1. In the arithmetic series with nt" term un, it is given that u4 = 7 and u9 = 22. Find the minimum value of n so that u1 + 2 + 43 + ... + un > 10 000. (Total 5 marks) 2. Find the sum of all three-digit natural numbers that are not exactly divisible by 3. 3. Consider the graphs y = e X and y = e " sin 4x, for 0 ? x ? 5Tt . 4 (Total 5 marks) 5Tt (a) On the same set of axes draw, on graph paper, the graphs, for 0 ? x ? - 4 Use a scale of 1 cm to 1100 on your x-axis and 5 cm to 1 unit on your y-axis. . (3) (b) Show that the x-intercepts of the graph y = e " sin 4x are 4 , n = 0, 1, 2, 3, 4, 5. (3) (c) Find the x-coordinates of the points at which the graph of y = e " sin 4x meets the graph of y = e *. Give your answers in terms of T. (3) (d) (i) Show that when the graph of y = e " sin 4x meets the graph of y = e , their gradients are equal. (ii) Hence explain why these three meeting points are not local maxima of the graph y = e * sin 4x. (6) IB Questionbank Mathematics Higher Level 3rd edition 1
(e) (i) Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local maxima of y = e " sin 4x for O ? x ? 5? . You do not need to show that they are maximum 4 values, but the values should be simplified. (ii) Show that y1, y2 and y3 form a geometric sequence and determine the common ratio r. (7) (Total 22 marks) 4. A geometric sequence has a first term of 2 and a common ratio of 1.05. Find the value of the smallest term that is greater than 500. (Total 5 marks) 5. A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum. (a) Write down an expression for the value of the investment after n full years. (1) (b) What will be the value of the investment at the end of five years? (1) (c) The value of the investment will exceed $10 000 after n full years. (i) Write an inequality to represent this information. (ii) Calculate the minimum value of n. (4) (Total 6 marks) 6. Consider the arithmetic sequence 8, 26, 44, .... (a) Find an expression for the nth term. (1) IB Questionbank Mathematics Higher Level 3rd edition 2
(b) Write down the sum of the first n terms using sigma notation. (c) Calculate the sum of the first 15 terms. (1) (2) (Total 4 marks) 7. Let f(x) = 4 - x2 . 4- Vx (a) State the largest possible domain for