> The monarch and their immediate family undertake various official, ceremonial, diplomatic and representational duties. As the monarchy is constitutional, the monarch is limited to functions such as bestowing honours and appointing the prime minister, which are performed in a non-partisan manner. The monarch is also Head of the British Armed Forces. Though the ultimate executive authority over the government is still formally by and through the royal prerogative, these powers may only be used according to laws enacted in Parliament and, in practice, within the constraints of convention and precedent. The Government of the United Kingdom is known as His (Her) Majesty's Government.
I wasn't aware that the monarch appointed the prime minister, but here you have the last one the Queen made:
> Liz Truss has became Britain’s next prime minister after meeting with Queen Elizabeth II, who asked her to form a new government.
https://www.aljazeera.com/news/2022/9/6/uks-johnson-bows-out...
The Wikipedia article later notes that prime minister appointment appears to fall into the ceremonial category:
> The sovereign has the power to appoint the prime minister. In accordance with unwritten constitutional conventions, the monarch appoints the individual who commands the support of the House of Commons, usually the leader of the party or coalition that has a majority in that House. The prime minister takes office by attending the monarch in a private audience, and after "kissing hands" that appointment is immediately effective without any other formality or instrument.[15] The sovereign also has the power to dismiss the prime minister, but the last time this power was exercised was in 1834, when William IV dismissed Lord Melbourne; since then, prime ministers have only left office upon their resignation, which they are expected to offer to the monarch upon losing their majority in the House of Commons.