Which of the following is true for the nucleus?

A) It has no definite structure
B) It is surrounded by membrane
C) It is the eukaryotic cell's energy-producing organelle
D) It is closely associated with the Golgi apparatus.
E) It is always bluish purple in colour

To determine which of the statements is true for the nucleus, we can go through each option:

A) It has no definite structure: This statement is not true. The nucleus has a definite structure with a double membrane called the nuclear envelope, which contains nuclear pores.

B) It is surrounded by a membrane: This statement is true. The nucleus is surrounded by a double membrane called the nuclear envelope.

C) It is the eukaryotic cell's energy-producing organelle: This statement is not true. The energy-producing organelle in eukaryotic cells is the mitochondria, not the nucleus.

D) It is closely associated with the Golgi apparatus: This statement is not true. The Golgi apparatus is a separate organelle involved in modifying, sorting, and packaging proteins, while the nucleus is involved in genetic material storage and controls cellular activities.

E) It is always bluish purple in color: This statement is not true. The color of the nucleus cannot be generalized as it can vary depending on the specific organism and staining techniques used.

Therefore, the correct answer is B) It is surrounded by a membrane.