Kindly answer the questions below with explanations for each

12.6 Estimate, by the trapezoidal rule or another suitable rule for approximate integration, the single premium for a temporary contingent assurance of 250,000 payable immediately on the death of Mrs Smith (aged 60), provided that this event occurs within 5 years and that her husband (aged 50) is alive at the date of her death. Mrs Smith is subject to the mortality of a (55) ultimate (females) and Mr Smith is subject to the mortality of A1967 - 70 ultimate. An interest rate of 7.5% p.a. is to be used, and allowance is to be made for expenses of 6% of the single premium. (Note A very accurate answer is not expected.) 12.7 Define the following functions in words, and give an expression for each of them in terms of an integral. (i) x41 (ii) A2 (iii) Al 12.8 The chief of a certain tribe holds that office until age 50 or earlier death, and may be succeeded only by a person aged from 36 to 45. (A person aged exactly 36 is eligible, but a person aged exactly 45 is not.) The customs of the tribe require that a chief's successor be his oldest eligible brother; if there is no eligible brother, the position of chief is given to someone from outside the previous chief's family, who are then permanently debarred from becoming chief. The present chief is aged exactly 47 and has two brothers, aged exactly 37 and 33 respectively. The chief and his brothers may be regarded as independent lives subject to the mortality of a given table. Obtain an expression, in terms of quantities of the form spy, no1 only, for the probability that (33) will become chief. 12.0 Your life office has been asked to quote a single premium for a contingent assurance policy providing 2300.000 immediately on the death of a woman now aged 80 within 15 years, provided that at the date of her death a man now aged 60 has died. Your office uses the following basis: mortality : males - a (55)ultimate (males) females - a (55) ultimate (females) interest : 8% per annum expenses : 10% of the single premium. (i) Assuming that the two lives are independent, write down a formula for the single premium in terms of an integral. (ii) State a suitable non-repeated rule of approximate integration for evaluating this integral. (You are NOT required to carry out the evaluation.) (iii) Would you subject the male life to stringent underwriting procedures? Give brief reasons for your