[0:00]IIT JEE mains aur JEE Advanced mathematics ki hum tayyari kar rahe hain. Hum 100% free of cost quality online education dete hain, Kinder Garden se 12th standard tak. Aap hamare notes, PDFs aur ebooks ke liye team se contact kar sakte hain. Iske liye aapko isi video ke description mein diye gaye link get notes par click karna hoga. Hum yeh tayyari jo kar rahe hain, ise majorly paanch chunks mein humne divide kiya hai. Humne pehle coordinate geometry se shuruat ki. Saare chapters kiye ab hum calculus par hain, phir hum vectors in 3D, phir algebra, trigonometry, is tarike se pura syllabus, saare chapters, har ek concept ko karenge. Unhe example se consolidate karenge. Practice questions karenge taaki pura chapter revise ho jaye aur phir JEE Advanced ke saare previous years questions us chapter ke karenge taaki aapko yakin dila sake ki cheeze aasaan hain. Matlab ek achi, majboot aur behtareen tayyari bilkul free of cost. Hum aaj baat karenge apne coordinate geometry khatam karne ke baad calculus ke first chapter Relations and Functions ke topic Modulus ki. Modulus mein aaj hum kya specifically baat karenge ki kisi bhi number ka modulus. Ab waise toh yeh bahut hi simple si baat hai ki sir mod toh hume pata hai. Par itni aasaan baat toh main karunga nahi toh kuch toh bahut saari baatein hongi. Toh bahut saari IIT JEE Mains aur JEE Advanced ke level ki hum modulus se related ek achha discussion continue karenge. Par pehle modulus ko lekar yeh naya ek perspective jo main aapko dena cha raha hu woh kya hai. Waise abhi tak hum kya samajhte hain ki sir -5 ka mod le lo 5, 3 ka mod le lo 3. Zero ka mod le lo zero. Par uska exact meaning kya hota hai? Use mathematically kaise define kiya jaata hai? Use aur alag-alag definition se kaise samjha jaata hai? Zara usko lekar thodi clarity lete hain. Gaur se suniyega. Pehli baat. We represent real numbers on real number line. Toh yeh aapka number scale hota hai. Yeh aapka real number line hoti hai, ispar hum numbers ko represent karte hain, hai na. Ek interesting feature hai real number line ki ki hum distances ko measure karne ke liye ek function ko use karte hain, ek ek ek functionality ko use karte hain. Woh kya hota hai? Gaur se sunna. Main ek choti si baat aapse kehna cha raha hu jaise ki dekho. Dhyan se sunna bahut kaam ki baat hai. Jaise yeh aapka raha number scale hai na. Agar yeh number scale hai aur ispar yahan kahin hai zero, yahan kahin hai one, phir two, phir three, aise chala jaa raha hai, hai na. Yeh chalte chale jayenge. Aise hi yahan par aapke -1, -2, -3 yeh chalte chale jayenge. Ab agar main aapse poochhu ki batao zara. Yeh jo -3 hai, yeh zero se kitna unit distance par hai? Toh aap kya kahoge sir dekho, zero se agar unit distance measure karo toh 1 unit, 2 unit aur 3 units. Toh -3 distance wise kitna distance par hai? -3 origin se 3 unit ke distance par hai. Similarly agar main +3 ko lekar aapse apna opinion maangu toh +3 kitni unit distance par hai? Isse dekho ek block, do block aur teesra block. Toh yeh bhi 3 unit distance par hai. Matlab signature ko bhi ek taraf jaane dein, direction ko ek taraf jaane dein ki left hand side jaa rahe hain, right hand side abhi yeh measure nahi kar rahe hain. Abhi bas main distance ke terms mein baat karu. Distance ke terms mein baat karne ka matlab hai ki woh origin se kitna door hai, kisi ek point ko reference maan liya jaise abhi filhaal origin aur usse woh point kitna door hai, humne ispar baat ki. Toh maine kaha ab chaahe origin se 3 tak jaayen, ya origin se -3 tak jaayen, distance wise aap same distance travel karte hain. Is baat ka ab significance samajh paa rahe ho kya? Main bas aapse yeh kehna cha raha hu ki agar aap chaahen toh -3 ki distance dekhen ya 3 ki, distance wise woh distance 3 blocks ya 3 units hi hai. Pehle toh aapko yeh baat samajh aa rahi hai kya? Main aapko ek ek nayi functionality, ek nayi mazedaar si baat sikhane ki koshish kar raha hu ki chaahe woh +4 ho ya -4, chaahe woh +5 ho ya -5. Distance wise agar origin se baat karein toh woh distance same hai. Aur us distance ko hi measure karne ke liye humne ek function use karna shuru kiya hai jise hum kehte hain Modulus Function ya phir Absolute Value Function. Absolute Value Function ko denote karne ka tarika hota hai mod x. Iske andar aap woh value likhte ho jiska aap absolute value ya mod nikalna chahte ho. Aur is mod x ka hamari bhasha mein abhi filhaal, main functions ki definition pe nahi ja raha hu, abhi filhaal toh main bas aapko abhi ek ek naye perspective ko dene ki koshish kar raha hu.
[4:27]Kya dene ki koshish ki ja rahi hai? Main bas yeh kehna cha raha hu ki filhaal distance ko lekar humara perspective hamesha se clear raha hai ki distance aapki hamesha positive ya at least non-negative hoti hai, zero bhi ho sakti hai kabhi. Toh main aapse yeh kahunga ki mod x hamesha aapka greater than equal to zero hoga aur sabse zaroori, sabse khaas baat ki mod x ka matlab abhi filhaal humare liye kya hoga? Distance. Kis ki? X ki. Kahan se? Origin, yaani zero se. Kya thodi bahut baatein samajh aa rahi hain? Aur clearly aayengi, bas sunte chale jaiye, samajhte chale jaiye, bahut acche se sunna. Pehli kuch teen-chaar zaroori baatein jo aapko yaad rahe toh bahut khushi hogi mujhe. Jaise ki kya? Ki kabhi bhi hum x square dekhenge ya mod x ka square dekhenge, humein jaisi zaroorat lagegi, hum interchangeably isko use karenge. Jaise maan lo kisi question mein mujhe kahin pe aisa dikh raha hoga, let's say ki x square + 3 times mod x, hai na, - 2. Agar mujhe aisa kuch dikh raha hoga koi expression aur mujhe lag raha hoga ki nahi sir, aap ek kaam karo, aap iska bhi kuch kar lo. Toh main kahunga ki dekho sahuliyat ke liye main is x square ko likh lunga mod x ka square + 3 times mod x - 2. Usse shayad meri problem shaayad asaan ho jaye ya mujhe kuch zaroorat lag rahi ho. Woh zaroorat kya hogi hum baat karenge. Par abhi main aapse bas yeh kehna cha raha hu, please dhyan se suniyega. Ki hum x square ko mod x ke square se ya mod x ke square ko x square se replace kar sakte hain, koi pareshani nahi hai. Kya aap meri baat samajh rahe hain? Main yeh kehna cha raha hu ki aap kisi number ka square lein, woh positive value aayegi, ya phir us number ke mod ka square lein. Arre woh number positive ho ya negative, uska square positive hi hoga na? Us number ko mod laga kar positive bana de aur phir uska square lein toh woh bhi woh positive hoga. Toh aap meri baat samajh paa rahe ho kya? Chaahe x square ho ya phir mod x ka square ho, yeh hamesha positive hoga. Achha, is baat ko lekar toh pareshani nahi hai na ki sir 3 ki origin se distance nikalo ya -3 ki, -3 ki, you know, zero se distance nikalo, woh dono hi kitni aati hain 3. Toh sabse pehli baat. Chaahe x ho ya -x ho, dono ka hi mod aapka ek same value hoti hai. Koi pareshani? Aur aage sunte jaiyega. Ek bahut important baat hai jise please yaad rakhna. Samjho baat ko. Main isko is tarike se samjhane ki koshish karta hu ki do values ka. Do values ka product ka mod unke individual mods ke product ke equal hota hai. Is baat ka justification bada aasaan sa hoga. Jaise aapko thik lage, sochiye. Ek baat batao, soch ke dekhna. Main magnitude wise baat kar raha hu, signature ki baat nahi kar raha hu kyunki mod lagte hi signature consideration se bahar ho jaata hai, hai na? Dhyan se samajhne ki koshish karna. Do numbers ka product liya. Ab chaahe woh dono positive hon, dono negative hon, ek negative ho, ek positive ho ya pehla positive ho, doosra negative ho, jo bhi combination ban raha ho. Magnitude wise toh number same banega na? Jaise main ek example leta hu 3 aur 5 aap le lein ya phir aap -3 aur -5 le lein ya phir aap 3 aur -5 le lein ya phir aap -3 aur +5 le lein. Kya aap meri baat samajh paa rahe hain? Magnitude wise yeh sab kitne banenge? 15, 15, 15, 15. Although numerically yeh 15, 15, -15, -15 banega. Par magnitude wise, magnitude wise ka matlab hai ki agar main aapse poochhu chaahe aap mujhe iski origin se distance bataayen, chaahe aap mujhe iski zero se distance bataayen, chaahe aap mujhe iski zero se distance bataayen, chaahe aap mujhe iski zero se distance bataayen. Zero se 15 utna hi door hai jitna zero se 15 hai, jitna zero se -15 hai, jitna zero se -15 hai. Matlab aap meri baat samajh paa rahe ho kya? Matlab in saari baaton ki ek baat ki inko aap likh sakte the kya? Inko aap likh sakte the aisa. Ya phir inko aap kaisa likh sakte the? Inko aap likh sakte the aisa.
[8:08]Ya phir inko aap chaahe toh inko aap likh sakte the aisa. Main aapse yeh kehna cha raha hu ki jaise hi mod picture mein aa jaa raha hai, humare liye direction ya phir filhaal abhi main kahu signature matter nahi karta hai. Yaani sirf modulus, yaani uski absolute value matter karti hai. Toh ab aap unko multiply karke unka mod lein ya phir mod lekar multiply karein, koi farak nahi padta hai. Kya yeh baat aap samajh paa rahe hain? Agar yeh baat aap samajh paayen toh isi ke maddenazar aap ek aur baat samajhiyega ki same rule applies for division as well. Maine abhi tak addition aur subtraction ko lekar kuch nahi kiya hai aur usko lekar main apna comment doonga, ek specific designated lecture mein but filhaal abhi usko lekar apna assumption mat bana lena ki sir dekho product aur division mein hua toh addition aur subtraction mein bhi ho raha hoga. Nahi maine aisa nahi kaha hai. Maine abhi sirf product aur division ki hi baat ki hai. Division ka matlab kya? Jaise ki agar main kal ko kahu aapse ki 10 ko 2 se divide karne hain. Toh haan sir aap 10 ko 2 se divide karke mod nikaliye ya -10 ko -2 se divide karke mod nikaliye, ya phir -10 ko +2 se divide karke mod nikaliye ya phir +10 ko -2 se divide karke mod nikaliye. Yeh sab kiske equal hoga? Yeh sab 10 ke mod ko 2 ke mod se divide karne ke, hai na? -10 ke mod ko -2 ke mod se divide karne ke, hai na? I hope yeh confusing nahi hai. -10 ke mod ko +2 ke mod se divide karne se ya phir 10 ke mod ko -2 ke mod se divide karne ke equal hoga.
[9:41]Yeh sab ek hi baatein hain. Yeh sab ek hi baatein hain. Aap kyu nahi samajh rahe ho? Ab signature matter hi nahi kar raha hai na? Toh multiplication aur division mein koi farak hi nahi padta hai. Kyunki magnitude impact hota hi nahi hai. Signature wise magnitude impact nahi hota hai. Signature ko agar hata dein toh magnitude ke terms mein baat karein toh product aur division mein aapka number wahi aata hai. Addition aur subtraction ek alag kahani hai jiski baat hum nahi kar rahe hain. Par yeh baat toh aap samajh rahe ho na? Yeh baat toh aap acche se samajh rahe ho bhai. I hope samajh rahe ho, hai na? Toh yahan tak cheeze agar clear ho rahi hain toh bas itni si baat hai jo bahut zaroori baat hai ki yeh charon terms aksar aapke kaam aa rahi hongi, ab aksar x square ko mod x ke square se. Mod x ko minus of mod x se. Mod of ab ko mod a mod b se aur mod of a upon b ko mod a upon mod b se interchangeably use kar rahe honge. Aur yeh bahut bahut valid aur bilkul sahi aur authentic baat hai jo ki aap mera yakin hai mante hue chalenge. Kya yeh baat aap yaad rakhenge? Koi confusion toh nahi hai? Chalo. Yahan se aage badhte hain, next point ko explore karte hain aur next point kya aata hai use sunte hain. Gaur se suniyega. Absolute values as square root function. Main aapko ek mazedaar si baat sikhane, samjhane ki koshish kar raha hu, bas gaur se jaldi se dekhna. Main is point par aa raha hu par pehle yeh dekho. Aap 2 ko kya 2 ke square ka under root likh sakte ho? 2 ke square ko kya main -2 ka square likh sakta hu? Main -2 ke square mein jo 2 ki power yeh jo under root hai isko 1/2 likh sakta hu kya? Toh yeh 2 se 1/2 cancel karke isko -2 bacha sakta hu kya? Toh main finally yeh prove kar de raha hu ki 2 -2 ke equal hai par yeh gadbad hai. Sir kuch bhi toh gadbad nahi hai, sab toh sahi hai. Pehli baat toh aap khud kahenge ki 2 -2 ke equal nahi hota hai toh aapne beech mein inmein se kisi step mein koi galti ki hai? Bilkul humne is step mein kahin na kahin is dauran koi na koi galti ki hai. Woh galti kya hai aur us galti ko ab nahi dohrana hai us bare mein aaj hum baat karenge. Suniyega dhyan se hai na. Kya hai woh galti? We know that square root of real number x exists only if x is zero or positive. Kya aap meri baat samajhte ho? Matlab of course hum yeh jaante hain, kisi negative number ka square root toh real numbers mein exist nahi karta hai. Imaginary world ya complex world mein jaayenge toh wahan pe kahani alag hogi. Par main agar real number scale par baat karu toh kisi bhi negative number ka under root exist nahi karta hai. Matlab agar kisi number ka under root exist karta hai real number par toh woh ya toh zero hoga ya positive hoga. Yeh baat aap sab jaante hain. Toh kya? Toh mera aapse kehna suniyega dhyan se ki aap yeh baat please maan ke chalo ki agar aapne koi positive number liya hai, koi positive number liya hai. Toh uska square root bhi kya hona chahiye? Uska square root bhi positive hona chahiye. Yeh kaun si baat hai ki sir kisi positive number ka square root negative hota hai? Phir toh woh ambiguity aa jayegi na. Phir toh woh ambiguity aa jayegi na ki woh hai kya, woh kis se associated hai, woh kis se related hai. Hai na? Toh isi baat ko consideration mein laane ke liye humne is problem ko door kiya. Kaise? Suniyega dhyan se. Humne uske liye ek ek simple si functionality introduce ki. Kaam ki baat hai sunte chale jaiyega. Kaam ki jo baat hai yeh ki aap samajhna -2 ka square jo ya +2 ka jo square likha tha woh toh 4 hi tha na? Yahan par jo likha hua tha woh toh 4 hi tha. Toh 4 ka under root jab aayega toh woh + -2 nahi hoga. 4 ka under root aapka 2 hi rehna chahiye. Main aapse bas yeh kehna cha raha hu. Kya aap meri baat samajh paa rahe ho? Toh aap please is baat ko yaad rakhna ki jab bhi aap under root se deal karo, jaise ki jab bhi aap kisi bhi expression ka under root nikalo, jaise ki aapne yeh nikala hai 4 ka under root, matlab 4 ke square ka under root ya jis bhi expression ka aap under root nikalo, aap is baat ko madde nazar rakhna ki woh jo under root wali value aayi hai woh positive aayi ho. Usse agar mujhe positive rakhna hai toh main kiski help le lunga modulus ki. Matlab main kahunga please dhyan se sunna. Main yeh kahunga ki agar mujhse kabhi bhi kaha jaye ki x square ka aapko under root lena hai. Toh main ise bas x nahi likh dunga. Usse bahut saari problems ho jayengi. Bahut saari problems kya jayengi? Abhi humne just discuss kiya jaise ki main aapko phir se dohra deta hu ki jaise mere paas likha hua hai jaise 9 ka square hai na. Mere paas likha hua hai 9 ka square. Ab main 9 ke square ko do tariko se likh sakta hu. 9 ke square ko ek baar under root se cancel kiya toh 9 mil jayega. Lekin 9 ke hi square ko main chaahu toh -9 ka bhi square likhu toh mujhe koi nahi rok sakta hai. Aur phir agar main mathematically apni baat check karu toh square se under root cancel toh yahan par mujhe -9 mil jayega. Matlab yahi expression ki value 9 bhi hai aur yahi expression ki value -9 hai aur aap prove kar rahe ho ki 9 -9 ke equal hai. Par gadbad yahin par yahi samajh aa rahi hai sir ki aisa nahi kar sakte. Aap jab bhi aisa karein jab bhi aap square ka under root se cancel kare toh yeh baat dhyan rakhein ki jis bhi value ka yeh square under root cancel hua hai woh value par mod laga ho. Taaki yeh absurdness, yeh ambiguity, yeh problem door ho jaye aur aap maths ko galat prove na kar dein. Maths mein har cheez logical hogi toh us logic ko hi ensure karne ke liye, validated rakhne ke liye aapne yeh likha ki kabhi bhi x square. Ka under root aap lein toh use x na likhe, use aap mod x likhe. Aap bas yeh ensure kar lein ki woh value positive ho. Aap use hamesha kya likhe? Mod x likhe. Kya aap meri baat samajh paye? Ab us case mein yeh dono hi kiski taraf lead karenge? 9 ki taraf. Kahani yeh hai ki negative numbers ka under root exist nahi karta hai. Real number scale par. Aur humne yeh concept ke saath kahani aage badhayi hai ki positive numbers ka under root positive hi hona chahiye. Toh jab aap under root lein toh aap yeh yaad rakhein ki woh ek modulus ke andar likhi gayi value ho. Koi pareshani toh nahi hai bhai is baat se? Dekho yeh alag-alag baatein hain. Agar aapko lag raha ho ki nahi sir coordinate mein toh aap kuch aur kehte the, matlab agar aisa likha hota tha x square = let's say 9 toh aap kya bolte the x = + - under root 9. Toh haan maine hamesha yeh kaha hai. Maine yeh nahi kaha hai ki aap yahan se aisa likh le rahe ho x = under root 9 aur phir aap likh rahe ho x = + - 3. Nahi, meri yeh process kabhi nahi rahi. Meri process hamesha se rahi hai ki x square 9 ke equal tha. Phir hum jab under root lete hain toh main kya lagata hu + -. Jab aap dono taraf under root lete hain toh aap kya lagate hain + -. Kyunki x kuch bhi ho sakta hai na? + -3 kuch bhi ho sakta hai. Toh usi baat ko ensure karne ke liye ab 9 ka under root + -3 nahi hoga. 9 ka under root 3 hoga. 9 ka under root 3 hi hoga. Toh x ki value kitni aa jaati hai? + -3. Yeh jo + - aaya hai yeh isliye aaya hai kyunki aapne dono taraf under root liya hai. Kya aapko yeh baatein digest ho rahi hain? Kya yeh har ek baat acche se samajh aa rahi hai? Bahut hi clearly har ek baat samajh aa rahi hai kya? I hope ab toh koi confusion nahi hai is baat ko lekar. Bas itni si baat hai ki kisi bhi number ka under root positive hota hai. Yaani ki x square ka under root agar lein toh aap use mod x likhiye, that's it. That's all, nothing important matters here in this case. Modulus ko lekar hum discussion kar rahe hain aur jitne bhi saare aapke mugalte hain hum door karne ki koshish kar rahe hain. Achha kuch zaroori baatein, kuch zaroori kya baatein hain? Suniyega dhyan se. Kabhi bhi koi kahe, kabhi bhi koi kahe ki a square b square se chhota hai toh hum kabhi bhi yeh nahi kahenge ki a b se chhota hai, hume nahi pata. Main phir se kehta hu jaise ki kabhi bhi koi kahe ki 9 jo hai. Matlab main a aur b par aa raha hu par main keh raha hu ki 9 jo hai 15 ki jagah main koi aur word le leta hu, koi aur number le leta hu jaise ki 9 jo hai woh 25 se chhota hai. 9 jo hai woh 25 se chhota hai. Ab 9 aur 25 mein mujhe nahi pata na ki a aur b kya the. Jaise ki 9 aur 25 aap kaise la sakte the? Meri marzi main kaise bhi le aaun. Jaise ki main 9 ko kaise la sakta tha? Jaise main 9 ko la sakta tha 3 square se aur 25 ko kaise la sakta tha? -5 ke square se. I hope aap meri baat samajh pa rahe ho. Toh main kabhi bhi aisa nahi kahunga ki 9 agar 25 se chhota hai toh 9 ka under root aur 25 ka under root mein bhi wahi relation hoga. Matlab I am really sorry, under root nahi, main isko aise likhna chah raha hu ki 9 aur 25 kya hain? 9 aur 25 yahan se ban rahe hain, 3 square aur -5 square se. Abhi filhaal isko bhool jao thodi der ke liye. Yeh maine aapke banane ke liye likha hai.
[17:54]Aap isko dekho. Please is baat ko samjho ki kahin par bhi power 2 lagi hui hai. Yahan se dekhna 9 aur 25 ko thodi der ke liye bhool jao. Jaise yahan par power 2 lagi hui hai. Aur yahan pe inke apne apne base hain. Toh aap keh rahe ho ki sir dekho power same hai toh base compare kiye ja sakte hain. Matlab jaisa relation ismein hai equation mein wahi relation inke base mein hoga kyunki inki power same hai. Ji nahi, aap khud dekh rahe ho. 3 ka square 9, -5 ka square 25 aur haan 25 9 se bada hota hai. Par hum yeh jaante hain -5 3 se bada nahi hota hai. Aapko meri baat problem samajh aa rahi hai? Problem mehez itni si hai ki agar aap isko karna chahein execute. Jaise ki yahan pe iska matlab kya hai? Iska matlab hai ki 9 jo hai woh 25 se chhota hai. Haan sir 9 25 se chhota hai toh iska yeh matlab hoga. Iska yeh matlab hoga ki aap bas yeh conclude karna. Aap kya conclude karna? Ki inmein jo relation hoga, matlab agar yeh diya hua hai toh ismein jo relation hoga woh yeh hoga. Ki yeh jo 3 aur -5 hain, yeh jo bhi numbers jinka aapne square kiya hai unke modulus mein wahi relation rahega. Unke modulus mein wahi relation rahega jo yahan tha. Par signature wise main baat nahi kar raha hu aapse modulus ki baat kar raha hu. Matlab main aapse kya kehna chah raha hu? Main aapse kehna chah raha hu 3 jo hai woh 5 se chhota hai. Kya aap meri baat samajh pa rahe ho? Modulus achha nahi lag raha hai toh aap un numbers ko le lo jo aapne 9 aur 25 liye the aur unke under root mein matlab in numbers ke jo squares ke baad jo value aa rahi thi unke under root mein matlab 9 ke under root aur 25 ke under root mein bhi wahi relation rahega jo relation diya gaya tha. Kyunki kyunki kyunki under root ki value negative positive nahi hoti. Under root se jo value nikal kar aati hai woh sirf aur sirf positive hoti hai. Toh 9 ka under root 3 aur 25 ka under root 5. Kya aap yeh baat samajh pa rahe ho? Yeh baat aapke liye samajhna, yaad rakhna bahut zaroori hai. Kya? Suniyega dhyan se. Ki kabhi bhi koi kahe, kabhi bhi koi kahe ki a square b square se chhota hai toh hum kabhi bhi yeh nahi kahenge ki a b se chhota hai, hume nahi pata. Main phir se kehta hu jaise ki kabhi bhi koi kahe. Yeh aapka number scale hai na, ispe kahin na kahin -3 hai, yahan par kahin 1 hai aur yahan beech mein kahin let's say zero hai. Ab jab x -3 se 1 ke beech hai. Jab x aapka -3 se 1 ke beech hai. Toh mujhe batao ismein aisa kaun sa point hai jo origin se minimum distance par hai, aur aisa kaun sa point hai jo origin se maximum distance par hai. Aapko point samajh aa raha hai kya? Toh -3 se 1 ke beech mein agar main dhoondhu, -3 se 1 ke beech mein dhoondhu, origin se matlab zero se minimum distance par kaun hai? Zero hi hai. Main phir se repeat karta hu, x jo hai sir aapka woh kahan se kahan tak hai? Equal kahan kahan laga hai bas woh dekh lete hain. Equal aapka 1 par nahi laga hai, hai na? Toh x jo hai aapka -3 se 1 tak hai. Please dhyan se sunna, bahut dhyan se sunna. X aapka -3 se 1 tak hai, maan liya. -3 se 1 ke beech mein aapke paas ek aisa point mil raha hai kya? Koi bhi dhoond lo aap, jo aapke zero se minimum distance par ho. Sir, zero hi mil raha hai. Matlab -3 se 1 ke beech mein zero hi dikh raha hai aur zero zero se minimum distance par hai kyunki woh zero distance par hai. Toh agar main ispar mod laga du, agar main ispar mod laga du toh mujhe minimum distance kitni milegi? Zero. Kya aap meri baat samajh pa rahe ho? Confusion toh nahi hai. Maximum distance kya hogi? Agar ab main maximum distance poochhu toh mujhe batao zero se door se door, in sab mein kaun hai? Toh sir is taraf gaye toh 1 thodi hi paas hai. Lekin agar is taraf gaye toh -3 sabse door hai. Aur zero se -3 kitni distance par hai? Sir zero se -3 3 unit distance par hai. Toh jab yeh keh raha hai, jab aapse yeh keh raha hai ki jo aapka yeh x hai jab x aapka -3 se 1 ke beech hai toh jo aapka modulus hoga janaab woh kahan se kahan tak hoga? Sir aapka mod x aapka zero se 3 ke beech hoga. Kya yeh baat ab aapko samajh aa rahi hai bhai? I hope ab toh koi confusion nahi hai. Yahan se aage badh sakte hain? Iske baad ab main next point par aata hu toh next question aapka kya aa raha hai bhai? Next question dhyan se dekhiyega.
[22:50]Next question mein woh keh raha hai ab usne pata hai x ko toh define kiya hai -5 se 7 lekin ab woh x ki baat nahi kar raha hai matlab woh mod x ki baat nahi kar raha hai. Ab woh baat kar raha hai x -2 ki toh mera yakin kariye yeh utna hi aasaan hai jitne aapke baaki saare questions. Sir aisa kyu keh rahe ho? Aap saari baatein chhodo pehle yeh likho.
[23:51]X kya hai sir? X hai aapka -5 se 7. Sunna dhyan se. X jo hai aapka woh kya hai bhai? Woh hai aapka -5 se 7. Ji haan hai toh sir. Achha x yeh hai toh x -2 kya hoga? Mod mat lagao abhi bas mujhe x -2 ki value batao. Toh x -2 kya hoga sir? X -2 ke liye aap teeno taraf 2-2 subtract kar do toh -5 -2 kitna? -7 aur 7 -2 kitna? 5. Aap meri baat samajh pa rahe ho kya? Agar aapko abhi cheeze aise nahi dikh rahi hain toh main aapko ek hint deta hu. Hint yeh deta hu gaur se sunna. Aapse woh kiski value poochh raha hai? Woh mod of x -2 ki value poochh raha hai. Toh thodi der ke liye maan lo ki jo aapka x -2 hai woh let's say k hai, hai na? Toh number scale pe plot karo aap k ko, hai na? Kis se kis ke beech mein? Aap -7 se 5 ke beech mein toh aapka question ban gaya hai ki -7 se 5 ke beech mein k ki values hain aur aapko mod of k define karna hai. Aapko mod of k define karna hai, hai na? Equal toh nahi laga tha kahin? Usmein ek cross check kar leta hu. Nahi laga tha. Ab sunna. Saadharan si baat hai. -7 se 5. Maan lo yeh hai kahin -7 aur yeh hai kahin 5 aur of course k kahin na kahin hoga jo ki of course kya hoga? Aapka zero se distance hum measure kar rahe hain toh hai na, toh zero kahin na kahin beech mein hoga jaise ki yahan kahin hoga. Yeh aap samajh pa rahe ho kya? Ab aap phir se wahi baat batao. K jo hai woh kya hai? Woh -7 se 5 ke beech mein aapki distance hai, maximum aur minimum, hai na? Toh main maximum aur minimum distance nikalu toh origin se maximum distance sorry minimum distance kitni hai? -7 se 5 ke beech mein koi aisa point dhoondho jo zero se minimum distance par ho. Sir, zero se minimum distance par zero hi hai. Toh main kya kahunga bhai ki iski minimum value kya hogi? Iski minimum value hogi zero. Kya aap meri baat samajh paye? Achha sir iske alawa.
[25:35]-7 aur 5 mein door se door, inmein inke beech mein jo bhi saare points hain, inmein sabse zyada door kaun hai aapke zero se? Main kahunga sir -7 jo hai distance wise sabse zyada door hai 7 units. Khair -7 pe equal nahi laga hai.
[25:52]Toh 7 unit distance par bhi equal nahi lagaunga. Toh aap meri baat samajh paaye kya? Yaani ki agar main keh pa raha tha, k by the way kya hai? x -2. Toh agar yeh pura scenario ban raha hai toh aap kahoge mod of x -2 jo hoga. Mod of x -2 jo hoga woh aapka kahan se kahan tak hoga? Woh aapka zero se 7 ke beech hoga. Kya yeh baat ab aapko samajh aa rahi hai bhai? I hope ab toh koi confusion nahi hai. Yahan se aage badh sakte hain?
[26:36]Iske baad ab main next point par aata hu toh next question aapka kya aa raha hai bhai? Next question dhyan se dekhiyega.
[26:50]Next question mein woh keh raha hai ab usne pata hai x ko toh define kiya hai -5 se 7 lekin ab woh x ki baat nahi kar raha hai matlab woh mod x ki baat nahi kar raha hai. Ab woh baat kar raha hai x -2 ki. Toh mera yakin kariye yeh utna hi aasaan hai jitne aapke baaki saare questions. Sir aisa kyu keh rahe ho? Aap saari baatein chhodo pehle yeh likho. X kya hai sir? X hai aapka -5 se 7. Sunna dhyan se. X jo hai aapka woh kya hai bhai? Woh hai aapka -5 se 7. Ji haan hai toh sir. Achha x yeh hai toh x -2 kya hoga? Mod mat lagao abhi bas mujhe x -2 ki value batao. Toh x -2 kya hoga sir? X -2 ke liye aap teeno taraf 2-2 subtract kar do toh -5 -2 kitna? -7 aur 7 -2 kitna? 5. Aap meri baat samajh pa rahe ho kya? Agar aapko abhi cheeze aise nahi dikh rahi hain toh main aapko ek hint deta hu. Hint yeh deta hu gaur se sunna. Aapse woh kiski value poochh raha hai? Woh mod of x -2 ki value poochh raha hai. Toh thodi der ke liye maan lo ki jo aapka x -2 hai woh let's say k hai, hai na? Toh number scale pe plot karo aap k ko, hai na? Kis se kis ke beech mein? Aap -7 se 5 ke beech mein toh aapka question ban gaya hai ki -7 se 5 ke beech mein k ki values hain aur aapko mod of k define karna hai. Aapko mod of k define karna hai, hai na? Equal toh nahi laga tha kahin? Usmein ek cross check kar leta hu. Nahi laga tha. Ab sunna. Saadharan si baat hai. -7 se 5. Maan lo yeh hai kahin -7 aur yeh hai kahin 5 aur of course k kahin na kahin hoga jo ki of course kya hoga? Aapka zero se distance hum measure kar rahe hain toh hai na, toh zero kahin na kahin beech mein hoga jaise ki yahan kahin hoga. Yeh aap samajh pa rahe ho kya? Ab aap phir se wahi baat batao. K jo hai woh kya hai? Woh -7 se 5 ke beech mein aapki distance hai, maximum aur minimum, hai na? Toh main maximum aur minimum distance nikalu toh origin se maximum distance sorry minimum distance kitni hai? -7 se 5 ke beech mein koi aisa point dhoondho jo zero se minimum distance par ho. Sir, zero se minimum distance par zero hi hai. Toh main kya kahunga bhai ki iski minimum value kya hogi? Iski minimum value hogi zero. Kya aap meri baat samajh pa rahe ho? Achha sir iske alawa.
[28:51]-7 aur 5 mein door se door, inmein inke beech mein jo bhi saare points hain, inmein sabse zyada door kaun hai aapke zero se? Main kahunga sir -7 jo hai distance wise sabse zyada door hai 7 units. Khair -7 pe equal nahi laga hai.
[29:07]Toh 7 unit distance par bhi equal nahi lagaunga. Toh aap meri baat samajh paaye kya? Yaani ki agar main keh pa raha tha, k by the way kya hai? x -2. Toh agar yeh pura scenario ban raha hai toh aap kahoge mod of x -2 jo hoga. Mod of x -2 jo hoga woh aapka kahan se kahan tak hoga? Woh aapka zero se 7 ke beech hoga. Kya yeh baat ab aapko samajh aa rahi hai bhai? I hope ab toh koi confusion nahi hai. Yahan se aage badh sakte hain?
