Practice Questions for CSIR NET Complex Analysis : Meromorphic Functions III

21. Determine the number of roots of the equation \(z^{5}-12 z^{2}+14=0\) that are located within the region \(\left\{z \in \mathbb{C}: 2 \leq|z|<\frac{5}{2}\right\}\).

(a.) 2
(b.) 3
(c.) 4
(d.) 5

22. Determine the number of zeros, counting multiplicities, of the polynomial \(z^{5}+3 z^{3}+z^{2}+1\) inside the circle \(|z|=2\).

(a.) 0
(b.) 2
(c.) 3
(d.) 5

23. Let \(f(z)=1+2 z+3 z^{2}+\ldots n z^{n}\) and let \(\gamma\) be a closed curve which contains all the zeros of \(f(z)\). Then \(\frac{1}{2 \pi i} \int_{\gamma} z^{2} \frac{f^{\prime}(z)}{f(z)} d z\) is equal to

(a.) 0
(b.) \(n\)
(c.) \(\frac{2 n-n^{2}+1}{n^{2}}\)
(d.) \(\frac{2 n-n^{2}-1}{n^{2}}\)

24. The value of \(\int_{C} \frac{f^{\prime}(z)}{f(z)} d z\) where \(f(z)=\frac{\left(z^{2}+1\right)^{2}}{\left(z^{2}+3 z+2\right)^{3}}\) and \(C\) is the circle \(|z|=3\) with positive sense

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

25. If \(f(z)=z^{2}+2\) then the minimum value of \(|f(z)|\) over the closed region \(|z| \leq 1\) is Fill the space from the following choices

(a.) 0
(b.) 1
(c.) 3
(d.) 2

26. Consider the polynomial \(p(z)=z^{5}+z^{3}+5 z^{2}+2\) then the number of zeros in \(1<|z|<2\) ?

(a.) 2
(b.) 3
(c.) 5
(d.) none

27. The number of zeros of \(p(z)=4 z^{3}-3 i z^{2}+i z-9=0\) in the a disc \(|z|<1\) is

(a.) 0
(b.) 1
(c.) 2
(d.) 4

28. If \(g(z)=z^{3}\) and \(f(z)=z^{3}-z+1\) then the value of \(\frac{1}{2 \pi i} \int_{C} g(z) \frac{f^{\prime}(z)}{f(z)} d z\) where \(C\) contains all the zeros of \(f(z)\) is

(a.) -1
(b.) 0
(c.) -3
(d.) -9

29. If \(f(z)\) is analytic on \(D\) where \(D=\{(x, y):|x| \leq a,|y| \leq b, a \geq b\}\) and if \(f(z)\) satisfies the inequality \(|f(z)| \leq M\) on the boundary of \(D\), then which can be taken as an upper bound of \(\left|f^{\prime}(0)\right|\)

(a.) \(\frac{2 M(a+b)}{\pi a b}\)
(b.) \(\frac{2 \pi a b}{M^{2}}\)
(c.) \(\frac{2 M^{2}}{2 \pi(a b)}\)
(d.) \(\frac{2 M(a+b)}{\pi b^{2}}\)

30. If \(C\) is the unit circle \(z=e^{i \theta},(0<\theta \leq 2 \pi)\), the value of \(\frac{1}{2 \pi i} \Delta \log \left(\frac{1}{z^{2}}\right)\) is

(a.) 0
(b.) 1
(c.) -1
(d.) -2

31. Suppose \(f\) and \(g\) are entire functions and \(g(z) \neq 0\) for all \(z \in \mathbb{C}\). If \(|f(z)| \leq|g(z)|\), then we conclude that

(a.) \(f(z) \neq 0\) for all \(z \in \mathbb{C}\)
(b.) \(f\) is a constant function.
(c.) \(f(0)=0\)
(d.) For some \(C \in \mathbb{C}\), \(f(z)=C g(z)\)

32. Let \(f(z)\) be the meromorphic function given by \(\frac{z}{\left(1-e^{z}\right) \sin z}\). Then

(a.) \(z=0\) is a pole of order 2 .
(b.) For every \(k \in \mathbb{Z}\), \(z=2 \pi i k\) is a simple pole.
(c.) For every \(k \in \mathbb{Z} \backslash\{0\}\), \(z=k \pi\) is a simple pole.
(d.) \(z=\pi+2 \pi i\) is a pole.