this is
1:38:54secant square xra
1:38:58right. So here what we get? So here
1:39:00student solution. So dy by dx what is dy
1:39:04by dx. So dy by dx is nothing but 1x
1:39:07something into secant square x. Okay.
1:39:09What is this something? This something
1:39:11is nothing but tan x.
1:39:13I hope this is clear.
1:39:16So it is cos x by sin x.
1:39:19So divided by 1x cos square x. So
1:39:22therefore it is 1x sin x into cos x.
1:39:25I hope this is clear.
1:39:28Did everyone understood this?
1:39:30Did everyone understood this?
1:39:33Hello.
1:39:37Yeah. Perfect. So
1:39:40watch here. Yeah. Suppose if I have
1:39:43given suppose if I have given students
1:39:45log of log of sin okay rootx log of sin
1:39:50rootx
1:39:52now do I have got only two functions.
1:40:00So
1:40:07which function is inside which function?
1:40:14What I said function
1:40:19g of x g of x is inside f
1:40:23g of x is inside f.
1:40:26Okay. Log x is inside s log x is inside
1:40:30s. Therefore g of x is log x and f is
1:40:32nothing but sin.
1:40:38So
1:40:50okay
1:40:51sin x² which function is inside which
1:40:54function x² is inside s x².
1:40:59So x² g of x and s f and tan x is inside
1:41:06log. So tan x is nothing but g of x and
1:41:10log is nothing but f. I hope this is
1:41:12clear.
1:41:16This is equivalent to what
1:41:19equivalent? There are three functions
1:41:21here.
1:41:22There are three functions here. f of g
1:41:26of h of x f of g of h of x equals to
1:41:31equals to this log of sin rootx.
1:41:35Okay. So
1:41:38h of x h of x is nothing but rootx.
1:41:41Okay. And then
1:41:45watch here. Watch here students. H of x
1:41:47is nothing but rootx g is nothing but
1:41:50sin. Okay. And then students and then f
1:41:53is nothing but it is log
1:41:57function.
1:42:00So what we get here? We get f is equal
1:42:02to log of something. Okay. And here it
1:42:06is G. G is nothing but G is nothing but
1:42:09sin of something and H of X is nothing
1:42:11but it is root X. I hope this is clear.
1:42:27Okay.
1:42:32So what we get? So now students watch.
1:42:35Now here students. So it is f G and then
1:42:39H of X. F G and H of X. F and K. F is
1:42:43nothing but log of something and it is
1:42:45sine of something and it is rootx. So
1:42:49what we get here? Here we get Fdash,
1:42:52Gdash and then Hdash of X. Okay. What is
1:42:55fdash? fdash is nothing but 1 by
1:42:58something.
1:42:59fdash is nothing but 1x something. So dy
1:43:03by dx dy by dx one by something. Okay.
1:43:07Then one then something.
1:43:16Okay. Next what we get?
1:43:20Cos of something. Next of something.
1:43:24cos of something and something again
1:43:27something this rootx
1:43:31okay and here hd of x what is this hdash
1:43:33of x it is nothing but 1x2 rootx so here
1:43:36we write 1x2<unk>x
1:43:39so what is the answer what is the answer
1:43:40students it is cotx
1:43:42it is cot rootx divided by it is 2 into
1:43:46rootx I hope this is clearly
1:43:55I hope this is clear
1:44:05it is not there since I'm teaching here
1:44:08the classes there it will be not there
1:44:14I hope this is clear studentsity
1:44:25cos by sign cos by sign. What is cos by
1:44:28sign?
1:44:30cos by s in cot cos cos rootx by sin
1:44:34rootx. It is nothing but cot
1:44:38cos rootx by sin rootx is nothing but
1:44:41cotx. I hope this is clear.
1:44:45I hope this is clear.
1:44:49You know
1:44:53I hope this is clear you're confused.
1:44:56Okay. So let me write this once one more
1:44:58step. Not it is nothing but cos rootx.
1:45:01It is cos rootx divided by sin rootx.
1:45:05Okay into 1x2x.
1:45:08So what is this? This is what cos by sin
1:45:10cos by sin is nothing but cot. It is
1:45:13cotx divided by 1x2x.
1:45:16So therefore it is cotx.
1:45:20Therefore it is cotx divided by it is
1:45:222x.
1:45:26I hope this is clear.
1:45:40What you got there?
1:45:43Explain first step.
1:45:47One by something.
1:46:02Differentation of rootx.
1:46:04What is the differentiation of rootx? 1x
1:46:062 roo<unk>x
1:46:08formula. You don't know this formula. So
1:46:11differentiation of rootx is nothing but
1:46:13d by dx of x^ 1x2. So it is nothing but
1:46:171x2 into x^ 1x2 minus 1. So what do we
1:46:21get? we get here it is 1x2 into x^ - 1x
1:46:252 it is 1x2 into okay 1x^ 1x2 so it is
1:46:311x
1:46:322<unk>x
1:46:34I hope this is clear
1:46:37so differentiation of rootx is nothing
1:46:39but it is 1x2x it is a standard formula
1:46:42this is a standard formula
1:46:44differentiation of rootx is nothing but
1:46:461x2 rootx so it will be available in the
1:46:50WhatsApp channel
1:46:53examin.
1:47:13So tell it to your daddy also.
1:47:1624 bar 7 24x7 car. Tell it to your daddy
1:47:19also and you will be a big question mark
1:47:21then. Okay.
1:47:2324x7 cars. Tell it to your father also
1:47:30and you will be a big question mark.
1:47:33Okay. Fine.
1:47:46Fine.
1:47:53Okay. So let us move to the next problem
1:47:55students. So our problem
1:47:58so
1:48:03Okay.
1:48:10foreign
1:48:19parents.
1:48:25So,
1:48:36so obviously there are some bad parents.
1:48:39Obviously there are some bad kids.
1:48:51Okay. Definitely. Definitely.
1:48:58education platform.
1:49:04It is not everybody's cup of tea.
1:49:21Okay. Sin of XY
1:49:24sin of XY.
1:49:27Yeah. Sin of XY. Watch here. Watch here
1:49:30students. Here f is nothing but G of X.
1:49:35What is F and what is G of X? Tell me
1:49:37first.
1:49:39What is F and what is G of X students?
1:49:44What is F and what is G of X students?
1:49:46Tell me first.
1:49:56What is f and what is g of x? Tell me
1:49:59first.
1:50:05Yeah. F is sine. Huh? F is sign and g of
1:50:10x is nothing but x into y.
1:50:14So fdash g dash of x.
1:50:23Yeah. Exactly.
1:50:26Exactly. Students it is cos so sine of
1:50:29something. It is nothing but cos of
1:50:31something. And here students it is x
1:50:33into dy by dx plus it is y into 1.
1:50:38Okay. So therefore we got
1:50:41therefore you got students here we get.
1:50:43Yeah. So dy by dx is nothing but it is
1:50:47cos of something. Okay cos of something
1:50:49into it is x into dy by dx
1:50:53plus y into 1. I hope this is clear and
1:50:57mlen xy that xy we get here. Okay that
1:51:00xy we get here. I hope this is clear. So
1:51:03what we get? We got here it is cos xy.
1:51:06Okay. into x into dy by dx into dy by dx
1:51:11plus it is y into cos xy
1:51:14and here it is dy by dx. So therefore dy
1:51:18by dx common. So dy by dx common
1:51:21students it is 1 - x into cos xy. Okay.
1:51:24So here it is y into cos xy.
1:51:28Okay. So therefore we get it is dy by dx
1:51:32equals to it is y into cos xy. So
1:51:36divided by it is 1 - x into cos xy I
1:51:40hope this is clear
1:51:46chain rule
1:51:49chain rule.
1:52:21So doubt. What doubt you got? Please
1:52:23tell me.
1:52:37What doubt you got? Please tell me.
1:52:41If you got any doubt, ask me. Okay.
1:52:46So, fine students on the 10 minutes
1:52:47break. Come back. On the 10 minutes
1:52:50break, come back students. Come back
1:52:52after 10 minutes.
1:52:54So, 10 minutes break students. Come
1:52:56back. Yes. Yes. I'm giving break. I'm
1:52:58giving break students. So 10 11 12. So
1:53:02let me give you 12 minutes break.
1:53:05Okay. So third step.
1:53:08So
1:53:11okay
1:53:13by dx
1:53:16dy by dx dy by dx
1:53:19what I have done. What I have done? I
1:53:21taken to this left hand side. left hand
1:53:23side
1:53:24dy by dx minus x into cos xy into dy by
1:53:30dx
1:53:32equals to y into cos xy.
1:53:35I hope this is clear.
1:53:50compositions.
1:53:52Composition
1:53:59functions.
1:54:03Yeah. Fine. So break. So take break
1:54:06students and come after 12 minutes.
1:54:11We can start with next concept. Come
1:54:13back. Come back. Come back after 12
1:54:15minutes.
2:06:24So welcome back students. Welcome back.
2:06:27Welcome back Mla. Welcome back.
2:07:06Yes, welcome back students. Welcome
2:07:08back.
2:07:14Welcome back my dear students. Welcome
2:07:16back.
2:07:34Yeah. So four marks four marks determin
2:07:41four marks determinants
2:07:58determin
2:08:03problems you can expect in determinance.
2:08:06Yeah. Determinance problems expect
2:08:10do you want these type of problems?
2:08:12So reply reply
2:08:16reply me students who are all watching
2:08:17the class.
2:08:21Which concepts do you all want? Yeah.
2:08:24Determinance. Okay. Fine. So
2:08:27concepts
2:08:30question
2:08:33marks. Okay.
2:08:37So four marks question. Four marks
2:08:39question.
2:08:44Yeah. Fine. So class will end at 10 or
2:08:48105.
2:08:49Class will end at 10 or 105. Okay. So
2:08:51watch here students.
2:08:54So, so watch your students.
2:08:59See given students given a square please
2:09:02watch your given a square. So rule
2:09:05system
2:09:08okay rule system
2:09:11first we need to show that
2:09:14okay using a students. So step there are
2:09:18only two things there are only two
2:09:19things step one. So step one students
2:09:22using a using matrix A using matrix A
2:09:26prove that okay using matrix A prove
2:09:29that prove that A² - 4 A + I =0
2:09:35this is step one what is step two
2:09:36students what is step two step two step
2:09:40two is nothing but okay using
2:09:44students a² - 4 a using a square - 4 a +
2:09:48i =0 find find a inverse find a inverse
2:09:55using a square - 4 a + i= to 0 find a
2:09:58inverse so there are only two steps to
2:10:00be done here so how to find
2:10:04first find a square here first find a
2:10:07square a square is 2 3 1 2 okay
2:10:10multiplied with 2 3 1 2 so what is this
2:10:14what do you get here we get it is 4 4 +
2:10:173 is 7 okay so next 6 + 6 12 then it is
2:10:212 + 2 4 and it is 3 3 + 4 7
2:10:28I hope this is clear 4 + 3 okay so 6 + 6
2:10:32and then 2 + 2 and then 3 + 4 fine so
2:10:36watch here everyone so consider consider
2:10:38LHS students so consider LHS so consider
2:10:43LHS what is LHS students LHS LHS is
2:10:46nothing but A² minus So consider LHS
2:10:49students a square - 4 a + i. So what is
2:10:53this? This is nothing but it is 71 12.
2:10:56Okay. 72 47 7 12 47. So minus minus
2:11:02students minus okay 72 47 minus okay - 4
2:11:07into it is a a is nothing but 72 47
2:11:11sorry a is nothing but a is nothing but
2:11:13it is 2 3 2 3 1 2 2 3 1 2 plus I. I is
2:11:18what? What student? It is 1 0 0 1 1 0 0
2:11:221. So what do we get here students?
2:11:29Yes.
2:11:33Jesus.
2:11:35So this is equal to this is equal to
2:11:37712. So I'm rewriting the students. I
2:11:40rewriting this 72 47 minus Yeah. 7 12 47
2:11:45and it is 8 8 12 48. Yeah. 8 to 12 48. 8
2:11:5112 48. Okay. 8 to 12 48. And then
2:11:55students it is plus plus 1 0 0 1 1 0 0
2:11:591. So what do we get? Elena Makla it is
2:12:01nothing but 7 - 8 7 - 8 + 1 12 - 12 + 0.
2:12:07It is 4 - 4 + 0 4 - 4 + 0 and it is 7 -
2:12:128 + 1. So we got it is 0 0. Okay. 0 0
2:12:17which is nothing but 0 which is nothing
2:12:19but your RHS which is nothing but
2:12:21students your RHS.
2:12:24So LHS consider I have considered LHS
2:12:27students. LHS consider I have considered
2:12:30LHS here and I have found RHS students
2:12:33and I have shown that it is equal to RS.
2:12:36So
2:12:44I taken LHS and I have shown that it is
2:12:46equal to RHS. So first part
2:12:50first part okay merger to show that LHS
2:12:54equal to RS to show that LHS equal to
2:12:57RHS.
2:13:02Okay. So next using this using this
2:13:05students I need to find a inverse. So
2:13:08consider so now watch your students
2:13:09consider the part two okay the part two
2:13:12students consider yeah consider students
2:13:14consider so a² - 4 a + i =0
2:13:20okay so here students premultiply okay
2:13:23premultiply by a inverse on both sides
2:13:25premultiply by a inverse on both sides
2:13:28makali what we get we get here it is a a
2:13:32inverse into a okay into a plus students
2:13:364 into A inverse into A. Okay. Plus
2:13:40students it is A inverse into I okay
2:13:42equals to zero. Okay. So what we get? We
2:13:46get here I into A + 4 into I plus
2:13:49students A inverse equals to 0 or it is
2:13:52A + 4 I plus A inverse equals to Z.
2:13:56Therefore students A inverse equals to
2:13:59it is 4 - 4 I.
2:14:03So a - 4. Okay, it is a square minus 4
2:14:08i.
2:14:11Okay, so it is a square. So it is a² -
2:14:16a² - 4 a and it is - 4 a and here it is
2:14:21-
2:14:234 into i
2:14:28okay 4 into i and then students here it
2:14:31is a okay a - 4 i okay and plus a
2:14:36inverse so here a inverse equals to a
2:14:39inverse equals to students it is 4 i
2:14:41minus a is
2:14:48Reply me fast
2:14:51even five questions.
2:15:20A into a inverse equals to
2:15:23a inverals to I. These are the basics
2:15:25required. Basics.
2:15:28Okay. These basics are required student.
2:15:30These basics are required. Okay. A into
2:15:32A inverse equals to I. This is first
2:15:33one. Second one. So second one students.
2:15:36A into I equals to A. Okay. Okay. So
2:15:40all these basics need to be known.
2:15:49So integrations
2:15:52question
2:15:583x3 matrix multiplication problem
2:16:02multiplication problem.
2:16:05Multiplication of matrices problem.
2:16:08Matrix multiplication problem. Should I
2:16:10teach it? Should I teach the matrix
2:16:12multiplication for you?
2:16:17Matrix multiplication. Let me teach you.
2:16:21Okay. So,
2:16:26a inverse equals to it is 4 into 10 01
2:16:30minus A. Yeah. A is nothing but it is.
2:16:34Okay. So a is nothing but it is 2 312.
2:16:37Yeah 2 3 1 2. So therefore it is 4 04
2:16:454 04 minus 2 312.
2:16:50Okay. Therefore a inverse equal to
2:16:52therefore a inverse equals to it is
2:16:554 - 2 0 - 3. Yeah. 4 - 2 0 - 3. And here
2:17:01it is 0 - 1 4 - 2 0 - 1 4 - 2. So here
2:17:07we got 2 - 3 -1 2. Okay, this is your a
2:17:12inverse. I hope this is clear.
2:17:25So I think you'll be getting from
2:17:27differentation
2:17:29six math question either from
2:17:30differentation or integration you'll be
2:17:32getting integrationally 7.4 Four.
2:17:36Okay. You can expect from there. So,
2:17:39okay. Let me teach the matrix
2:17:40multiplication. Please watch.
2:17:43Please watch here. See, matrix
2:17:45multiplication AB is defined. Correct.
2:17:48Matrix multiplication.
2:17:52Matrix multiplication.
2:17:55Okay.
2:17:57AB is defined. Matrix multiplication AB
2:17:59is defined students. AB is defined.
2:18:02Matrix multiplication AB is defined. If
2:18:05if students
2:18:07what do you mean by defined
2:18:11exist
2:18:13so it will exist we can do okay it
2:18:16existed
2:18:23we will get the output
2:18:26okay is defined if
2:18:29number of columns
2:18:31number of columns of a okay Number of
2:18:34columns. Yes. Number of columns of
2:18:37matrix A. Number of columns of A is
2:18:39equal to number of rows of B.
2:18:43Number of rows of B. Okay. Number of
2:18:46columns of A equals to number of rows of
2:18:48B. Not really. Watch here. Everyone
2:18:51watch your students. If A is equal to
2:18:53suppose students, if A is equal to 2 3
2:18:554. Okay. 5 6 7 or 5 6 or 5 2 1. Okay.
2:19:01And B is equal to Watch here everyone.
2:19:04B is equal to this is 31 3 okay 3 1 2 4
2:19:091 6
2:19:11suppose if you have got like this so how
2:19:12many rows and how many columns we have
2:19:14students
2:19:20so
2:19:22maximum question you can easily get 60
2:19:25out of 60 you can easily get 60 out of
2:19:2860 if you have completed the model
2:19:29questions
2:19:30okay MCQs
2:19:36If you complete full complete model for
2:19:3980 you'll get 75 for both exam exam and
2:19:43like exam board exams 75 if you complete
2:19:48all five model.
2:19:52Yeah. So watch this students
2:19:55order. What is the order of a students?
2:20:00row. What is the order of this?
2:20:04So tell me students what is the order of
2:20:06this?
2:20:08What is the order of this? Tell me
2:20:09first. What is the order of this matrix?
2:20:11Students tell me first. Yeah. Three.
2:20:14Yeah. Three rows. Three rows or two
2:20:15columns. Seriously.
2:20:18Seriously. Three rows. It is two rows
2:20:22and three columns. And here it is three
2:20:23rows and two columns.
2:20:31Okay. Horizontal lines of elements.
2:20:36It is horizontal line of elements.
2:20:39Horizontal line of elements are called
2:20:41as rows. Horizontal line of
2:20:44elements
2:20:47are called as rows.
2:20:51Horizontal line of elements are called
2:20:53as rows. Okay. So
2:20:59elements these are rows
2:21:05lines. How many horizontal lines are
2:21:09two horizontal lines? There are two
2:21:11horizontal lines. There are two
2:21:13horizontal lines.
2:21:18Three horizontal sorry three horizontal
2:21:20lines. There are three horizontal lines.
2:21:22So we get here three rows is equal to
2:21:24three. Okay. What are columns students?
2:21:27Columns what are columns? Columns the
2:21:30vertical line of elements. Columns the
2:21:33vertical line of elements.
2:21:35The vertical line of elements. Yeah. The
2:21:39vertical line of elements are called as
2:21:41the vertical line of elements
2:21:46are called as columns
2:21:51are called as columns.
2:21:55Okay. So vertical line of elements we
2:21:58get like this.
2:22:02Okay. The vertical line of elements are
2:22:03called as columns. So,
2:22:08so what we get here is
2:22:10the vertical line of elements. See how
2:22:12many vertical line of elements are
2:22:13there. So, here we get one, two and then
2:22:16three. So, therefore we are getting
2:22:18three columns. Okay. There are two
2:22:21columns. One, two. So, therefore we have
2:22:23got here two columns.
2:22:40Yeah. Thank you for your wishes,
2:22:41students, teachers.
2:22:44Yeah, you'll get this PDF.
2:22:49You'll get this PDF PDF after the live
2:22:51session.
2:22:54live session.
2:23:00[clears throat] So watch your students.
2:23:01Let me take this problem. So A is
2:23:04nothing but Yeah. A is nothing but
2:23:06students. It is 2 3 4 5 2 1. So it is 2
2:23:103 4 5 2 1. Okay. And B is nothing but
2:23:13students. And B is nothing but Okay. AB.
2:23:17So AB is nothing but students. Here it
2:23:18is. Here it is 312 416. Yeah, it is 312
2:23:23416. 312 416. Okay. So, order of A is
2:23:28what? What is order of A students? Order
2:23:31of A is nothing but Yeah. Order of A is
2:23:33nothing but students it is two rows and
2:23:35three columns. And order of B is nothing
2:23:38but order of B is nothing but it is
2:23:40three rows and two columns. So this and
2:23:43this should be same.
2:23:46Both of this should be same.
2:23:50Okay. columns column
2:23:56number of columns of A number of columns
2:23:58of A should be equals to number of rows
2:24:00of B. Okay. Therefore number of columns
2:24:03number of columns of A should be equals
2:24:04to number of rows of B and order of A
2:24:10order of A will be this and this 2 into
2:24:12two it is 2x2 elements elements
2:24:16elements. See watch here. We get here
2:24:19students only four elements.
2:24:21We get 1 2 3 and then four.
2:24:27So how do we get watch?
2:24:33Okay.
2:24:37Okay.
2:24:41Because these people are blind.
2:24:47Blind people
2:24:50what they will do? What they will do?
2:24:52Students here this fellow two two is
2:24:55holding and of three. Okay. Two three
2:24:59one and four two two.
2:25:03Okay.
2:25:05See two is holding two is holding
2:25:07students the end of two is holding the
2:25:08hand of three plus three is holding the
2:25:11hand of one plus four is ending holding
2:25:13the hand of two. I hope this is clear.
2:25:17Did everyone understood this?
2:25:26Okay. Did everyone understood this?
2:25:28Reply me fast.
2:25:33What do we get?
2:25:36So these people will again hold the hand
2:25:37of these two these three people. Okay.
2:25:39Again students these three people will
2:25:42hold the hand of these three. So 2 into
2:25:454 plus 3 into 1 + 4 into 6.
2:25:51Okay. So
2:25:55these these three people are holding the
2:25:57end of these three and again these three
2:25:58people are holding the end of these
2:26:00three that is done.
2:26:04These three people will come.
2:26:06So these three people will hold the end
2:26:07of these three people. So 5 into 3 + 2
2:26:11into 1 + 1 into 2. Okay. Next again
2:26:15either. So again these three people will
2:26:17hold the end of these three. So it is 5
2:26:19into 4 + 2 into 1 + 1 into 6. So what we
2:26:23get? Helena makla we get here it is 6. 6
2:26:26+ 3 9. 9 + 8 is 17 and it is 8. 8 + 3
2:26:3011. 11 + 24. 11 + 24 is nothing but 35
2:26:34and 15. 15 + 2 17 17 19 and yes 20 22 22
2:26:40uh it is 28. Yeah. So answer is 28. I
2:26:43hope this is clear.
2:26:46Reply me first.
2:26:57So
2:27:01composition problems
2:27:06composition
2:27:30So sir Omar sir will be true. He has
2:27:33joined some other place. Okay fine.
2:27:38So
2:27:40did everyone understood this? Reply me
2:27:41first.
2:27:47Yes. Yes.
2:27:53Will you all try this? Will you all try
2:27:56this student?
2:27:58Shall we try this problem? This is a
2:28:00four marks problem. This is a four marks
2:28:02problem. Students, shall we try this?
2:28:05Shall we try this four marks problem?
2:28:07Tell me first.
2:28:19Okay. So given students a is equal. So
2:28:22given a is equal to students it is 31
2:28:25okay -12.
2:28:27So therefore a square is equal therefore
2:28:30a square is equal to it is 3 1 -12
2:28:34* 3 1 so -12.
2:28:38So we got here it is see it is nothing
2:28:41but 9 - 1. Okay, this is 9 - 1 3. Okay,
2:28:47so 3 + 2 and it is -3 - 2 and it is -1 -
2:28:524. So we got here it is 8 5. Okay, 8 5 -
2:28:565 and then - 5. I hope this is clear.
2:28:58This is your a square. Do you all agree
2:29:00with this students?
2:29:023 - 1 is 8. Sorry, 9 - 1 is 8. 3 + 2 5.
2:29:06Okay. -3 -2 - 5 and -1 - 4. So, -1 + 4.
2:29:12Okay. -1 + 4. Students, it is not -4.
2:29:15So, -1 + 4. It is -1 + 4. It is nothing
2:29:19but 3.
2:29:24Okay. Fine. So now students consider
2:29:28consider LHS. Yeah. Consider students
2:29:31LHS. What is the LHS students? LHS know
2:29:34LHS is nothing but A square. Yeah. A
2:29:37square minus 5 A + 7 I.
2:29:40Okay. Which is equal to Yeah. Which is
2:29:42equal to students it is 8. Okay. - 53 -
2:29:485 into it is 3 1 3 1 - 1 2 + 7 I it is 1
2:29:560 0 1. So we got here it is 8 5 okay - 5
2:30:023 minus students it is 15
2:30:05okay - 5 10 and then students plus it is
2:30:087 0 0 1 07 7 0 07 so we got here
2:30:15this is nothing but 8 - 15 + 7 so 5 - 5
2:30:19+ 0 okay so - 5 + 5 + 0 it is 3 - 10 + 7
2:30:25so we got here it is 0 0. So this is
2:30:28what this is 0 0 0.
2:30:31So which is nothing but equal to 0
2:30:33students which is RHS. This [snorts] is
2:30:35nothing but RHS.
2:30:38I hope this is clear. LHS equal to RHS.
2:30:42Hence proved.
2:30:45I hope this is clear.
2:30:47Reply me first.
2:30:59Reply
2:31:02everyone understood this. Reply me
2:31:04first.
2:31:07Okay. So therefore students consider. So
2:31:10consider yeah consider students. Next
2:31:12one is consider. Yeah. Consider is a
2:31:15square - 5 a + 7 i. Consider a² - 5 a +
2:31:207 i equals to 0. Okay. So premultiply by
2:31:23premultiply by a inverse on both sides.
2:31:26Premultiply
2:31:28by a inverse on both sides.
2:31:32So what we get? We get here it is a
2:31:35inverse into a okay into a + 5 into a
2:31:40inverse sorry - 5 into
2:31:43it is - 5 into a inverse into a + 7 into
2:31:46a inverse into a i equals to z. So we
2:31:51got here it is I into A - 5 I + 7 into A
2:31:55inverse equals to Z or students.
2:32:07Okay. Therefore here we got
2:32:11Okay. Okay. Therefore students, here we
2:32:13got 7 into a inverse 7 into a inverse
2:32:16equals to 5 I - I I sorry this is I - 5
2:32:25A - 5 I + 7 into A inverse equals to
2:32:30zero. Therefore 7 into A inverse equals
2:32:33to it is 5 I - A. Therefore a inverse
2:32:37equals to 1x 7 into 5 I - a.
2:32:45So therefore a inverse equals to it is
2:32:481x 7 into okay 5 into 1 0 0 1.
2:32:53Okay it is 1 0 0 1. So minus a what is a
2:32:57students? A and reu what is a? A is
2:33:00nothing but 3 1 - 1 2 it is 3 1 - 1 2 it
2:33:04is 3 1 - 1 2 okay so therefore we got
2:33:08here it is 1x 7 yeah 1x 7 into it is
2:33:13yeah it is students five okay yeah it is
2:33:16five okay 5 - 3 5 0 - 5 sorry 0 - 1 so 0
2:33:23- of -1
2:33:25okay and then it is students and then it
2:33:28is I - 2. Okay. So 1x 7 into 1x 7 into
2:33:34students 2 - 1 1 and then 3. So this is
2:33:37your a inverse. This is your a inverse.
2:33:40I hope this is clear.
2:33:43Did everyone understood this? Reply me
2:33:44first.
2:33:57Oh hello. Hello prana.
2:34:07Hi Samarth. Hi. How are you? Kanda, how
2:34:09are you?
2:34:14How's your preparations going on?
2:34:19preparation
2:34:27only
2:34:29for KCD there is only lines
2:34:34fine
2:34:50computer science.
2:34:56Yes.
2:35:04Yeah. Same year, summer, same year.
2:35:09So tell hi if he is there
2:35:13tell hi to him from my end. Okay.
2:35:16So next what do you what do you want
2:35:18students? Do you want this proof?
2:35:24Yeah. Good. Good. Yeah. Good. All the
2:35:27best from from my end. All the best from
2:35:32my end. So I mean now in recent days I'm
2:35:35giving this example for most of the
2:35:36students.
2:35:38So how should how we should why should
2:35:40we not ignore exams?
2:35:49See image reflection image image of the
2:35:53perpendicular problem
2:35:56they can they might ask it comes under
2:35:58lines
2:36:00not only in planes
2:36:03of the perpendicular problems they might
2:36:05ask it.
2:36:10Okay. So do you want this proof students
2:36:12proves
2:36:15do you want this proof?
2:36:17Tell me students, do you want this
2:36:18proof? E proofs bea
2:36:22maka.
2:36:30Tell me students, do you want this
2:36:31proofs? Tell me first.
2:36:38Reply me. Makla. He proves bac and row.
2:36:40Reply mga.
2:36:42You want this proof? Okay. Fine. Fine.
2:36:45So prove that it is a three marks proof
2:36:47ml. This is a three marks proof. This is
2:36:49a three marks proof. So prove that for
2:36:52any square matrix a okay a plus a
2:36:54transpose is symmetric and a minus a
2:36:56transpose is q symmetric. Proof
2:36:59okay proof of this. So given a is a
2:37:02square matrix given a is a square
2:37:05matrix.
2:37:07Given a is a square matrix.
2:37:11Okay. So let let students let P is equal
2:37:15to A + A transpose.
2:37:18Okay. Let's let P is equal to A plus A
2:37:20transpose. Okay. So transpose on both
2:37:22sides.
2:37:24So transpose on both sides. Transpose.
2:37:28So we got here it is P transpose. Okay.
2:37:30Equals it is A + A transpose old
2:37:33transpose.
2:37:35Okay. So we know that we know that
2:37:36students X into Y. Sorry X + Y old
2:37:39transpose. What is x + y old transpose?
2:37:41It is xrpose plus yrpose. Okay. And then
2:37:45there is next one more thing that is x
2:37:47xrpose all transpose is nothing but x.
2:37:50So both of these conditions are very
2:37:52much required.
2:37:54So both of these conditions are very
2:37:56much required. So three marks this is
2:37:58for three marks. So here we get a
2:38:01transpose plus it is a transpose old
2:38:04transpose.
2:38:05So we got here a transpose plus a. Okay.
2:38:08So prpose equals to A transpose plus A
2:38:11or we can also write. So we can also
2:38:13write students P transpose equals to it
2:38:15is A plus A transpose. So which is
2:38:17nothing but P. So P transpose equals to
2:38:20P.
2:38:22This condition is for what?
2:38:26This condition is for what?
2:38:31Okay.
2:38:35The conditionals
2:38:38this is the condition and symmetric.
2:38:42Okay.
2:38:45What is the condition for SK symmetric?
2:38:48So the condition is nothing but A
2:38:49transpose equals to minus A. A transpose
2:38:52equals to minus A. This is the
2:38:54condition. So okay P transpose equals to
2:38:58P. I2 therefore therefore therefore P is
2:39:01equal to A + A transpose is symmetric
2:39:08I hope this is clear
2:39:13okay so second proof this is first part
2:39:16second part students second part let Q
2:39:18is equal to let Q is equal to A minus A
2:39:21transpose okay so transpose on both
2:39:24sides transpose on both sides
2:39:29transpose on both sides. So therefore we
2:39:31get qranspose equals to it is a minus a
2:39:34transpose old transpose.
2:39:37Okay. Therefore we got here it is a
2:39:38transpose minus a transpose old
2:39:41transpose equals to a transpose minus a.
2:39:45Okay. Therefore minus common it is a
2:39:47minus a transpose. Okay. Here it is
2:39:50qrpose. Therefore, therefore students
2:39:52qrpose equals to it is minus q.
2:39:57So therefore, therefore students
2:39:59therefore q equals to it is a minus a
2:40:02transpose is q symmetric. I hope this is
2:40:05clear.
2:40:07Hence the proof.
2:40:09Hence proved.
2:40:15I hope this is clear.
2:40:20So thank you Adita. Thank you Panda.
2:40:33Thank you Si. Sorry. Thank you Sam.
2:40:37Thank you. Thank you. Thank you so much.
2:40:41Thank you. Thank you students. Thank
2:40:42you.
2:40:55Yeah, these proofs are very sorry these
2:40:56questions three marks questions
2:41:00important very very very very important
2:41:04important three marks for three marks
2:41:07these questions are very very important
2:41:09most repeated questions exams repeat
2:41:13these questions are repeated for exams
2:41:15many many many times
2:41:19So, do you want me to solve these
2:41:20questions or you know it
2:41:29doubt?
2:41:36Yeah. Thank you, Satya. Thank you. Thank
2:41:38you.
2:41:43So, I'm getting old. I am getting old.
2:41:49Not only beards, even airs.
2:41:56Nowadays, even small kids get like the
2:41:58airs. What is there in that? It is a
2:42:01symbol that I'm getting old and I'm
2:42:03getting more mature.
2:42:09Okay.
2:42:18You know this.
2:42:22You know this. Okay. Fine. So
2:42:26these are required for four. Sorry.
2:42:27These questions are asked for four
2:42:29marks.
2:42:30Four mark questions. You all know these
2:42:33questions. Four marks questions.
2:42:50Thank you. Thank you.
2:43:13They all need to care about it.
2:43:18I am paid for that. So I need to
2:43:19concentrate on my subject more. Okay.
2:43:28Engineering mathematics basics teach
2:43:31basic for engineering mathematics. Basic
2:43:33integration. Sorry. Basic different.
2:43:36Basication integration.
2:43:40Yes. Correct. Correct. Okay. So, you all
2:43:44know this students. You all know this.
2:43:46You all know this. Reply me fast. You
2:43:49all know this.
2:43:53These are four mark questions. Start
2:43:55integration. Integration and inve.
2:43:58So, previous one question solve
2:44:02matrices.
2:44:04sum of symmetric and symmetric matrices.
2:44:07Okay, Rosie, you're asking about this
2:44:13sum.
2:44:17So about this want this problem. Rosie
2:44:21want this problem. Okay. So please watch
2:44:24here. So solution for first one. First
2:44:27one let a =
2:44:31a = 3 1 - 1 okay so a transpose
2:44:38a transpose at least you need to know
2:44:40some basics
2:44:43so thank you thank you so at least you
2:44:46need to know some basics
2:44:49you know the basics
2:44:57This is 3 1 - 1
2:45:04column second row second column first
2:45:07row first column second row second
2:45:09column got it
2:45:13you understood this next okay next let
2:45:16me write here okay let let students
2:45:21P is equal to 1 by 2 into Okay, please P
2:45:26+ P transpose.
2:45:29Okay, let P=
2:45:34Okay,
2:45:39A.
2:45:41So first Apose find first find a
2:45:45transpose. So what you will do? What you
2:45:48will do students here? Okay.
2:45:53Let Pal
2:45:55let P= A plus A transpose by 2 and Q is
2:46:00equal to students and let Q is equal to
2:46:02it is Q minus sorry A minus A transpose
2:46:05by 2.
2:46:08Okay. So next so show that show that
2:46:12prpose equals to p and here student show
2:46:15that show that qranspose equals to q
2:46:17minus q. Okay show that yeah prpose
2:46:21equals to p prpose equals to p and here
2:46:24qrpose equals to minus q.
2:46:28So what do we get here? So this is
2:46:30nothing but this is nothing but p is
2:46:32symmetric. This is nothing but p is
2:46:34symmetric.
2:46:36Okay, this is nothing but Q is skew
2:46:38symmetric. This is nothing but Q is skew
2:46:41symmetric.
2:46:55Okay, lastly therefore so show that show
2:46:57that students P + Q is equal to A. Show
2:47:00that P + Q is equal to A and right
2:47:02conclusion. So right conclusion.
2:47:07So this is the step involved. This is a
2:47:10step involved.
2:47:13These are the steps involved here.
2:47:17Okay. These are the steps involved.
2:47:20Okay.
2:47:21P= 1 by 2. So
2:47:25find a transpose
2:47:28A + Apose by 2. Okay. Okay. So 1x2 into
2:47:32P + P transpose.
2:47:34So 1x2 into what is P? Okay. What is P
2:47:38here? P is nothing but 15. Okay. 1 5 - 1
2:47:412 plus what is P transpose? P transpose
2:47:44A sorry A sorry A plus A transpose.
2:47:48Sorry it is A plus A transpose. It is A
2:47:51+ A transpose.
2:47:55Fine. So a transpose it is 3 1 3 1 5 -
2:48:001. Okay. So watch here. So this is
2:48:03written as 1x 2 into Yeah. 1x2 into it
2:48:06is
2:48:081x 2 into it is 4 6. Okay. 4.
2:48:18I did wrong. I did wrong. I did wrong.
2:48:28Okay. So three this is three. Okay. So
2:48:32now here we get
2:48:363 + 3 6 5 + 1 6 in.
2:48:42Okay. Okay. Okay. I did wrong. Just give
2:48:45me a moment. So it is 3 5 1 - 1 1 - 1.
2:48:48Yeah. So it is 66 66 6 - 2. So therefore
2:48:53it is yeah therefore it is students this
2:48:56is nothing but 3 3 - 1. So this is P. So
2:49:00therefore students what is prose here?
2:49:03What is prose? PRpose equals to 3 3 - 1.
2:49:08Okay. So this implies
2:49:11pr
2:49:13so therefore therefore students yeah
2:49:15therefore here we get therefore here
2:49:18here students prpose equals to P. This
2:49:21implies P is symmetric. This implies
2:49:24students P is symmetric.
2:49:28I hope this is clear.
2:49:38Rectangle problem important. Rectangle
2:49:41problem important.
2:49:43Okay. Rate of change of rate of change
2:49:46of area. Rate of change of perimeter
2:49:48question.
2:49:50Rate of change of area rate of change of
2:49:51perimeter of the rectangle problem go
2:49:54through that and students let let Q is
2:49:57equal to let Q is equal to students half
2:50:00into it is A minus A dash so half into
2:50:04it is yeah half into students A what is
2:50:06A? It is nothing but 3 1 - 1 3 5 1 - 1
2:50:11and it is what is a transpose? It is 3 5
2:50:141 - 1. Okay. So therefore we got it is
2:50:171x 2 into it is 0. Okay. So 4 - 4 and
2:50:22then 0. So here it is 02. Okay. Min - 2
2:50:260. This is your P. Yeah. Q. So therefore
2:50:29qrpose. Okay. So qranspose it is nothing
2:50:33but 02. Okay. Minus 2 0. Fine. So here
2:50:37we get qranspose equals to. So take
2:50:40minus common minus common. Take minus
2:50:43common here. Take minus common. Take
2:50:46negative sign common. Take negative sign
2:50:48common.
2:50:51Okay.
2:50:54We get yeah here we get student 0. Okay.
2:50:572 and it is minus 2 0. So what is this?
2:51:01This is nothing but therefore qranspose
2:51:04equals to minus q.
2:51:07Therefore therefore q is sk symmetric.
2:51:11Therefore Q is Q symmetric.
2:51:15I hope this is clear. Q is Q symmetric.
2:51:18So next what do we get? So now now
2:51:22students now consider now consider P
2:51:25plus Q. So now consider students P plus
2:51:28Q. So which is nothing but what is P?
2:51:31Okay. What is P students? P.
2:51:34Yeah this is P. P this is P. This is
2:51:37your P. This is your P. Okay. Okay. So 3
2:51:403 3 3 - 1. Okay. This is 3 3 3 - 1. And
2:51:45what is Q? Q is what? Q is nothing but
2:51:4802 0 to - 2 0. So we get it is 3.
2:51:54Okay. So 1 and then minus one. So this
2:51:57is nothing but a. So p + a.
2:52:04So conclusion. Conclusion is very
2:52:05important. Conclusion is very important
2:52:08students. Next page. Conclusion. So next
2:52:11page Makla open the next page and
2:52:12conclusion is there. So conclusion is
2:52:14very important students. Conclusion is
2:52:16very important. What is that conclusion?
2:52:18Conclusion you know every square matrix
2:52:21can be expressed as So what is the
2:52:23conclusion? Every square matrix every
2:52:26square matrix can be expressed as a sum
2:52:28of every square matrix.
2:52:31Okay. Every square matrix can be
2:52:33expressed as every square matrix.
2:52:37Every square matrix can be expressed
2:52:40yeah can be expressed as sum of so can
2:52:43be expressed as sum of symmetric and sum
2:52:47of symmetric and sku symmetric matrices.
2:52:51sum of symmetric and sku symmetric
2:52:53matrices.
2:53:03I hope this is clear.
2:53:06replay
2:53:23techn No facts.
2:53:26So you need to make more simple and more
2:53:27concepts and questions.
2:53:42It's a different thing.
2:53:48So,
2:53:56okay. So,
2:53:58so what do you want students next?
2:53:59Integration proofs.
2:54:01Do you want students integration proofs
2:54:03next?
2:54:08What do you want students next? Do you
2:54:09want integration proofs
2:54:12or you want application of derivatives?
2:54:13Y.
2:54:15What do you want students? Do you want
2:54:16application of derivatives or do you
2:54:18want proofs of integration? Yik, which
2:54:20you want
2:54:24ad maka.
2:54:31Tell me first.
2:54:50make
2:54:52application of derivatives.
2:55:09Yes.
2:55:12See this session is for clearing your
2:55:14doubts.
2:55:19So how how can I expect 4our session 4
2:55:23hour session can you expect I can cover
2:55:2465 marks expect
2:55:44you can take more than 75
2:55:49Okay. Fine.
2:55:52So
2:55:54rate of change. Okay. So rate of change.
2:55:58These problems are very important.
2:56:01These problems are important. You want
2:56:03me to solve this?
2:56:06Do you want to solve me? This important.
2:56:09This is very very important. Very very
2:56:11important. Rate of change. Rate of
2:56:14change. Important question. These
2:56:16questions are all important. Important.
2:56:19This question is important.
2:56:22Yes. This ladder problem. Ladder
2:56:24problem. Ladder problem is not required.
2:56:28Okay. So, yeah. Stone is dropped.
2:56:32Okay. Stone is dropped. E problem
2:56:35important. Okay. Circle problem
2:56:38important.
2:56:44So yes pra you can start you can get you
2:56:47can get you can definitely get okay so
2:56:49let me start solving the problems really
2:56:53okay so let me start from here two
2:56:55questions so two question
2:57:02we starting we'll be starting
2:57:05yeah so we'll be starting No.
2:57:13Okay. So coming to this. Okay. So what
2:57:16is the volume of sphere students? Volume
2:57:18V
2:57:20of sphere. Volume V of sphere. What is
2:57:23the volume of the sphere students? A
2:57:24vega it is 4x3 into p<unk> r cube. And
2:57:28surface area. What is the surface area
2:57:30students? What is the surface area?
2:57:33Surface area S is nothing but it is 4 pi
2:57:36R².
2:57:39It is 4 pi R²
2:57:44a balloon which always remains sperical
2:57:46on inflation on inflation.
2:57:50Okay. And
2:57:52okay
2:57:549 900 cm. So this is the unit. This is
2:57:58the unit. So this is the unit students.
2:58:00This unit tells us that. So unit. So
2:58:04given so given students given dv by dt.
2:58:08So dv by dt equals to it is 900 cm.
2:58:13Okay. 900 cm per second. Okay. 900 cm
2:58:19per second.
2:58:21900 cm per second.
2:58:28Okay. So find the rate at which the
2:58:29radius of the balloon increases when the
2:58:31radius is 15 cm. Yes.
2:58:34So to find to find what students see and
2:58:36find out mle to find d by dt to find d
2:58:40by dt when when students when r is equal
2:58:43to 15 cm
2:58:47when r is equal to 15 cm
2:58:50okay so we know that students we know
2:58:52that we know that volume v okay equals
2:58:56to is 4x3 pi r cq so differentiate with
2:58:59respect to t so differentiate with
2:59:02respect to t students
2:59:04So with respect to t differentiate.
2:59:07So what do we get here?
2:59:09We get here dv by dt equals to dv by dt=
2:59:12is 4x3 into 3<unk>i r².
2:59:17Yeah 3 p<unk> r² into d r by dt. Okay.
2:59:21So now your 3 and 3 get cancel. So it is
2:59:244 into pi into okay r² r² is nothing but
2:59:2715 square. Okay. And it is 900 here
2:59:31equals to into it is d r by dt. Okay. So
2:59:34therefore we got I hope this is clear
2:59:37students. So then 900 equals to it is 4
2:59:41into pi into 225. Okay. Into d by dt dt.
2:59:45So it is 900 equals to 4 into sorry 900
2:59:49pi. This is 900 pi into d by dt. So what
2:59:53we get? Therefore d r by dt equals to it
2:59:56is 1x pi. Okay. Okay. So it is cm/s.
3:00:01This is cm/s. Unit is must. Okay. You
3:00:05need to write mention. Yeah. You need to
3:00:07mention unit maka. You need to mention
3:00:09unit in the answer. You need to mention
3:00:11unit in the answer. You need to mention
3:00:15unit
3:00:17in the answer.
3:00:20So answer mention
3:00:23you need to mention unit in the answer
3:00:25at any cost. Makla.
3:00:29Did everyone understood this?
3:00:56I hope this is clear.
3:01:30Okay. Coming to this
3:01:33fine.
3:01:43So we are actually So we were supposed
3:01:46to have holiday but
3:01:54we have sacrificed our holiday for your
3:01:56wellness.
3:01:58So we have sacrificed our holidays for
3:01:59your wellness.
3:02:02So it is Sunday also. Sunday we'll be
3:02:05taking alternate holidays.
3:02:11Yeah, definitely.
3:02:17Which college are you from in Vijaya and
3:02:21brother student? Who is your brother?
3:02:39Important questions list after the
3:02:40class.
3:02:42So yeah.
3:02:45So what doubt you got? Tell me.
3:02:50Okay. So now coming to this
3:02:54decreasing.
3:02:56See decreasing we use negative sign. We
3:02:59use here negative sign. Negative sign.
3:03:02So please watch solution. So given
3:03:05students given so please watch here.
3:03:07Given given students dx by dt equals to
3:03:11dx by dt equals to minus it is 5 cm per
3:03:15minute. It is 5 cm per minute. Okay. And
3:03:21width y okay and width y that is dy by
3:03:24dt equals to it is 4 cm per minute.
3:03:30Okay. So now students to find out ml.
3:03:33Yeah. So to find first one okay given
3:03:36also given students also given x= 8 cm
3:03:41and y = it is 6 cm
3:03:45fine. So
3:03:47first one perimeter.
3:03:49So first one perimeter
3:03:51perimeter
3:03:54if x and y are the sides. If X and Y are
3:03:58the sides of a rectangle. If X and Y are
3:04:00the sides of a rectangle.
3:04:04If X and Y are the sides of a rectangle,
3:04:07students.
3:04:09Okay. Perimeter. First one. Perimeter.
3:04:12What is a perimeter student? Perimeter.
3:04:14Perimeter is nothing but 2 into X + Y.
3:04:17Okay. And area is nothing but and area
3:04:20is nothing but it is X into Y.
3:04:28Okay. Oh, from excellent college. Okay.
3:04:30Fine.
3:04:48Fine. Fine. Fine.
3:04:57So last.
3:05:04Okay.
3:05:06So problem. Okay.
3:05:09So perimeter is nothing but 2 into x +
3:05:11y. Okay. So
3:05:17e question
3:05:23they will ask perimeter or they will ask
3:05:25area.
3:05:27Okay either they will ask perimeter
3:05:28students or they will ask area.
3:05:31So I will come to that.
3:05:34So differentiate with respect to t.
3:05:37Differentiate with respect to t.
3:05:38Therefore we get here it is dp by dt. dp
3:05:41by dt equals to it is 2 into it is dx by
3:05:44dt plus it is dy by dt.
3:05:49So what we get we get here 2 into it is
3:05:52dx by dt is nothing but minus y okay
3:05:56plus it is 4. So here we get it is 2
3:05:59into minus1 so it is -2 okay cm per
3:06:04minute -2 cm per minute this is the
3:06:07answer I hope this is clear
3:06:18what else you will get here I am finding
3:06:21what they asking find the rate at which
3:06:24the radius of the ball is balloon is
3:06:26increases
3:06:28So dt.
3:06:31Okay. So you get your dt equals to 1x.
3:06:36So you'll be getting the PDF
3:06:40after the class you'll be getting the
3:06:41PDF. Don't worry about it.
3:06:44PDF don't worry.
3:06:49Exam difficulty don't worry. Exam
3:06:52difficulty don't worry. And students
3:06:54area coming to the area here. Yeah.
3:06:57Coming to the area students.
3:06:59So area is nothing but area equals to x
3:07:02into y.
3:07:03So differentiate with respect to t. So
3:07:06differentiate with respect to t.
3:07:10So what we get here we get it is da d d
3:07:13d d d d d d d d d d d d d d d d d d da
3:07:13by dt. It is da by dt equals to it is x
3:07:16into dy by dt
3:07:18plus y into dx by dt.
3:07:22Okay. Okay. So therefore we get here we
3:07:25get it is x into 8 into so this is 8
3:07:28into dy by dt is nothing but 4
3:07:33+ y into okay y is nothing but 6 6 into
3:07:36- 5 6 into - 5. So here we got 32 - 30
3:07:41which is equal to 2. So da by dta equals
3:07:44to 2 square cm per minute
3:07:49we are using the yeah we are using what
3:07:51square cm per minute
3:08:00so kav you can join our batch kanda why
3:08:04why you didn't take our batch
3:08:07so rohan hello no rohan what are you
3:08:10doing rohan
3:08:12Rohan, where are you?
3:08:22Important questions.
3:08:27You can see that video important
3:08:30questions video.
3:08:33Okay. So square cm it is square cm
3:08:36because area it is okay perimeter unit
3:08:40perimeter unit. So perimeter area
3:08:49what will be the unit students see.
3:08:52So
3:08:56so what will be the unit? Unit
3:08:59will be it is cime per minute and this
3:09:02is square cm per minute.
3:09:05You understood this.
3:09:17Yeah. Morning I have discussed MCQs.
3:09:19MCQs.
3:09:31problem. Watch this. So solution.
3:09:36So given students, okay, so given Yeah.
3:09:40Given students given
3:09:43ycoordinate
3:09:45ycoordinate is changing eight times as
3:09:48fast as x coordinate
3:09:50given ycoordinate
3:09:53given ycoordinate
3:09:55is changing
3:09:58is changing
3:10:00eight times as x coordinate eight times
3:10:02as
3:10:04xcoordinate.
3:10:06So what do you mean by this?
3:10:12What do you mean by this? This means
3:10:14this means dy by dt equals to okay it is
3:10:188 * dx by dt
3:10:21okay this is given student this is given
3:10:23so ycoordinate is changing ycoordinate
3:10:26is changing students 8 * x coordinate
3:10:30so given given given student 6 into y =
3:10:34x² + 2 so differentiate with respect to
3:10:37t differentiate with respect to t
3:10:42so therefore we get it is 6 into 6 into
3:10:45dy by dt okay = to 2 into x into dx by
3:10:50dt okay equals to 0 sorry plus 0 plus 0
3:10:55I even don't want to write it so what we
3:10:57get 6 into what is this it is 8 into
3:11:01yeah 8 into students dx by dt
3:11:05okay 8 into dx by dt equals to it is 2x
3:11:11equals to it
3:11:132x into it is dx by dt.
3:11:16Okay, dx by dt. So dx by dt dx by dt get
3:11:21cancel. So it is 48 = to 2x. So x equal
3:11:25to 24. I hope this is clear.
3:11:29x= 24. So now we have
3:11:38[clears throat]
3:11:39So here students here here 6 y = to x² +
3:11:432. Okay. So therefore
3:11:47okay therefore 6 y = to it is 24 square
3:11:52+ 2.
3:11:57Okay. Therefore it is 6 y = to it is 576
3:12:01+ 2. So it is 6 y = to it is 578.
3:12:06Okay. So y equals to
3:12:09y y y y y y y y y y y y y y y y y y y y
3:12:09y = to students 578 divided by 8 6. So
3:12:1354 38.
3:12:16So what we get here 2 3 and two students
3:12:20it is okay. Y = 27 it is 8 and 189. 289
3:12:26by 33 bar makla. So you get here 289 by
3:12:303. Therefore the point is therefore
3:12:33therefore coordinate is coordinate in
3:12:36barat makla. Therefore coordinate is
3:12:40so coordinate in makla it is nothing but
3:12:42x y which is equal to 24 it is 289
3:12:46divided by 3. So is this correct?
3:12:54Is this correct students? Tell me fast
3:12:56any doubt here.
3:12:58Is this clear? Any doubt students?
3:13:10Tell me fast. Head out here.
3:13:24Any doubt students tell me fast. Okay.
3:13:27Question.
3:13:30Yeah, that doubt I had there is a Yeah,
3:13:33there is a problem with the question.
3:13:36The question is wrong.
3:13:39The question is wrongly typed dear.
3:13:42Okay. Question wrong. The question is
3:13:44wrong. I anticipated it. Okay. The
3:13:46question is wrong students.
3:13:49Just give me a moment. Just give me a
3:13:51moment students. The question is wrong
3:13:53here.
3:13:56The question is wrong. The question is
3:13:59wrong here. Just give me a moment.
3:14:04Question. Just give me a moment. It is 6
3:14:07x cub. So 6 y = xq + 2. Okay. So here it
3:14:11is 3x².
3:14:13Fine. So it is 6 into. Okay. So it is
3:14:173x² into dx by dt. Okay. So here it is.
3:14:22This is what this is 8 into 8 into dx by
3:14:26dt. Okay. So here we get so dx by dt dx
3:14:30by dt cancels 48 = to 3x².
3:14:34Okay. So therefore x² equals to it is 48
3:14:38/ 3 which is equal to 16. So x= to plus
3:14:42or minus 4.
3:14:46Okay. x is equal to plus or minus 4. So
3:14:48coming to the next page.
3:14:52Okay. So here here y = 6 y = x² + 2.
3:14:59Okay. x² + 2. If x = 4,
3:15:05if x= 4 students
3:15:09h
3:15:11okay xq + 2. This is xq + 2. So here we
3:15:16get it is yeah. So xq + 2. Therefore
3:15:19here x if x is equal to 4. If x is equal
3:15:21to 4 students then it is 6 y = it is 4
3:15:24cq + 2 it is 64
3:15:28this is 64 + 2 so therefore it is 66 6 y
3:15:32= to 66 y = 11 okay therefore therefore
3:15:37coordinate okay therefore coordinate
3:15:40this is nothing but 4a 11
3:15:434a 11
3:15:46okay and then and then students second
3:15:48one if x is equal to if X equ= -4.
3:15:52Okay. 6 Y = It is -4 whole cube + 2 it
3:15:58is nothing but - 64 + 2 it is - 62. So 6
3:16:02Y. So Y = it is - 62 divided by 6 it is
3:16:07nothing but Y = it is - 31 divided by 3.
3:16:12Okay. Therefore coordinate is nothing
3:16:14but
3:16:15therefore coordinate is nothing but
3:16:21it is -4 - 3 1 by 3 I hope this is clear
3:16:29reply me fast be reply m
3:16:35did everyone understood this reply me
3:16:37fast.
3:16:46Yes,
3:16:48thank you. Thank you.
3:16:54I hope this is clear.
3:17:00So now the things are fine.
3:17:06Fine.
3:17:07So problems these problems should be
3:17:09learned.
3:17:11So learn this problem students. So learn
3:17:14this and cube problem. So cube problem
3:17:17important.
3:17:21Yes. Vishba you can do it. You can do
3:17:24it. Vishba.
3:17:27If you try definitely you can get the
3:17:28marks.
3:17:315 minutes break. I want some break. Yeah
3:17:34I want some break students. I will give
3:17:36some 5 minutes break and we can continue
3:17:38further. So yeah dhoni tala for reason
3:17:42I'm giving seven 7 minutes break
3:17:44students come back come back after 7
3:17:46minutes.
3:25:01Yeah. So, welcome back students. Welcome
3:25:03back.
3:25:11Welcome back students.
3:25:19Do you want me to solve this?
3:25:21Yeah. Do you want me to solve the
3:25:23students? Cub
3:25:25volume
3:25:28is nothing but x cube. Okay. And then
3:25:32surface area. What is the surface area
3:25:35students? What is the surface area?
3:25:37Surface area is nothing but it is 6 x².
3:25:41Okay. So we need to use this
3:25:44where x is nothing but edge. Where x is
3:25:47nothing but edge.
3:25:58So Nikil already I have done MCQs you
3:26:00can check in the live session. So live
3:26:02session MCQ is already please go and
3:26:05check there. MCQ is already please go
3:26:08and check there.
3:26:12Yeah
3:26:16I have seen it twice. I'm waiting for
3:26:18the English version. English version.
3:26:22Okay. Fine. So,
3:26:25so coming to this solution.
3:26:28Yes. An edge of the variable cube is
3:26:29increasing. Edge is increasing.
3:26:33Okay. Edge is increasing at the rate of
3:26:353 cm/s. If x is the edge, if x is the
3:26:39edge of the cube, if x is the student's
3:26:42edge of the cube. Okay. So given given
3:26:47dx by dt equals to given dx by dt equals
3:26:51to student it is 3 cm per second. Okay.
3:26:55So how fast is the volume of the cube?
3:26:57So to find yeah to find students dv by
3:27:01dt to find dv by dt when x is sorry when
3:27:05x is equal to 10 cm.
3:27:08Fine. So,
3:27:10so we know that we know that volume of
3:27:12the cube is nothing but we know that
3:27:14volume V is equal to X cube. So,
3:27:17differentiate with respect to T. Yeah.
3:27:19Differentiate with respect to T
3:27:21students. So, we got here it is dv by dt
3:27:23equals to we get here dv by dt equals to
3:27:25it is 3x² into dx by dt.
3:27:30Okay. So therefore we got it is 3 into
3:27:3210² okay into 3. So therefore it is 9
3:27:36into 100 which is nothing but 900. Okay
3:27:40900. So
3:27:43what is the unit we get? It is centime
3:27:46cm per second. It is cubic cm per
3:27:49second.
3:27:52I hope this is clear.
3:27:55So important questions.
3:28:00So you need to wait for it.
3:28:04Don't worry
3:28:06you will get it.
3:28:10So fine. So any questions
3:28:14any other question students which you
3:28:15want tell me first.
3:28:33You want any other questions students
3:28:34tell me first reply makla?
3:28:46You want any other questions students
3:28:48or shall I give the important questions?
3:28:50Shall I end this session?
3:28:53Shall I end this session students and
3:28:54give you the important questions?
3:29:01Allah the important questions. I will
3:29:03give the important questions then. So
3:29:05any other topic my doubty makla have you
3:29:07got doubts on any other topics tell me
3:29:09first
3:29:15so already I have said it
3:29:19important question we can't complete
3:29:21everything
3:29:28if you have doubt in any yeah important
3:29:30questions
3:29:33you will get a So
3:29:36integral integral important questions
3:29:40my student. So important question.
3:29:48So actually record
3:29:51actually
3:29:52there was some issue with that video. So
3:29:54therefore it has not been published but
3:29:56it will be coming soon. Yes.
3:30:01So
3:30:04let me complete there are a lot of
3:30:05questions
3:30:08you ask me which questions you have got
3:30:10doubt let me clear those
3:30:16question
3:30:20you cannot cover by chapters okay
3:30:26you need to go mark wise
3:30:30First to six marks questions, next to
3:30:33four marks questions,
3:30:35five marks questions,
3:30:37selected selected three and two marks
3:30:39questions. So first try to get 30 marks
3:30:42and then 50 marks and then 60 marks and
3:30:46then finally try to cover the MCQs and
3:30:48fill in the blanks. So
3:30:52So learning gaining knowledge gaining
3:30:54knowledge is different from getting
3:30:56marks.
3:30:58Knowledge
3:31:00knowledge
3:31:05You will waste lot of unnecessary time.
3:31:08You will waste lot of unnecessary time.
3:31:10Don't cover each and every questions
3:31:12from your textbook which has which are
3:31:14unnecessary
3:31:16knowledge
3:31:18but
3:31:19complete only the important questions
3:31:21which we will be giving
3:31:24easily you can get out of.
3:31:27Are you getting it students? Same guy.
3:31:34Are you all getting it?
3:31:44Everything will be given. All important
3:31:46questions will be given. Even LP
3:31:47questions also will be given. LP
3:31:49questions also will be given.
3:31:56Don't worry. Okay. Huh?
3:31:59Is that fine?
3:32:06Okay.
3:32:11Everything will be given. Don't worry.
3:32:14Every questions will be given. Don't
3:32:15worry.
3:32:17Fine.
3:32:19Let me end this session.
3:32:22So
3:32:25it will be uploaded in in the morning
3:32:27itself.
3:32:29Yes. Yes. Yes. It will be uploaded in
3:32:30the morning itself.
3:32:32Yes.
3:32:34No.
3:32:38By morning 10 11 it will be uploaded.
3:32:40Okay. Bye. Take care. See you all. Good
3:32:42night. Good night my dear students. Good
3:32:45night. It will be uploaded soon. Okay.
3:32:46Bye Makla. Take care. Good night. Good
3:32:48night. Good night students. for

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