3
nihit.desai
·2009-04-01 09:50:06
1st que :
given expression is same as coefficient of x^6 in the expansion of :
((1+x)10 - 1)5
Expand and get the answer...
341
Hari Shankar
·2009-04-01 19:04:24
yeah. the equality condition in AM-GM requires the summands to be equal.
Here's a method that uses only Cauchy Schwarz only
(1^2+1^2+1^2+1^2)(4x^2+4x^2+y^2+z^2) \ge (2x+2x+y+z)^2 by CS
Further (2x+2x+y+z) \left(\frac{1}{2x} + \frac{1}{2x} + \frac{1}{y} + \frac{1}{z} \right)\ge (1+1+1+1)^2 = 4^2 again by CS [u can also use AM-HM or AM-GM)
Therefore we have
4(4x^2+4x^2+y^2+z^2) \left(\frac{1}{2x} + \frac{1}{2x} + \frac{1}{y} + \frac{1}{z} \right)^2 \ge \left[ \left(2x+2x+y+z) \left(\frac{1}{2x} + \frac{1}{2x} + \frac{1}{y} + \frac{1}{z} \right) \right]^2 \ge 4^4
and hence the inequaliyty holds
3
nihit.desai
·2009-04-01 13:53:22
othrwise i had thought abt sumthing else...
if u c carefully, in the first expression, equality holds whn 8x2 = y2 = z2 = some k
in the second one, equality holds whn x-1 = y-1 = z-1 = some constant
now 54 is the min value only if both are satisfied simultaneously, ie for the same values of x, y and z...which is clearly not possible...
so i think 64 is 100 % correct ans..
3
nihit.desai
·2009-04-01 13:50:22
hmmm....gud yaar...i think its now clear !!!
11
Subash
·2009-04-01 10:39:13
I think i found it
in nihit's solution
Second part
squaring the inequality can reverse the inequality because its value can be <1 also
so that solution will not be valid
:D
1
The Race begins...
·2009-04-01 10:32:35
hm ! even the explanation given in solution booklet matches with the one given by prophet sir. :)
But still, why is AM-GM not working here !?
11
Subash
·2009-04-01 10:26:49
I think he meant at what values do we get the value 54?
the values of x,y,z
1
The Race begins...
·2009-04-01 10:24:36
well, i'm sorry. Even the condition x,y,z>0 is given in the question. But still, answer is given as 64, not 54 ! dont know why?
prophet sir, please explain ur statement : "Also the equality condition is not satisfied, so that 54 is definitely a lower bound , but not the greatest lower bound for this expression."
i think this statement has everything that declares 64 as correct answer. !
but i couldn't get it clearly. please explain sir !
3
nihit.desai
·2009-04-01 10:13:38
surely the inequality u've used is far more sound mathematically...but whr is the concept going wrng with AM - GM thing ??
3
nihit.desai
·2009-04-01 10:11:27
see...
(8x2 + y2 + z2)/3 ≥ (8x2 * y2 * z2 )1/3
now calculating..v get :
(8x2 + y2 + z2) ≥ 6 (xyz)2/3 ... (1)
Second part :
(x-1 + y-1 + z-1 )/3 ≥ (1/xyz)1/3
again shfting 3 to rite and squaring...v get
(x-1 + y-1 + z-1 )2 ≥ 9 (1/xyz)2/3 ... (2)
multiplying (1) and (2) we get the result naa...strangely it comes out to be independent of x, y and z...
plzz sum1 temme wats wrong...
341
Hari Shankar
·2009-04-01 10:06:54
Well, I dont know if there is any other method. the answer 54 is got by applying AM-GM on the two factors. But that needs x,y and z to be positive. Also the equality condition is not satisfied, so that 54 is definitely a lower bound , but not the greatest lower bound for this expression.
3
nihit.desai
·2009-04-01 10:05:25
hey...u welcum..but wait !...
i m still not convinced y that ans cannot b 54...
i mean, even i got that whn i workd with AM GM inequality...i cant find my mistake...
1
The Race begins...
·2009-04-01 10:01:28
thanx nihit :)
well, it's done then.
Thanq everyone who involved in this forum and also the one's who tried these problems. Thanx once again :)
11
Subash
·2009-04-01 07:43:10
Q3
my answer B(:D)
tanAtanBtanC=6
tanA+tanB+tanC=6
tanB=2/3 tanC
Substitute tanB in the first two equations
and solve for tanA
1
The Race begins...
·2009-04-01 09:50:02
thanx a lot sir. :)
but if that's not applicable for jee syllabus, how to solve such problems within jee level.?
This question was asked in our "iit-jee grand test".!
Did they expect us to solve it by this method or is there any other simple method too (which is within jee level) ?
341
Hari Shankar
·2009-04-01 09:44:53
For the inequality what is needed is an inequality known as Hölder Inequality. It is a generalisation of Cauchy Schwarz inequality
I will give it in a form that is useful in this problem:
\left(x_{11}^3 + x_{12}^3 + x_{13}^3 \right)^{\frac{1}{3}} \left(x_{21}^3 + x_{22}^3 + x_{23}^3 \right)^{\frac{1}{3}} \left(x_{31}^3 + x_{32}^3 + x_{33}^3 \right)^{\frac{1}{3}} \ge \left(x_{11}x_{21}x_{31} + x_{12}x_{22} x_{32} + x_{13}x_{23}x_{33} \right)
So when you apply it, we get
\left(8x^2+y^2+z^2 \right) \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right)\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \ge (2+1+1)^3 = 64
Note: In JEE Cauchy Schwarz itself may not find any application. So Hölder may be overkill as far as JEE preparation is concerned.
1
The Race begins...
·2009-04-01 09:34:02
The answer given is 64 itself, but most of my friends argued it to be 54.!
i couldn't have a detailed discussion with them, so posted it here if anyone could solve it in an easier/compatible way.!
brother, u post ur generalization thought, it may be right as the answer is correct :)
but please, post it in a detailed way, as i'm newly exposed to this cauchy schwarz !
62
Lokesh Verma
·2009-04-01 09:09:27
The Race...
I seem to have lost my own arguement..
I think what i have done is wrong..
what is the right answer??
For a moment i thought that the generalization of cauchcy schwarz wud work.. but now i am not feeling the same..
1
The Race begins...
·2009-04-01 08:26:06
@Nishant bro. Thanx for the links. I got the inequality now. :)
but, i didn't get ur soluton yet ! please post 2-3 intermediate steps at the point where u wrote "now apply cauchy schwarz". that will be helpful for everyone, especially me ! please
11
virang1 Jhaveri
·2009-04-01 08:16:20
=50!/6!44! - 5!/4!.40!/6!34! + 5!/2!3!.30!/6!24! - 5!/3!2!.20!/6!14! + 5!/4!.10!/6!4!
=1/6!(50!/44!- 5!/4!.40!/34! +5!/2!3! . 30!/24! - 5!/3!2!.20!/14! + 5!/4!.10!/4!)
=1/6!([50*49*48*47*49*45]-[40*39*38*37*36*34] + [2*5*30*29*28*27*26*25] -[5*2*20*19*18*17*16*15] + [10*9*8*7*6*5])
10/6!([5*49*48*47*49*45]-[4*39*38*37*36*34] +[*30*29*28*27*26*25] -[*20*19*18*17*16*15] + [*9*8*7*6*5]
50/6!([49*48*47*46*45]-[4*39*38*37*7.2*34] +[6*29*28*27*26*25] -[4*19*18*17*16*15] + [*9*8*7*6])
200/6!([49*12*47*46*45]-[39*38*37*7.2*34] +[6*29*7*27*26*25] -[19*18*17*16*15] + [*9*2*7*6])
1800/6!([[49*12*47*46*5]-[13*38*37*2.4*34] +[6*29*7*3*26*25] -[19*6*17*16*5] + [**2*7*6])
=14400/6!([49*3*47*23*5]-[13*38*37*0.3*34] +[3*29*7*3*13*12.5] -[19*6*17*2*5] + [3.5*3])
i am getting mad
11
Subash
·2009-04-01 08:10:47
bhaiyya
how do you take it from there in post two
im not that good at this inequality
but isnt that a whole square in the second term
so explain a bit more :)
62
Lokesh Verma
·2009-04-01 08:06:23
http://targetiit.com/iit_jee_forum/posts/22nd_december_2008_1146.html
http://targetiit.com/iit_jee_forum/posts/inequalities_3234.html
read these two..
62
Lokesh Verma
·2009-04-01 08:04:07
sorry i explained the above very clumsily.. not much was clear..
but let me try to find some good explanation..
1
The Race begins...
·2009-04-01 08:01:37
@Nishant bro, can u briefly explain the basic definition of cauchy schwarz and explain it's applications in jee level.
I've done a lot of google search but couldn't get any conclusion. so please explain me it in ur words.!
62
Lokesh Verma
·2009-04-01 07:54:05
(8x^{2}+y^{2}+z^{2})(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})^{2} \\=(4x^{2}+4x^2+y^{2}+z^{2})(\frac{1}{2x}+\frac{1}{2x}+\frac{1}{y}+\frac{1}{z})^{2}
>=
now apply cauchy schwarz..
8x^3=y^3=z^3
2x=y=z=t
4t^2 x (4/t)^2 = 64
1
The Race begins...
·2009-04-01 07:50:20
u r correct :)
solve the other two also. please!