MTH632 Quiz 1

Q1. ${\rm{The}}\,\,{\rm{domain}}\,\,{\rm{of}}\,\,\,{\rm{the}}\,\,{\rm{function }}f{\rm{(}}z{\rm{)}}\, = \,\frac{1}{{z\, - \,\left( {1 + i} \right)}}\,\,{\rm{is}}\,\,\,\_\_\_\_\_\_\_\_\_\_.$

1. $C\, \,\{ {\,1 + i\,} \}\,$
2. $C\, \,\{ {\,\,0\,} \}\,$
3. $C\, \,\{ {\,1 - i\,} \}\,$
4. $C\, \,\{ {\,\,1\,} \}\,$

Correct answer: $C\, \,\{ {\,1 + i\,} \}\,$

Q2. $$If\,\,we\,\,write\,\,the\,\,function\,\,w = {z^2} - z\,\,as\,\,w = u(x,y) + iv(x,y),\,\,then\,\,u(x,y) =$$

1. $$x{}^2 + {y^2}$$
2. $$x{}^2 - x - {y^2}$$
3. $$x{}^2 - {y^2}$$
4. ${{None}}\,\,{{ of}}\,\,{{ these}}$

Correct answer: $$x{}^2 - x - {y^2}$$

Q3. ${\rm{The}}\,\,{\rm{function }}\,f{\rm{(}}z{\rm{) }} = {\rm{ }}\frac{{\rm{1}}}{z}\,\,\,\,\,{\rm{is }}\,{\rm{\_\_\_\_\_\_\_\_\_\_}}\,{\rm{.}}$

1. not a polynomial
2. a polynomial

Correct answer: not a polynomial

Q4. ${\rm{If }}\,\,{\rm{the}}\,{\rm{ }}\,{\rm{5th}}\,\,{\rm{ roots }}\,\,{\rm{of }}\,\,{\rm{unity}}\,{\rm{ }}\,{\rm{are }}\,\,\,\_\_\_\_\_\_\_\_\_\_\_\_.$

1. $5\,e{\,^{i\,\,( {{{2k }}{5}} )}},\,\,\,k\,\, = \,\,0,\,\,1,\,\,2,\,\,3$
2. $e{\,^{i\,\,( {{{2k }}{5}} )}},\,\,\,k\,\, = \,\,0,\,\,1,\,\,2,\,\,3,\,\,4$
3. $e{\,^{i\,\,( {{{2k }}{5}} )}},\,\,\,k\, = \,0,\,1,\,2,\,3$
4. $2\,e{\,^{i\,\,( {{{2k }}{5}} )}},\,\,\,k\,\, = \,\,0,\,\,1,\,\,2,\,\,3,\,\,4$

Correct answer: $e{\,^{i\,\,( {{{2k }}{5}} )}},\,\,\,k\,\, = \,\,0,\,\,1,\,\,2,\,\,3,\,\,4$

Q5. Let z = 7 i, then Arg(z) = ________.

1. ${π 6}$
2. ${π 3}$
3. ${π 4}$
4. ${π 2}$

Correct answer: ${π 2}$

Q6. ${\rm{The }}\,{\rm{values }}\,{\rm{of }}\,\,{\left( {{\rm{2}}i} \right)^{\frac{1}{3}}}\,\,\,{\rm{are }}\,\,{\rm{\_\_\_\_\_\_\_\_\_\_\_}}{\rm{.}}$

1. $2{\,^{{1}{3}}} e{ ^{i\,\,( {\,{{i\, }}{6} + {{2k }}{3}} )}},\,\,\,\,\,\,\,k\, = \,0,\,1$
2. $2{\,^{{1}{3}}} e{ ^{i\,\,( {\,{{i\, }}{6} + {{2k }}{3}} )}},\,\,\,\,\,\,\,k\, = \,0,\,1,\,2$

Correct answer: $2{\,^{{1}{3}}} e{ ^{i\,\,( {\,{{i\, }}{6} + {{2k }}{3}} )}},\,\,\,\,\,\,\,k\, = \,0,\,1,\,2$

Q7. In the complex valued function w = f ( z ), the independent variable is __________.

1. z
2. w

Correct answer: z