Summer Review for Mathematical Studies Rising 12 th graders Due the first day of school in August 2017. Please show all work and round to 3 significant digits. A graphing calculator is required for these 15 problems. The Math Studies Formula Sheet can also be used. 1. Five pipes labeled, 6 meters in length, were delivered to a building site. The contractor measured each pipe to check its length (in meters) and recorded the following; 5.96, 5.95, 6.02, 5.95, 5.99. (i) Find the mean of the contractor s measurements. Calculate the percentage error between the mean and the stated, approximate length of 6 meters. Calculate 3.89 5 8.73 0.5, giving your answer (i) correct to the nearest integer; in the form a 10 k, where 1 a > 10, k. (i)...... (i)...... (Total 6 marks) 1
2. Let A = 4.5 10 3 and B = 6.2 10 4. Find AB; 2(A + B). Give your answers in the form a 10 k, where 1 a < 10 and k is an element of..... (Total 4 marks) 3. Ten students were asked for their average grade at the end of their last year of high school and their average grade at the end of their last year at university. The results were put into a table as follows: Student High School grade, x University grade, y 1 2 3 4 5 6 7 8 9 10 90 75 80 70 95 85 90 70 95 85 3.2 2.6 3.0 1.6 3.8 3.1 3.8 2.8 3.0 3.5 Total 835 30.4 Given that s x = 8.96, s y = 0.610 and s xy = 4.16, find the correlation coefficient r, giving your answer to two decimal places. Describe the correlation between the high school grades and the university grades. (c) Find the equation of the regression line for y on x in the form y = ax + b. (Total 6 marks) 2
2 4. Let U = { 4,, 1, 5, 13, 26.7, 69, 10 33 }. 3 A is the set of all the integers in U. B is the set of all the rational numbers in U. List all the prime numbers contained in U. List all the members of A. (c) List all the members of B. (d) List all the members of the set A intersection B....... (c)... (d)... (Total 8 marks) 3
5. 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. Answer:.. (Total 4 marks) 4
6. The graph below shows the cumulative frequency for the yearly incomes of 200 people. 200 180 160 140 120 100 Cumulative frequency 80 60 40 20 0 0 5000 10 000 15 000 20 000 25 000 30 000 35 000 Annual income in British pounds Use the graph to estimate (c) the number of people who earn less than 5000 British pounds per year; the median salary of the group of 200 people; the lowest income of the richest 20% of this group....... (c)... (Total 4 marks) 5
7. The length of one side of a rectangle is 2 cm longer than its width. If the smaller side is x cm, find the perimeter of the rectangle in terms of x. The perimeter of a square is equal to the perimeter of the rectangle in part. Determine the length of each side of the square in terms of x. The sum of the areas of the rectangle and the square is 2x 2 + 4x +1 (cm 2 ). (c) (i) Given that this sum is 49 cm 2, find x. Find the area of the square....... (c) (i)...... (Total 6 marks) 6
8. In the diagram, triangle ABC is isosceles. AB = AC, CB = 15 cm and angle ACB is 23. Diagram not to scale A C 23º 15 cm B Find the size of angle CAB; the length of AB..... (Total 4 marks) 7
9. The mean of the ten numbers listed below is 5.5. 4, 3, a, 8, 7, 3, 9, 5, 8, 3 Find the value of a. Find the median of these numbers..... (Total 4 marks) 8
10. The following table shows the age distribution of teachers who smoke at Laughlin High School. Ages Number of smokers 20 x < 30 5 30 x < 40 4 40 x < 50 3 50 x < 60 2 60 x < 70 3 Calculate an estimate of the mean smoking age. On the following grid, construct a histogram to represent this data... (Total 4 marks) 9
11. The heights (cm) of seedlings in a sample are shown below. 6 3, 7 key 6 3 represents 63 cm 7 2, 5, 8 8 3, 6, 6, 8, 8 9 2, 5, 7, 8 10 3, 6, 6 11 2, 2 State how many seedlings are in the sample. Write down the values of (i) the median; the first and third quartile. (c) (d) Calculate the range. Using the scale below, draw a box and whisker plot for this data. 60 70 80 90 100 110 120... (i)...... (c)... (Total 6 marks) 10
12. The following diagram shows a triangle ABC. AB = 8 m, AC = 14 m, BC = 18 m, and B ÂC = 110. Diagram not to scale A 110º 8 m 14 m B 18 m C Calculate the area of triangle ABC; the size of angle A ĈB....... (Total 4 marks) 11
13. For his Mathematical Studies Project a student gave his classmates a questionnaire to fill out. The results for the question on the gender of the student and specific subjects taken by the student are given in the table below, which is a 2 3 contingency table of observed values. History Biology French Female 22 20 18 (60) Male 20 11 9 (40) (42) (31) (27) The following is the table for the expected values. History Biology French Female p 18.6 16.2 Male q r 10.8 Calculate the values of p, q and r. (3) The chi-squared test is used to determine if the choice of subject is independent of gender, at the 5% level of significance. (i) State a suitable null hypothesis H 0. (iii) Show that the number of degrees of freedom is two. Write down the critical value of chi-squared at the 5% level of significance. (3) (c) The calculated value of chi-squared is 1.78. Do you accept H 0? Explain your answer. (Total 8 marks) 12
14. Vanessa wants to rent a place for her wedding reception. She obtains two quotations. The local council will charge her 30 for the use of the community hall plus 10 per guest. (i) Copy and complete this table for charges made by the local council. Number of guests (N) 10 30 50 70 90 Charges (C) in On graph paper, using suitable scales, draw and label a graph showing the charges. Take the horizontal axis as the number of guests and the vertical axis as the charges. (3) (iii) Write a formula for C, in terms N, that can be used by the local council to calculate their charges. The local hotel calculates charges for their conference room using the formula: C = 5N + 500 2 where C is the charge in and N is the number of guests. (i) Describe, in words only, what this formula means. (iii) Copy and complete this table for the charges made by the hotel. Number of guests (N) 0 20 40 80 Charges (C) in On the same axes used in part, draw this graph of C. Label your graph clearly. (c) Explain, briefly, what the two graphs tell you about the charges made. (d) Using your graphs or otherwise, find the cost of renting the community hall if there are 87 guests. (Total 16 marks) 13
15. In an experiment a vertical spring was fixed at its upper end. It was stretched by hanging different weights on its lower end. The length of the spring was then measured. The following readings were obtained. Load (kg) x 0 1 2 3 4 5 6 7 8 Length (cm) y 23.5 25 26.5 27 28.5 31.5 34.5 36 37.5 Plot these pairs of values on a scatter diagram taking 1 cm to represent 1 kg on the horizontal axis and 1 cm to represent 2 cm on the vertical axis. (4) (i) Write down the mean value of the load ( x ). Write down the standard deviation of the load. (iii) Write down the mean value of the length ( y ). (iv) Write down the standard deviation of the length. (c) Plot the mean point ( x, y ) on the scatter diagram. Name it L. (d) (i) Write down the correlation coefficient, r. Comment on this result. (e) Find the equation of the regression line of y on x. (f) Draw the line of regression on the scatter diagram. (g) (i) Using your diagram or otherwise, estimate the length of the spring when a load of 5.4 kg is applied. Malcolm uses the equation to claim that a weight of 30 kg would result in a length of 62.8 cm. Comment on his claim. (Total 18 marks) 14