PRACTICE WORKSHEET FOR MA3ENA EXAM

Transcription

1 PRACTICE WORKSHEET FOR MA3ENA EXAM What you definitely need to know: - definition of an arithmetic and geometric sequence - a formula for the general term (u n ) of each of these sequences - a formula for the sum of the first n terms of each of these sequences (S n ) - the condition for the existence of an infinite sum of the terms of a geometric sequence ( 1 < r < 1) and a formula for the sum (S = u 1 1 r ) - the definitions of mode, median, mean, range, lower and upper quartiles, interquartile range - how to draw a (precise) box and whisker plot - understand the concept of cumulative frequency - enter data in the calculator (Lists and spreadsheets) and apply (and understand the output of) One-variable statistical calculations - draw a tree diagram or a Venn diagram to represent a probability problem (occasionally both are possible) - understand the structure of a tree diagram (conditional probabilities on branches, etc.) - the formula to calculate conditional probability - the definition of ( n ) or nck (n choose k) and the corresponding command on the calculator k - calculate the probability of repeated events (probabilities may or may not change at each trial) What you should definitely keep in mind: You have to show your work even if it is not written in each question. Giving a result is by far not enough. Use your calculator wisely. If you can use it to solve an equation, etc., you do not need to make a reference to it.

2 Calculator part practice problems 1. The first term of an arithmetic sequence is -3. Its 7 th term is 9. a. Calculate the common difference of this sequence. The general term of an arithmetic sequence is u n = u 1 + (n 1)d. Now u 1 = 3, u 7 = 9. Using the above formula for u 7 yields the equation 9 = 3 + (7 1)d 1 = 6d = d That is, the common difference is. (Note: feel free to solve the equation 9 = 3 + (7 1)d with the calculator. In this case you simply have to write down the solution for d after the equation. This is actually an important thing: if you set up an equation or inequality that your calculator can solve, you can simply write down the solution(s) without any further comment.) b. Write down the first 7 terms of the sequence. -3, -1, 1, 3, 5, 7, 9 c. Determine the nd term of the sequence. u = 3 + ( 1) = = 439 d. Calculate the sum of the first terms of the sequence. The formula for the sum of the first n terms of an arithmetic sequence is S n = u 1 + u n n = u 1 + (n 1)d n Now S = = A company pays a certain amount of money (called dividends) to its stockholders every year. In 001, the company paid 450 euros to a stockholder. Since then each year it pays 4 of the 5 previous year s dividends. a. Show that the yearly dividends form a geometric sequence. To calculate the 4 of a number, you have to multiply by 4. That is, it is a geometric 5 5 sequence with common ratio 4 5. (Note that having a common ratio is exactly the same as always multiplying by the same number.) b. How much dividend do they pay to this stockholder in 018? This is a geometric sequence with u 1 = 450 and q = 4 5. If 001 corresponds to u 1, 018 corresponds to u 18. To calculate u 18, use the formula for the general term of a geometric sequence: u n = u 1 q n 1 Here u 18 = 450 ( 4 5 )

3 That is, the company pays euros in 018. c. How much money did he get from the company from 001 to 018? Use the formula for the sum of the first n terms of a geometric sequence: q S n = u n 1 1 if q 1 q 1 Now we talk about the first 18 years, so S 18 = 450 (4 5 )18 1 = That is, he gets a total of euros. d. In which year will the company pay less than 10 euros of dividends for the first time? We have to solve the inequality u n < 10. That is, 450 ( 4 5 )n 1 < Solving it with the calculator yields n > Thus, this will first happen in the 6 th year, that is, in 06. e. If the company will exist forever, how much will they pay to the stockholder altogether? (Find the infinite sum.) Since q = 4 is between 1 and 1, the infinite sum can be calculated with the 5 formula: S = u 1 1 q = = 450 = That is, the stockholder will get 150 euros (if he lives infinitely long). (General note to the whole problem: feel free to use 0.8 instead of 4 if you feel 5 more comfortable with it.) 3. Three consecutive terms of a geometric sequence are x, x + 1, x + 6. Find these three numbers. The ratio of consecutive terms is constant (always the same) for a geometric sequence, that is, x + 1 x + 6 = x x + 1 Solve this equation with the calculator to get x = 6 or x = 4. That is, there are two solutions. The three numbers are 6, 6, 6 or 4, 36, 54 (Not part of this question, but note that the common ratio of the first sequence is 6 = 6 1, the common ratio of the second sequence is 36 = 3 = 1. 5.) 4 4. Statistical data shows that 55% of 6th year European school students follow the 3-period course (the rest follow the 5-period course). At the end of the 6th year 70% of the pupils who follow the 3-period course pass their exam. Only 10% of the pupils who follow the 5-period course fail their exam.

4 A 6th year student is selected at random at the end of the year. Find the probability of the following events: a. draw an appropriate tree diagram; b. the student follows the 3-period course and passed his exam; c. the student passed his exam; d. the student failed his exam given that he follows the 3-period course; e. the student follows the 3-period course given that he passed his exam. 0.7 passed 3 per failed per passed failed P(follows the 3 period course and passed his exam) = = P(passed) = P(3 period and passed OR 5 period and passed) = = P(failed 3 period) = = 0. 3 P(3 period passed) = P(3 period and passed) P(passed) = George takes 15 sandwiches to school for lunch. In his rather sizeable lunchbox, there are 8 tuna sandwiches, 4 ham sandwiches and 3 cheese sandwiches. During the first 5-minute break, he already eats 3 of his sandwiches at random. Find the probability of the following events. a. He eats only tuna sandwiches. b. He eats only ham sandwiches. c. He eats at least one cheese sandwich. d. He eats tuna sandwiches and one cheese sandwich.

5 P(only tuna) = = 8 ( 0. 13) 65 P(only ham) = = 4 ( ) 455 P(at leas one cheese) = 1 P(no cheese) = = 1 13 P( tuna and a cheese) = ( 3 1 ) = 1 ( ) = 47 ( ) 91 (Note: in the last question you have to multiply by ( 3 ) or simply by 3 because the cheese 1 sandwich could be the first one, the second one or the third one.) 6. In a supermarket, certain customers were asked to evaluate the supermarket with a score from 1 to 6. The histogram below shows the results of the survey. a. How many customers evaluated the supermarket? = 1 1 customers evaluated the supermarket. b. Calculate the mean, mode, median, range, lower and upper quartiles and interquartile range of the scores. mean: 3 mode: 5 median: +3 =. 5 range: 5 lower quartile: 1 upper quartile: 5 interquartile range: 4 (Note: feel free to use the One-variable statistics command of your calculator.) 7. The following table shows the times taken by a group of walkers to complete a 15-mile walk. Their times have been recorded to the nearest hour. Time (hours) Frequency 5 x Cumulative frequency y

6 a. Find the value of x. 7 + x = 19 yields x = 1 b. Find the value of y. y = = 37 c. Calculate the mode, the median and the mean of the walkers time. mode: 5 median: 5 mean: 5.51

7 Non-calculator part practice problems 1. The first row of a theatre has 1 seats. Each row after the first one has 3 more seats than the previous row. There are 13 rows of seats in the theatre. a. Show that the number of seats in each row forms an arithmetic sequence. Since the difference between consecutive rows is 3, it is an arithmetic sequence with u 1 = 1 and d = 3. b. How many seats are there in the last row? There are 13 rows, so we need u 13 = 1 + (13 1) 3 = 48. There are 48 chairs in the last row. c. How many seats are there in the theatre? Use the formula for the sum of the first n terms of an arithmetic sequence with n = 13. S 13 = = 390. That is, there are 390 seats in the theatre.. The first term of a geometric sequence is 4, the second term of this sequence is. a. Find the common ratio of the sequence. Since u u 1 = 4 = 1 = 0 5, the common ratio is 0.5 (or 1 ) b. Give the next 4 terms of the sequence. 4,, 1, 0. 5, 0. 5, (fractions are also perfect here) c. Justify that the sum of this infinite sequence exists. Since for q = 0. 5, 1 < 0. 5 < 1, the sum exists. d. Calculate the sum of the infinite sequence. S = = 8

8 3. There are 30 people in a room. 0 of them have a dog. 16 of them have a cat. 4 of them have neither a dog, nor a cat. A randomly selected person leaves the room. a. Draw an appropriate diagram. b. Calculate the probabiltiy the person who leaves has a dog and a cat. c. Calculate the probability that (s)he has a dog but no cat. d. Calculate the probability that (s)he has a dog if we know that (s)he has a cat. dog x 0-x 16-x cat 4 Since 0 x + x + 16 x + 4 = 30, we get 40 x = 30 and so x = 10. P(dog and cat) = = P(dog and no cat) = = = 1 3 P(dog and cat) P(dog cat) = = 10 P(cat) 16 = There are 5 students in s6ma3ena. In a draw, two of them can win a trip to the World Cup final in July. How many different outcomes are possible? (that is, how many pairs of students can be formed in the class?) For any students (5) we can select anyone from the remaining 4, that is we get 5 4 possible pairs. It does not matter, however, who gets the first ticket for the match. There are two possible orders, so the number of pairs is actually 5 4 = In a park that has several basketball courts, a student counts the number of players playing basketball each day over a two week period and records the following data. 10, 90, 30, 0, 50, 30, 60, 40, 70, 40, 30, 60, 80, 0 a. Calculate the mode, median, range, lower and upper quartiles and interquartile range of the number of players per day. rearrange the data in ascending (or descending order) to answer the questions: 10, 0, 0, 30, 30, 30, 40, 40, 50, 60, 60, 70, 80, 90

9 mode. 30 median: = 40 range: 80 lower quartile: 30 upper quartile: 60 interquartile range: 30 b. Draw a box-and-whisker plot of the data. Do not forget to use an appropriate scale