*** Q3 is dbt only.....rest i have done... ***
Q1 Determine sign of roots of equation
x2+(3-2a)x-3a+2=0
Q2 If a,b,c are odd integers prove that both roots of ax2+bx+c=0 can not be rational
Q3 ax2+bx+c=p has both roots integers .Show that ax2+bx+c=2p can necer ahve both roots inetgers where p is
prime no.≠a and a≠1
Q4 integral sons of x4-y4=3789106
39come on dude 1 and 2 have been done many many no: times over here:
2) let the roos be p/q and r/s
then the quadratic become s
(x-p/q)(x-r/s)=0
comparing this with ax2+bx+c we get
qs=a -(ps+rq)=b and pr=c
this means q,s should be odd because their product is an odd integer a.
even p,r should be odd in a similar way
this mean s ps+rq will be evn .. but this is equal to -b whihc is an even integer..
hence a contradictoon..
therefore roots cant be positive integers
39can u post till where u could move on with the solution of `1`.
ar atleast post waht is the problem u r facing in q 1?
24got Q1,2,4......wasnt able to reach ans in morning ..so psoted here...
but i am still looking for Q3...plz help guys...
1The new equation is ax2 + bx + c - 2p = 0
product of the roots -- ( c - 2p ) / a = [ ( c - p ) / a ] - [ p / a ]
an integer if roots are both integers .
NOw , ( c - p ) / a = product of the roots of ax2 + bx + c = p ,
so ( c - p ) / a = integer since the roots are integers.
so it follows that p / a should be an integer also.
But that cannot happen as p ≠a and a ≠1 . So both roots cannot be integers.
341(@eureka, are a,b,c natural numbers. You should have stated this)
Solution: From the given statement we have
ax^2+bx+c-p = a(x-m)(x-n) for integers m and n
Now, if for some integer q, we have aq^2+bq+c=2p, then
aq^2+bq+c-p = a(q-m)(q-n) = p \Rightarrow a|p which
is not possible if a \notin \{p,1\}
24Thanks sir and math musing
@Sir,I posted exact question...maybe the question setters could ahve missed that
341which is usually the case when the problem has been cut 'n' paste by someone who hasnt bothered to solve it him/herself
341i dont have much idea of which coaching centres do this. But in the pressure to make their tests seem "tough" as compared to others in the business they seem to be randomly inserting olympiad problems
This confuses the students on two counts:
(1) The level of problems to be expected - Olympiad orientation is different from IIT's. Naturally because olympiads are testing you in only one subject. For IITs they need proficiency in three subjects. So they factor in this lack of depth in subjects. Also the topics they cover like coord geo, vectors, real analysis are typically not in olympiads. Likewise inequalities, number theory etc. dont find a place in JEE/AIEEE etc.
Since the toughness level also has a bearing on how much time a student needs to devote to the subject, those who design test series must keep this in mind.
Now, there certainly is a case for well-chosen olympiad problems that can be discussed in a class because they can help clarify many concepts, improve one's logical facilities and really bring the subject alive (Pigeonhole principle is a sure-fire spine tingler according to me).
(2) Time gets "wasted" on the problem - This involves my pet peeve that usually the guys who set the papers have never really attempted to solve the problem themselves. In emergencies, the fully worked out solution is always close at hand. Otherwise they would know the time it would take for the student to work out the problem. More seriously, they are not teaching the student how to think.
Frankly speaking, the IIT-JEE is a heavily dumbed down version of what it used to be in the 1990's. For us old timers some of the qns are almost oral. I feel the coaching centres have really hyped up things to make it look as if students could never attempt the exam without their help!
24In full agreement with you Sir.....
thats what Fiitjee used to do till last yr..(this yr they have improved a bit..just a bit) .....but this yr its BMAT which is following fiitjee's path......
its such a disgusting feeling to spend 5k on all this test series thing and get only and only demotivated from them at the end ...just becoz the standard of ques isnt upto the mark[2]
23"Frankly speaking, the IIT-JEE is a heavily dumbed down version of what it used to be in the 1990's. For us old timers some of the qns are almost oral. I feel the coaching centres have really hyped up things to make it look as if students could never attempt the exam without their help!"
Sir apne to mere mu ki baat cheen li
Professors of IIT B themselves are agreeing that most of the crowd coming in since past 5 yrs isnt upto the level of IIT of 90's.
1the best format was previous one of screening and mains...
MCQ mey "tukka" factor aa jaata hai.. ;)
waisey this yr maths paper of jee looked lik a typical fiitjee aits paper...with some
putnam probs dumped into it !
1''rankly speaking, the IIT-JEE is a heavily dumbed down version of what it used to be in the 1990's. For us old timers some of the qns are almost oral'''
wery demoralizing....