Answer the following questions.

A mortality investigation covered the period 1 January 2007 to 1 January 2008. Time is measured in years from 1 January 2007 and P.() denotes the number of lives at time / aged x last birthday. The following data were recorded for each x : d = number of deaths aged x next birthday P. (0) and P.(1) (i) Obtain an expression for the initial exposed to risk in terms of the available census data that may be used to estimate the initial rate of mortality q,+ stating your assumptions.The disreputable insurance company Honest Sid's Mutual had mixed fortunes in the year 2006. At both the start and the end of the year 547 policies were in force in respect of policyholders aged 40, but these figures do not tell the whole story. There was adverse publicity early in the year linking the company's investment managers with a gambling syndicate. As a result, many policyholders "took their money elsewhere". Following a successful marketing campaign offering a free toaster to all applicants, the number of policyholders aged 40 rose from 325 at 1 June 2006 to 613 at 1 September 2006. Calculate the value of EF approximately.(a) Assume the following information represents the National Income model of an "Utopian' economy. Y= C+I+G Cza+b( Y- T) T= d+ tY G =G. Where a >0;0 0; 0