A small town has three gyms: Fit, Gymie and Hip. Each year Fit retains 70% of its members with 20% moving to Gymie and the rest to Hip. Gymie retains 60%, losing the rest in equal numbers to its rivals. Hip retains just 30% with 40% moving to Fit and 30% moving to Gymie. There are no other changes to the gym membership in the town. (a) If x= [1000, 800, 800] is a 3 x 1 matrix for the current numbers, show that it is possible to work out next year's membership figures by performing a matrix multiplication Ax, where A is a 3 x 3 matrix which should be stated. (b) Use matrix multiplication to work out the number of members in Fit 2 years from now. (a) A =

Question 2 (6 marks) A Bauman string is a string of letters that satisfies the following two conditions: Each letter in the string is A, B, C, D, or E. No two adjacent letters in the string are the same. For example, AECD and DBCEC are Bauman strings of length 4 and length 5, respectively. However, ABBC and DAEEE are NOT Bauman strings. .

We can find the remainder for the quotient using either substitution or synthetic division as discussed in class. Using either method, explain whether or not we can conclude that X - 5 is a factor of the polynomial ƒ (x) = (x³ — x² – 17x - 15). (Your answer must include what that remainder value is and what it indicates.)

Q1 a) Find the length of the line segment joining the points (6,5) and (2,2) b) Find the equation of the straight line that passes through these points, giving your answer in the form ax+by+c=0. (5 marks)

Q2 The coordinates of points A and B are (3,5) and (4,-5) respectively. Find the coordinates of the mid-point of AB, and the gradient of AB. (5 marks)

Q3 P is the point (7,5) and Lr is the line with equation 3x + 4y = 16. a) Find the equation of the line L2, which passes through P and is perpendicular to Lr. b)Find the point of intersection of the lines L1 and L₂. c) Find the perpendicular distance of P from the line L₁.

a) If you know that you ARE supposed to be able to do the integral using substitution, give an example of an expression under the coffee that would make it possible. (There's more than one possibility) b) What would be an example of an expression underneath that coffee that would make it NOT possible to integrate with substitution?

Q.2) Find the amount to which $1,500 will grow if interest of 6.75% is daily for 10 years? (Note: Assume 360 days in a year). compounded

Please give an exact answer (which means you're not touching your calculator and your answer has no decimals) for the integral below:

Q.1) Find the total amount due on a loan of $1600 at 7% simple interest at the end of 9 months?