IIT JEE , AIEEE Forum - Mathematics , Algebra

neeraj_agarwal_1990

If

,

are the roots of x²+px +q=0 and also of x²ⁿ+pⁿqⁿ+qⁿ=0 and if

/

,

/

are the roots of xⁿ+1 +(x+1)ⁿ=0,then n is:

(a) an odd integer

(b) an even integer

(c) any integer

(d) none

puneet

hiiiii Neeraj ..

The question has not come up properly due to some problem .. u know these crosses.. what is in there place .. it is not clear .

I was wondering if u can post this qusetion again ..

I hope u will not mind ...

cheers

neeraj_agarwal_1990

alpha ,beta, alpha/beta ,beta/alpha

vash

would u plz put it once again i am not gettin u

vash

ok i got ur problem

i think in the second equation it's xⁿinstead of pⁿ

+

= -p

= q

the next equation has roots

ⁿ &

ⁿ

therefore

ⁿ +

ⁿ= -p ( let xⁿ=y ) ok

ⁿ=k and

ⁿ= m ( disgustin typin all that)

km=qⁿ

now (

/

)^n + (

/

+1)^n + 1 (since

/

is its root)

=k/m + (

+

)^n/

^n + 1 ( where k and m hav their usual meanings)

takin l.c.m

= {-p^n + (-p)^n}/

ⁿ

now if it is a factor then the above term should add upto zero therefore n is even

now that was a tough job to pen down all those xponentials huh..... plzzzzzzz give me rates if i deserve.

if u didn't get this i am there to sort it out

neeraj_agarwal_1990

But why are the roots of the second equation alpha^n and beta^n?

Its roots are given alpha and beta?I've understood the rest of the solution

Your answer is correct
thanks

ramyadiamond

since

and

are the roots of the second equation, hence the roots of "y"(=xⁿ) must be equal to (alpha)ⁿ and (beta)ⁿ, because y=xⁿ, and x=alpha,beta.

vash

ya i agree with ramya by symmetry if

&

are roots of equation x^2 + ax + c=0, then

^n &

^n should b the roots of the other equation.

neeraj_agarwal_1990

WHY???

If first polynomial is represented by 'y' ,then the other equation is not
y^n .

then where comes the symmetry??

ramyadiamond

but u r taking xⁿ=y and when u substitute this in the second equation, u get two roots of y,mind u, not of xⁿ, and u equate this to (alpha)ⁿ and (beta)ⁿ, bcoz y=xⁿ, and from the first equation, we know, x= alpha and beta

neeraj_agarwal_1990

I got it....thanks

vash

ok let me put it the other way

see alpha and beta are the roots of the second equation so alpha^2n + p^n*alpha^n +q^n =0 since it satisfies it. now this is a quadratic with roots alpha^n & beta^n

. so the roots are alpha^n and beta^n. i think u got my point!!!!!!

Ask & Discuss Questions with Community & Experts