Practice Questions for CSIR NET Complex Analysis : Unit disk Automorphism, Schwarz Lemma III

21. Let \(w=f(z)\) be the bilinear transformation that maps \(-1,0\) and 1 to \(-i, 1\) and \(i\) respectively. Then \(f(1-i)\) equals

(a.) \(-1+2 i\)
(b.) \(2 i\)
(c.) \(-2+i\)
(d.) \(-1+i\)

22. The bilinear transformation as which maps the points \(0,1, \infty\) in the \(z\)-plane onto the points \(-i, \infty, 1\) in \(w\)-plane is

(a.) \(\frac{z-1}{z+i}\)
(b.) \(\frac{z+i}{z+1}\)
(c.) \(\frac{z+i}{z-1}\)
(d.) \(\frac{z+1}{z-i}\)

23. The function \(f(z)=z^{2}\) maps the \(\mathrm{I}^{\text {st }}\) quadrant onto

(a.) Itself
(b.) Upper half plane
(c.) Third Quadrant
(d.) Right half plane.

24. Let \(w=\frac{a z+b}{c z+d}\) and \(f=\frac{\alpha z+\beta}{r z+s}\) be bilinear (Mobius) transformation, then the following is also a bilinear transformation

(a.) \(f(z) w(z)\)
(b.) \(f\{w(z)\}\)
(c.) \(f(z)+w(z)\)
(d.) \(f(z)+\frac{1}{w(z)}\)

25. The fixed points of \(f(z)=\frac{2 i z+5}{z-2 i}\) are

(a.) \(1 \pm i\)
(b.) \(1 \pm 2 i\)
(c.) \(2 i \pm 1\)
(d.) \(i \pm 1\).

26. The function \(w(z)=-\left(\frac{1}{z}+b z\right),-1

(a.) A half plane
(b.) Exterior of the circle
(c.) Exterior of an ellipse
(d.) Interior of an ellipse

27. The transformation \(w=e^{i \theta}\left(\frac{z-p}{\bar{p} z-1}\right)\), where \(p\) is a constant, maps \(|z|<1\) onto

(a.) \(|w|<1\) if \(|p|<1\)
(b.) \(|w|>1\) if \(|p|>1\)
(c.) \(|w|=1\) if \(|p|=1\)
(d.) \(|w|=3\) if \(p=0\)

28. The conjugate (also called symmetric) point of \(1+i\) with respect to circle \(|z-1|=2\) is

(a.) \(1-i\)
(b.) \(1+4 i\)
(c.) \(1+2 i\)
(d.) \(-1-i\)

29. The bilinear transformation \(w=\frac{2 z}{z-2}\) maps \(\{z:|z-1|<1\}\) onto.

(a.) \(\{w: \operatorname{Re} w<0\}\)
(b.) \(\{w: \operatorname{Im} w>0\}\)
(c.) \(\{w: \operatorname{Re} w>0\}\)
(d.) \(\{w:|w+2|<1\}\)

30. The transformation \(w=\frac{1}{2}\left[z+\alpha^{2} z^{-1}\right], \alpha \in \mathbf{R}\) maps the circle \(|z|=r(r \neq \alpha)\) into

(a.) Circle
(b.) Ellipse
(c.) Hyperbola
(d.) Line