Answer the questions below with explanation for each

12.1 Which (if any) of the following statements are correct? A. Arr = 2A1 B. Afr = 2A, C. Arr = 2A, - A, D. 12.2 Adams (aged 40) and Brown (aged 50) are two business partners. Adams wishes to provide for the sum of 280.000 to be paid immediately on Brown's death if Brown predeceases him within ten years, and effects a policy providing this benefit by single premium. The life office issuing the contract employs the following basis: Mortality (both lives) : A1967 - 70 ultimate Interest : 6% Expenses: 2% of the single premium. Using Simpson's rule, or otherwise, estimate the single premium payable by Adams. 12.3 Estimate the value of a91 on the basis of A1967 - 70 ultimate mortality. (Assume /109 = 0, so that the integral is over a range of 24 years. Break this into 3 sub-intervals and use Simpson's Rule over each.) 12.4 (i) Express Ay in terms of as, any and the rate of interest. (ii) Smith and Jones are both aged 60. A life office has been asked to issue a special joint-life assurance policy providing 210.000 at the end of the year of death of the first to die of these two lives. In addition, if Smith is the second to die, a further 25.000 will be payable at the end of the year of his death. The policy is to have annual premiums payable during the joint lifetime of Smith and Jones. Calculate the annual premium on the following basis: A1967 - 70 ultimate mortality 4% interest expenses are 5% of all premiums, with an additional initial expense of 6100. (iii) Write down (but do NOT evaluate) formulae for the reserve at duration 10 years (imme- diately before payment of the premium then due) on the premium basis, if (a) both Smith and Jones are alive; and (b) Jones has died but Smith is alive. 12.5 A policy providing the sum of 2100,000 immediately on the death of (r) if she dies before (y) is to be issued by a life office to a group of trustees. (i) Ignoring expenses, write down an expression for the single premium in terms of an integral. (ii) The trustees suggest that level annual premiums should be payable in advance until the death of the last survivor of (r) and (y)