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This post was updated on Jun 07, 2014; 9:29pm.
57) S(alpha) is a point on the curve which means that it is an ordered pair in the coordinate system.
Now consider the following. S1=(1/x) S2=(1/x^2) S3=(1/x^3) . . . . S(n)=(1/x^n) Now consider the points in S1={(x,y) s.t xER, y=(1/x)} Similarly S2={(x,y) s.t. xER and y=(1/x^2)}. Also S3={(x,y) s.t. xER and y=(1/x^3)} . . . for S(n)={(x,y) s.t. xER and y=(1/x^n)} Now if u consider the set S1⋂S2⋂S3⋂...⋂S(n) the only common point is for sufficiently large x, y=0 which means that the intersection contains only one point which is the common point of convergence of each series i.e 0, thus the only elements are is (inf,0) and (-inf,0), however the answer given is a single elements...Plz do check whether the procedure seems alright or not and where am i wrong..??
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OK thanks subhayu I got it now....and what about q55...I m not getting interpretation of g(1,0/n)...does fn= n*1 if xEA and fn=n*0 if x not belongs to A...? Am I going right?
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In reply to this post by Granpa Simpson
Yes Vaibhav i think the procedure for 57 is ok..it will be a single point only because the question says "for positive real number x"...
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In reply to this post by Dreyfus
I think you are going in the right way...!!!!!
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56..i am getting constant sequence(which is the key ans)
for any arbitary integer k, fk (x)=k if 0<x<1/k = 0 otherwise integration of fk over its domain is itgretate from 0 to 1/k of kdx=1. so the sequence will be 1+1+1+1....a cosntant sequence guys please help me with the working of question 57 of 2012 ..i am getting option a though the ans is c. |
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Pls explain 55 ...m stuck
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http://discussion-forum.2150183.n2.nabble.com/DSE-2012-Q-57-td7591897.html#a7591969
Akshay solved question 55 here |
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In reply to this post by Dreyfus
Q57: The different possible sets are 1/x, 1/x^2, 1/x^3,...
To find the intersection we equate 1/x = 1/x^2 which gives x=1. If we do it for any of the sets above we get x^n=1 ie x=1 (as the domain is defined for positive a). So that single point is x=1 |
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In reply to this post by Granpa Simpson
Hey Subhayu.. I follow your logic for the rest, but how do you know this point is unique? How can we be certain these curves dont intersect elsewhere as well?
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Yes Ridhika dats a valid question..??infact i missed (1,1)..this will be common for all the sets..so by this logic it shud contain exactly two points, but not any other point except dat..because consider two curves y=(1/x^2) and y=(1/x^3)..except x=1, you cannot find any value for which the ordered pair will be same for both the curves..similarly if u consider for all the curves you will find the same..however according to my logic there will be exactly two points..!!!
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Hey Subhayu does (infinity, 0) is really a point? I think that single point mentioned by you is _(1,1)
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Yes Vaibhav actually (inf,0) cannot be a point in a rectangular ordered coordinate system..so (1,1) is the only point....
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Okay guys.. I see why 1,1 is a point of intersection. But again how are we sure that no point corresponding to x E (1,infinity ) is also an intersection?
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From: "subhayu [via Discussion forum]" <ml-node+[hidden email]> To: "Ridhika" <[hidden email]> Subject: DSE 2008 Q55, Q56, Q57 Date: Wed, Jun 11, 2014 12:39 AM Yes Vaibhav actually (inf,0) cannot be a point in a rectangular ordered coordinate system..so (1,1) is the only point....
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Ridhika even (1,infinity) is not a point in the coordinate system...!!!!
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Yaya ! I know that ! I'm just asking how to show that (1,1) is unique?
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From: "subhayu [via Discussion forum]" <ml-node+[hidden email]> To: "Ridhika" <[hidden email]> Subject: DSE 2008 Q55, Q56, Q57 Date: Wed, Jun 11, 2014 12:51 AM Ridhika even (1,infinity) is not a point in the coordinate system...!!!!
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Ridhika I have just used the logic that since the exponents of x are different you cannot get a point except (1,1) which is common to all the sets..may be a counter example can disprove this fact..but i dont think there exists one..however there must be a rigorous mathematical proof related to this fact which i am unaware of..but if you insist for such a proof i will surely look in to it and let u know (obviously if it is within the scope of my brain)...
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Hehe .. no no I don't insist! I just kept thinking I'm missing something! Thanks a lot :)
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From: "subhayu [via Discussion forum]" <ml-node+[hidden email]> To: "Ridhika" <[hidden email]> Subject: DSE 2008 Q55, Q56, Q57 Date: Wed, Jun 11, 2014 1:17 AM Ridhika I have just used the logic that since the exponents of x are different you cannot get a point except (1,1) which is same..may be a counter example can disprove this fact..but i dont think there exists one..however there must be a rigorous proof
related to this fact which i am unaware of..but if you insist for such a proof i will surely look in it and let u know...
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In reply to this post by Granpa Simpson
Gracias @ Subhayu
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Mrittik: Si...
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