TargetIIT

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

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

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

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

yup i saw u posted der :P

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]

thanks a lot prophet sir and che..

these were the unit test questions at aakash for another batch....(1 yr course)

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