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        riddles >> medium >> 3 JEE matrix problems
        
(Message started by: towr on Jul 14^th, 2018, 1:08pm)

Title: 3 JEE matrix problems
Post by towr on Jul 14^th, 2018, 1:08pm

On behalf of navdeep1771 I'm posting the follow problems from the 2010 Joint Entrance Exam

Title: Re: 3 JEE matrix problems
Post by navdeep1771 on Jul 15^th, 2018, 4:08am

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).

Title: Re: 3 JEE matrix problems
Post by towr on Jul 15^th, 2018, 7:18am

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.
[hide]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[/hide]

Title: Re: 3 JEE matrix problems
Post by navdeep1771 on Jul 18^th, 2018, 12:30am

[hide] That's Correct.
Well mathematical proof goes like this:-
We must have a^2 − b^2 = kp
⇒ (a + b) (a − b) = kp
⇒ either a − 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 ⇒ a, b has p − 1
∴ Total number of matrices = p + p − 1
= 2p − 1. [/hide]

But what about '43' and '44'?

Title: Re: 3 JEE matrix problems
Post by towr on Jul 19^th, 2018, 10:55pm

[hide]
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
[/hide]
;D

Title: Re: 3 JEE matrix problems
Post by navdeep1771 on Jul 20^th, 2018, 8:15am

[hide] @towr
You are insane. [/hide]