A subset $S$ of a ring $R$ is said to be multiplicative if $ab \in S \;\forall \; a,b \in S$. Show that $P$ is a prime ideal of a commutative ring $R$ if and only if $R-P$ is multiplicative.

Let $I$ be the principal ideal $\langle x \rangle$ of the ring $\mathbb{Z}[x]$. Prove that $I$ is prime but not maximal.

Let $R$ be a commutative ring with identity in which the set of all non-units forms an ideal $M$. Show that $M$ is a maximal ideal of $R$ and it contains every proper ideal of $R$. Also show that for each $r \in R$ either $r$ or $1-r$ is a unit.

Let $R$ be a commutative ring. Show that $a|b \Leftrightarrow Ra \supset Rb, \forall a,b \in R$. Further if $R$ contains the identity, show that for any $a \in R$, if $Ra$ is maximal, then $a$ is an irreducible element of $R$.

Let $R$ be a non-trivial ring with identity in which every subring is an ideal. Show that $R \cong \mathbb{Z}$ or $\mathbb{Z_n}$ for some $n .$

Write parametric equation of hypocycloid, epicycloid, epitrochoid, trochoid.

Let $C$ be circle $|z| =R(R >1)$  oriented counterclockwise. Show that

$\left | \int_C \frac {Logz^2}{z^2}dz \right | \le 4\pi \left ( \frac{\pi + logR}{R} \right )$ and hence   $\lim_{R \to \infty} \int_C \frac{Logz^2}{z^2}dz =0$ .

Without evaluating the integral, show that $\left | \int_C \frac {1}{\overline{z}^2 + \overline{z} +1}dz \right| \le \frac{9\pi}{16}$  where $C$ is the arc of circle $|z|=3$ from $z=3$ to $z=3i$ lying in the first quadrant.

Find the zeros and poles of

$f(z) = \frac{1}{(z-a)(z-b)(e^{z-a} -1)}$

and hence

$\int_{-\infty}^{\infty} \frac{1}{(x-a)(x-b)(e^{x-a} -1)}dx$

where $a,b$ are real .

Let $f$ be the function defined by $f(z)=\frac{(z+5)(2z+1)}{z^2-4}$ . Find the zero and poles of $f,\frac{1}{f}, f^\prime$ and $\frac{f^\prime}{f}$ .

Show that the function $f$ defined by the series $f(z)=\sum ^{\infty}_{n=1}\frac{1}{(z+n)^2}$ is meromorphic on every bounded subset of $\mathbb{C}$, and find the residues at its poles.