Evaluate: $$\int\limits_{\text{c}} {\frac{{{\text{dz}}}}{{{\text{z}}\sin {\text{z}}}}} ,$$   where c is x² + y² = 1.

f(z) = u(x, y) + iv(x, y) is an analytic function or complex variable z = x + iy where $${\text{i}} = \sqrt { - 1} ,$$   u(x, y) = 2xy, then v(x, y) may be expressed as

A. -x² + y² + constant

B. x² - y² + constant

C. x² + y² + constant

D. -(x² + y²) + constant

The product of complex numbers (3 - 2i) and (3 + i4) results in

A. 1 + 6i

B. 9 - 8i

C. 9 + 8i

D. 17 + 6i

If a complex number $${\text{z}} = \frac{{\sqrt 3 }}{2} + {\text{i}}\frac{1}{2}$$   then z⁴ is

A. $$2\sqrt 2 + 2{\text{i}}$$

B. $$\frac{{ - 1}}{2} + \frac{{{\text{i}}{{\sqrt 3 }^2}}}{2}$$

C. $$\frac{{\sqrt 3 }}{2} - {\text{i}}\frac{1}{2}$$

D. $$\frac{{\sqrt 3 }}{2} - {\text{i}}\frac{1}{8}$$

If C is a circle of radius r with centre z₀, in the complex z-plane and if n is a non-zero integer, then $$\oint {\frac{{{\text{dz}}}}{{{{\left( {{\text{z}} - {{\text{z}}_0}} \right)}^{{\text{n}} + 1}}}}} $$   equals

A. 2πnj

C. $$\frac{{\pi {\text{j}}}}{{2\pi }}$$

D. 2πn