341no i didnt. but i remember the qn from my student days. was in TMH if my memory serves me right
The following is a system of quadratic equation with a,b,c being real and a≠0.....
ax12+bx1+c=x2
ax22+bx2+c=x3
ax32+bx3+c=x4
ax42+bx4+c=x5
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axn-12+bxn-1+c=xn
axn2+bxn+c=x1
Answer the following questions....
Q1) The no of real solutions if (b-1)2<4ac ...
Q2) The no of real solutions if (b-1)2>4ac ...
Q3) The no of real solutions if (b-1)2=4ac ...
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16 Answers
62this is an awesome question...
If i remember correctly, it was solved by prophet sir, sometime back.. let me dig the post out!
34162it is from Arihant..
SOmeone asked me recently.. I had a hard tiem to explain why this is not obviously 1)
and it took me a lot of trouble to make him understand that the question was not obvious!!
3sir adding the eqns i substitute y=Σxi
we will have a quadratic eqn the quadratic eqn in y have no roots
62no sankara.. this is not the right way..
see the expression again..
11The main hitch in solving this question is identifying that (b-1)2-4ac is the discriminant....
62@adiya .. yes...
But that alone will not help!
what I mean that is what gave me a tough time to explain to one of the students that (b-1)2-4ac does not mean that none of these equations have a solution!
Because what he was arguing is that none of these equations have a solution.. hence the whole set of these equations cannot have a solution!
11yes definitely... But any hints on solving this toughie ?
62aditya... express teh sum of all the expressions above as a sum of quadratics
11axi2+bxi+c=xi+1
If (b-1)2-4ac<0....then subtract xi from both sides....
Now add all the equations after this operation.
L.H.S is not 0...(meaning it's either >0 or <0, depending on a)....while R.H.S=0
Thus no solutions.