Hello Brilliant !

You're most welcome to leave messages for me.

You can share your ideas, thoughts, complaints, suggestions, inspirations, and any other constructive stuffs for me.

Thanks a lot !

**With love**

**Sandeep Bhardwaj**

Hello Brilliant !

You're most welcome to leave messages for me.

You can share your ideas, thoughts, complaints, suggestions, inspirations, and any other constructive stuffs for me.

Thanks a lot !

**With love**

**Sandeep Bhardwaj**

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## Comments

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TopNewest!\(\ddot\smile\). I guess You are Solving question of our community Brillaint.org in This Pic , ain't You ? – Deepanshu Gupta · 2 years ago

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– Sandeep Bhardwaj · 2 years ago

Yeah, You're absolutely right. Doing so makes me feel alive. ! :PLog in to reply

– Deepanshu Gupta · 2 years ago

So True , It is Really Great Site , which offers Such an great Platform for Interacting with incredible Students and Awesome Professors Like You . I'am Proud To be member of This Family ! And Also Thank you for Your So much Contribution To our Community !Log in to reply

– Prakash S · 1 year, 7 months ago

How i increase my mathematical knowledgeLog in to reply

Sir I am predicting the future -

Yours and agnishom's messageboard will get atleast 100 reshares – Megh Choksi · 2 years ago

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@megh choksi – Sandeep Bhardwaj · 2 years ago

A lot of thanks for your wishes.Log in to reply

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The worst way to teach mathematics is tell the students to use calculator in all algebra problems, to draw exact figure, then measure the lengths in geometry problems, to count all the cases in case of combinatorics problems. This will frustrate the students and they may have a tendency to hate Maths. This is the worst way, so it must be avoided. AND use of the word "obviously" on directly telling the result without answering the question of how and why will make it more weird.

I think that the golden rule that every mathematics teacher should follow that he/she should not tell only the solutions to the problems, but also the approach (matters most) and what stuffs made that approach stroke in his/her mind. On doing so, students will start thinking about the problems,then Maths won't look something magical or strange to them anymore. And this will enhance the interest of students in Mathematics. Being a JEE-Advanced Maths Faculty, this golden rule is subjected to the problem solving, as Maths is much more about computational approach than the conceptual one for JEE.

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– Agnishom Chattopadhyay · 2 years ago

Thanks for replying! :)Log in to reply

– Kushal Patankar · 2 years ago

May be I could answer your first question - worst way would be to just read out the NCERT in the classroom with no more extra explanation.Log in to reply

How do you relate yourself with 'Po' the Panda in the movie Kungfu Panda? @Timon – Sanjeet Raria · 2 years ago

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– Sandeep Bhardwaj · 2 years ago

"I eat when I'm frustrated." this relates me with PO in that movie. And always remember, the people who love to eat are always the best people. :P @pumba \(\ddot\smile\).Log in to reply

– Sanjeet Raria · 2 years ago

& what about our tea @ ganesh?Log in to reply

– Sandeep Bhardwaj · 2 years ago

Po used to take soup, not tea. And I think thats what differs me from PO. Our tea @ganesh is just pain-relieving, that's one of my Awesome things.Log in to reply

What drives your ambition? – Calvin Lin Staff · 2 years ago

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@Sanjeet Raria, Always keep myself ready to face any challenge (any test, discussion, presentation, interview, lecture etc.), all these drive my ambition. @Calvin Lin – Sandeep Bhardwaj · 2 years ago

My keenness to learn Mathematics and share that knowledge, posting and solving problems on Brilliant, always having a desire to give my students the best of me : a lot of discussion, analysis, problem making, elegant approaches, how to face the exam, how to overcome some depressing moments in the exam, discussing Mathematical ideas and our future plans with my best friendLog in to reply

\[\huge \mathbb{HAPPY} \quad \mathbb{ BIRTHDAY} \quad \mathbb{ SIR!}\]

– Rajdeep Dhingra · 1 year, 8 months agoLog in to reply

@Rajdeep Dhingra – Sandeep Bhardwaj · 1 year, 8 months ago

thank you very very much \(\ddot \smile\)Log in to reply

– Soumo Mukherjee · 1 year, 8 months ago

Lol the 2nd gif is so funny.Log in to reply

I can't thank you enough for how much of an inspiration you are to me along with all the others are Brilliant. I'm hopefully striving to reach your (and others) levels of math skills by the time I graduate from high school or by the time the school year here ends. – Tytan Le Nguyen · 2 years ago

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@tytan le nguyen Keep doing problems and keep learning from your mistakes. Always try to look all the given solutions to any problem and analyse each of those solutions, that will enhance your skills very rapidly. Great. Keep it up. – Sandeep Bhardwaj · 2 years ago

All the best Mr.Log in to reply

Feeling great to talk directly to you!

Sir please add more sequence and series problems. – Megh Choksi · 2 years ago

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@megh choksi – Sandeep Bhardwaj · 2 years ago

I am also feeling awesome. Nowadays, I am unable to give my time to the Brilliant. I will be very much active within a week, then I will add a lot of awesome problems of Sequence and Series. And off course, of many other topics too. You will definitely enjoy the problems.Log in to reply

Happy Birthday to you\[\] The sky is Blue\[\] No clouds \[\] Sun shining bright for u!!\[\] HAPPY BIRTHDAY SIR!! – Adarsh Kumar · 1 year, 8 months ago

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– Sandeep Bhardwaj · 1 year, 8 months ago

Thank you very much \(\ddot \smile\)Log in to reply

On which topics did you write your thesis paper?

Oh.. I also have a doubt:

\(\Huge Is\quad i>0,\quad where\quad { i }^{ 2 }=-1.\quad Why\quad Not?\) – Soumo Mukherjee · 2 years ago

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– Sandeep Bhardwaj · 2 years ago

Inequalities are not applicable in case of complex numbers. You can't even say that \(2i>i\).Log in to reply

I also wanted to know why? – Soumo Mukherjee · 2 years ago

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So whole I want to say is that in Mathematics anything is accepted universally when it doesn't violate the laws of mathematics (it is conceptually true) and it has further applications. Probably there are not applications further associated with the inequalities in complex numbers or may be it violates some basic laws of maths when proceeded with this concept. @Math Philic – Sandeep Bhardwaj · 2 years ago

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– Soumo Mukherjee · 2 years ago

Sorry, if I sounded rude. I have deleted that comment.Log in to reply

– Sandeep Bhardwaj · 2 years ago

It's ok dude. I disapprove of what you say, but I will defend to the death your right to say it. I replied to your comment with a hope that you will take my words in positive way. Read my comment again, may be there is something good that I wanted to convey.Log in to reply

@Sandeep Bhardwaj , I have revisited it and went through it again.

OkThanks for replying. – Soumo Mukherjee · 2 years ago

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– Radhesh Sarma · 10 months, 1 week ago

Because complex numbers like i cannot be represented on the number line.Numbers like 1,2,3,-1,-2 can be represented on the number line and hence can be compared.Log in to reply

– Prajwal Kavad · 1 year, 10 months ago

(its simple) since i is a imaginary number then how can u compare it .Log in to reply

\[\LARGE \color{red}{\text{H}} \color{blue}{\text{A}} \color{yellow}{\text{P}} \color{cyan}{\text{P}}\color{green}{\text{Y}}\]

\[\LARGE \color{green}{\text{B}} \color{red}{\text{I}} \color{brown}{\text{R}} \color{magenta}{\text{T}} \color{blue}{\text{H}} \color{yellow}{\text{D}} \color{cyan}{\text{A}} \color{green}{\text{Y}}\] – Ashish Siva · 8 months, 1 week ago

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– Sandeep Bhardwaj · 8 months, 1 week ago

Thanks, Ashish! :)Log in to reply

– Ashish Siva · 8 months, 1 week ago

Many many happy returns of the night XD, Am I right one to wish you :PLog in to reply

Belated

– Soumo Mukherjee · 1 year, 8 months agoLog in to reply

@Soumo Mukherjee – Sandeep Bhardwaj · 1 year, 8 months ago

thanks a lot \(\ddot \smile\)Log in to reply

And as usual , your selection of pics is the best :D – Azhaghu Roopesh M · 1 year, 8 months ago

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– Soumo Mukherjee · 1 year, 8 months ago

Lol...I didn't notice the time ...But I also wished him on Facebook. Perhaps he is busy enjoying. When I logged in here I found Sandeep's message board reshared, thought also to mention here. I didn't notice what's was the time.Log in to reply

– Azhaghu Roopesh M · 1 year, 8 months ago

K . Fb huh ? Looks like I have to finally create an account there .Log in to reply

– Soumo Mukherjee · 1 year, 8 months ago

Are you in google+?Log in to reply

Here's the link to my profile . – Azhaghu Roopesh M · 1 year, 8 months ago

Yes I am .Log in to reply

here – Soumo Mukherjee · 1 year, 8 months ago

Mine isLog in to reply

And btw I don't have a mobile currently , so I'll wait till I buy one(probably next month b4 joining college) , to create an fb account . – Azhaghu Roopesh M · 1 year, 8 months ago

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\(HAPPY~BIRTHDAY ~SIR!\) – Nishant Rai · 1 year, 8 months ago

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@Nishant Rai – Sandeep Bhardwaj · 1 year, 8 months ago

thanks a lot \(\ddot \smile\)Log in to reply

\[\huge HAPPY BIRTHDAY SIR!\]

May you live many years to celebrate your birthday, and educate people so that humanity reaches it's potential! – Sravanth Chebrolu · 1 year, 8 months ago

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@Sravanth Chebrolu – Sandeep Bhardwaj · 1 year, 8 months ago

Thanks a ton \(\ddot \smile\)Log in to reply

\[ \huge HAPPY BIRTHDAY SIR! \]

Do you like carrots? :P – Nihar Mahajan · 1 year, 8 months ago

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However, I don't want to comment about the mystery of Carrots right now. :P

@Nihar Mahajan – Sandeep Bhardwaj · 1 year, 8 months ago

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\(\huge \mathbb {HAPPY BIRTHDAY} Sir!\) – Rohit Ner · 1 year, 8 months ago

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@Rohit Ner – Sandeep Bhardwaj · 1 year, 8 months ago

thank you very much \(\ddot \smile\)Log in to reply

HAPPY BIRTHDAY SIR!! – Ninad Akolekar · 1 year, 8 months ago

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@Ninad Akolekar – Sandeep Bhardwaj · 1 year, 8 months ago

Thank you very much \(\ddot \smile\)Log in to reply

\[\huge \text{Happy Birthday sir !!} \] – Azhaghu Roopesh M · 1 year, 8 months ago

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– Sandeep Bhardwaj · 1 year, 8 months ago

Thank you very much \(\ddot \smile\)Log in to reply

Sir , I have a suggestion . Seeing that you like to change your profile pic every 2-3 days , can you update the pic on this message board ?

This pic is from 2014 , please post one from 2015 :D

It's just a suggestion sir !! – Azhaghu Roopesh M · 1 year, 10 months ago

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Thanks for the suggestion.

@Azhaghu Roopesh M – Sandeep Bhardwaj · 1 year, 10 months ago

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Well , there's one way . If the pic on this message board has been added using Markdown , then I guess you can add another one by simply deleting the previous code and uploading another pic , but if you have added the pic using the "Attach Image" , then I'm not too sure about it . Sorry sir . – Azhaghu Roopesh M · 1 year, 10 months ago

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I've attached the image there. I also don't know the way of resolving it.

\(\ddot \smile\) Yeah , the sky has been touched.

I also want to delete some of the pics. But there is no such option there.

Btw if I upload the image on message-board, it's gonna buried deep after some time, as it was some days ago.

@Azhaghu Roopesh M – Sandeep Bhardwaj · 1 year, 10 months ago

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Hi sir , I just completed your set "Expected JEE-MAINS-2015" .

Marks obtained : 115 Time Taken : 1 h 27 minutes Questions Attempted : 30 Number of Guesses : 3

I enjoyed going through this set and am eagerly awaiting your next set that you have promised to @Cody Martin !!!! – Azhaghu Roopesh M · 1 year, 10 months ago

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@Azhaghu Roopesh M – Sandeep Bhardwaj · 1 year, 10 months ago

Your score is very good. Keep it up. You can get in top 10 All India Rank, if you perform like this in all the three subjects in JEE syllabus. Now in Maths, you need to reduce the time consumed. Ultimately, to increase your speed with accuracy. For that I will suggest you to memorize all the formulas instead of deriving those in the exam time, especially over co-ordinate geometry, for eg. equation of tangents, normals, parametric form, intercept-formulas etc. I wish you all the very best. Keep your efforts on and focus on more improvement.!!Log in to reply

I guess I have to start remembering the formulae then \(\ddot\smile\)

Thanks a lot sir for your best wishes \(\ddot\smile\) – Azhaghu Roopesh M · 1 year, 10 months ago

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@Azhaghu Roopesh M – Sandeep Bhardwaj · 1 year, 10 months ago

Yeah, you should start. On reading your reply, I realized that I've attained numerous priceless qualities of an Ideal teacher through one year experience of teaching JEE students. Thanks for making me this realization. Keep it up.Log in to reply

please solve https://brilliant.org/problems/can-you-solve-it-9/?group=j4P7dyAoNhAb&ref_id=568173 – Visakh Radhakrishnan · 2 years ago

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How did you find Brilliant? :D – Marc Vince Casimiro · 2 years ago

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@Marc Vince Casimiro – Sandeep Bhardwaj · 2 years ago

My friend Sanjeet Raria told me about Brilliant, I signed up then on Brilliant, and I am doing it almost regularly since June 2014 (approx).Log in to reply

@Sanjeet Raria must be friends/brothers because "sandeep" and "sanjeet" rhyme very nicely :P – Nihar Mahajan · 1 year, 8 months ago

From first day only i knew that you andLog in to reply

– Sandeep Bhardwaj · 1 year, 8 months ago

hahahaha, Nicely observed. :PLog in to reply

Do you like carrots? – Agnishom Chattopadhyay · 2 years ago

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– Sandeep Bhardwaj · 2 years ago

Yeah I like carrots. Infact, I like everything to eat. \(\ddot\smile\).Log in to reply

Super hero by night :P what do you mean :P xD – Krishna Sharma · 2 years ago

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– Sandeep Bhardwaj · 2 years ago

The whole day moves on with learning and teaching. But life is not all about this, So all other stuffs (especially my interests, mentioned in my profile) left for the NIGHT, are to be accomplished like a SUPER-HERO. I would like to add here that the nights have the credit for making of the "real me".Log in to reply

– Azhaghu Roopesh M · 1 year, 7 months ago

When do you sleep ? Are you a vampire ? :P :DLog in to reply

@Azhaghu Roopesh M – Nihar Mahajan · 1 year, 7 months ago

Solve the tic tac toe problems by me :PLog in to reply

– Sandeep Bhardwaj · 1 year, 7 months ago

It's not fixed when I sleep. I assume that I ain't an vampire :P :D But not too much sure I am !Log in to reply

Sir a typo on this page shoukdnt it be 4 out of 5 people have trouble with fractions instead of 5 out of 4?

– Ashish Siva · 8 months agoLog in to reply

BTW thanks for caring about this. – Sandeep Bhardwaj · 8 months ago

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– Ashish Siva · 8 months ago

It is intentional?!?! Oh my god!Log in to reply

Sir , I request you to make a new note of the JEE ADVANCED MATHEMATICS CONTEST-'16, as it is taking time to load the contents. – Rohit Ner · 8 months, 4 weeks ago

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– Sandeep Bhardwaj · 8 months, 4 weeks ago

Yeah, I already have this point in mind. But I'm waiting for the contest to grab 50 problems in the current note and then forward it to the new note. :)Log in to reply

Sir I 'm Atanu a COMMERCE student. Please help me to refer me some books. That will help me to be master means that will help me very much to understand the depth of COMMERCE(Accounts, Business study & Economics) & MATHEMATICS also. PLEASE HELP ME!

And i am very grateful to you that you have created this note. Please help me sir! – Atanu Ghosh · 9 months, 3 weeks ago

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hai – Prakash S · 10 months, 1 week ago

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Why don't you start posting problems for jee mains? Just like last year. – Aditya Kumar · 10 months, 1 week ago

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@Aditya Kumar – Sandeep Bhardwaj · 10 months, 1 week ago

Do you want me to post the problems for JEE this year too?Log in to reply

– Aditya Kumar · 10 months, 1 week ago

Yes. That will help JEE aspirants. :)Log in to reply

– Sandeep Bhardwaj · 10 months, 1 week ago

Alright. I'll defnitely go for this one. Thanks!Log in to reply

– Aditya Kumar · 10 months, 1 week ago

I should thank you :)Log in to reply

https://brilliant.org/discussions/thread/doubt-5/?ref_id=898502 – Radhesh Sarma · 1 year, 5 months ago

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Hello Sandeep Bhardwaj

During school days Math was not my favorite subject, I felt its difficult and it was not explained properly .

But Now its inverse, I joined Brilliant and realized that its interesting. This made me think what is fascinating me to solve Math problems?

Now i want to understand Maths properly and in depth. So i need your help for the same. I want to get trained by Maths experts like you. Please let me know how can i seek that? – Suneet Bendre · 1 year, 7 months ago

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Sir, I'm a student in class 11 at the moment, and plan on giving JEE in 2017. My coaching class starts day after tomorrow, and I had a question: How much sleep do you recommend? And from when to when? Others are free to comment too. – Omkar Kulkarni · 1 year, 7 months ago

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friend how increase my mathematical knowldge – Prakash S · 1 year, 7 months ago

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Hello sir need ur hlp its urgent sir – Reena Rathode · 1 year, 7 months ago

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– Sandeep Bhardwaj · 1 year, 7 months ago

Yeah, tell me..What help do you need ?Log in to reply

Happy b'day sandeep sir – Nishu Sharma · 1 year, 8 months ago

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@Nishu sharma – Sandeep Bhardwaj · 1 year, 8 months ago

thank you very much \(\ddot \smile\)Log in to reply

Sandeep Bhardwaj Sir . I wanted to to show you my approach to this problem . I have my work scanned . Should i post it here or do i mail it to you ? If so please tell me your id. Thanks ! \(:)\) – Keshav Tiwari · 1 year, 8 months ago

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Can you please post some more problems on Conic sections and binomial theorem?? – Ninad Akolekar · 1 year, 8 months ago

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Sir could you help here – Rajdeep Dhingra · 1 year, 9 months ago

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sir i attempted EXPECTED JEE MAINS B i am getting only 25 marks in 2 and half an hour i got 10 questions correct and 15 questions wrong guessed only 2 problems but i learnt all the mistakes i did and i got 29 marks in the first set – Prajwal Kavad · 1 year, 10 months ago

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Hello Sir,

I wanted to ask you for a suggestion.

Sir, I have just finished integration from RD Sharma class 12th. Could you suggest me what to do next \(\ddot\smile\). – Rajdeep Dhingra · 1 year, 10 months ago

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@Rajdeep Dhingra – Sandeep Bhardwaj · 1 year, 10 months ago

Enhance the skills of integration from some other book whose level is higher than that of R.D. Sharma. You can go for the books recommended for JEE-Advanced.Log in to reply

– Rajdeep Dhingra · 1 year, 10 months ago

Any specific book ?Log in to reply

Sir please can you make one final set for jee mains 2015 Thanks :) – Cody Martin · 1 year, 10 months ago

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@Cody Martin – Sandeep Bhardwaj · 1 year, 10 months ago

I don't promise for that, but I will try.Log in to reply

Check tis out!

Please participate and reshare. – Rajdeep Dhingra · 1 year, 10 months ago

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Please make 1 more set for jee mains – Cody Martin · 1 year, 11 months ago

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@Rajdeep Dhingra – Sandeep Bhardwaj · 1 year, 11 months ago

Good initiative. Surely It will get a good response. My best wishes for that !Log in to reply

out!

Check tisPlease participate and reshare. – Rajdeep Dhingra · 1 year, 10 months ago

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Just a suggestion :) – Azhaghu Roopesh M · 1 year, 11 months ago

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Will you participate ? Are the questions fine ? – Rajdeep Dhingra · 1 year, 11 months ago

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The questions are fine , I'd have loved it if you could add some Number Theory problems in the next edition :) – Azhaghu Roopesh M · 1 year, 11 months ago

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– Rajdeep Dhingra · 1 year, 11 months ago

On it.Log in to reply

– Sandeep Bhardwaj · 1 year, 11 months ago

Off course, So be ready. Expected JEE-Mains-2015-II coming soon...All the best !Log in to reply

Your set on Expected JEE main 2015 has only 3 questions. But you say there are 30.

Imgur

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@Rajdeep Dhingra – Sandeep Bhardwaj · 1 year, 11 months ago

I'm posting the questions, Please wait. It will be ready up to 6pm today. Please have a little patience.Log in to reply

At Sandeep Bhardwaj : Thankyou your advice is invaluable – Atif Farooq · 2 years ago

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When you multiply 6 by an even number, they both end in the same digit.

Example: 6×2=12, 6×4=24, 6×6=36, etc

Why does this happen?? – Harshvardhan Mehta · 2 years ago

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@Harshvardhan Mehta – Sandeep Bhardwaj · 2 years ago

It's just an observation. I think, at this level at which the question is, rules are made from the observations, that is theories are based on experiments(observations). It's just like asking \(1\) comes after \(0\), why ? So, there is no answer to this question. So whole I want to say is that such things just happen, It will be beneficial for you if you take it just as an observation and use it somewhere.Log in to reply

I agree with you! The

whyquestions can probably never be answered.However, the statement can be proven by induction – Agnishom Chattopadhyay · 2 years ago

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@Agnishom Chattopadhyay – Sandeep Bhardwaj · 2 years ago

I love this quote very much as posted by you ; "In mathematics you don't understand things, you get used to them. - By John von Newumann".Log in to reply

Salutations Sir,

i have found brilliant to be an excellent source for enhancing problem solving skills and i have recommended it to many friends i am an engineering student in my freshmen year but i enjoy mathematics more my question is that how should i proceed going about learning the more rigorous kind of mathematics like analysis ,topology etc after i take the introductory courses like linear algebra differential equations etc as engineers in my university are not allowed to take these courses and unfortunately switching to math major is not a feasible option for me.

i understand this may not be the appropriate forum to ask these questions but any help would be much appreciated

Thank you – Atif Farooq · 2 years ago

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Every person in the world has 24 hours a day. But the thing that creates the difference is how you utilize that time.All the best @Atif Farooq – Sandeep Bhardwaj · 2 years ago

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Can any one say wats the 5th dimension – Prajwal Kavad · 2 years ago

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– Sandeep Bhardwaj · 2 years ago

Such things are not universally accepted yet. These have a multiple issues. You can search wikipedia for that.Log in to reply

In which college did you study sir ? Just, my curiosity's getting the better of me . – Azhaghu Roopesh M · 2 years ago

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@Azhaghu Roopesh M – Sandeep Bhardwaj · 2 years ago

Actually I didn't go to the college. :PLog in to reply

I want to know how to solve calculus and algebra easily, can you help me ? – Anish Harsha · 2 years ago

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@Anish Harsha – Sandeep Bhardwaj · 2 years ago

Problem solving can be learnt only by solving the problems. Keep doing problems and learn from every problem. After some time, you will find that you've been leveled up and you can solve the problems more easily. All the best.Log in to reply

– Marc Vince Casimiro · 2 years ago

I agree. I learned almost all Number Theory, Calculus and Combinatorics here at Brilliant!Log in to reply

– Anish Harsha · 2 years ago

Thnx, I too learn like this. At start, I knew only little. But soon, I knew how to solve problems and soon level uped in many topics.Log in to reply

– Prajwal Kavad · 2 years ago

Calm down ,there is lots of time left and all u have to do is just solve a lots of problems and find the reason behind why ur solving in a particular way , why not a easier step. And then whenever u learn a formula or get to see any results from problems then find how the formula was created and how the inventer or dicoverer would have reached to the result ; that is enough to take u to great heights; all the bestLog in to reply

Sir m preparing for jee It's only about 6-7 months left of mains nd I hv to prepare for my 12 boards also Would you give me some suggestion on how should I study to balance both things? Thanks – Abhimanyu Singh · 1 year, 7 months ago

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Hai All the best – Prakash S · 1 year, 7 months ago

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Sir, I would like to ask you how to find the no. of digits in the expansion of a given number when the power is not an integral multiple or power of 10.. For example we can use logarithms to find the no.of digits in \(8^{1000}\)

\(10^{(x-1)}\) will have x digits and \(10^{(x-1)}\) = \(8^{1000}\)

And by taking log on both sides, (x-1) = log \(8^{1000}\) x-1 = 1000 \times log 8

And, x=904 (on Approximation)

But, Consider this question, If \(x^{2}\) + x + 1 = 0 then the no.of digits if number \(x^{3}\) + 1/\(x^{3}\) the whole raised to 2014.... Please Solve it ... – Anirudh Krishna · 2 years ago

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We can see x=0 will not be a factor of it, thus it is legal to divide it by x.

\( \implies x + \dfrac{1}{x} = -1\)

\( -1^3 = x^3 + \dfrac{1}{x^3} + 3\times x\times \dfrac{1}{x}(x + \dfrac{1}{x})\)

\( x^3 + \dfrac{1}{x^3} = 2\)

Now do the same as you proceeded for \(8^{1000}\)

Can you explain me why taking logarithms helps us to find the number of digits? – Megh Choksi · 2 years ago

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– Sandeep Bhardwaj · 2 years ago

Nice one. Also \(\omega\) is the root of the equation \(x^2+x+1=0\) and \(\omega^3=1\) \(\implies x^3+\dfrac{1}{x^3}=2\).Log in to reply

\( 10^{1} = \) The first appearance of a 2 digit number

. . . .

So we basically equate the number of digits by taking log on both sides .

Hope this was satisfactory :) – Azhaghu Roopesh M · 2 years ago

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– Megh Choksi · 2 years ago

Nice logic +1Log in to reply

@megh choksi @Azhaghu Roopesh M for your valuable responses to the problem. – Sandeep Bhardwaj · 2 years ago

ThanksLog in to reply

– Anirudh Krishna · 2 years ago

I hope @Azhagu Roopesh M explained it already .... ( However It was me who was late :P ) Anyway Thanx for the solution bro _ /\ _ !Log in to reply

out!

Check tisPlease participate and reshare. – Rajdeep Dhingra · 1 year, 10 months ago

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Sir, where do you teach?? – Prashant Ramnani · 2 years ago

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@Prashant Ramnani – Sandeep Bhardwaj · 2 years ago

I teach at an IIT-JEE institute named SAKSHAM IIT PMT, New Delhi.Log in to reply

– Prashant Ramnani · 2 years ago

okay,thanksLog in to reply

https://brilliant.org/problems/how-to-make-a-triangle/?group=ZTtB5mdNQtMx&ref_id=539978 – Visakh Radhakrishnan · 2 years ago

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– Sandeep Bhardwaj · 1 year, 7 months ago

Yup, what can I do for you ? What's the problem you're facing ?Log in to reply

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– Sandeep Bhardwaj · 1 year, 7 months ago

Well, so what do you want to know about that ?Log in to reply

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– Sandeep Bhardwaj · 1 year, 7 months ago

You are most welcomed to comment here, and tell me your problem and I assure you that you will overcome that problem very soon.Log in to reply

– Sakshi Rathore · 1 year, 6 months ago

sir i m very weak in geometry and i can't complete my mcq paper on time how could i improve my speed as well as my geometryLog in to reply

– Rajdeep Dhingra · 1 year, 6 months ago

My speed in Geometry is less but not very. I want to improve it.Log in to reply

@Rajdeep Dhingra – Sakshi Rathore · 1 year, 5 months ago

oh but u are in level 5..u r a genius in geometry can u tell me some important theorum that help me improve my geometryLog in to reply