1. An exam consists of four questions. If Adam spends time , on Question i, he expects to receive mark wi (1-e /Pi ), where w; > 0 is the number of marks assigned to Question i, and 8; > 0. The total time length of the exam is T. (a) [5 marks] How does Adam's expected mark from Question i depend on By; is it increasing or de- creasing in B,? What is an intuitive meaning of the parameter S;? (b) [5 marks] Formulate Adam's problem of the optimal time allocation (among the four exam questions). His goal is to maximise his expected mark from the whole exam. (c) [5 marks] Show that Adam's problem always has a solution