MATHEMATICAL STUDIES SL YEAR 2 SUMMER PACKET

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1 MATHEMATICAL STUDIES SL YEAR 2 SUMMER PACKET Congratulations, you ve made it through year one of Hillside High School IB Diploma Program. It s been a lot of work, but I guarantee that if you make a commitment to persevere through one more year, it s worth it! Over the summer, there are several topics that you need to be comfortable with. These are skills that you should have mastered during this past year. This packet is meant to help you review those topics, because we will not review this material once the year begins. To complete this packet, a graphing calculator is required (our class uses a TI-84), which is a requirement for the course. If you have trouble with any of the topics in this packet, you have permission to use outside resources to help you, however, you must show work neatly on a separate sheet of paper for all of the questions. If you turn in a packet with only answers written in it, you will receive a zero. The packet will be your first grade in the class and is due on the first day of class, August 28, You can me at latoya.clay@hillisidehornets.net with questions and I will respond as I have time over the summer. 1. Calculate , writing your answer correct to two decimal places; (i) correct to three significant figures; (ii) in the form a 10 k,where 1 a < 10, k. 2. A woman deposits $100 into her son's savings account on his first birthday. On his second birthday she deposits $125, $150 on his third birthday, and so on. How much money would she deposit into her son's account on his 17th birthday? How much in total would she have deposited after her son's 17th birthday? 3. Let m = and n = Express each of the following in the form a 10 k, where 1 a < 10 and k. mn; m. 1

2 4. Arthur needs to calculate a value from a trigonometric formula. He uses his calculator to find the value of r given by r = sin(86 ) sin(85 ). (c) Calculate the value of r, correct to three significant figures. Arthur makes the mistake of rounding both of the sines to three significant figures before taking their difference. Calculate the value of r found by Arthur. Call this value r A. Calculate the percentage error E in Arthur's calculation, given by the formula E = 100(r r A ). (Total 8 marks) 5. Sketch a graph of y = 2 x for 10 x 10. Hence write down the equations of the horizontal and vertical asymptotes. (Total 6 marks) 6. A basketball is dropped vertically. It reaches a height of 2 m on the first bounce. The height of each subsequent bounce is 90% of the previous bounce. What height does it reach on the 8th bounce? What is the total vertical distance travelled by the ball between the first and sixth time the ball hits the ground? (4) (Total 6 marks) 7. Factorise the expression 2x 2 3x 5. Hence, or otherwise, solve the equation 2x 2 3x = The height of a vertical cliff is 450 m. The angle of elevation from a ship to the top of the cliff is 23. The ship is x metres from the bottom of the cliff. Draw a diagram to show this information. Diagram: Calculate the value of x. 2

3 9. The following diagram shows the graph of y = 3 x + 2. The curve passes through the points (0, a) and (1, b). Diagram not to scale. Find the value of (i) a; (ii) b. y Write down the equation of the asymptote to this curve. (0, a) (1, b) O x (Total 8 marks) 10. The diagrams below show the graphs of two functions, y = f(x), and y = g(x). y 2 1 y = f(x) y x y = g(x) 360º 180º º 360º x State the domain and range of the function f; the function g. (Total 8 marks) 3

4 11. A small manufacturing company makes and sells x machines each month. The monthly cost C, in dollars, of making x machines is given by C (x) = x 2. The monthly income I, in dollars, obtained by selling x machines is given by I (x) = 150x 0.6x 2. Show that the company's monthly profit can be calculated using the quadratic function (c) P (x) = x x The maximum profit occurs at the vertex of the function P(x). How many machines should be made and sold each month for a maximum profit? If the company does maximize profit, what is the selling price of each machine? (4) (d) Given that P (x) = (x 20) (130 x), find the smallest number of machines the company must make and sell each month in order to make positive profit. (4) (Total 12 marks) 12. A student has drawn the two straight line graphs L 1 and L 2 and marked in the angle between them as a right angle, as shown below. The student has drawn one of the lines incorrectly. y 3 L x L1 1 Consider L 1 with equation y = 2x + 2 and L 2 with equation y = 1 x + 1. (c) (d) Write down the gradients of L 1 and L 2 using the given equations. Which of the two lines has the student drawn incorrectly? How can you tell from the answer to part that the angle between L 1 and L 2 should not be 90? Draw the correct version of the incorrectly drawn line on the diagram. (Total 8 marks) 4

5 13. The line L 1 shown on the set of axes below has equation 3x + 4y = 24. L 1 cuts the x-axis at A and cuts the y-axis at B. Diagram not drawn to scale y L 1 B L 2 M O C A x Write down the coordinates of A and B. M is the midpoint of the line segment [AB]. Write down the coordinates of M. The line L 2 passes through the point M and the point C (0, 2). (c) Write down the equation of L 2. (d) Find the length of (i) (ii) MC; AC. (e) The length of AM is 5. Find (i) the size of angle CMA; (3) (ii) the area of the triangle with vertices C, M and A. (Total 15 marks)

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7 14. The diagram shows a cuboid 22.5 cm by 40 cm by 30 cm. H G E F 40 cm A Calculate the length of [AC]. Calculate the size of GAC ˆ. D 22.5 cm B C 30 cm 15. Find the volume of the following prism. 8 cm Diagram not to scale 5.7 cm 42 6

8 16. An office tower is in the shape of a cuboid with a square base. The roof of the tower is in the shape of a square based right pyramid. The diagram shows the tower and its roof with dimensions indicated. The diagram is not drawn to scale. O H 10 m G E F 40 m D C A 6 m B Calculate, correct to three significant figures, (i) (ii) (iii) (iv) the size of the angle between OF and FG; the shortest distance from O to FG; the total surface area of the four triangular sections of the roof; the size of the angle between the slant height of the roof and the plane EFGH; (3) (3) (v) the height of the tower from the base to O. A parrot's nest is perched at a point, P, on the edge, BF, of the tower. A person at the point A, outside the building, measures the angle of elevation to point P to be 79. Find, correct to three significant figures, the height of the nest from the base of the tower. (Total 14 marks) 7