BOSWELL-BÈTA James Boswell Exam MathemaBcs A VWO Date: Time: Number of exercises: 5 Number of parts: 23 Number of appendices: 1 Number of points: 80 • Write your name on every sheet of paper that you hand in. • Use a separate sheet of paper for each exercise. • For each exercise, show how you obtained your answer either by means of a calculaQon or, if you used a graphing calculator, an explanaQon. No points will be awarded to an answer without an explanaQon. • Make sure that your handwriQng is legible and write in ink. No correcQon fluid of any kind is permiRed. Use a pencil only to draw graphs. • You may use the following: • Overview of formulas • Graphing calculator • Drawing utensils • DicQonary, subject to the approval of the invigilator Exercise 1: Fair game? Peter and QuenQn play a game of dice. Two fair dice are thrown (it does not maRer who throws them). If both dice show the same number of dots (rolling doubles), Peter gets a point. In all other cases, QuenQn gets a point. 2p 5 a. Show that the probability that QuenQn gets a point is 6 . The first player to reach a predetermined number of points, wins the game. But the game is clearly unfair: QuenQn's chance of geSng a point is five Qmes that of Peter. That is why they decide to change the rules and agree that when the dice show doubles, Peter gets five points. QuenQn sQll gets one point if no doubles are thrown. Assume that Peter and QuenQn have agreed that the first player to reach 5 points, wins the game. 3p b. Show that the probability that QuenQn wins the game is less than 0.5. Peter and QuenQn do not realize that QuenQn has a smaller chance to win the game than Peter. They keep score as they agreed, so when doubles are rolled, Peter gets five points and when no doubles are rolled, QuenQn gets one point. Because they agreed that the first player to reach five points, wins the game, the game could be decided aBer just one throw. If doubles are rolled, Peter gets five points and he wins. However, it is also possible that it takes several throws to have a winner. You can calculate the number of throws it takes, on average, to have a winner. For this, we use the table below. This table has also been printed on a separate sheet at the end of this exam. Number of throws it takes 1 2 3 4 5 Probability 5p c. Fill out the table on the separate sheet at the end of this exam, and use it to calculate the expected number of throw it takes to have a winner. Round your final answer to one decimal place. Peter and QuenQn feel like playing some more, and decide to stop when one of them reaches 10 points. Examples of a game trajectory where Peter wins are: Q-‐P-‐Q-‐Q-‐Q-‐Q-‐Q-‐P and P-‐Q-‐Q-‐Q-‐P. An example of a game trajectory where QuenQn wins is: Q-‐Q-‐Q-‐Q-‐P-‐Q-‐Q-‐Q-‐Q-‐Q-‐Q. d. Calculate the number of different game trajectories where QuenQn wins. 4p Exercise 2: DifferenBaBon DifferenQate the following funcQons. Rewrite your final answer to a form without any negaQve or fracQonal exponents. Simplify your answer if possible. 3p a. f ( x )=19 x−12−20 x 13 3p 3p b. g ( x)=(19 x−12)(−20 x+ 13) c. h( x)= 1 x + x 13 20 d. Determine exactly the local extreme value of 9p k ( x)=6 √ x 2−38 x+ 365 and explain whether it is a minimum or a maximum. Exercise 3: The orbital period of the Earth The Qme it takes for the Earth to make one complete orbit around the Sun is called a solar year. We oBen say that a year consists of 365 days, but that is not enQrely accurate. Each calendar year divisible by 4 is called a leap year, consisQng of 366 days. A leap year is a correcQon, indicaQng that the actual period of a solar year is not 365 days. 1p a. Explain whether the period of a solar year is more or less than 365 days. Each calendar year divisible by 100 is called a century year. Each century year not divisible by 400 (so 1400, 1500, 1700 etc.) is skipped as a leap year. MulQples of 400 (1600, 2000, etc.) do remain leap years. The period we now obtain for the orbit of the Earth around the Sun is accurate to a few tenths of a second. 3p b. Calculate the orbital period of the Earth around the Sun, based on these facts. We consider the Qme of sunrise in the Netherlands (52° N, 5° E) without using daylight saving Qme. On December 21st, the shortest day of the year, the sun rises at 8:48 and on June 21st, the longest day of the year, it rises at 5:20. We consider d =0 to be January 1st at 0:00. 6p c. Show that the harmonic funcQon Z , describing the Qme of sunrise in the Netherlands, is given by Z ( d )=7.07+ 1.73⋅sin (0.0172(d + 102.3)) , where d is the day of the year. Most people like to travel to work when the sun has already risen. 6p d. Calculate on which days of the year this is possible at 7:30. Exercise 4: Bell peppers Each year a farmer plants bell peppers on part of his land. The bag of seeds for green bell peppers says that 5% of the seeds might be of a different color (red, orange or yellow). 2p a. Calculate the probability, accurate to four decimal places, that in a row of 100 bell pepper plants, 2 will end up having a different color. 3p b. Calculate the probability, accurate to four decimal places, that in a row of 100 bell pepper plants, more than 5 will end up having a different color. Last year, in a row of 100 bell pepper plants, the farmer found 9 bell pepper plants of a different color! That was more than he had expected. 2p c. How many bell pepper plants of a different color did he expect? The farmer decided to test staQsQcally whether seeing 9 out of 100 bell pepper plants of a different color really was a significantly deviaQon. 2p d. State the hypotheses for this test. 5p e. Perform the test and draw a conclusion about the percentage of bell pepper seeds of a different color. Use a significance level of 5%. Exercise 5: Paper cups Many school restaurants serve coffee in paper cups since disposable cups are considered to be more hygienic than real cups that have to be washed to be reused. Supposedly, the producQon costs of paper cups are also less than the costs of washing real cups. However, drinking from a paper cup does not feel as good as drinking from a real cup. The difference is mainly due to the edge: the edge of a paper cup is much thinner than that of a real cup. That is why Company B-‐Cur makes paper cups with an edge that is slightly curled. Their cups are 9 cm in height and the edge is 6 mm thick. Fortunately, a stack of paper cups takes up much less space than a stack of real cups: each cup can be inserted into the next, where the edge of each cup rests on the edge of the cup below. See the figure. 1p a. How high is the stack of 6 cups shown in the figure? 2p b. Let us denote the height of a stack of n cups (in cm) with h( n) . Then h( 1) , h(2) , h(3), ... forms a sequence. What kind of sequence and why? 3p c. Provide both the recursive formula as well as the direct formula for h(n) . At the beginning of the academic year, the restaurant offers a 50% discount on the price of a cup of coffee. To aRract people, 20 stacks of cups are aligned in the window, from high to low: first a stack of 20 cups, then one of 19, 18, etc. From the outside, this kind of looks like an arrow, poinQng towards the coffee machine. 4p d. How many cups were needed to make the 20 stacks? Provide an exact calculaQon. Actually, no two cups are exactly alike. To create the cups, the company uses two machines: one that cuts a piece of paper and another one that curles the edge. Machines are preRy good at cuSng paper, but curling remains a preRy difficult thing to do. A representaQve of the company says that it is true that the height of each cup is 9 cm on average, but also that the height is normally distributed with a standard deviaQon of 0.5 cm. Similarly, the edge does have an average thickness of 6 mm, but it is normally distributed with a standard deviaQon of 1 mm. 3p e. What is the probability that a cup is higher than 9.75 cm? 5p f. What is the probability that the stack of 6 cups from the figure is not higher than 12.5 cm? The End Appendix Name:.................................................. Exercise 1c Number of throws it takes Probability 1 2 3 4 5
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