If `omega` is any complex number such that `z omega=|z|^(2)` and `|z-barz|+|omega+baromega|=4`, then as `omega` varies, then the area bounded by the locus of `z` is
A. `4` sq. units
B. `8` sq. units
C. `16` sq. units
D. `12` sq. units
Correct Answer - B
`(b)` `zomega=|z|^(2)`
`implieszomega=zbarz`
`impliesomega=barz`
`:. |z-barz|+|z+barz|=4`
`:. |x|+|y|=2`
which is a square `:.` Area `=8`sq. units
Given `|int_(a-t)^(a)f(x)dx|=|int_(a)^(a+t)f(x)dx|AAtepsilonR`
or `int_(a-t)^(a)f(x)dx=-int_(a)^(a+t)f(x)dx`
[since `f(a)=0` and `f(x)` is monotonic]
or `f(a-t)=-f(a-t)` (differentiating both side w.r.t `t`)
or `f(a+t)=-f(a-t)=x` (say)
or `t=f^(-1)(x)-a`..........2
and `t=a-f^(-1)(-x)` ............3
From equations 3 and 2...
It is a point on hyperbola `x^(2)-y^(2)=1`.
Area `(PQRP) = 2 underset(1)overset(e^(t_(1))+e^(-t_(1)))overset(2)(int)ydx = 2 underset(1)overset(e^(t_(1))+e^(-t_(1)))overset(2)(int)sqrt(x^(2)-1)dx`
`= 2[x/2sqrt(x^(2)-1)-1/2l(x+sqrt(x^(2)-1))]_(1)^(e^(t_(1))+e^(-t_(1))) = (e^(2t_(1))-e^(2t_(1)))/(4)-t_(1)`
Area of `DeltaOPQ = 2 xx 1/2((e^(t_(1))+e^(t_(1)))/(2))((e^(t_(1))-e^(t_(1)))/(2)) = (e^(2t_(1))-e^(-2t_(1)))/(4)`
`:.` Required...
Correct Answer - C
`(c )` Let `z=r(costheta+isintheta)`
Then the given equation is `r^(2)+(4)/(r^(2))-2.2cos2theta-16=0`
`implies r^(4)-4r^(2)(cos2theta+4)+4=0`
`implies r^(2)=2(cos2theta+4)+2sqrt((cos2theta+4)^(2)-1)`
The maximum value is obtained when `cos2theta=1`
`:.` The maximum value of `r^(2)=10+2sqrt(24)`...
Correct Answer - C
`(c )` As `|(aalpha+bbeta+cgamma)/(a beta+bgamma+calpha)|=1`
`implies` When `alpha`, `beta`, `gamma` are different, then number of triplet `(alpha,beta,gamma)=` permutation of `1`, `omega` and `omega^(2)=6` and when `alpha-beta=gamma`, number...
Correct Answer - C
`(c )` We have `||z_(1)|-|z_(2)|| le |z_(1)-z_(2)|` and equality holds only when `argz_(1)=argz_(2)`
`implies||z-w|-|z-w^(2)|| le |w^(2)-w| le sqrt(3)` and equality canhold only when `|z|=2` and not when...
Correct Answer - C
`(c )` Given `(1+barz)/(z)` is real `implies(1+barz)/(z)=(1+z)/(z)`
`impliesbarz+barz^(2)=z+z^(2)implies(barz-z)+(barz-z)(barz+z)=0`
`implies(barz-z)(1+barz+z)=0`
So either `barz=z(z ne 0)` or `z+barz+1=0`
`implies y=0` or `x=(-1)/(2)` but excluding origin.
Correct Answer - C
`(c )` Centre `(5,0)`, radius `=5sqrt(2)`
`:.` Equation of the circle `(x-5)^(2)+y^(2)=50`
For `x=1`, `y^(2)=34`.
Total number of integral points `=(5+5+1)`.
Similarly for `x=2,3,…..12`
Finally number of...