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

[00:01:02]Hi students. Good evening. Good evening all. Hello. Hi.

[00:01:07]Hello Alpa.

[00:01:16]Yeah. Happy evening Nish. Hi Abby. Yes.

[00:01:19]Good evening also. Good evening.

[00:01:21]So yeah. Yes. Good evening. Good evening.

[00:01:26]Good evening. So chapter yes. Good evening. Yes.

[00:01:42]Yes.

[00:01:45]Yes. Good evening. Good evening. Yes.

[00:01:52]So chapter of equations and chapter yes.

[00:02:20]So just Okay. Hi. Okay.

[00:02:36]Time.

[00:02:41]Hi. Hello. Good evening. Okay.

[00:02:47]Easy compulsory questions. Compulsory questions. Okay.

[00:03:06]equation. So polinomial equation different types of polinomial equation.

[00:03:24]So different types of polinomial equation equations.

[00:03:30]P= So P of X P of X is equal to A X^ N and AUS X^ NUS so like that it goes on right so A1 X + A finally you get the constant value equation Happy evening. Yes. Hi. Okay. Yes. Clear function.

[00:04:11]So function.

[00:04:14]Yes. So this one is a polinomial function x^us 1. So, so a + function equation.

[00:04:38]Okay.

[00:04:42]Okay. Hi. Good evening. Okay. Clear.

[00:04:46]Yes.

[00:05:00]a x² + b x + c = 0. So what is a quadratic formula?

[00:05:06]Yes. Good evening. So quadratic formula.

[00:05:16]So the very first okay yes okay equation yes okay so - b plus or minus roo<unk> b² - 4 a c 2 a quadratic formula.

[00:05:54]So square root of value.

[00:06:12]So b²us 4 a discriminant value.

[00:06:19]Yes.

[00:06:25]Okay.

[00:06:41]So value equal to root and greater than less than.

[00:07:01]Okay.

[00:07:03]H.

[00:07:06]So equal to roots roots are equal and distinct and distinct.

[00:07:22]No real roots.

[00:07:25]Okay. No real roots.

[00:07:33]3.3.

[00:07:40]Okay. Good.

[00:07:42]11th 12th class 12th. Okay. So, next formula and formula for quad equation sum of the roots and product of the roots.

[00:08:02]x²us of sum of the roots product of the roots.

[00:08:06]Sum of the roots or product of the roots.

[00:08:20]Okay.

[00:08:24]Sum of the roots or product of the roots or coefficient sum of the roots. So x²us sum of the roots sum of the roots into x. Okay. So plus product of the roots product of the roots which is equal to zero. So square - alpha + beta into x plus alpha beta = 0.

[00:09:11]Sum of the roots.

[00:09:19]Sum of the roots.

[00:09:32]Good evening.

[00:09:50]minus by a hit like button. Okay. Yes.

[00:10:07]So product of the roots it is constant coefficient of x². Okay.

[00:10:16]by first example 3.1.

[00:10:27]If alpha and beta are the roots of the quadratic equation roots and beta.

[00:10:37]Okay.

[00:10:39]Construation whose roots are alpha + 2 beta equation.

[00:10:51]Okay. So given equation 17 x² + 43x - 73 = Alsign.

[00:11:42]roots of the roots.

[00:11:49]Yeah. Yes.

[00:11:52]Okay.

[00:12:04]A sum of the roots.

[00:12:08]What is the sum of the roots?

[00:12:15]Sum of the roots.

[00:12:27]So okay. So - 43ID 17 product of the roots C by A - 73 / 17 roots new roots.

[00:12:47]So new and beta sum of the roots everyone. Good. Good.

[00:12:56]Good.

[00:12:58]Sum of the roots.

[00:13:13]Alpha plus beta already we find the value right. Okay.

[00:13:30]Okay. So 43 + 4. So now take LCM - 43 + 17 into 4 LCM I think 68.

[00:13:56]Okay. Yes. So 68.

[00:14:09]Yes.

[00:14:11]Okay. Good. 68.

[00:14:16]It is 25. So 25.

[00:14:19]Sorry. 17.

[00:14:28]product of the roots.

[00:14:31]Product of the roots alpha + 2 into beta + 2.

[00:14:40]So into 2 into 2 alpha beta 2 alpha + 2 beta plus 2 sorry four okay plus beta 2.

[00:15:07]So alpha + beta + 4.

[00:15:11]Okay. Yes. So already it is - 73.

[00:15:18]Correct. 73.

[00:15:22]Okay. So 43 / 17. So plus 4. So yes.

[00:15:41]So -73 - 86. So -7.

[00:15:51]So plus okay sorry - 156 - 156 Okay sorry 156 159 is where sign changes 43.

[00:16:31]Yes. Clear. Okay. So 17. So next 159 + 68 8 - 91 - 91 product of the roots x²us of sum of the root sum of the roots 25 17 x - 91 = 0.

[00:17:39]So 17 x² - 25x so - 91 = equation equation. Did you understand?

[00:18:03]Okay. So the next question of the roots and product of the roots.

[00:18:30]Yes.

[00:18:40]Yes.

[00:18:42]Next question.

[00:18:56]So you try it. Okay. Yeah. Yes. So the next equation sorry a square b sorry a x cq bx² + c x + d = 0ic square= minus of minus value alpha plus beta plus gamma. So coefficient of alpha beta and beta. So gamma minus alpha beta and okay yes sorry this one is for yeah one second formula for polinomial equation of degree Okay.

[00:20:19]Yes.

[00:20:27]Yes. So these are the formula. Okay.

[00:20:29]Coefficient of x^ 2 3 and nus one upreater.

[00:20:45]Okay. Already yes. Okay. So next summation of alpha 1 alpha plus beta plus gamma delta single single alpha beta summation alpha beta alpha dela beta dela terms.

[00:21:38]Okay. Yes. So the next question example 3.5 find the condition that the roots of the cubic equation equation ratio condition.

[00:22:04]Okay.

[00:22:08]Okay. Yes. Okay.

[00:22:13]Seventh standard 12th max.

[00:22:19]Okay.

[00:22:31]P lambda P lambda lambda. P lambda and Q lambda and R lambda roots.

[00:22:44]Okay.

[00:22:50]Okay. So, P lambda, Q lambda and R lambda which is First negative summation.

[00:23:26]Okay.

[00:23:33]So which is equal to plusal plus okay summation p lambda q lambda and r lambda= okayh Okay.

[00:24:15]So first lambda Q lambda r lambda= minus a lambda.

[00:24:27]So lambdalus which is equal to minus lambda = a / p + r.

[00:24:44]Okay.

[00:24:49]Yeah. Okay. Bye. Take care. Okay. So next equation. Sorry. Third equation.

[00:24:56]Second third equation. So third equation lambda Q lambda lambda. So which is equal to lambda. Yes.

[00:25:24]One second.

[00:26:11]Okay.

[00:26:20]So lambda= lambda is equal to lambda = minus c PQ.

[00:26:36]Yes. So the equation number four.

[00:26:59]fourth equation.

[00:27:03]So substitute four and five.

[00:27:15]Okay.

[00:27:18]value. So minus a p + q + r the whole cube which is equal to minus pqr multiplication.

[00:27:42]Okay.

[00:27:43]- get cancel. So A / P + Q + R the whole cube. So C by PQ. So now cross multiply.

[00:27:55]So A= C into P + Q + R the whole. This is your condition.

[00:28:03]Okay.

[00:28:07]Okay.

[00:28:10]Okay.

[00:28:34]Okay. Yes.

[00:28:36]Are you clear? So next question pama next to papa.

[00:28:48]Okay ma.

[00:28:54]Yeah. Okay. Done. So the next question.

[00:28:59]So the example 3.7. If P is real, discuss the nature of the roots of the equation in terms of P. Simply you have to find out the roots value if P is real in terms of P. So 4 X² 4 P X + P 2. Okay.

[00:29:31]A c. So what is a value here? A ce of the root discriminant= square.

[00:29:55]Yes.

[00:29:59]A value and c.

[00:30:07]Yes. Danish and gautami.

[00:30:12]Okay. Good. So, a = 4 and b = 4 p and c= p + 2.

[00:30:20]4 p² - 4 into 4 into c value + 2. Okay.

[00:30:26]16 p². Yes. 16 p². Next - 16 into p + 2.

[00:30:34]Correct. 16 p² - 16.

[00:30:42]Yes.

[00:30:51]16 p² - 2 16.

[00:30:59]So 16 into p²us pus.

[00:31:22]So multiplyus add 1 + so the delta is less than z condition. So -1 less than p less than 2 p= condition.

[00:32:07]So minus infinity less than p less than minus one or it is 2 greater than p greater than infinity. Sorry less than less than p less than infinity.

[00:32:19]Okay.

[00:32:24]Okay.

[00:32:26]Yes.

[00:32:29]So the discriminant value if =1 or p = 2al + Okay.

[00:33:03]Infinity.

[00:33:08]Okay.

[00:33:09]Yes. Okay.

[00:33:12]So next delta discriminant value equal roots and distinus less than less and equal roots equal roots if p= minus1 or p= okay distinct real roots so distinct real roots if minus infinity less than p less than minus1 or 2 less than p less than infinity in the in the equation of the nature of the roots.

[00:34:19]Yes. Okay.

[00:34:23]So the next question so 3.1 fourth question solve the equation if the product of the Okay. Yes. So solve the equation 3x 3x - 16 + 23x - Okay. So alpha beta root for 3x - 16 / 3 into x². So 23 into x - 6x 3 = 0 into x² 3 into x and - 2 = 0 First sum of the products.

[00:36:14]Okay, we'll do. Okay. Yes.

[00:36:41]Okay. So alpha plus beta plus gamma equal to yes 16 by 3.

[00:37:04]Okay.

[00:37:08]So okay 16 + 16 equation number one product of the roots alpha beta beta and gamma which is equal to the next 23.

[00:37:50]Yes. Alpha beta gamma= number two terms product of two roots is one beta Sorry.

[00:38:32]So the gamma= 2.

[00:38:36]Yes. Gamma= 2. Okay.

[00:38:46]Okay.

[00:38:57]First summation alpha + 1 + 1.

[00:39:23]Okay. Max.

[00:39:28]Yes. Okay.

[00:39:47]So ala square + 1 / so 16x 3 - 2 so - 15 + 1 by 3 sorry 10 by 3. So sorry it is 16 no sorry 6 10 by 3 sorry okay so 3 2 16 sorry 6 so 16 - 6 so it is 10 okay so ala square + 1 ala= 10 now take multiplication 3 into ala square + 1 = 10 alpha square + 3 which is equal to 10.

[00:40:37]So - 10 + 3 = 0.

[00:40:53]Okay.

[00:41:03]If suppose okay so don't worry about that. Okay. So answer will be so minus - so - 3 - 1 by 3 okay = 1 by 3 and three.

[00:41:52]So therefore the roots are Sleeping mode 3 1 by 3 2.1 fth question find the sum of squares of roots of the equation. Okay. Sum of squares of the roots of the equation.

[00:42:41]Yes.

[00:42:51]Okay.

[00:42:55]Four roots beta gamma delta four roots alpha summation alpha beta summation okay so alpha plus beta gamma plus delta which is equal to - 8 by 2 sorry minus of - 8x2 2 minus of - 8 by 2 correct. So which is equal to 4.

[00:43:32]Okay. So alpha plus beta plus gamma plus delta= 4 equation number first equation number.

[00:43:50]Okay.

[00:43:52]Question number two multiplication. So alpha beta beta gamma gamma delta and delta. Okay which is equal beta alpha beta alpha gamma and alpha delta.

[00:44:20]So beta gamma beta gamma and beta delta. So beta sorry.

[00:44:30]Okay done.

[00:44:44]Okay. So which is equal to 6 by3 sorry 6x2 which is equal to 3 equation number.

[00:44:58]So alpha plus delta square alpha square + beta square gamma square and delta square + 2 into yes. So sum of squares of the roots. Sum of squares of the roots.

[00:45:47]Sum of the squares of the roots.

[00:45:52]So alpha square beta square gamma square and delta square is equal to alpha + beta + gamma plus delta the whole square and okay yes clear value.

[00:46:24]Okay. So 4² + 2 into 3 + 16. So the sorry - okay so which is equal to 10us.

[00:46:57]Yes sir.

[00:47:02]in a dou.

[00:47:09]Okay.

[00:47:24]Okay.

[00:47:26]Okay.

[00:47:30]question.

[00:47:36]Okay. So if it is given that two of its roots are in the ratio.

[00:47:49]Okay.

[00:47:53]roots.

[00:47:56]If two of its roots are in the ratio 3 is total 3 toal 3 lambda and when beta= = 2 lambda beta sum of the roots.

[00:48:36]So minus by a minus 9. So alpha + beta= 9. So which is equation number equation number two.

[00:48:58]Okay.

[00:49:05]So roots alpha and beta sorry 3 sorry alpha equal to 3 lambda and bal scalar it is scalar. Sorry it is.

[00:49:23]Okay. So it is a scal it can be any value.

[00:49:35]Okay. So which is lambda 3 lambda 2 lambda plus 9.

[00:49:55]So 5 lambda plus sorry 5 lambda + gamma= 9. So therefore gamma= 9 - 5 lambda. So okay so product of the terms yes so that is alpha beta beta gamma and gamma which is equal to c by 14 okay that is alpha beta gamma = minus 24 constant value which is equation number four= 3 lambda and beta = 2 lambda and gamma= sorry gamma = 9us Okay.

[00:51:08]3 lambda into 2 lambda. And next one 2 lambda into what is gamma value? 9 - 5 lambda and next 9 - 5 lambda into 3 lambda.

[00:51:25]Which is equal to which is equal to account important questions already.

[00:51:41]So 6 lambda square and 18 lambda and 10 lambda square lambda - 15 lambda square which is equal to 14 lambda square - 15 lambda square lambda square + 18 + 27 into lambda. Okay. Which is equal to 14.

[00:52:23]Okay.

[00:52:28]Okay.

[00:52:31]So - 25.

[00:52:33]So - 19 lambda square - 19 lambda square plus 45 lambda + 45 lambda 14.

[00:52:48]So 14= multipus 19 lambda square - 45 lambda + 14 = 0.

[00:53:05]Okay.

[00:53:14]Plus roo<unk> of b² - 4 a c / 2 aus.

[00:53:27]So a value. So a 19 bus and ces plus or minus square roo<unk> of - 45 the whole square and - 4 into 19 into 14 divided by 2 into 19.

[00:53:55]Okay.

[00:54:00]test future learning.

[00:54:09]Okay.

[00:54:18]Other than that you have to study in the CA. Okay. Yes. So next one second.

[00:54:29]Okay.

[00:54:33]So 45 square value 45 square value 2025. So 2025 - 4 into 19 into 14 correct. Okay.

[00:54:59]Yes.

[00:55:08]19 into 14 into 4.

[00:55:17]So take your time. No issues.

[00:55:20]1064 1064ID 38 again.

[00:55:27]So plus 2025,64 Yes. 961 961 381 which is sorry 31. So 31ID 38 31ID 38.

[00:55:56]Okay.

[00:56:02]So 45 + 31id 38. So 45 - 31id 38 76id 38 148 14 you get 17 by 19. Okay.

[00:56:42]Yes.

[00:56:48]So root.

[00:56:53]So the real root fraction for when lambda equal 2 lambda correct and next= 9 lambda. So 9 - 2 h. So 9 - 10 = -1.

[00:57:58]Okay. Lambda = 6, beta = 4 and gamma = -1 value when lambda = 7 by 19= 3 into 17 by 19. Sorry 7 by 19 21 into 7 by 19. So 14 gamma. So 9 - 5 lambda 9 - 5 into 7 by 90. So which is equal to 9 - 35.

[00:58:43]Correct. Okay. So simplify. So 9 into 19. What is the value?

[00:58:51]the 12th standard 19 into 9.

[00:58:58]So value 19 into 91 179. So ends with 91.

[00:59:19]So 171us 35 / 19= 136 196 19. So solution 21 14id 19 and 136 by 19 or 6 4 - 1. Okay. the final answer.

[00:59:55]So see you next class. Okay. Thank you.

[00:59:58]Thank you all. Good night. Take care.

[01:00:00]Okay.