IIT JEE , AIEEE Forum - Mathematics , Algebra

Conjurer

1) If ax^2 + (b-c)x + a-b-c =0 has unequal real roots for all c

R then prove that either b<a<0 or b>a>0.

2) Prove that for -1/4 <a <0 , the equation ^{[2 ]}

(a + ^{[2 ]}

(a+x) )= x has exactly two real solutions of which one is 1 + ^{[ 2]}

(4a +1) /2

3) Show that if p,q,r and s are real no.s and pr=2(q+s) then at least one of the equations x^2 +px +q =0 , x^2 +rx +s =0 as real roots.

arpan1

all these ques. can b easily solved by use of calculus..

pls try them urselves

if not possible, i will post the solutions...

i give u the outlines of approach..

prove the first by counter-examaple

assume b<a<0 .... and prove it graphically..

second ques.. can b done via same approach

if u donot get anything, nudge me....

at the moment, i am busy for boards and will help u as soon as possible

Conjurer

My approach in 1 and 3 is the use of factorization but i cant seem to make factors such that I can use the given conditions.

And I have no idea about the second question.And I dont know how we can use calculus in the questions.

connectR

how to do??

Conjurer

I solved the last 2.Can anyone please help me with the first.If anyone wants the solutions to the last 2, I can post.

But here is the logic anyways:

2) Assume a+x = y and replace all x with y-a and square.Assume quadratic in a and solve for a.You will then get quadratic in y by squaring.

3) Squaring pr = 2(q+s) and then applying AM GM for q^2 and s^2 , we get p^2r^2 >= 16qs .

arpan1

rate me first and then i will do it

computer001

EDITED:
{hey pls dun rate all this mate!! i value mah rates :) n i think/hope tht they shud come 4 more valuable contributions...}

yeah arpan1 i was jus wonderin when u were gonna ask to be rated

arpan1

its my style of doing problems...hope u got it

anandghegde

begging for rates before solving the problem is a style?????? Razz

Conjurer

Does boink count as rate?

elastiboysai

that was funny conjurer
edited 2 avoid further disturbance

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