12th Maths Chapter 3 Important Questions Part 4 for first Mid term 2026

[00:00:38]Hi students, good evening. Good evening all. Hello. Hi. Sorry for the delay. Hi.

[00:00:43]Hi. Hi. Good evening.

[00:00:48]Yes. Good evening. Good evening.

[00:00:56]Yes. Good evening.

[00:01:02]How are you? So extremely sorry.

[00:01:16]So last problem is there anyone?

[00:01:29]Uh yes. Hi Raja Lakshmi. Hello. Yes.

[00:01:34]So remaining problems.

[00:01:46]Okay.

[00:01:50]Which is easy or difficult? Okay. So multiple problems of problems.

[00:02:09]So it will be easy. Okay. Otherwise it is difficult only. Okay.

[00:02:26]Good evening.

[00:02:28]Hi.

[00:02:30]Okay. So, problem for question.

[00:02:58]So first question example 310. Okay.

[00:03:04]Form a polomial equation with coefficients. So root.

[00:03:17]Okay. Yes. Okay.

[00:03:26]Okay.

[00:03:27]Important question.

[00:03:32]Hi. Good evening.

[00:03:35]11th max. 11th max. Okay. So question number 12.

[00:03:53][snorts] So what is the given? So square root of 1 second.

[00:04:05]So the square root of <unk>2 xeic x x - 3 = 0 x = +alus important question for okay so yes social science import.

[00:05:08]Okay.

[00:05:30]So first product of the roots five and six already.

[00:05:46]Okay. One second. Okay.

[00:05:55]Example 1.1 it is already done. Okay.

[00:06:04]So x + so square roo<unk> of <unk>2 / <unk>3 into x - the square root of <unk>2.

[00:06:14]Okay. So which is equal to a + a square of square root.

[00:06:40]So the remaining values x² - <unk>2 / <unk>3.

[00:06:51]Yes.

[00:06:53]Okay.

[00:06:58]Yes. So already important questions max.

[00:07:06]Okay.

[00:07:08]Fore.

[00:07:25]Okay.

[00:07:38]Okay. X² <unk>2 <unk>3 into X² +<unk>2 /<unk>3 X^ 4 - 2X 3 Okay, Okay. Okay.

[00:08:18]Okay.

[00:08:31]This frame x ^ 4 - 2x 3 which is = 0= 0 3x ^ 4 - 2 / Okay.

[00:09:09]Okay. Yes.

[00:09:33]we get the polinomial equation. Okay.

[00:09:36]Yes. So the next question that the equation 2x² - x + 7 = 0 cannot be satisfied cannot be satisfied by any real values.

[00:09:48]imaginary value.

[00:09:58]Okay. Okay.

[00:10:17]Okay. First given equation 2x² - 6 x + 7 = 0 x² + b x + cement.

[00:10:38]So what is the formula?

[00:10:46]Thank you.

[00:10:48]Okay. Yes. So, how did you find the value? What is the formula formula?

[00:10:55]Yes. Yes. Good. So, discriminant = b² - 4 a value.

[00:11:11]So a value and c value important question.

[00:11:24]So value it is - 6. Okay. So the next one what is c value? It is okay button. Okay.

[00:11:52]So, okay. So b² - 6 the whole square and -4 into 2 into 7. So 36 - 8 into 7. So what is the value? 36 - 56. So final answer.

[00:12:18]Yes.

[00:12:25]Chapter three.

[00:12:28]already it is done.

[00:12:31]Okay.

[00:12:34]Okay.

[00:12:37]Okay. It isUS 20.

[00:12:42]Okay. So it is minus 20.

[00:12:45]Okay ma okay okay honeyus.

[00:13:03]Yes.

[00:13:10]So roots are what is the nature of the roots?

[00:13:18]It is obviously imaginary.

[00:13:22]Okay. Roots are imaginary. So okay. Yes. So the next question has equal roots equal roots.

[00:13:44]So equal roots what is the condition?

[00:13:48]Same discrimin value and C value.

[00:14:05]So equation number one physics important patients already uploaded.

[00:14:25]Yeah. Okay. Thank you.

[00:14:29]Okay.

[00:14:36]It is equal to greater than real and unequal roots. Correct. Real and unequal.

[00:14:51]discriminant less than it is. Okay.

[00:15:02]Yes.

[00:15:06]Okay. Yes.

[00:15:12]The terms 2 into k + 2² - 4 into a into citution 4 into k + 2² - 4 into 9.

[00:15:38]Okay. Okay. Okay. Thank you. 50.

[00:15:50]Yes.

[00:15:53]So the 4 into the square 36k = 0 36 opposite side 4 into k + 2 the whole square= 36k.

[00:16:16]Okay. K + 2² = 36K / 4. So answer.

[00:16:34]Okay.

[00:16:39]Okay. H. So 9 k + 2 square which is equal to 9.

[00:16:58]K² 2 K 2 2 2 K into 2 square which is equal to 9K. So K² + 4K + 4 = 9K. Okay.

[00:17:16]Okay. So k² + 4k + 4 - 9k = 0. So this equation becomes k² - k + 4 = 0.

[00:17:43]2.1.

[00:17:53]Okay. Hi 2.1 it is -4 and 1 -4 -1 -4 and -1. Okay.

[00:18:13]Yes.

[00:18:15]It is kus 4 and kus =0. Therefore k = 4a 1. Okay. 4a 1. So the next question.

[00:18:38]Hi. Good evening.

[00:18:40]Yes. Chra. Hi.

[00:18:42]Okay.

[00:18:54][snorts] Okay. So, next next question.

[00:19:00]Exercise 3.2. Second question. So, find the polomial equation of minimum degree.

[00:19:06]minimum degree with rational coefficients having 2 + roo<unk>3 I exercise 3.2 Exercise 3.23 roo<unk>3.

[00:19:31]Okay. root. So again you have to assume another root another 2us roo<unk>3 I okay this roo<unk>2us roo<unk>3 alpha = 2 + roo<unk>3 I and beta = 2us roo<unk>3 of the roots equation with minimum degree okay Yes. Okay.

[00:20:07]So with the minimum degree products assume.

[00:20:38]Okay. Again I explain. So okay sum of the roots sum of the roots it is alpha plus beta alpha plus beta.

[00:20:56]So 2 +<unk>3 I + 2 -<unk>3 I cancel. So the remaining term alpha plus beta= next.

[00:21:16][snorts] Okay.

[00:21:20]Okay. Good.

[00:21:23]of the roots.

[00:21:25]So 2 +<unk>3 I into 2 -<unk>3 I a + formula 2 2² minus of <unk>3 I² <unk>3 I² I square value what is I square Okay. Alpha into beta = 3. Sorry. 4 + 3 it is 7. Okay. It is 7. Okay.

[00:22:20]X². So sum of the roots. Sum of the roots into X plus product of the roots= 0. X²us 4X + 7= 0.

[00:22:35]required equation 10th class.

[00:22:46]Okay. So x² - 4x + 7 = equation. Okay. Is that clear?

[00:22:55][snorts] Okay.

[00:23:03]Okay.

[00:23:06]Fourth question again you have to assume another root.

[00:23:13]So roo<unk> and roo<unk>3.

[00:23:29]Is that clear? No.

[00:23:31]Hi there. Good evening.

[00:23:34]Okay.

[00:23:35]Okay. Okay. Thank you.

[00:23:42]Okay.

[00:23:43]So easy sum of the roots of the roots.

[00:23:51]So the roots of higher degree pol.

[00:24:07]So example 3.1 example 3.1.

[00:24:12]So science important question already slow is that No. Okay.

[00:24:51]2x + 11 x² - 9 x - 18.

[00:24:57]So is there any idea? So the equation number Okay. So, first sum of sum of the coefficients of odd powers.

[00:25:30]So first question, good evening.

[00:25:46]Social important questions already uploaded. So please check the shots.

[00:25:50]Okay. So sum of the coefficients of even powers.

[00:25:56]So 2 and 9 obviously next.

[00:26:22]So even power It is 11 - 18 - 7.

[00:26:51]in the problem of equations.

[00:26:57]So even you have to take the constant value also I mean x² like 5x whateveral for equation.

[00:27:39]So the coefficients of power rant.

[00:27:50]So sum of so coefficients of odd power odd power equal to so power equal to even power values.

[00:28:01]Okay.

[00:28:03]Two more. One more.

[00:28:06]So, x + 1 is a factor.

[00:28:15]I'm not sure previous 11.

[00:28:28]So x + x - and x + 2 x - x.

[00:28:53]Okay. So x + one. So which is equal to division.

[00:29:18]Okay.

[00:29:20]equation 2x² sorry 2xqub + 11 x² - 9 x Okay.

[00:29:54]Okay.

[00:30:00]like okay so 2x² into x 2x + 2x² the remaining 9x - 9x so + 9 x² + 9 x daily.

[00:30:47]Okay. Yes.

[00:30:51]Okay.

[00:31:07]student.

[00:31:18]So within two 1 minute which class you are studying. Okay. So again you have to multiple which is equal to next or subtract.

[00:32:33]Yes.

[00:32:40]add.

[00:32:42]So - 36 - 36 and + 9 22 3R how to multiply?

[00:33:02]Yes, it is 12 and 3. Are you sure? + 12 and minus 3. Okay. + 12 andus 3. So yeah. Okay. So you get the value 36 and okay already - 6 and - 3x2. Correct.

[00:33:38]Okay. Correct. X = 6 or 3 by 2. Okay.

[00:33:45]3x2 root value.

[00:33:57]Wrongus.

[00:34:05]Okay.

[00:34:07]Okay. So so all the roots of the given polomial.

[00:34:17]So given polomial are yes beginning x + one is a factor of this polus 3x 2.

[00:34:37]So these are the roots of the given polic.

[00:34:53]Okay. Yes. Clear.

[00:34:56]So next example 3.1 obtain the condition that the roots of this equation numbers sequence.

[00:35:20]for section.

[00:35:38]So, APse.

[00:35:51]So, three consecutive terms.

[00:35:58]Three consecutive terms.

[00:36:04]Yes.

[00:36:07]So the consecutive terms roots. So let the roots.

[00:36:14]So let the roots be as beta max.

[00:36:28]So, a minus b a sorry a minus d a minus d a + d.

[00:36:36]Okay. So, what is d here?

[00:36:43]H.

[00:36:52]It's okay.

[00:36:58]difference.

[00:36:59]Okay.

[00:37:05]Sum of the roots. So sum of the roots equal to value by a minus b by a minus d + a + a + d = - by a. So what is b value here? B value a = b = Cal what is the value? [snorts] Okay. So a = 1, p sorry b= p and c= d= rsus sorry a minus d a + d which is equal to minus d get cancelled right okay so 3 a = 3 a = so therefore a= minus p by 3 Okay.

[00:38:32]Okay.

[00:38:46]Okay. So, science class.

[00:38:51]Okay. So, pol of the pol.

[00:39:09]Okay.

[00:39:13]substation rearrange. So, a + p a² - a plus r plus = Okay. So, Fore.

[00:40:03]Are you clear?

[00:40:06]Okay. Yes.

[00:40:10]Substitution - P / 3 + P into - P / 3² + Q into - P by 3 + R = 0.

[00:40:27]So - pq / 27 + p² / 9 - pq / 3 + r = 0 - p3 27 and p / Yes. So 3= 0. So now we have to take the LCM.

[00:41:04]Hi good evening 27 and n portions. No, no.

[00:41:25][snorts] So it becomes 27 27us pip.

[00:41:38]So 3 pubus 9. So 9 pq and here 27 r= 07 2 pq. So 2 pq - 9 pq + 27 r= the term rearrange.

[00:42:07]So 9 pq = 2 pq + 27 rar.

[00:42:19]Anyone please guess ninth portion.

[00:42:34]for no abia then you said write statement moon scene h no so What is the condition A + 2 B= A + C.

[00:43:48]Simple required condition.

[00:43:56]Okay.

[00:44:14]square=.

[00:44:18]Okay, done. So, next find the condition that the roots of a x + bx² + c x + d=0 or in geometric progression assume these values are not equal to zero.

[00:44:37]H find out law same.

[00:44:55]Okay.

[00:45:00]Okay. And roots of the roots.

[00:45:14]Okay.

[00:45:18]Easy.

[00:45:22]Okay.

[00:45:26]Yes. So the next question. So solve the equation if the roots form an arithmetic progression.

[00:45:34]So same nth question.

[00:45:39]Okay. So same arithmetic progression.

[00:45:43]So you have to take a d a sum of the roots of the roots.

[00:45:58]Solve the equation.

[00:46:04]Previous value of the roots.

[00:46:35]portion which class you are studying.

[00:46:45]So theory of equations max.

[00:47:09]Okay.

[00:47:13]Okay.

[00:47:16]Then why did you ask about nth standard?

[00:47:18]Oh, okay.

[00:47:22]Okay. So, the next one. So, solve the equation.

[00:47:31]So, progression similar value.

[00:47:42]So s of the roots of the roots.

[00:47:46][gasps] Okay. Yes. So finally you have to find out the value of x.

[00:47:52]So easy.

[00:48:04]Basic questions.

[00:48:10]Fourth chapter.

[00:48:26]Okay. Yeah, we'll solve. No issues.

[00:48:29]Okay. So, first 3x - 26.

[00:48:41]Okay. So, + 52x - 24 = 0.

[00:48:48]So let roots be a by r a and a r. So sum of the okay sum of the roots. So sum of the roots right a [snorts] r a= a value. So what is B value and A value here? So a value, b value and c value a by r a + a r are equal to so a 3 b - 26 and next c 52 okay a r + a + a r = 26.

[00:49:55]Okay. Okay. Good.

[00:50:00]Right. Okay.

[00:50:03]So, = 26.

[00:50:16]So product of the sum of the product of the two terms taken at a time.

[00:50:32]So sum of the product of the two terms at a time. Okay. So which is a by r into a and a into a r then a r into a by r.

[00:50:45]Okay. So which is equal to next C by A C by A. Okay. So C by 52. So 52 product of the roots.

[00:51:08]Okay.

[00:51:14]Sum of the product of the two terms at a time. Two roots at a time by cus dummation alpha summation alpha beta summation alpha beta gamma for summation terms.

[00:52:04]Okay.

[00:52:09]A² by R. Okay. A square A square.

[00:52:25]So 1 R + R + 1 which is 52 by 3.

[00:52:34]product of the roots.

[00:52:36]Okay.

[00:52:38]So, product of the roots sum of product of two roots at a time.

[00:52:53]So, statement of the roots which is a r into a into a r okay so which is equal to -4 okay r get cancel so the remaining max so a = 24 / you get what is a value.

[00:53:46]Okay.

[00:53:53]Biology class into 1 + 1. So ral which is equal to 26 value.

[00:54:28]Yes. 26.

[00:54:33]So 1 + r². So r² + r / r = 26 3 into 1 by 2.

[00:55:02]Okay. Okay. Okay.

[00:55:05]Yes.

[00:55:08]So 1 + r² + r= 13 miplication.

[00:55:18]So cross you get this value 3 + 3 r² + 3 r = 13 rar.

[00:55:33]So rearrange you get this value. Okay. So again 3 r² so - 10 r + 3. So, so same question chance. Okay.

[00:56:20]So it becomes 3 andus.

[00:56:24]Okay.

[00:56:26]So the value becomes so r = 1 by 3 or 3.

[00:56:35]Okay.

[00:56:38]When r = 1 by 3 and a = 2 root when r= set of roots final So within 2 minutes max= to this value. Okay. So substitute so you get three values.

[00:57:25]Okay. So see you next class. Okay. Bye.

[00:57:28]Take care. Good night.

[00:57:32]3.3.

[00:57:39]Okay. Thank you. Thank you so much.

[00:57:46]Thank you. Thank you so much.