Q1. A biologist is researching the growth of a certain species of hamster. She proposes that the length, x cm, of a hamster t days after its birth is given by x=15-12 e 14 (a) Use this model to find: (i) the length of a hamster when it is born; (1) (ii) the length of a hamster after 14 days, giving your answer to three significant figures. (2) (b) Show that the time for a hamster to grow to 10 cm in length is given by t = -14In(%) where a and b are integers. (3) (ii) Find this time to the nearest day. (1) (Total 7 marks) Q2. The average weekly pay of a footballer at a certain club was 80 on 1 August 1960. By 1 August 1985, this had risen to 2000. The average weekly pay of a footballer at this club can be modelled by the equation P = Ak where P is the average weekly pay t years after 1 August 1960, and A and k are constants. (a) (i) Write down the value of A. (1) (ii) Show that the value of k is 1.137411, correct to six decimal places. (2) (b) Use this model to predict the year in which, on 1 August, the average weekly pay of a footballer at this club will first exceed 100 000. (3) Q3. (a) Find the value of x in each of the following: (i) log9 x = 0; 1 (ii) log9 x = 2. (b) Given that find the possible values of n. = 2 logan loga 18+ loga (n = 4) (1) (1) (5) (Total 7 marks) Q4. (a) Sketch the graph of y = 3*, stating the coordinates of the point where the graph crosses the y-axis. (b) Describe a single geometrical transformation that maps the graph of y = 3: (i) onto the graph of y = 32; (ii) onto the graph of y = 3*+1. (c) (i) Using the substitution Y = 3*, show that the equation 9* -3x+1+2=0 can be written as (Y-1)(Y-2) 0 (2) (2) (2) (2) (ii) Hence show that the equation 9* - 3*+1 + 2 = 0 has a solution x = 0 and, by using logarithms, find the other solution, giving your answer to four decimal places. (4) (Total 12 marks) Q5. A particle, of mass 3 kg, moves along a straight line. At time t seconds, the displacement, s metres, of the particle from the origin is given by s = 813 + 15 (a) Find the velocity of the particle at time t. (b) Find the magnitude of the resultant force acting on the particle when t = 2. (2) (4)