\[{\text{For any complex number z, }}\frac{d}{{dz}}{\text{sinh }}z{\text{ = ___________}}{\text{.}}\]

• A) \[\cosh z\]
• B) \[\tanh z\]
• C) \[{\csc ^2}z\]
• D) \[\sec {h^2}z\]

{\text{For any complex number z, cosh }}z{\text{ = _____________}}{\text{.}}

• A) \frac{{{e^{iz}} - {e^{ - iz}}}}{{2i}}
• B) \frac{{{e^{iz}} + {e^{ - iz}}}}{2}
• C) \frac{{{e^z} - {e^{ - z}}}}{2}
• D) \frac{{{e^z} + {e^{ - z}}}}{2}

\[\begin{gathered} {\text{For z = x + i y, we have}} \hfill \\\ {\text{sin z = ____________}}{\text{.}} \hfill \\\\ \end{gathered} \]

• A) \[\sin x\cos y + i\cos x\sin y\]
• B) \[\cos x\sin y + i\sin x\cos y\]
• C) \[\cos x\sinh y + i\sin x\cosh y\]
• D) \[\sin x\cosh y + i\cos x\sinh y\]

\[{\text{For any complex number z, }}\frac{d}{{dz}}{\text{coth }}z{\text{ = ____________}}{\text{.}}\]

• A) \[ - \csc {h^2}z\]
• B) \[\sec {h^2}z\]
• C) \[ - \csc hz\coth z\]
• D) \[\sinh z\]

\[{\text{For the power series }}\sum\limits_{n = 0}^\infty {{Z^n}} {\text{ is equal to __________}{\text{.}}\]

• A) 1/Z
• B) Z
• C) 1/(1+Z)
• D) 1/(1-Z)

\[{\text{For the power series }}\sum\limits_{n = 0}^\infty {(n + 1){Z^n}} {\text{ is equal to _________}}{\text{.}}\]

• A) \[\frac{1}{{{{(1 - Z)}^2}}}\]
• B) \[\frac{1}{{{Z^2}}}\]
• C) \[\frac{1}{{{{(1 + Z)}^2}}}\]
• D) \[\frac{1}{Z}\]

\[{\text{For any complex number z, cosh }}z{\text{ = _____________}}{\text{.}}\]

• A) \[\frac{{{e^z} + {e^{ - z}}}}{2}\]
• B) \[\frac{{{e^{iz}} + {e^{ - iz}}}}{2}\]
• C) \[\frac{{{e^{iz}} - {e^{ - iz}}}}{{2i}}\]
• D) \[\frac{{{e^z} - {e^{ - z}}}}{2}\]

\[{\text{For any complex number z, }}\frac{d}{{dz}}{\text{cosh }}z{\text{ = ____________}}{\text{.}}\]

• A) \[\sec {h^2}z\]
• B) \[\sin z\]
• C) \[\tanh z\]
• D) \[\sinh z\]

\[\frac{d}{{dz}}(lo{g_\alpha }z) = \_\_\_\_\_\_\_\_\_\_,{\text{ for z = r }}{{\text{e}}^{i\theta }},{\text{ }}\alpha < \theta < \alpha + 2\pi .\]

• A) z
• B) 1/z
• C) 1/(1+z)
• D) iz

\[{\text{For any complex number z, tanh }}z{\text{ = ____________}}{\text{.}}\]

• A) \[\frac{{{e^z} + {e^{ - z}}}}{{{e^z} - {e^{ - z}}}}\]
• B) \[\frac{{{e^{iz}} + {e^{ - iz}}}}{{{e^{iz}} - {e^{ - iz}}}}\]
• C) \[\frac{{{e^{iz}} - {e^{ - iz}}}}{{{e^{iz}} + {e^{ - iz}}}}\]
• D) \[\frac{{{e^z} - {e^{ - z}}}}{{{e^z} + {e^{ - z}}}}\]

\[\begin{gathered} {\text{For z = x + i y, we have}} \hfill \ {\text{cos z = ___________}}{\text{.}} \hfill \\\ \end{gathered} \]

• A) \[\sin x\sinh y + i\cos x\cosh y\]
• B) \[\cos x\cosh y - i\sin x\sinh y\]
• C) \[\sin x\cosh y + i\cos x\sinh y\]
• D) \[\cos x\cosh y + i\sin x\sinh y\]

\[{\text{For any complex number z, }}\frac{d}{{dz}}{\text{(tan z) = ________}}{\text{.}}\]

• A) \[\tan z\sec z\]
• B) \[ - {\csc ^2}z\]
• C) \[ - \sec z\tan z\]
• D) \[{\sec ^2}z\]

$$\begin{gathered} {\text{If }}\left( {{\text{G}}, \bullet } \right){\text{ and }}\left( {H,*} \right){\text{ are two groups}},{\text{ the function f:G}} \to {\text{H is called a group homomorphism if }} \hfill \\ {\text{f}}\left( {{\text{a}} \bullet {\text{b}}} \right){\text{ }} \ne {\text{ f}}\left( {\text{a}} \right) * {\text{f}}\left( {\text{b}} \right),{\text{ for all a}},{\text{b }} \in {\text{G}}. \hfill \\ \end{gathered}$$

• A) True
• B) False
• C)
• D)

$$The\,quotient\,group\,R/Z\,is\,{\text{isomorphic}}\,{\text{to}}\,{\text{the circle}}\,{\text{group}}\,{\text{W = }}\left\{ {{{\text{e}}^\theta } \in C|\theta \in R} \right\}.$$

• A) False
• B) True
• C)
• D)

$${\text{Let H be a subgroup of a group G}},{\text{ H is said to be a normal subgroup of G if}},{\text{ gH}} = {\text{Hg}},{\text{ }}\forall {\text{g }} \in {\text{G}}.$$

• A) False
• B) True
• C)
• D)

$$Let\,\left( {G,} \right)\,be\,a\,group\,with\,subgroup\,H.\,For\,a,\,b \in G,\,a\,is\,congruent\,to\,b\,\bmod ulo\,H,\,and\,written\,a \equiv b\bmod H\,iff\,a{b^{ - 1}} \in H.$$

• A) False
• B) True
• C)
• D)

$$\begin{gathered} {\text{Let K be a Kernal of the group morphism f:G}} \to {\text{H}}.{\text{ Then}}\,G/K\,{\text{is isomorphism to the image of}}\,f,{\text{ and the isomorphism}}{\text{is defined by}} \hfill \\ \psi {\text{: }}G/K \to \operatorname{Im} \,f\,is\,defined\,by\,\psi (Kg) = f(g). \hfill \\ \end{gathered}$$

• A) True
• B) False
• C)
• D)

$$The\,quotient\,group\,R/Z\,is\,{\text{isomorphic}}\,{\text{to}}\,{\text{the circle}}\,{\text{group}}\,{\text{W = }}\left\{ {{{\text{e}}^{i\theta }} \in C|\theta \in R} \right\}.$$

• A) True
• B) False
• C)
• D)

$$\begin{gathered} {\text{If }}\left( {{\text{G}}, \bullet } \right){\text{ and }}\left( {H,*} \right){\text{ are two groups}},{\text{ the function f:G}} \to {\text{H is called a group homomorphism if }} \hfill \\ {\text{f}}\left( {{\text{a}} \bullet {\text{b}}} \right){\text{ }} = {\text{ f}}\left( {\text{a}} \right) * {\text{f}}\left( {\text{b}} \right),{\text{ for all a}},{\text{b }} \in {\text{G}}. \hfill \\ \end{gathered}$$

• A) False
• B) True
• C)
• D)

$$Let\,\left( {G,} \right)\,be\,a\,group\,with\,subgroup\,H.\,For\,a,\,b \in G,\,a\,is\,congruent\,to\,b\,\bmod ulo\,H,\,and\,written\,a \equiv b\bmod H\,iff\,{a^{ - 1}}b \in H.$$

• A) True
• B) False
• C)
• D)