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

31. The magnification factor of the conformal mapping \(W=\sqrt{2} e^{\pi i / 4} z(1-2 i)\) is

(a.) 1
(b.) 2
(c.) 3
(d.) 10

32. The circle \(a(x^{2}+y^{2})+b x+c y+d=0\) is transformed by \(f(z)=\frac{1}{z}\) into a circle. The center of the circle is given by:

(a.) \(\left(\frac{2 a b}{b^{2}+c^{2}}, \frac{2 a c}{b^{2}+c^{2}}\right)\)
(b.) \(\left(\frac{-2 a b}{b^{2}+c^{2}}, \frac{2 a c}{b^{2}+c^{2}}\right)\)
(c.) \(\left(\frac{2 a b}{b^{2}+c^{2}}, \frac{-2 a c}{b^{2}+c^{2}}\right)\)
(d.) \(\left(\frac{-2 a b}{b^{2}+c^{2}}, \frac{-2 a c}{b^{2}+c^{2}}\right)\)

33. If \(f(z)=e^{2 x}-2 i z+3\) then which of the following is incorrect

(a.) \(f(z)\) has infinitely many critical points
(b.) Scale factor of \(f(z)\) at every real number is defined
(c.) Scale factor at every point on the imaginary axis is defined
(d.) \(f(z)\) is not conformal

34. Which region/curve of the \(w\)-plane by the transformation \(z=a \cos \omega+i b \sin \omega\) is mapped into the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

(a.) The real axis of \(w\)-plane
(b.) The imaginary axis of \(w\)-plane
(c.) An ellipse in \(w\)-plane
(d.) The complete \(w\)-plane

35. The rectangular region \(R\) bounded by \(x=0, y=0, x=2, y=3\) in \(z\)-plane is mapped into the rectangular region \(R\) of \(w\)-plane through the transformation \(w=\sqrt{2} e^{j / 4} z\). This transformation performs

(a.) A rotation
(b.) A magnification
(c.) A translation and magnification
(d.) A rotation and magnification

36. Choose the correct statements

(a.) Let \(z_1, z_2, z_3, z_4\) are fixed complex number then we can define at most 6 different cross ratios.
(b.) The cross ratio is invariant under B.T. and this property can be used in obtaining specific B. T. mapping three points into three other points
(c.) A B.T. which maps \(z=6, i,-i\) into \(w=i, 1,0\) as \(w=-\frac{i(1+z)}{(z+1)}\)
(d.) \(\left(z_1, z_2, z_3, z_4\right)=\frac{\left(z_1-z_2\right)\left(z_3,-z_4\right)}{\left(z_1-z_4\right)\left(z_3-z_2\right)}\)

37. Let \(A=\{z \in \mathbb{C}|z|>1\}, B=\{z \in \mathbb{C} \mid z \neq 0\}\) Which of the following are true?

(a.) There is a continuous onto function \(f: A \rightarrow B\).
(b.) There is a continuous one to one function \(f: B \rightarrow A\).
(c.) There is a nonconstant analytic function \(f: B \rightarrow A\).
(d.) There is a nonconstant analytic function \(f: A \rightarrow B\).

38. Let \(H=\{z=x+i y \in \mathrm{C}: y>0\}\) be the upper half plane and \(D=\{z \in C:|z|<1\}\) be the open unit disc. Suppose that \(f\) is a Mobius transformation, which maps \(H\) conformally onto \(D\). Suppose that \(f(2 i)=0\). Pick each correct statement from below.

(a.) \(f\) has a simple pole at \(z=-2 i\).
(b.) \(f\) satisfies \(f(i) \overline{f(-i)}=1\).
(c.) \(f\) has an essential singularity at \(z=-2 i\).
(d.) \(|f(2+2 i)|=\frac{1}{\sqrt{5}}\).

39. Let \(D\) be the open unit disc in C. Let \(\boldsymbol{g}: \boldsymbol{D} \rightarrow \boldsymbol{D}\) be holomorphic, \(g(0)=0\), and Let \(h(z)=\left\{\begin{array}{c}g(z) / z \quad z \in D, z \neq 0 \\ g^{\prime}(0), \quad z=0\end{array}\right.\) Which of the following statements are true?

(a.) \(h\) is holomorphic in \(\boldsymbol{D}\).
(b.) \(h(\boldsymbol{D}) \subseteq \overline{\boldsymbol{D}}\).
(c.) \(\left|g^{\prime}(0)\right|>1\)
(d.) \(|g(1 / 2)| \leq 1 / 2\).

40. Let \(a, b, c\) be non-collinear points in the complex plane and let \(\Delta\) denote the closed triangular region of the plane with vertices \(a, b, c\). For \(z \in \Delta\), let \(h(z)=|z-a| \cdot|z-b| \cdot|z-c|\) The maximum value of the function \(h\):

(a.) is not attained at any point of \(\Delta\).
(b.) is attained at an interior point of \(\Delta\).
(c.) is attained at the centre of gravity of \(\Delta\).
(d.) is attained at a boundary point of \(\Delta\).

41. The maximum modulus of \(e^{z^2}\) on the set \(S=\{z \in \mathbb{C}: 0 \leq \operatorname{Re}(z) \leq 1,0 \leq \operatorname{Im}(z) \leq 1\}\) is

(a.) \(2 / \mathrm{e}\).
(b.) \(\mathrm{e}\).
(c.) \(\mathrm{e}+1\).
(d.) \(\mathrm{e}^2\).

42. Let \(T\) be the closed unit disk and \(\partial T\) be the unit circle. Then which one of following? holds for every analytic function \(f: T \rightarrow C\).

(a.) \(|f|\) attains its minimum and maximum on \(\partial T\).
(b.) \(|f|\) attains its minimum on \(\partial T\) but need not attain its maximum on \(\partial T\).
(c.) \(|f|\) attains maximum on \(\partial T\) but need not attain its minimum on \(\partial T\).
(d.) \(|f|\) need not attain its maximum on \(\partial T\) and also it need not attain its minimum on \(\partial T\).

43. Let \(f(z)=2 z^2-1\) then the maximum value of \(|f(z)|\) on the unit \(\operatorname{disc} T=\{z \in \mathbb{C}:|z| \leq 1\}\) equals.

(a.) 1
(b.) 2
(c.) 3
(d.) 4