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towr
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 3 JEE matrix problems   iit-jee-2010-1.png « on: Jul 14th, 2018, 1:08pm » Quote Modify

On behalf of navdeep1771 I'm posting the follow problems from the 2010 Joint Entrance Exam
 « Last Edit: Jul 14th, 2018, 1:09pm by towr » IP Logged

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navdeep1771
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 Re: 3 JEE matrix problems   « Reply #1 on: Jul 15th, 2018, 4:08am » Quote Modify

I was unable to post this image file. As it requires some minimum number of posts.
So towr posted this from my side.
Thanks towr
Well these 3 problems are new to everyone (including towr).
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towr
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 Re: 3 JEE matrix problems   « Reply #2 on: Jul 15th, 2018, 7:18am » Quote Modify

I hadn't heard of skew-symmetric matrices before. But from the definition I'd say it's either a red herring or we're missing a statement that we're operating modulo p.
On the other hand, it doesn't really matter. For odd p, a has to be 0 in a skew-symmetric matrix, because only then a=-a, and b = -c, so det(Tp)=b^2, which isn't divisible by p.
Now if we try p=5, the only answer that fits turns out to be D: 2p-1

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navdeep1771
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 Re: 3 JEE matrix problems   « Reply #3 on: Jul 18th, 2018, 12:30am » Quote Modify

That's Correct.
Well mathematical proof goes like this:-
We must have a^2 &#8722; b^2 = kp
&#8658; (a + b) (a &#8722; b) = kp
&#8658; either a &#8722; b = 0 or a + b is a multiple of p
when a = b; number of matrices is p
and when a + b = multiple of p &#8658; a, b has p &#8722; 1
&#8756; Total number of matrices = p + p &#8722; 1
= 2p &#8722; 1.

But what about '43' and '44'?
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towr
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 Re: 3 JEE matrix problems   « Reply #4 on: Jul 19th, 2018, 10:55pm » Quote Modify

Since a < p and p is odd, then 2a is only divisible by p when a = 0
If a= 0 then det (A) is a multiple p only if b=0 or c=0, which are 2p - 1 cases

This seems irrelevant, because it's exactly what's not asked in either question 43 or 44

However, together with 43 and 44 it covers all possible A,  so answer(43) + answer(44) + (2p - 1) must be p^3
Which means the answers to 43 and 44 should be C and D

 « Last Edit: Jul 19th, 2018, 10:57pm by towr » IP Logged

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navdeep1771
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 Re: 3 JEE matrix problems   « Reply #5 on: Jul 20th, 2018, 8:15am » Quote Modify

@towr
You are insane.
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