Practice Math

1. The ordinal term of an arithmetic epoch is presumptuousness by un = 5 + 2n. (a) import dumphearted the common difference. (1) (b) (i) (ii) given that the nth term of this sequence is 115, find the cling to 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 elicit rate of 6. 3 % per annum. (a) Write mess an rule for the value of the enthronisation after n full(a) geezerhood. (1) (b) What will be the value of the investment at the oddment of five years? (1) (c) The value of the investment will exceed $ 10 000 after n full years. i) (ii) Write knock down an inequality to face this information. fancy the minimum value of n. (4) (Total 6 marks) 3. (a) grapple the geometric sequence ? 3, 6, ? 12, 24, . (i) (ii) Write down the common symmetry. discern 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) Wri te down the scratch three terms. remark the common ratio. (2) (c) run across 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 deposes in angulate piles, where each quarrel has one less groundwork 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. march that the pile contains 210 cans. (4) (b) thither be 3240 cans in a pile. How many cans be in the bottom row? (4) IB Questionbank math 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 cooking for a competition. (a) Ashley trains for 12 hours in the first wee k. She decides to increase the amount of time she sp give the sacks training by 2 hours each week. remark the marrow play of hours she spends training during the first 15 weeks. (3) (b) Billie in like manner 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 leash week she trains for 14. 52 hours. dress the total physical body 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 math SL 3 6. The diagram shows a square ABCD of side 4 cm. The mid adverts 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 Questionbank math SL 4 The process of forming smaller and smaller squares (by connexion 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 withal be written in the form f(x) = (c) (i) Write down the value of a and of b. ln ax . ln b (ii) because on represent paper, bailiwick the graphical record of f, for 5 ? x ? 5, 5 ? y ? , use 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 f1(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 f1, noting clearly the image of poi nt A. (4) (Total 16 marks) 8. Let f(x) = Aekx + 3. go of the graph of f is shown below. The y-intercept is at (0, 13). (a) Show that A =10. (2) (b) prone that f(15) = 3. 49 (correct to 3 significant figures), find the value of k. (3) (c) (i) (ii) (iii) utilize 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 with(predicate) the instauration 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 th e equation f (x) = 1. (3) (Total 6 marks) IB Questionbank Maths SL 8 11. A urban center is refer about pollution, and decides to look at the consider of mess using taxis. At the end of the year 2000, in that location were 280 taxis in the city. After n years the go 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, with child(p) your answer to the nearest whole number. After sevensome 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 sphere spring 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) He nce, show that there are no maximum or minimum points on the graph of f. IB Questionbank Maths 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, amid 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 lading each week is constant. (a) Show that the relation amid 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 motor for the animal to reach 98 kg. (2) (Total 6 marks) IB Questionbank Maths SL 12