Numerical Integration - Trapezoidal Rule, Simpsons 1/3 & 3/8 Rule

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  • Опубликовано: 27 янв 2025

Комментарии • 1,8 тыс.

  • @gajendrapurohit
    @gajendrapurohit  5 лет назад +240

    Thanks for your comment, your appreciation always motivate us to do such kind work.
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    • @yakshichaudhary8086
      @yakshichaudhary8086 5 лет назад +2

      Sir ... I am not able to express my gratitude to you in words
      You are too good
      Your explanation is soooooo affective that I have seen all the videos of numerical analysis of yours

    • @boomusic5658
      @boomusic5658 5 лет назад +2

      Numerical integration, integration and newton quadrature same h ya difference🤔. Please reply me sir🙏 I have many confusion 😥

    • @Jansaurajpartyfanclub
      @Jansaurajpartyfanclub 4 года назад +1

      Hiiii sir
      Numerical method ke liye kaun sa book reference me le plz sir

    • @travelfreak6222
      @travelfreak6222 4 года назад

      Thanku sir for ur efforts 😀

    • @swatbeauty5423
      @swatbeauty5423 4 года назад

      sir what is the difference between analytical solution and semi analytical solution

  • @deanwinchester7802
    @deanwinchester7802 4 года назад +1705

    I go to college just for attendance and come back home and learn from your videos :) Thank you so much sir for all your efforts .

  • @hydermusa3747
    @hydermusa3747 Год назад +120

    covering a whole unit in one video .This man is awesome

  • @devrajbhattacharya7912
    @devrajbhattacharya7912 2 года назад +81

    if there were many teachers like you,then the world of education will be better❤️Nice video sir

  • @pradeep.kshirsagar
    @pradeep.kshirsagar 5 лет назад +75

    Sir my back subject is clear for your unbelievable teaching
    I am proud of u,
    Very thankful

  • @anshtanwar1813
    @anshtanwar1813 2 года назад +131

    Yes you are correct =>The answer to Q4 is 1.187

  • @rafiakhalid4999
    @rafiakhalid4999 4 года назад +24

    Thank you Sir! There is my exam after 2 to 3 days...and these topics are included.. and you make me understand these rules with in no time!

  • @sarfarazshaik3288
    @sarfarazshaik3288 Год назад +47

    I have scored 15/15 marks in GATE Maths by learning from your channel. Thank you Sir

    • @secularbanda1808
      @secularbanda1808 Год назад +1

      congratulations

    • @Rockydhruva3
      @Rockydhruva3 9 месяцев назад

      Please send playlist for my preparation gate mathematics

    • @sombohare3132
      @sombohare3132 7 месяцев назад

      Hello sir I want some guidance from you

  • @ankit_yadav11
    @ankit_yadav11 2 года назад +7

    Guruji U r great bas ek din me aapke lectures se 2.5 moths ka syllabus pdh kr aaj exam dekr aaya hun midsem aur 28 marks ka banaya hun out of 30 , u r great, dhanyavad dil se aapko ❤️❤️❤️ last waqt pe bhi aap ek ajib kism ki confidence utpann kar dete ho body me aur mai aag lgakr aa jata hun ♥️♥️

  • @sakshi1480
    @sakshi1480 3 года назад +20

    Teachers like you are a blessing ❤️✨🤞

  • @Alliswell-vw7hx
    @Alliswell-vw7hx 5 лет назад +158

    East or West GP sir is the best 😍

    • @gajendrapurohit
      @gajendrapurohit  5 лет назад +13

      Hello VEERENDRA P uploads
      Thanks for Subscribing!, your appreciation always motivate us to do such kind work.
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    • @eddyr978
      @eddyr978 5 лет назад +5

      Yrr kisi ko pta h sir ki numerical differentiation vali video kha h?
      Plz muje bta do ksi ko pta h to

    • @tejasviprabhat2161
      @tejasviprabhat2161 4 года назад +3

      @@gajendrapurohit okay sir i visit you on unacademy

    • @nirliptaniharikaprusty3551
      @nirliptaniharikaprusty3551 4 года назад +1

      @@eddyr978 numerical analysis ke playlist mein hai

    • @varunrawat599
      @varunrawat599 4 года назад

      @@eddyr978 nahi hai bhai

  • @jyothirmaimididoddi7134
    @jyothirmaimididoddi7134 3 года назад +12

    U just completed my whole chapter with in 30 minutes and I'm confident for tomorrow's exam
    Thanks alot Sir❤🙏

  • @ROHANIN
    @ROHANIN 5 лет назад +6

    Sir thank you soo much for telling me that radian and degree thing. My mostly question got wrong as I didn't put that degree and soo my answers went wrong. Thanks for mentoring that topic in my mind.

  • @MayankVashishth
    @MayankVashishth 2 года назад +4

    Sir, u r the bestest teacher of maths, i scored 89/100 in maths in BCA 5th sem. Thankyou so much sir bcz this is possible only because of you.

  • @rahulgorai03
    @rahulgorai03 Год назад +11

    clear and crisp concepts , thanks for saving me in my semester/CA3

  • @MrJorwal95
    @MrJorwal95 2 года назад +5

    💖I did not understand all these three methods for the whole year, but today I understood all three in one day.
    thank you very much sir.....💝

  • @tushargautam8775
    @tushargautam8775 4 года назад +13

    You are superb sir...
    Amazing Talent and knowledge, anyone can learn in single attempt from your lectures.

  • @mikhaildbjornsson99
    @mikhaildbjornsson99 9 месяцев назад +3

    One of the best maths teachers out there. If only the other subjects teachers were like you.

  • @kookiescream1539
    @kookiescream1539 4 года назад +13

    Thank you sir..,! These days when the lockdown is here. Your study videos are amazing. 👍🏻

  • @p.harshaharsha8640
    @p.harshaharsha8640 Год назад +2

    Greatest explanation ever i have seen in my life such a simplified one😊

  • @gigachadkartik
    @gigachadkartik 4 года назад +135

    Note: In calculus always understand logx as lnx and values in radians.

    • @humane5218
      @humane5218 2 года назад +1

      please tell me why?

    • @munjee2
      @munjee2 2 года назад +1

      @@humane5218 there's no specific reason it's just what the examiners had in mind when designing the questions typically

    • @bhubaneswarsingha6607
      @bhubaneswarsingha6607 Год назад +1

      ​@thisismarketif it is calculator fx 82ms will it be different

    • @bhubaneswarsingha6607
      @bhubaneswarsingha6607 Год назад +1

      It's same

  • @krishnachauhan5977
    @krishnachauhan5977 Год назад +2

    🙏🏻😎Sir aapka compair kisi bhi mathmatics ke Teacher se nahi sakta hai. Kyoki Sir aap unique ho🙏🏻🙏🏻

  • @shaileshaanand
    @shaileshaanand 5 лет назад +11

    Sir, you save me in every endsem exam, Thank You

  • @kanaksinghal9748
    @kanaksinghal9748 17 дней назад

    Very Nice Tutorial . Beginner Freindly , Short and Standard Question for Exam Preparation.Though sometimes sime errors are present but Overall Very Great Tutorials.

  • @keertikadevi8635
    @keertikadevi8635 4 года назад +19

    These tutorials are very useful in this quarantine days,and we will also have a grip in the subject by remembering your classes.Thank-you

    • @Paaaaaaaaaaal
      @Paaaaaaaaaaal 4 года назад +4

      You studying because of quarantine lol. I am doing it because I have exam in just four hours.

  • @MANSIDUBEY-mx1jf
    @MANSIDUBEY-mx1jf Год назад +1

    Awesome explanation sir!!!!!🤩

  • @sumedha_b
    @sumedha_b 4 года назад +6

    Thank u so much sir!!U have not only made my concept stronger but also made my preparation smooth and easy!!!This video has really helped me.

  • @engineerspot-umangmishra2405
    @engineerspot-umangmishra2405 3 года назад +1

    Bahut hi easy language me samjha diya sir aapne to
    Pta hi nhi chala ye itna hard topic itna easily cover ho gya😘😘🙏🙏

  • @shubhambehera4125
    @shubhambehera4125 4 года назад +6

    Kya sir 3 hours topic 30 mins ma with full details.Impressive sir.

  • @utkarshsingh1635
    @utkarshsingh1635 Год назад +1

    thnku sir for your help... you are best math teacher for university purpose....

  • @shashankkataria6530
    @shashankkataria6530 5 лет назад +17

    Great lectures. Pls also include lectures on abstract algebra and calculation of errors in numerical methods.

  • @Abhay_yadav5_5_5_5
    @Abhay_yadav5_5_5_5 Год назад +2

    You are the best teacher for exam preparation

  • @talhamuneer2599
    @talhamuneer2599 5 лет назад +35

    Sir Kamal Samjhatey Hain Ap. Thank You So Much!! Seriously Concept Bohat Clear Ho Gaye Hain. And Sir Aik Correction Karni Thi,Video Me 25:26 par jab ap trapezoidal rule se solve kar rahe hain (1/x) ko toh apne final answer ko h se multiply nai kiya huya steps me. I thought apko bta du takey students confuse na hon.

    • @gajendrapurohit
      @gajendrapurohit  5 лет назад +6

      Hello I Adam
      Thanks for Subscribing!, your appreciation always motivate us to do such kind work.
      Please Share it With Your Friends
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      Topic :- Solution of Paper of Maths & Science CBSE Exam 2019 Class 10th
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      Please share it with your Friends
      Thanks
      Dr.Gajendra Purohit

    • @AbdurRahman-zu3sm
      @AbdurRahman-zu3sm 4 года назад +2

      Sir ne toh sirf questions solve karke bataya hai , concept toh tab clear hote hai jab pta chale ki formula kaise derive hua .

    • @syedaatifshakoor6054
      @syedaatifshakoor6054 3 года назад

      @@AbdurRahman-zu3sm syllabus mey nhi bro .

    • @arihantratdiya319
      @arihantratdiya319 3 года назад +1

      Thank you I have also confused cause my answer was different

    • @SourajeetRaha
      @SourajeetRaha 9 месяцев назад

      yes bro 1/h multiply hoga or whapar ans hoga 6.3

  • @Tricksiq
    @Tricksiq Год назад +3

    Tomorrow is my paper and I lost all hopes that I can understand these 3 rules but in one lecture I complete all three topics in just 30 minutes thanks a lot sir🙏💞

  • @monikakundu2351
    @monikakundu2351 5 лет назад +29

    Thankyouuuu so much sir for helping us by making such amazing videos . The way u explain everything is outstanding . 😘 All your videos are so helpful . Thanks a lot .

    • @gajendrapurohit
      @gajendrapurohit  5 лет назад +6

      Hello Monika kundu
      Thanks for Subscribing!, your appreciation always motivate us to do such kind work.
      Please Share it With Your Friends
      Want to get notes and link of videos with latest update then Connect me on Telegram Link:- t.me/gajendrapurohit
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      Topic:- Matrics and Linear Algebra
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      Topic:-Fourier Series
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      Topic:-Differential Equation of First Order and First Degree
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      Topic:- Higher Order Differential Equation
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      Topic:- Laplace Transform
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      Topic:-Asymptote
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      Topic:-Fourier Transform
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      Topic :- Simultaneous Differential Equation
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      Please share it with your Friends
      Thanks
      Dr.Gajendra Purohit

  • @JoyfulJourney73
    @JoyfulJourney73 3 года назад +1

    Thanks

  • @jitendrakumartailor7894
    @jitendrakumartailor7894 3 года назад +6

    Thanks a lot sir ,
    We can learn through your videos in such situations COVID-19, and also helpful for open University students.

  • @sweetysarova3348
    @sweetysarova3348 3 месяца назад

    Thanku sir for this video
    It was very helpful
    Explanation is very easy to understand 🙌👌👏☺️

  • @saurabhsharma7847
    @saurabhsharma7847 2 года назад +6

    Thanks a lot sir❤ because you helped lot of students like me🙏💕

  • @ravikumarchhipa8198
    @ravikumarchhipa8198 13 дней назад +1

    Aapka bahut bahut dhanyabad sir ji jay hind🇮🇳❤😊❤❤

  • @tenmeh
    @tenmeh 2 года назад +4

    Thanks a lot sir!
    You are a life saver!
    Wish you more and more success!

  • @happyeevlogs
    @happyeevlogs Месяц назад

    Hello sir, your lessons teach us very well. I have passed all my college Mathematics problems by learning with you. We need teachers like you here at NIT Jaipur. Always grateful sir. Thank you!

  • @deepakraghuwanshi7135
    @deepakraghuwanshi7135 4 года назад +5

    Love u sir u changed my life .I am actually having interest in statistics and going to take this sub as my carrier option otherwise I used to be a pubg addict ..but now I am addicted to u sir..thank u for coming...🥰

  • @tanvivishindasani24
    @tanvivishindasani24 Месяц назад

    Thank you so much sir , I've always struggled a lot with maths and I'm intimidated by it but your explanation really helped 🙌🏾

  • @fawadali955
    @fawadali955 3 года назад +4

    Started to watch your vides from the 1st semester of my EE Degree and now I am in my final year, still watch your videos to understand all of my mathematics courses. Thank you Sir, love from Pakistan

  • @fahimislam8982
    @fahimislam8982 3 года назад

    Sir countless respect and love for you from Bangladesh

  • @prernasingh3300
    @prernasingh3300 4 года назад +4

    Thank you so much Sir. the way you teach, amazing : )

  • @m____i.....8564
    @m____i.....8564 2 года назад

    Ap bohot achchhe se padate ho,,, sab kuchh clear ho Gaya. God bless you 🥰🥰

  • @priyanka__verma
    @priyanka__verma 2 года назад +3

    Your videos are always helpful Sir! I had a query regarding the last question of (sin x)^(1/2). When we are taking the limits of integration ( 0 to pi/2) then 'h' is also in radians but we have changed x into degrees and then calculated corresponding Y(x) so won't 'h' also needs to be changed into degrees?

  • @TechtonicThinker
    @TechtonicThinker 2 года назад +2

    Thankyou sir
    Thankyou so much for teaching numerical analysis 🙏🙏

  • @shivamshrivastva4126
    @shivamshrivastva4126 5 лет назад +19

    In trapezoidal rule one thing is missed
    I.e 2(y1+y2+y3+......+y(n-1))

    • @gajendrapurohit
      @gajendrapurohit  5 лет назад +5

      Hello Shivam Shrivastva
      Thanks for Subscribing!,
      Preparing CSIR NET ? Then Follow me on Unacademy Link:-unacademy.com/user/Dr-GajendraPurohit
      Please Avoid Mistakes, I am Very Happy that you Find Mistakes, it shows that you are watching my videos seriously. these are my old videos so there is some silly mistakes. Now from last 50 videos there is no mistakes.
      Once again Thanks n keep Watching videos and share it with your friends
      Want to get notes and link of videos with latest update then Connect me on Whasapp Group Link:- chat.whatsapp.com/Fck1k8gzunMLVL70DdG4eK

    • @jasminesoni8225
      @jasminesoni8225 5 лет назад

      No nothing is missed ,...

    • @harshvardhan-so1bt
      @harshvardhan-so1bt 5 лет назад +1

      Bhai sir ne 2 se eq ko multiply ker diya

    • @harshvardhan-so1bt
      @harshvardhan-so1bt 5 лет назад

      @@backchodinfluencer Bhai ques. Konsa h

    • @harshvardhan-so1bt
      @harshvardhan-so1bt 5 лет назад

      And which method

  • @rajamalik7536
    @rajamalik7536 3 года назад +1

    Thanks sir 🙏 may God bless you

  • @vrdmathsclasses6505
    @vrdmathsclasses6505 5 лет назад +4

    पहले तो प्रणाम सर, आपके अध्यापन की तारीफ करना तो समुद्र के पानी को एक बूंद में बताने जैसा है अगर हम आपके अध्यापन की कोई तारीफ करें सच में आपके लिए तो तौहीन के बराबर होगी, आपका अध्यापन कार्य बहुत ही लाजवाब है यह इतना अच्छा है कि हमने अपनी मैथमेटिक्स को आपके संरक्षण में ही बहुत अच्छा किया आपको तहे दिल से बहुत-बहुत शुक्रिया इतना अच्छा कार्य करने के लिए thank you so much sir 🙏🙏🙏🙏

  • @ankursingh7820
    @ankursingh7820 4 года назад +1

    lagatar 3rd time in this Month is video ko dekh chuka hu jitna baar dekhta hu utni baar concept aur acche se clear hita hai
    Thank you Sir

  • @adarshmaurya3315
    @adarshmaurya3315 2 года назад +9

    One of the finest guider and teacher of my list..u are one of the saviour for us ..thank u sir for ur efforts and making mathematics quite interesting 🙏🙏

  • @amit9790singh
    @amit9790singh 3 года назад +1

    Thanku sir .
    You are teaching marvelous!

  • @mahendrraimahendrrai6234
    @mahendrraimahendrrai6234 3 года назад

    Sir apka ka padhaya gya easily se smjh aa jata h aur formulae learn bhi ho jate h easily, thanks alot of u, sir🙏🙏🙏🙏🙏

  • @VarunK1386
    @VarunK1386 4 года назад +7

    These exams after covid were relieved by your videos sir... Thanks for explaining clearly

  • @UnboxingAnime-x2m
    @UnboxingAnime-x2m Год назад

    i am from bangladesh....thanks for giving these videos ...our full department is your fan sir.....

  • @rohit7590
    @rohit7590 3 года назад +7

    Your video's are very useful for us sir. Thank you so much sir ♥️🙏. Students are lucky to have you as there professor ♥️

  • @surajkumar-oj3wx
    @surajkumar-oj3wx 3 года назад +1

    Your lecture is awesome 😎😎

  • @WeSmart83
    @WeSmart83 2 года назад +3

    Great explaination

  • @jaspalkaur1756
    @jaspalkaur1756 Год назад

    what a wonderful teaching way 👏👏

  • @preetam4048
    @preetam4048 Год назад +14

    Kal exm hai 😁

  • @electricalsujitgope
    @electricalsujitgope 3 года назад +1

    Thanks you for your support and help many students studying in btech you are god for us who always understand easily method .I am appreciating you that you are god of engineering🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏

  • @dhruvsharma7365
    @dhruvsharma7365 2 года назад +5

    I've a query in example 3 how "1.5/2+3/4+3/5 = 0.70" when 1.5/2=0.75, 3/4=0.75, 3/5=0.6 simple mistake or my mind is breaking :(

  • @tibroru8773
    @tibroru8773 3 месяца назад

    From Rajshahi University completing whole course in one night with your videos ....A lot of respect sir

  • @THENGINFINITY
    @THENGINFINITY Месяц назад +6

    Main toh exam centre pe aakar dekh rha

  • @shubhamparte9045
    @shubhamparte9045 Год назад

    U r one of the best teacher for online classes we are all doubts are clear understand when watch your video sir love you 😙

  • @theyuvrajyadavv
    @theyuvrajyadavv Год назад +15

    legends are watching this at exam time 😁😂

    • @UNKNOWN-ui8yp
      @UNKNOWN-ui8yp 9 месяцев назад

      legends nahi gawar dekhte hey exam time per

    • @lavishchauhan2732
      @lavishchauhan2732 9 месяцев назад

      Up corse bro 😂

    • @lavishchauhan2732
      @lavishchauhan2732 9 месяцев назад

      ​@UNKNOWN-ui8yp chl

    • @snehadey883
      @snehadey883 7 месяцев назад

      Correct 💯

    • @theyuvrajyadavv
      @theyuvrajyadavv 7 месяцев назад

      @@UNKNOWN-ui8yp haa vahi gawar phir ek din me pd kr tum jeso se jyada number le aate he.

  • @himanshik9934
    @himanshik9934 8 месяцев назад

    Whatever you teach is understood in one go. Thank you sir. ❤😊

  • @AnkitRaj-tk2rr
    @AnkitRaj-tk2rr 9 месяцев назад +26

    25 min me exam hai 😅😅

  • @shivanker12
    @shivanker12 7 месяцев назад

    Great work sir
    Complex ko easy krke sikha dya 🎉🎉🎉

  • @shinchannohara7653
    @shinchannohara7653 6 месяцев назад +3

    Only Legends are watching lecture vidios before the exam time thats why they always score higher max in the bottom 😂😂😂

    • @mmuzammal760
      @mmuzammal760 6 месяцев назад

      @@shinchannohara7653
      You are absolutely Right ✌🏻😂💯

  • @sumitkumarshaw8333
    @sumitkumarshaw8333 2 года назад +1

    Amazing teacher ❤️...
    For just u i got to cover my syllabus

  • @joshibhuwan4991
    @joshibhuwan4991 9 месяцев назад +6

    Legend's-bhai kl exam h abhi dekh leta hu 😂😂

  • @RamChandra-pd7pv
    @RamChandra-pd7pv 2 года назад +2

    Great work sir. Lots of love form Nepal. Thanks a lot.

  • @recepton.
    @recepton. 2 года назад +3

    Today is my exam at 8:30am and I am watching this video at 3am.

    • @vinnu181
      @vinnu181 2 года назад +1

      Hope you did well.... Kal Mera h😂

    • @recepton.
      @recepton. 2 года назад

      @@vinnu181 yes bro bach gaya lekin en 3 rules mai se kuch aya hi nahi😅

  • @anupamamishra8105
    @anupamamishra8105 2 года назад

    Best video in RUclips regarding such topic

  • @danishzia6442
    @danishzia6442 4 года назад +4

    Allah Pak ap ko Iman Ki daulat Sa BHI mala maal farmae ameen

  • @AlishbaShoaib-k8y
    @AlishbaShoaib-k8y 15 дней назад

    ❤❤ so good that clear my concept thank you sr

  • @nafiurhasanniaz2595
    @nafiurhasanniaz2595 4 месяца назад +5

    Exam in an hour. Hahaha.

  • @ra______brothers....2582
    @ra______brothers....2582 2 года назад

    Thank you so much sir🙏🙏...I am so happy after watching your videos...
    Sir,the way of teaching of your so amazing
    I feel very delight when I watch your videos
    Again and again thank you so much sir 🙏🙏🙏

  • @abhishekfitnessworld3865
    @abhishekfitnessworld3865 3 года назад +1

    Sir ,,u have a unique way of teaching method 😍😍,,thanx a lot sir 🙏

  • @sahilhassan7149
    @sahilhassan7149 2 года назад

    Sir you are so awesome
    Only for you i understand B sc maths
    Thank you very much sir for making this videos
    You are one of greatest teacher and my favourite teacher 💕💕💕💕

  • @kiddykine4301
    @kiddykine4301 Год назад

    sir Ur so good at teaching .... thank u so much, sir ........❣🙏

  • @devyanimeena5089
    @devyanimeena5089 8 месяцев назад

    Thank you sir u r great teacher for mathematics your videos are very helpfull 😊😊

  • @fathekhan6788
    @fathekhan6788 2 года назад

    The best teacher of the world ☺

  • @malikshahid2316
    @malikshahid2316 Год назад

    Thanks alot sir your videos always help me alot to slove the problems
    Thanks alot sir ❤❤❤

  • @nasimbegam1406
    @nasimbegam1406 Год назад +1

    sir your videos are very accurate and to the point for our syllabus but sometimes you miss something like you don't explain something clearly.due to which I take help from others but it's okay.amd one more thing sir (avoid negativity)
    thanks a lot sir

  • @SOMASHRISAMARMISHRA
    @SOMASHRISAMARMISHRA 3 года назад +1

    thank you! my all doubts are cleared and yess camera quality is very nice!

  • @shahtaj3277
    @shahtaj3277 Год назад

    Thanks, Thanks so much for your Great efforts and I really like your way of teaching. May God Bless you with good health, wealth, Prosperity , Success and Happiness Amen😊.
    And please add the error calculating topic to these questions please.😇

  • @hermanyt3235
    @hermanyt3235 7 месяцев назад

    Thank u sir..... your video saved my day....🤩

  • @a-046nilotpalchatterjee3
    @a-046nilotpalchatterjee3 3 года назад +1

    Awesome explanation sir

  • @asmitagupta395
    @asmitagupta395 Год назад

    Thanku sir... I always watch ur vedios 1 night before my exam.. & it is helping me a lot

  • @karmachand7094
    @karmachand7094 3 года назад

    Dhamakedaar sir 👌👌

  • @vikashdwivedi2015
    @vikashdwivedi2015 3 года назад +1

    Thankyou so much sir you r great❤️🙏

  • @apurvakhapekar8888
    @apurvakhapekar8888 2 года назад

    Sir ,,Apke sbhi vedios bhot beneficial hai ..thankyou very much sir ..

  • @akramshaikh807
    @akramshaikh807 3 года назад

    Thankyou so much sir.
    I was forgotten and very confused in these three method.
    Now I am clear.
    Thankyou again

  • @VinodKumar-g7h2h
    @VinodKumar-g7h2h 2 месяца назад

    Thanks Sir for many efforts in questions solving