12. (i) A student states that "n2 - n + 5 is a prime number for all n EN " Show, by counter example, that the student's statement is not true. (1) (ii) Prove that tan O + - 1 O 0 # (90n)n EN tan O sin O cos (3) (Total for Question 12 is 4 marks) 13. A treatment is used to reduce the concentration of nitrate in the water in a pond. The concentration of nitrate in the pond water, N ppm (parts per million), is modelled by wl] the equation .booge inslentoo is sail Idgistle s ni elover) leod A O nigiN = 65 -30.1/ s tERS - + 2 0 roitieoq end tod olt visitinl O of svitslot mud (id where t hours is the time after the treatment was applied. i) noitieoq esd trod erdt estunim 02 ToffA brurot ed of fanlenoo Bei q sTow d mol ElV q al sod off to booga ard isdi work? (D) (E) Use the equation of the model to answer parts (a) and (b). (a) Calculate the reduction in the concentration of nitrate in the pond water in the tsod edT first 8 hours after the treatment was applied. O to less drion oub ai X tell nov3) than 20 ppm. For fish to survive in the pond, the concentration of nitrate in the water must be no more (d) (b) Calculate the minimum time, after the treatment is applied, before fish can be safely introduced into the pond. Give your answer in hours to one decimal place. (c) State a limitation of this model. (3) (1) (Total for Question 13 is 7 marks)