Express g(x) in the form f(x) + ln a, where a (4)

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1 SL 2 SUMMER PACKET 2013 PRINT OUT ENTIRE PACKET, SHOW YOUR WORK FOR ALL EXERCISES ON SEPARATE PAPER. MAKE SURE THAT YOUR WORK IS NEAT AND ORGANIZED. WORK SHOULD BE COMPLETE AND READY TO TURN IN THE FIRST DAY OF SCHOOL. 1. In an arithmetic sequence, u 1 = 2 and u 3 = 8. Find d. Find u 20. (c) Find S Let f(x) = 3 ln x and g(x) = ln 5x 3. Express g(x) in the form f(x) + ln a, where a +. The graph of g is a transformation of the graph of f. Give a full geometric description of this transformation. (Total 7 marks) 3. The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counterclockwise) direction. diagram not to scale The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres above the water level. After t seconds, the height of the bucket above the water level is given by h = a sin bt + 2. Show that a = 4. The wheel turns at a rate of one rotation every 30 seconds. π Show that b =. 15 IB Questionbank Maths SL 1

2 In the first rotation, there are two values of t when the bucket is descending at a rate of 0.5 m s 1. (c) Find these values of t. d) Determine whether the bucket is underwater at the second value of t. (6) (Total 14 marks) 4. A scientist has 100 female fish and 100 male fish. She measures their lengths to the nearest cm. These are shown in the following box and whisker diagrams. Find the range of the lengths of all 200 fish. Four cumulative frequency graphs are shown below. Which graph is the best representation of the lengths of the female fish? (Total 5 marks) IB Questionbank Maths SL 2

3 5. In a school with 125 girls, each student is tested to see how many sit-up exercises (sit-ups) she can do in one minute. The results are given in the table below. Number of sit-ups Number of students Cumulative number of students p 18 q (i) Write down the value of p. (c) (ii) Find the value of q. Find the median number of sit-ups. Find the mean number of sit-ups. (Total 7 marks) 6. The following table gives the examination grades for 120 students. Grade Number of students Cumulative frequency p 4 q Find the value of (i) p; (c) (ii) q. Find the mean grade. Write down the standard deviation. (1) (Total 7 marks) IB Questionbank Maths SL 3

4 7. Let f(x) = log 3 2 x + log3 16 log 3 4, for x > 0. Show that f(x) = log 3 2x. Find the value of f(0.5) and of f(4.5). The function f can also be written in the form f(x) = ln ax. ln b (c) (i) Write down the value of a and of b. (ii) Hence on graph paper, sketch the graph of f, for 5 x 5, 5 y 5, using a scale of 1 cm to 1 unit on each axis. (iii) Write down the equation of the asymptote. (d) Write down the value of f 1 (0). (6) (1) 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. (Total 16 marks) 8. The diagram below shows a quadrilateral ABCD with obtuse angles A Bˆ C and A Dˆ C. diagram not to scale AB = 5 cm, BC = 4 cm, CD = 4 cm, AD = 4 cm, B ÂC = 30, A Bˆ C = x, A Dˆ C = y. Use the cosine rule to show that AC = 41 40cos x. (1) Use the sine rule in triangle ABC to find another expression for AC. (c) (i) Hence, find x, giving your answer to two decimal places. (ii) Find AC. (6) (d) (i) Find y. (ii) Hence, or otherwise, find the area of triangle ACD. (5) (Total 14 marks) IB Questionbank Maths SL 4

5 9. Let f(x) = cos 2x and g(x) = 2x 2 1. Find π f. 2 π Find (g f). 2 (c) Given that (g f)(x) can be written as cos (kx), find the value of k, k. (Total 7 marks) 10. There is a vertical tower TA of height 36 m at the base A of a hill. A straight path goes up the hill from A to a point U. This information is represented by the following diagram. The path makes a 4 angle with the horizontal. The point U on the path is 25 m away from the base of the tower. The top of the tower is fixed to U by a wire of length x m. Complete the diagram, showing clearly all the information above. Find x. 11. Solve the equation 2cos x = sin 2x, for 0 x 3π. (Total 7 marks) (Total 7 marks) 12. The following diagram shows triangle ABC. AB = 7 cm, BC = 9 cm and A Bˆ C = 120. diagram not to scale IB Questionbank Maths SL 5

6 Find AC. Find B ÂC. 13. Let W = and P = Find WP. Given that 2WP + S = 26 12, find S Solve the following equations. log x 49 = 2 log 2 8 = x (c) log 25 x = 1 2 (d) log 2 x + log 2 (x 7) = 3 (5) (Total 13 marks) 15. The following diagram shows a sector of a circle of radius r cm, and angle at the centre. The perimeter of the sector is 20 cm. Show that = 202r. r The area of the sector is 25 cm 2. Find the value of r. IB Questionbank Maths SL 6

7 16. Let A = and B = Write down A 1. Solve AX = B. (Total 5 marks) 17. The mass m kg of a radio-active substance at time t hours is given by m = 4e 0.2t. Write down the initial mass. The mass is reduced to 1.5 kg. How long does this take? Working: Answers: Let A =. 1 3 Find A Solve the matrix equation AX = There were 1420 doctors working in a city on 1 January After n years the number of doctors, D, working in the city is given by D = n. (i) How many doctors were there working in the city at the start of 2004? IB Questionbank Maths SL 7

8 (ii) In what year were there first more than 2000 doctors working in the city? At the beginning of 1994 the city had a population of 1.2 million. After n years, the population, P, of the city is given by P = (1.025) n. (i) Find the population P at the beginning of (ii) Calculate the percentage growth in population between 1 January 1994 and 1 January (iii) In what year will the population first become greater than 2 million? (7) (c) (i) What was the average number of people per doctor at the beginning of 1994? (ii) After how many complete years will the number of people per doctor first fall below 600? (5) (Total 15 marks) 20. Let f(x) = 3x x +1, g(x) = 4cos Let h(x) = (g f)(x). Find an expression for h(x). Write down the period of h. (c) Write down the range of h. (1) 21. The following diagram shows a triangle ABC, where A ĈB is 90, AB = 3, AC = 2 and B ÂC is. Show that sin = Show that sin 2 = (c) Find the exact value of cos 2. IB Questionbank Maths SL 8

9 22. Let f (x) = 4 tan 2 π π x 4 sin x, x. 3 3 On the grid below, sketch the graph of y = f (x). Solve the equation f (x) = Let f(x) = k log 2 x. Given that f 1 (1) = 8, find the value of k. Find f (Total 7 marks) 24. In the triangle PQR, PR = 5 cm, QR = 4 cm and PQ = 6 cm. Calculate the size of P Qˆ R ; the area of triangle PQR. 25. Let f (x) = 6 sin x, and g (x) = 6e x 3, for 0 x 2. The graph of f is shown on the diagram below. There is a maximum value at B (0.5, b). y B x IB Questionbank Maths SL 9

10 Write down the value of b. On the same diagram, sketch the graph of g. (c) Solve f (x) = g (x), 0.5 x 1.5. Working: Answers: The following is a cumulative frequency diagram for the time t, in minutes, taken by 80 students to complete a task. IB Questionbank Maths SL 10

11 (c) Write down the median. Find the interquartile range. Complete the frequency table below. Time (minutes) Number of students 0 t < t < t < t < t < t < 60 6 (1) 27. A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m, a second side, [AB], is 65 m and the angle between these two sides is 60. Use the cosine rule to calculate the length of the third side of the field. Given that sin 60 = 3, find the area of the field in the form 3p 3 where p is an integer. 2 Let D be a point on [BC] such that [AD] bisects the 60 angle. The farmer divides the field into two parts A 1 and A 2 by constructing a straight fence [AD] of length x metres, as shown on the diagram below. IB Questionbank Maths SL 11

12 (c) (i) Show that the area of A 1 is given by 65x. 4 (ii) Find a similar expression for the area of A 2. (iii) Hence, find the value of x in the form q 3, where q is an integer. (7) (d) (i) Explain why sinadˆ C sinadˆ B. (ii) Use the result of part (i) and the sine rule to show that BD 5. DC 8 (5) (Total 18 marks) 28. A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given below. Write down the number of students who scored 40 marks or less on the test. The middle 50 % of test results lie between marks a and b, where a < b. Find a and b. IB Questionbank Maths SL 12

13 29. Let f(x) = 7 2x and g(x) = x + 3. Find (g f)(x). Write down g 1 (x). (1) (c) Find (f g 1 )(5). (Total 5 marks) 30. A machine was purchased for $ Its value V after t years is given by V =100000e 0.3t. The machine must be replaced at the end of the year in which its value drops below $1500. Determine in how many years the machine will need to be replaced. Working: Answers: The following diagram shows a pentagon ABCDE, with AB = 9.2 cm, BC = 3.2 cm, BD = 7.1 cm, A ÊD =110, A Dˆ E = 52 and A Bˆ D = 60. Find AD. Find DE. IB Questionbank Maths SL 13

14 (c) The area of triangle BCD is 5.68 cm 2. Find D Bˆ C. (d) (e) Find AC. Find the area of quadrilateral ABCD. (5) (Total 21 marks) 32. Let f(t) = a cos b (t c) + d, t 0. Part of the graph of y = f(t) is given below. When t = 3, there is a maximum value of 29, at M. When t = 9, there is a minimum value of 15. (i) Find the value of a. (ii) Show that b = 6 π. (iii) Find the value of d. (iv) Write down a value for c. (7) 1 The transformation P is given by a horizontal stretch of a scale factor of, followed by a translation of Let M be the image of M under P. Find the coordinates of M. IB Questionbank Maths SL 14

15 The graph of g is the image of the graph of f under P. (c) Find g(t) in the form g(t) = 7 cos B(t C) + D. (d) Give a full geometric description of the transformation that maps the graph of g to the graph of f. (Total 16 marks) 33. Let f(x) = 3x 2. The graph of f is translated 1 unit to the right and 2 units down. The graph of g is the image of the graph of f after this translation. Write down the coordinates of the vertex of the graph of g. Express g in the form g(x) = 3(x p) 2 + q. The graph of h is the reflection of the graph of g in the x-axis. (c) Write down the coordinates of the vertex of the graph of h. 34. The following diagram shows part of the graph of a quadratic function f. The x-intercepts are at ( 4, 0) and (6, 0) and the y-intercept is at (0, 240). Write down f(x) in the form f(x) = 10(x p)(x q). Find another expression for f(x) in the form f(x) = 10(x h) 2 + k. (c) Show that f(x) can also be written in the form f(x) = x 10x 2. IB Questionbank Maths SL 15

16 A particle moves along a straight line so that its velocity, v m s 1, at time t seconds is given by v = t 10t 2, for 0 t 6. (d) (i) Find the value of t when the speed of the particle is greatest. (ii) Find the acceleration of the particle when its speed is zero. (7) (Total 15 marks) 35. Consider f(x) = 2kx 2 4kx + 1, for k 0. The equation f(x) = 0 has two equal roots. Find the value of k. (5) The line y = p intersects the graph of f. Find all possible values of p. (Total 7 marks) IB Questionbank Maths SL 16