let us assume that let us assume that assume that 6 + < tk2 is rational 6 plus2 r not equal to Z okay let us assume that 6 + < tk2 is IR that cannot be okay R if it is 6 plus2 is rational we can express 6 Plus 6+ < tk2 in the form of a by B and then A and B are A and B are okay A and B that is < tk2 equal to which implies < tk2 equal to a minus 6 B [Music] by r rational which implies that since a comma B comma 6 are integers the whole term that is Aus 6B by B is also rational clear which implies that root2 is rational root2 is a 6us Aus 6 B by B is RCT given statement root2 is irrational this contradict this contradict this contradict the given statement that this contradict the given statement that statement statement that root2 is root2 is irrational root2 this contradiction this contradiction has arisen has arisen due to our due to our wrong assumption that wrong wrong assumption that wrong assumption 6 + < tk2 is assumtion can conclude 6 + < tk2 is irrational hence the proof [Music] clear okay okay2 us Aus 6 by R number 6 okay sorry a equal to 1us H 6 into B value 2 divided by 2 so 1 minus 12 by 2 - 11 by 2- 11 by notal equal so also Ral this contradict the given statement that root2 is irrational therefore this contion has arisen due to our wrong assumption that 6 Plus root2 is rational therefore we can conclude that 6 + < tk2 is irrational clear C good evening and the simplification okay Arjun hello hello sir good evening yes good evening sir conclusion this contradict root is rational given statement this contion has arisen because of a wrong assumption therefore it is irrational okay sent you are 10th standard cbsc students no yes yes sure yeah this is not much lengthy okay the next one [Music] okay root it is irrational okay okay yes next one Shani [Music] Shivani contct the given statement given statement this contradict the fact that root2 is irrational understood [Music] understood [Music] okay if it is not given in the statement as a statement so we can conclude that it is a fact okay clear okay next one Rohan LS is not equal toal therefore contradiction has AR okay is also rational so L is not rational because of the fact that okay yes fine next turn Okay j j j j Jo okay good evening [Music] St given statement that root2 is irrational because of our wrong assumption okay the next one okay next one Teresa yes good evening number like that and they don't give that iral and it's for that is no no Need No need no need this must is enough okay okay next one B Bia B yes Bia [Music] fast okay okay next one a aul aul or aul a can you speak it louder B canc how many times b in a tell me if you canceling b b b and you will also get this one a minus B by B A minus B by 6 okay we expanded this one once we got the numerical value then only we can cancel it otherwise the common denominator we can write the equation as a by B minus enough okay okay okay next one okay next Dia yes good [Music] evening okay okay next one Shar okay next one ah this is this is this Contra contradict [Music] contradict okay right next one yes good [Music] evening it is contradict the fact that okay we know that it is irrational okay okay the next one [Music] Richard yes that's why I calling you that's why we call you tell [Music] me which one [Music] this method is not enough in in in the expression of root a by B simply it is Define that it is a by B and we are Co Prime okay not mention that it is integers okay with uh by the method of we are going in that kind of problem we are going to contradict the statement that the copre numbers okay that's why we are uh um following that that manner okay next one in the question if it is root2 is IR rational statement is given we should need to prove that many times I answer to the question uh this is the method we are going to adopt okay no need to metion no need to prove that root2 is irrational Jo is it clear the next one gaana gaana yes sir tell me [Music] Ral already iral 6 Plus root2 is even okay okay thenn next question sirn rational number definition B because rational number in the form of a by B where A and B are integers B not equal to zero clear okay the next one prove that 5us root3 is irrational given that root3 is irrational 53 is irrational M1 5us root3 is IR rational same way let us assume that let us assume that let us assume that 5 - < tk3 is rational okay R if it is rational 5us root3 equal to a by B where A and B are integers and and B are there is no common factor standard okay okay 5 - A by B = > okay 53 = a by B then minus < tk3 = a by B 5 Aus 5 byus always express the denominator is 5 B minus Aus ausus okay so 5 that a by 5us a / B since a comma B comma 5 are integ B is Ral okay therefore which implies that root3 is Ral okay statement given stat root contradict this contradict this contradict the fact that the fact that or given statement that and fact that root3 is irrational root3 irrational statement this contion has arisen this is due to this is this contradiction is Horizon due to wrong assumption due to wrong assumption that wrong assumtion that 53 is rational therefore we can conclude that 5 - < tk3 is irrational okay 5us root3 and irrational [Music] yes [Music] okay okay next one next one hello yes [Music] [Music] okay okay okay okay next one same format next one [Music] way rational number definition a ah suppose bid by five 4 by yes okay that is 3 + 2un 7 is irrational given that root 7 is irrational okay let us assume that let us assume that let us assume that 3 + 2un is rational R 3 + 2un 7 Ral if it is rational three + 2un 7 a by B format where A and B are integers and coprime numbers okay no common factor yes okay then 3 + 2un 7 equal to a by B listen 2un that is equal to a by B which is equal that is Aus 3 cross multiplication by < TK 7 = Aus 3 B / denomination goes to divis okay a comma B comma 3 comma 2 are integers Aus 3B / 2B Val therefore root7 is rational this contradict this contradict contradict the given statement stct the fact the fact that this contradict the fact that root2 is sorry root 7 is this contradiction has arisen this contradiction contradiction has arisen Arison due to our wrong assumption wrong assumption that wrong assumption that root 7 sorry wrong assumption that 3 + 2 root7 is rational Ral therefore we can conclude that 3 + root 7 2un 7 is irrational hence the proof clear then next question okay next one okay ason ason ason ason ason or ason ason ason given [Music] statement yes sir [Music] [Music] okay [Music] C okay 5- 7 < tk2 is irrational let us assume that let us assume problem okay okay assume that 5us 2un 7 root2 okay 5 minus 7 < tk2 is R 5- 7 root2 can be written as in the form of a by B where A and B are integers A and B are at the same time it is standard form it is okay thenot 7un equal a by B A by B that is equal a - 5 b / by B A minus 5 b / byun2 - into < tk22 Aus 5 5B minus a by 7 B for always expressed as the positive okay which is equal to since since a comma B comma 5 comma 7 is [Applause] in 5 B minus a / by 7 B is and run2 is Ral this contradict this contradict the fact or given statement that fact that given statement the fact fact that root2 is this condtion has arise this contradiction this contradiction has Arison has arisen due to our wrong assumption due to our wrong assumption that due to our wrong assumption that R to our wrong assumption that 5 - 7 < tk2 is rational therefore 5- 72 is 7 < tk2 is irrational we can conclude it like this okay hence the proof okay next one [Music] multiplication posi positive denominator always expressed as the [Music] positive 8 by always express the numat denominator as posi okay next one [Music] ad final answer okay yes next one nand Nanda yeah of course no problem you can write it okay after right after from this step you can going to skip to this one okay yes okay no problem yes next one [Music] always express the denominator as positive then next one next [Music] one for ah addition subtraction subtraction subtra add multiplication into okay add and subtraction next [Music] one x equal to 2 4 by two which is equal to [Music] 2 Richard so uh do you have any doubt uh based on this question no sir no I will explain to at the end of the session okay fine next one an an good evening sir yes good evening a a minus a minus okay minutes 10 minutes break okay after that we can continue okay 7un five is irrational given that root five is rational speed up speed up time over okay asse that let us assume that let us assume that 7 < tk5 is 7un 5 is rational okay 7un 5 rational Ral if it is rational 7 root5 can be written as a by B where A and B are and integers and they are coprime numbers also okay Co primes okay then are integers problem integers a by 7 B is rational a by 7 B is rational root 5 equal to okay root 5 equal to therefore root five is root five is RCT this contradict the fact that or given statement that fact that root five is root five is irrational irrational this has arisen due to our wrong assumption that 7 root five is rational therefore we can conclude that 7un 5 is irrational understood jug jug nice name jug jug very good then we are moving to the next question next of question be very careful okay yes [Music] uhad has AR due to our [Music] that H therefore we can conclude that irrational okay we are moving to the next question moving to the next question > 7+ root2 is irrational otk 7 + < tk2 is irrational otk 7 + < tk2 is irrational okay us assume that +k2 isun 7 + < tk2 equal to a forun 7 equal a by B minus < tk2 okay otk 7 = to a by B minus < tk2 and then squaring on both side [Music] okaying on both s side squaring on both side squaring on both side okay please put your pens down all of them and listen in the class all the students please put your pens down and listen in the class listen in the class confus a a s 2 into a a by B into bk22 plus a squ A squ plus - 2 a into b + B squ < TK to the whole square root to the whole Square which is equal to a² by b² minus 2 a by 2 aun2 equal which implies that Aus b square a s- 2 a + [Music] b² okay then a square by b square - 2 a - 2 a by into so 7us 2 = 5 5al a² by b² - 2 a byun2 5us a² b² - 2 a by B into < tk2 and St okay 7 = a² - 2 a b + b² 7 - 2 = 5 so a square by b square - 2 a by aun2 okay 5us a s by b square s Square Aus Square expansion ausans [Music] 2 into a into b l minus 2 a l Aus 2 into a byun2 plus root2 Square b square plus b [Music] [Music] square n n Nan good evening yes good evening fine fine G thank you and what about you that's why I told you in the beginning you can square on this side I doing like this okay that method we can discuss this later otherwise it is totally confused okay that's fine next [Music] oneya next one ah okay and we are and we are copright thank you b² minus a² ided b² equal to equal to - 2 a by B into < tk22 Ro 5 b² minus a yes fracp with the that is B by b byus 2 a b byus 2 a with the reciprocal reciprocal multip in order to divide by the fraction equal to [Music] s okay yes 5 b - a² okay divided by - 2 Aus 2 a okay equal tok2 yes since a b 2 comma 5 are integers because in rational number definition it is integers are speed up speed up [Music] [Music] okay multiplication [Music] okay then Geral Gerald [Music] Geral always stand okay okay and then [Music] denominator minus 2 a into B minus 2 a okay that is which implies 5 b² - a² / - 2 is which implies that < tk2 is rational okay root2 is rational root2 [Music] RCT the fact that do contradict the fact that root2 is irrational it is irrational stct this contradiction has Aris this contradiction this contradiction has arisen has Arison due to our wrong assumption that due to our wrong assumption due to our wrong assumption that wrong assumption that < tk7 + < tk2 is R we can conclude that < tk7 + < tk2 is irrational is irrational root2 root 7 + root2 is irrational is irrational Okay g or Giver Giver Giver is it clear ah yes what about the first one Anna [Music] question is prove that < tk2 plus root2 and prove that < TK 5 minus < tk2 is irrational hunk 5 + < tk2 is irrational root 5 + < tk2 is irrational uh root2 minus root 5 so otk 5 minus root2 otk 5 - < tk2 is irrational otk 5- root two is irrational okay different problem exam exam