IB MATH SL2 SUMMER ASSIGNMENT review of topics from year 1. We will be quizzing on this when you return to school. This review is optional but you will earn bonus points if you complete it. Questions? Email me at laichej@pwcs.edu Here is a link to the formula booklet: http://debbiebaker.wikispaces.com/file/view/testing+formulaes.pdf PART 1 With Calculator 1. In an arithmetic sequence, u 1 = 2 and u 3 = 8. Find d. Find u 20. (c) Find S 20. 2. The first three terms of an infinite geometric sequence are 32, 16 and 8. Write down the value of r. Find u 6. (c) Find the sum to infinity of this sequence. (Total 5 marks) 3. Expand 2 7 r 4 r as the sum of four terms. (i) Find the value of 2 30 r 4 r. (ii) Explain why 4 2 r r cannot be evaluated. (6) IB Questionbank Maths SL 1
4. Consider the expansion of (x + 2) 11. Write down the number of terms in this expansion. Find the term containing x 2. (4) (Total 5 marks) 5. Expand (2 + x) 4 and simplify your result. Hence, find the term in x 2 in (2 + x) 4 1 1 2. x 6. Find the term in x 4 in the expansion of 2 2 5 3x. x 7. Let g(x) = 2 1 x sin x, for 0 x 4. Sketch the graph of g on the following set of axes. (4) Hence find the value of x for which g(x) = 1. IB Questionbank Maths SL 2
8. A formula for the depth d metres of water in a harbour at a time t hours after midnight is d P Q cos t, 6 0 t 24, where P and Q are positive constants. In the following graph the point (6, 8.2) is a minimum point and the point (12, 14.6) is a maximum point. d 15 (12, 14.6) 10. 5 (6, 8.2) 0 6 12 18 24 t Find the value of (i) Q; (ii) P. Find the first time in the 24-hour period when the depth of the water is 10 metres. (c) (i) Use the symmetry of the graph to find the next time when the depth of the water is 10 metres. (ii) Hence find the time intervals in the 24-hour period during which the water is less than 10 metres deep. (4) 9. 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. IB Questionbank Maths SL 3
10. The following diagram shows triangle ABC. diagram not to scale AB = 7 cm, BC = 9 cm and A Bˆ C = 120. Find AC. Find B ÂC. 11. The following diagram shows the triangle ABC. The angle at C is obtuse, AC = 5 cm, BC = 13.6 cm and the area is 20 cm 2. diagram not to scale Find A ĈB. (4) Find AB. IB Questionbank Maths SL 4
12. The diagram below shows a circle centre O, with radius r. The length of arc ABC is 3 cm and 2π A ÔC =. 9 Find the value of r. (c) Find the perimeter of sector OABC. Find the area of sector OABC. 13. The diagram below shows a sector AOB of a circle of radius 15 cm and centre O. The angle at the centre of the circle is 2 radians. Diagram not to scale A B O Calculate the area of the sector AOB. Calculate the area of the shaded region. (Total 4 marks) 14. The population of a city at the end of 1972 was 250 000. The population increases by 1.3 per year. Write down the population at the end of 1973. Find the population at the end of 2002. IB Questionbank Maths SL 5
PART 2 WITHOUT CALCULATOR 15. 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. 16. Let f(x) = 8x 2x 2. Part of the graph of f is shown below. Find the x-intercepts of the graph. (4) (i) Write down the equation of the axis of symmetry. (ii) Find the y-coordinate of the vertex. IB Questionbank Maths SL 6
17. Let f(x) = 2x 2 + 4x 6. Express f(x) in the form f(x) = 2(x h) 2 + k. Write down the equation of the axis of symmetry of the graph of f. (c) Express f(x) in the form f(x) = 2(x p)(x q). 18. Let f(x) = 7 2x and g(x) = x + 3. Find (g f)(x). Write down g 1 (x). (c) Find (f g 1 )(5). (Total 5 marks) 19. The quadratic equation kx 2 + (k 3)x + 1 = 0 has two equal real roots. Find the possible values of k. (5) Write down the values of k for which x 2 + (k 3)x + k = 0 has two equal real roots. IB Questionbank Maths SL 7
20. Let f(x) = cos 2x and g(x) = 2x 2 1. π Find 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. 21. Let f(x) = log 3 x, for x > 0. Show that f 1 (x) = 3 2x. Write down the range of f 1. Let g(x) = log 3 x, for x > 0. (c) Find the value of (f 1 g), giving your answer as an integer. (4) 22. Solve log 2 x + log 2 (x 2) = 3, for x > 2. IB Questionbank Maths SL 8
23. 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) 24. Find log 2 32. Given that log 2 32 y 8 x can be written as px + qy, find the value of p and of q. (4) (Total 5 marks) 25. Show that 4 cos 2θ + 5 sin θ = 2 sin 2 θ + 5 sin θ + 3. Hence, solve the equation 4 cos 2θ + 5 sin θ = 0 for 0 θ 2π. (5) 26. Solve the equation 2cos x = sin 2x, for 0 x 3π. IB Questionbank Maths SL 9
3 27. The straight line with equation y = x 4 makes an acute angle θ with the x-axis. Write down the value of tan θ. Find the value of (i) sin 2θ; (ii) cos 2θ. (6) 28. Consider g (x) = 3 sin 2x. Write down the period of g. On the diagram below, sketch the curve of g, for 0 x 2. y 4 3 2 1 0 1 2 3 π 2 π 3π 2 2π x 4 (c) Write down the number of solutions to the equation g (x) = 2, for 0 x 2. IB Questionbank Maths SL 10
29. A data set has a mean of 20 and a standard deviation of 6. [6 marks] a) Each value in the data set has 10 added to it. Write down the value of (i) the new mean; (ii) the new standard deviation. b) Each value in the original data set is multiplied by 10. Write down the value of (i) the new mean; (ii) the new standard deviation. 30. WITH CALCULATOR!!!!!!! The following table shows the amount of fuel ( litres) used by a car to travel certain distances ( km). Distance (x) 40 75 120 150 195 Fuel used (y) 3.6 6.5 9.9 13.1 16.2 This data can be modelled by the regression line with equation. a) Write down the value of and of. b) Explain what the gradient represents. c) Use the model to estimate the amount of fuel the car would use if it is driven km. d) Give the correlation coefficient and explain what it means (direction and strength) IB Questionbank Maths SL 11
31. The weekly wages (in dollars) of 80 employees are shown in the cumulative frequency curve below. a) (i) Write down the median weekly wage. (ii) Find the interquartile range of the weekly wages. b) The box-and-whisker plot below displays the weekly wages of the employees. Find a, b, and c. IB Questionbank Maths SL 12