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Share/Save/Bookmark Login/ Register to Bookmark Topic : "algebra" Started by Asish

Asish

#1 Posted 8:25pm 03-02-10  

algebra

Q1. The value of
[im]http://codecogs.izyba.com/gif.latex?%5E%7B100%7DC_%7B0%7D%5C%3B%20%5E%7B200%7DC_%7B100%7D%20-%20%5E%7B100%7DC_%7B1%7D%5C%3B%20%5E%7B199%7DC_%7B100%7D%20+%20%5E%7B100%7DC_%7B2%7D%5C%3B%20%5E%7B198%7DC_%7B100%7D-%5E%7B100%7DC_%7B3%7D%5C%3B%20%5E%7B197%7DC_%7B100%7D+.......+%5E%7B100%7DC_%7B100%7D%5C%3B%20%5E%7B100%7DC_%7B100%7D[/im]

Q2. If [im]http://codecogs.izyba.com/gif.latex?%5Cleft%7Caz+b%5Cbar%7Bz%7D+c%20%5Cright%7C%20%3D%20%5Calpha%20%5Cleft%7Ca%20%5Cright%7C%5Cleft%7Cz-z_%7B1%7D%20%5Cright%7C[/im]
a≠0, a=[o]b[/o],c ε R, where z[ss]1[/ss] is a given non-zero complex number and α ε R[p]+[/p], then locus of z may be
(a) Parabola  (b) Ellipse   (c) Hyperbola    (d) Circle

Q3. If z lies on the unit circle with centre at origin then the maximum value of  [im]http://codecogs.izyba.com/gif.latex?I_%7Bm%7D%281-%5Cbar%7Bz%7D+z%5E2%29[/im] is

sadly life is not masti in IITK with MTH 101 making first yr hell fr all freshers (no exception)... even got seniors repeating the course  Edited on 09:39am 04-02-10    

Che

#2 Posted 10:34am 04-02-10  

Re: algebra

2)
parabola

ellipse

hyperbola ??

since here a=[o]b[/o]
so [im]http://codecogs.izyba.com/gif.latex?%5Cleft%7Ca%20%5Cright%7C%3D%5Cleft%7Cb%20%5Cright%7C%3D%5Cleft%7C%5Cbar%7Bb%7D%20%5Cright%7C%3D%5Cleft%7C%5Cbar%7Ba%7D%20%5Cright%7C[/im]
and since
the given eq is

[im]http://codecogs.izyba.com/gif.latex?2%5Cleft%28%5Cfrac%7B%5Cleft%7C%5Cbar%7Bb%7Dz+b%5Cbar%7Bz%7D%20+c%5Cright%7C%7D%7B2%5Cleft%7Cb%20%5Cright%7C%7D%20%5Cright%29%3D%5Calpha%20%5Cleft%7Cz-z_%7B1%7D%20%5Cright%7C[/im]

[im]http://codecogs.izyba.com/gif.latex?%5Cleft%28%5Cfrac%7B%5Cleft%7C%5Cbar%7Bb%7Dz+b%5Cbar%7Bz%7D%20+c%5Cright%7C%7D%7B2%5Cleft%7Cb%20%5Cright%7C%7D%20%5Cright%29[/im]=distance of z from line [im]http://codecogs.izyba.com/gif.latex?%5Cbar%7Bb%7Dz+b%5Cbar%7Bz%7D%20+c[/im]

and [im]http://codecogs.izyba.com/gif.latex?%5Cleft%7Cz-z_%7B1%7D%20%5Cright%7C[/im] is distance of point z from z[ss]1[/ss]

hence we can clearly see here [im]http://codecogs.izyba.com/gif.latex?%5Cfrac%7B2%7D%7B%5Calpha%20%7D[/im] is the eccentricity of conic

and z[ss]1[/ss] is teh focus and  [im]http://codecogs.izyba.com/gif.latex?\bar{b}z+b\bar{z}+c=0[/im] is the directrix

now [im]http://codecogs.izyba.com/gif.latex?e%3D%5Cfrac%7B2%7D%7B%5Calpha%20%7D[/im]

wen [im]http://codecogs.izyba.com/gif.latex?%5Calpha%20%3D2[/im] it is a parabola

wen [im]http://codecogs.izyba.com/gif.latex?%5Calpha%20%3E2[/im] it is an ellipse

wen [im]http://codecogs.izyba.com/gif.latex?%5Calpha%20%3C2[/im] it is a hyperbola
Dont Mess With Che~!  Edited on 11:07am 04-02-10    

Che

#3 Posted 10:46am 04-02-10  

Re: algebra

3) put z=cosθ+isinθ

then u hav to find max value of  2sin(3θ/2)cos(θ/2)

i guess i m getting 1.76 something
Dont Mess With Che~!  Edited on 11:09am 04-02-10    

theprophet

#4 Posted 11:01am 04-02-10  

Re: algebra

Where are these qns from? BTW, I've just seen a beautiful soln to #1 on Mathlinks - http://www.mathlinks.ro/viewtopic.php?t=329319

Do read it.

Someone's posted Q3 is disguised form asking what is max of sin θ + sin 2θ
   

Che

#5 Posted 11:39am 04-02-10  

Re: algebra

yup i saw u posted der :P
Dont Mess With Che~!    

Che

#6 Posted 1:13pm 04-02-10  

Re: algebra

check this for another way

for 1st q

http://www.mathlinks.ro/viewtopic.php?p=1762870#1762870

it turns out to be Vandermonde's Convulution identity

got to know abt it today only [3]

[hide]btw this aint related to jee by miles[/hide]
Dont Mess With Che~!    

Asish

#7 Posted 4:46pm 04-02-10  

Re: algebra

thanks a lot prophet sir and che..

these were the unit test questions at aakash for another batch....(1 yr course)
sadly life is not masti in IITK with MTH 101 making first yr hell fr all freshers (no exception)... even got seniors repeating the course    

mavarick

#8 Posted 11:24am 05-02-10  

Re: algebra

[b]can someone solve [im]http://codecogs.izyba.com/gif.latex?sin%5Ctheta%20+sin2%5Ctheta[/im] with the help of Trigonometry[/b]
   
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