Provide step by step solution to the following questions. Kindly be precise.

A man takes out a loan over 25 years. He makes monthly repayments in arrears. The first payment is $50 and the remaining payments increase by $5 each time until the payments reach $770. The remaining payments are then all $770. The annual effective rate of interest is 7.5%. (i) Show that the amount of money that the man borrowed is $66,767. [6] (ii) Calculate the amount of capital outstanding after 6 years. [4] (iii) Calculate the amount of capital repaid in the 6th year. [3] (iv) Calculate the interest paid in the 73rd payment. Comment on your answers to parts (i), (ii), (iii) and (iv). [3] (v) At the end of the sixth year (after the payment then due) the man decides to reschedule the loan over a term of ten years with level monthly payments. He negotiates a deal with the loan company to reduce the interest he pays by 0.1% pa. Calculate how much he saves (compared to his original loan) under this arrangement. [7] [Total 23]The integrated hazard for mortality for a group of lives over the period (0,?), where r is measured in weeks, is being modelled by the function: H(t) =1- {1 + exp[(t -2)/3];-1 (i) Find an expression for h(t), the hazard function at time f. [2] (ii) Sketch a graph of h(t). [5] (iii) Suggest a context where a hazard function with this shape might be appropriate. [2] [Total 9]