In the PRINCE2 methodology, the 'project mandate' is the term used for initial documentation that triggers the start of a project, typically provided by corporate or program management. It can often come in the form of a request for proposal received by an external supplier, setting out the initial definition of the project and what it aims to achieve, which then leads to more detailed project definitions in the 'starting up a project' process.
PRINCE2 7 Foundation resources on project initiation documentation.

When tailoring PRINCE2 to suit the project environment, changes can be made to the terminology used to better align with the organization's existing standards, policies, or practices. This helps in integrating PRINCE2 within the organization more seamlessly without altering the core principles and themes which are foundational to the methodology's effectiveness. References: PRINCE2 Foundation Guide, Tailoring PRINCE2 section

The principle of continued business justification in PRINCE2 means that a project must remain justified throughout its life cycle. The key factor is the project's ongoing justification, not solely its ability to deliver financial benefits. Projects may continue for reasons other than financial gain, such as strategic importance, compliance, or public safety. Therefore, stopping a project based only on financial criteria does not reflect this principle accurately. References: PRINCE2 Foundation manual, PRINCE2 official resources

The statement that "The Closing a Project process provides a fixed point to review if the objectives set out in the original Project Initiation Documentation have been achieved" is true. This process is crucial for ensuring that the project has met its objectives and that all remaining project deliverables are acceptable to stakeholders. The first statement is incorrect as the receipt of completed Work Packages is typically handled during the 'Managing a Stage Boundary' process rather than in the 'Closing a Project' process. References:
PRINCE2 Foundation manual, PRINCE2 official resources

The purpose of the plans theme in PRINCE2 is to facilitate the management and control of the project by establishing whether the project's targets are achievable. This involves setting out how and when the project's objectives are to be achieved by defining the necessary resources, activities, timescales, and costs. References:
PRINCE2 Foundation Handbook, Plans Theme section.