Remember that all three types of RNA are synthesized from DNA molecules. Since DNA molecules can’t leave the nucleus, the synthesis of RNA has to occur in the nucleus.

On the other hand, proteins are synthesized in the cytosol and rough endoplasmic reticulum through translation. Proteins cannot be synthesized directly in the nucleus.

mRNA, or messenger RNA, is used as the template to synthesize proteins in translation. rRNA, or ribosomal RNA, make up a structural components of ribosomes. tRNA, or transfer RNA, facilitate the synthesis of proteins from mRNA. Though these molecules function in the cytosol, they are all synthesized in the nucleus.

The cell’s nucleus, often termed the control center of the cell, is a very important organelle that contains the DNA. The nucleus is surrounded by two cell membranes, made up of two phospholipid bilayers, called the nuclear envelope; therefore, statement I is true.

The nucleus also contains a nucleolus, which is the site of rRNA synthesis. rRNA, or ribosomal RNA, are the building blocks for ribosomes and are assembled into ribosomal subunits in the nucleolus. Statement II is true.

The nucleus does contain nuclear pores, however, DNA cannot exit the nucleus. Messenger RNA (mRNA), on the other hand, can exit the nucleus and enter the cytosol, where it can bind to ribosomes and undergo translation; therefore, statement III is false.

The nucleolus is an area inside of the nucleus that stains dark purple on most cell slides. It is responsible for the creation of ribosomal subunits, as well as the transcription of rRNA.

The smooth endoplasmic reticulum synthesizes lipids, lysosomes digest macromolecules (particularly proteins), and mitochondria generate cellular energy in the form of ATP.

The nucleolus is the region of the nucleus that is engaged in ribosome synthesis. The nucleolus contains genes encoding ribosomal RNA, newly synthesized ribosomal RNA, and ribosomal proteins.

Chromatin consists of DNA and its associated proteins. Proteins are organic molecules compose of one or more chains of amino acids and serve a variety of important biological functions. Ribosomes are the site of protein synthesis. ATP, or adenosine triphosphate, is the major energy carrier in cells. 

The nucleus is a membrane-bound container for DNA. The nucleus is called the "brain" of the cell because the DNA contained within directs all of the cell's activities. The nucleus houses all genomic DNA.

The nucleolus is the site of ribosome synthesis. The endoplasmic reticulum synthesizes lipids. The mitochondria produce energy by aerobic metabolism. The Golgi apparatus packages lipids and proteins for distribution.

Note that the mitochondria and chloroplasts each contain small amounts of DNA, but this is considered extragenomic DNA.

The nucleus of a cell contains the genetic information and is the organelle most easily seen when viewing cells under a light microscope. The nucleus is usually spherical and is generally the largest organelle in eukaryotic cells. The mitochondrion is a smaller organelle, and is responsible for making the energy (ATP) in the cell. Lysosomes are small organelles that contain hydrolytic enzymes used to degrade and recycle macromolecules. The peroxisome is an organelle responsible for catabolism of lipids, detoxification of certain compounds, and regulation of cellular metabolism.

Nuclear pores are complex structures made up of proteins which allow movement of certain molecules into and out of the nucleus. Flagella are long tail-like projections found on some cells that function in cell motility. The rough endoplasmic reticulum has ribosomes on it, which are the sites of protein synthesis. Lysosomes are organelles that contain hydrolytic enzymes used to recycle/destroy macromolecules.

Prokaryotes have a single circular DNA molecule, and eukaryotes have multiple linear chromosomes.  The four nucleotides for DNA are in both prokaryotes and eukaryotes.

A double phospholipid bilayer membrane makes up the nuclear envelope. The outer layer of this nuclear envelope is continuous with the endoplasmic reticulum, also known as cytoplasm's interior membrane system. The cell's plasma membrane, and the membranes surrounding various organelles (except the nucleus, and mitochondria which both have double phospholipid bilayers) are single phospholipid bilayers. Cell walls are found outside the plasma membrane of plant, bacterial, and fungal cells. Peptidoglycan is the main component of the cell walls of bacteria. 

A double phospholipid bilayer membrane makes up the nuclear envelope. The outer layer of this nuclear envelope is continuous with the endoplasmic reticulum, also known as cytoplasm's interior membrane system. The cell's plasma membrane, and the membranes surrounding various organelles (except the nucleus, and mitochondria which both have double phospholipid bilayers) are single phospholipid bilayers. Cell walls are found outside the plasma membrane of plant, bacterial, and fungal cells. Peptidoglycan is the main component of the cell walls of bacteria.