Answer the questions below with explanations

1 (i) Define the following symbols in words, and give a formula in terms of an integral for each of them: (a) A 1 (b) A2 (c) ayr (ii) Consider the following sets of payments: (1) fl immediately on the death of (y) if (y) dies before (x), and (2) an income of fo p.a. payable continuously to (x) after the death of (y), plus fl immedia ately on the death of (x) if this occurs after that of (y). Prove that the present values (at force of interest o p.a.) of (1) and (2) are equal. Hence write down a relationship involving A ,, A2, and ayr. 2 A special life policy on 2 lives aged a and y respectively provides cash sums of f10,000 and f20,000 immediately on the first and second deaths respectively. In addition, an annuity at the rate of f1,000 per annum will be paid continuously, commencing immediately on the first death and ceasing immediately on the second death. Obtain an expression for the mean present value of the benefits in terms of joint-life and single- life annuity functions and the force of interest. Ignore expenses. 3 An office issues a policy on the lives of a woman aged 60 and her husband aged 64. Under this policy, level premiums are payable annually in advance for 20 years or until the first death of the couple, if earlier. On the first death of the couple, the survivor will receive an annuity of f10,000 per annum, payable weekly, beginning immediately on the first death. Calculate the annual premium if the office uses the basis given below: Mortality males: a(55) males ultimate females: a(55) females ultimate Expenses: 20% of the first premium 5% of each premium after the first Interest: 6% per annum. 4 A special annuity, payable yearly in arrear, is effected on the lives of a man aged x and his wife aged y. The conditions of payment are: (a) so long as both survive the rate of payment will be 23,000 per annum; (b) if the wife dies first, the rate of payment will be f2,000 per annum until the man's death; (c) payments at the rate of 23,000 per annum will continue for six years certain after the death of the husband, the first payment being at the end of the year of his death, and will be reduced thereafter to f1,500 per annum during the lifetime of the wife. Obtain an expression for the present value of this annuity in terms of single and joint-life annuity factors, life table and compound interest functions. Assume that the same (non- select) table of mortality is appropriate for the two lives