# My Rough Work during the interview for CSA Research IISc

This is the rough work I did during the interview, the question were added in a separate post (https://gateoverflow.in/blog/10800/iisc-csa-research-interview-questions)  as I did not want to reveal hints in the original post (perhaps open the questions side by side to make sense of the following rough work)

CSA prof 1 (not everything is correct in my solution):

CSA prof 2 (not everything is complete in my rough work):

Feel free to ask in the comments, I am sure, some of the answers are not complete as the prof. just asked to explain some bits instead of writing everything down.

posted Aug 25, 2020
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Can you please explain what happened after $k \geqslant \frac{a^{2}+ b^{2} + 2ab}{a^{2} + b^{2}}$ in the third problem of the second image.

Thanks.

Yeah, I was struggling and told him that I don't think it can be simplified any further. Then I said if $k$ were 1, it can never be $true$ . Then I said if $k$ were 2, then..$2ab <= a^2 + b^2$ ,  the professor then said, I will give you a hint now, take it to the other side. I said okay, then it became $(a-b)^2 >= 0$  which was always $true$. So, for $k=2$ , the inequality is always correct.

On the rhs (a + b)^2 should become (a^2 + b^2 + 2ab). Why have you written (a^2 + b^2)? Not able to understand.

@nvs16 It was a different question, the first line was a simple question, where the answer was for any $k$ greater than equal to $1$, the inequality holds. So, then the professor asked me to change the problem to $a^2 + b^2 + 2ab <= k(a^2 + b^2)$ .. its a different problem, I have written it in the post as well https://gateoverflow.in/blog/10800/iisc-csa-research-interview-questions , perhaps you missed it. There is no perfect demarcation in the rough work I did hence I can understand the confusion though

How is this possible .. $k*(a+b)^2 = k*(a^2+b^2)$
@Priyansh Singh, k on both sides? why?

he did like this above

$a^2+b^2+2ab \leqslant k(a+b)^2$

$a^2+b^2+2ab \leqslant k(a^2+b^2)$

now look at only RHS in the equation how the square got split inside ?

They are two very different problems.

First problem was so easy that it didn't need any calculation. Second one starting on the second line is harder.