Although the words that you have written in your question are all very similar to one another, they are actually a lot more different than you think. If they all meant the same thing, then they would all be called one word. The difference between each of the definitions is sometimes very slight.
Below are the definitions of each the words that you stated in your question.
A problem statement is the result of the first step of the research process, where the problem to be investigated is chosen. To begin with, the researcher will have a general topic and then over time they will narrow it down to a more specific problem, which will be written out in a problem statement. This will consist of a detailed description of the actual problem and its importance.
A purpose statement is more commonly known as a personal statement and is usually written by graduates when they are applying to go on to higher education. It is normally a brief yet focused essay that specifies a person's career goals or what they aspire to do with their life. It may also contain information on previous work that is related to their prospective course.
The definition of a research question is very basic. When a researcher starts the research process, they begin with a broad topic and will narrow it down until they have a specific topic. The research question simply states what is being researched.
A hypotheses is defined as an explanation that is proposed for a phenomenon. Once a hypotheses has been developed, it will be researched and then can either be proved or disproved. During the case of the latter, a new hypotheses will be proposed.
An objective is also known as a learning outcome and is something that is expected to be achieved at the end of something e.g. A course or lesson.
