Practice Math

1. The nth term of an arithmetic sequence is given by un = 5 + 2n. (a) Write down the common difference. (1) (b) (i) (ii) Given that the nth term of this sequence is 115, find the value of n. For this value of n, find the sum of the sequence. (5) (Total 6 marks) 2. 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) (ii) Write down an inequality to represent this information. Calculate the minimum value of n. (4) (Total 6 marks) 3. (a) Consider the geometric sequence ? 3, 6, ? 12, 24, †¦. (i) (ii) Write down the common ratio. Find the 15th term. (3) Consider the sequence x ? 3, x +1, 2x + 8, †¦. IB Questionbank Maths SL 1 (b) When x = 5, the sequence is geometric. (i) (ii) Write down the first three terms. Find the co mmon ratio. (2) (c) Find the other value of x for which the sequence is geometric. (4) (d) For this value of x, find (i) (ii) the common ratio; the sum of the infinite sequence. (3) (Total 12 marks) . Clara organizes cans in triangular piles, where each row has one less can than the row below. For example, the pile of 15 cans shown has 5 cans in the bottom row and 4 cans in the row above it. (a) A pile has 20 cans in the bottom row. Show that the pile contains 210 cans. (4) (b) There are 3240 cans in a pile. How many cans are in the bottom row? (4) IB Questionbank Maths SL 2 (c) (i) There are S cans and they are organized in a triangular pile with n cans in the bottom row. Show that n2 + n ? 2S = 0. Clara has 2100 cans. Explain why she cannot organize them in a triangular pile. 6) (Total 14 marks) (ii) 5. Ashley and Billie are swimmers training for a competition. (a) Ashley trains for 12 hours in the first week. She decides to increase the amount of time she spends training by 2 hou rs each week. Find the total number of hours she spends training during the first 15 weeks. (3) (b) Billie also trains for 12 hours in the first week. She decides to train for 10% longer each week than the previous week. (i) (ii) Show that in the third week she trains for 14. 52 hours. Find the total number of hours she spends training during the first 15 weeks. (4) (c)In which week will the time Billie spends training first exceed 50 hours? (4) (Total 11 marks) IB Questionbank Maths SL 3 6. The diagram shows a square ABCD of side 4 cm. The midpoints P, Q, R, S of the sides are joined to form a second square. A Q B P R D (a) (i) (ii) Show that PQ = 2 2 cm. Find the area of PQRS. S C (3) The midpoints W, X, Y, Z of the sides of PQRS are now joined to form a third square as shown. A W Q X B P Y S R Z D C (b) (i) (ii) Write down the area of the third square, WXYZ. Show that the areas of ABCD, PQRS, and WXYZ form a geometric sequence. Find the common ratio of this sequence. 3) IB Questi onbank Maths SL 4 The process of forming smaller and smaller squares (by joining the midpoints) is continued indefinitely. (c) (i) (ii) Find the area of the 11th square. Calculate the sum of the areas of all the squares. (4) (Total 10 marks) 7. Let f(x) = log3 (a) x + log3 16 – log3 4, for x > 0. 2 Show that f(x) = log3 2x. (2) (b) Find the value of f(0. 5) and of f(4. 5). (3) The function f can also be written in the form f(x) = (c) (i) Write down the value of a and of b. ln ax . ln b (ii) Hence on graph paper, sketch the graph of f, for –5 ? x ? 5, –5 ? y ? , using a scale of 1 cm to 1 unit on each axis. (iii) Write down the equation of the asymptote. (6) (d) Write down the value of f–1(0). (1) IB Questionbank Maths SL 5 The point A lies on the graph of f. At A, x = 4. 5. (e) On your diagram, sketch the graph of f–1, noting clearly the image of point A. (4) (Total 16 marks) 8. Let f(x) = Aekx + 3. Part of the graph of f is shown below. The y-inte rcept is at (0, 13). (a) Show that A =10. (2) (b) Given that f(15) = 3. 49 (correct to 3 significant figures), find the value of k. (3) (c) (i) (ii) (iii) Using your value of k, find f? (x).Hence, explain why f is a decreasing function. Write down the equation of the horizontal asymptote of the graph f. (5) IB Questionbank Maths SL 6 Let g(x) = –x2 + 12x – 24. (d) Find the area enclosed by the graphs of f and g. (6) (Total 16 marks) 9. Consider the function f(x) = px3 + qx2 + rx. Part of the graph of f is shown below. The graph passes through the origin O and the points A(–2, –8), B(1, –2) and C(2, 0). (a) Find three linear equations in p, q and r. (4) (b) Hence find the value of p, of q and of r. (3) (Total 7 marks) IB Questionbank Maths SL 7 10. Let f (x) = 4 tan2 x – 4 sin x, ? a) ? ? ? x? . 3 3 On the grid below, sketch the graph of y = f (x). (3) (b) Solve the equation f (x) = 1. (3) (Total 6 marks) IB Questionbank Maths SL 8 11. A city is concerned about pollution, and decides to look at the number of people using taxis. At the end of the year 2000, there were 280 taxis in the city. After n years the number of taxis, T, in the city is given by T = 280 ? 1. 12n. (a) (i) (ii) Find the number of taxis in the city at the end of 2005. Find the year in which the number of taxis is double the number of taxis there were at the end of 2000. (6) (b)At the end of 2000 there were 25 600 people in the city who used taxis. After n years the number of people, P, in the city who used taxis is given by P= (i) (ii) 2 560000 . 10 ? 90e – 0. 1n Find the value of P at the end of 2005, giving your answer to the nearest whole number. After seven complete years, will the value of P be double its value at the end of 2000? Justify your answer. (6) (c) Let R be the ratio of the number of people using taxis in the city to the number of taxis. The city will reduce the number of taxis if R ? 70. (i) (ii) Find the value of R at the end of 2000.After how many complete years will the city first reduce the number of taxis? (5) (Total 17 marks) IB Questionbank Maths SL 9 12. The function f is defined by f(x) = 3 9 ? x2 , for –3 < x < 3. (a) On the grid below, sketch the graph of f. (2) (b) Write down the equation of each vertical asymptote. (2) (c) Write down the range of the function f. (2) (Total 6 marks) IB Questionbank Maths SL 10 13. Let f (x) = p ? 3x , where p, q? x ? q2 2 + . Part of the graph of f, including the asymptotes, is shown below. (a) The equations of the asymptotes are x =1, x = ? , y = 2. Write down the value of (i) (ii) p; q. (2) (b) Let R be the region bounded by the graph of f, the x-axis, and the y-axis. (i) (ii) Find the negative x-intercept of f. Hence find the volume obtained when R is revolved through 360? about the x-axis. (7) (c) (i) Show that f ? (x) = 3 x 2 ? 1 ?x ? 2 ?1 ? 2 ?. (8) (ii) Hence, show that there are no maximum or minimum points on the graph of f. IB Questionbank Mat hs SL 11 (d) Let g (x) = f ? (x). Let A be the area of the region enclosed by the graph of g and the x-axis, between x = 0 and x = a, where a ? . Given that A = 2, find the value of a. (7) (Total 24 marks) 14. Two weeks after its birth, an animal weighed 13 kg. At 10 weeks this animal weighed 53 kg. The increase in weight each week is constant. (a) Show that the relation between y, the weight in kg, and x, the time in weeks, can be written as y = 5x + 3 (2) (b) (c) (d) Write down the weight of the animal at birth. (1) Write down the weekly increase in weight of the animal. (1) Calculate how many weeks it will take for the animal to reach 98 kg. (2) (Total 6 marks) IB Questionbank Maths SL 12