Summer Assignment Cover Sheet Course: IB Higher Level Math 2 Teacher Names: Assignment Title: Mrs. Short IB Higher Level Math 2 – Summer Assignment Assignment Summary/Purpose: This assignment contains selected review problems on previously taught material. The purpose of this assignment is for student review/retention of learned concepts. Due date Assigned during: Due Date: June 2013 First day of class in September 2013 Estimated time needed to complete the assignment: Approximately 3-5 hours Description of how the assignment will be assessed: The assignment will be collected and graded for completion and accuracy. The assignment will then be returned to the student to make corrections. After approximately two class sessions the assignment must be turned in again and will be graded for accuracy. The overall grade on the assignment will be a combination of the first “completion” grade and the second “accuracy” grade. For the Internal Assessment: Come up with a topic that you would like to write about….see last page of summer packet! Grade impact to overall course grade: The problem set assignment will count as a quiz grade in the first quarter gradebook. Tools/resources needed to complete the assignment: Graphing calculator, Precalculus book (already given to students), your IBHL 1 notebook/material. Contacts: Name E-mail Mrs. Short mary.short@fcps.edu IBHL2 – Summer Assignment Name __________________________ This assignment is due the FIRST day of class in September 2013! These problems are all based on things that you learned in IBHL Math 1. So it is a review of things that you should know.  You may have to look back at your notes or tests/quizzes, etc. to help you remember! Work on this a little bit every week – don’t leave it until the last week of summer! If you need a bit of help with just a couple of problems or if you would like to check your solutions with me you may come by my room the week before school starts…email me first to make sure when I will be there.  Mrs. Short (mary.short@fcps.edu) Show all work in a neat and orderly manner! Use your calculator where appropriate  1. When the function q(x) = x3 + kx2 – 7x + 3 is divided by (x + 1) the remainder is seven times the remainder that is found when the function is divided by (x + 2). Find the value of k. (Total 5 marks) 2. If P(A) = 1 1 5 , P(B) = , and P(A  B) = , what is P(A′ / B′)? 6 3 12 (Total 6 marks) 3. (a) Solve the equation z3 = –2 + 2i, giving your answers in modulus–argument form. (6) (b) Hence show that one of the solutions is 1 + i when written in Cartesian form. (1) (Total 7 marks) 4. Use mathematical induction to prove that 5n + 9n + 2 is divisible by 4, for n  + . (Total 9 marks) 5. In the right circular cone below, O is the centre of the base which has radius 6 cm. The points B and C are on the circumference of the base of the cone. The height AO of the cone is ˆ C is 60°. 8 cm and the angle BO ˆC. Calculate the size of the angle BA diagram not to scale (Total 6 marks) 6. There are 30 students in a class, of which 18 are girls and 12 are boys. Four students are selected at random to form a committee. Calculate the probability that the committee contains (a) two girls and two boys; (3) (b) students all of the same gender. (3) (Total 6 marks) 7. Solve sin 2x = 2 cos x, 0 ≤ x ≤ π. (Total 6 marks) 8. Solve the equation 22x+2 – 10 × 2x + 4 = 0, x  . (Total 6 marks) 9. The cumulative frequency graph below represents the weight in grams of 80 apples picked from a particular tree. (a) Estimate the (i) median weight of the apples; (ii) 30th percentile of the weight of the apples. (2) (b) Estimate the number of apples that weigh more than 110 grams. (2) (Total 4 marks) 10. A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence. The angle of the largest sector is twice the angle of the smallest sector. Find the size of the angle of the smallest sector. (Total 5 marks) 11. The functions f and g are defined as: 2 f (x) = e x , x  0 1 , x   3. g (x) = x 3 (a) Find h (x) where h (x) = g ◦ f (x). (2) (b) State the domain of h−1 (x). (2) (c) Find h−1 (x). (4) (Total 8 marks) 12. A student arrives at a school X minutes after 08:00, where X may be assumed to be normally distributed. On a particular day it is observed that 40 % of the students arrive before 08:30 and 90 % arrive before 08:55. (a) Find the mean and standard deviation of X. (5) (b) The school has 1200 students and classes start at 09:00. Estimate the number of students who will be late on that day. (3) (c) Maelis had not arrived by 08:30. Find the probability that she arrived late. (2) At 15:00 it is the end of the school day, and it is assumed that the departure of the students from school can be modelled by a Poisson distribution. On average, 24 students leave the school every minute. (d) Find the probability that at least 700 students leave school before 15:30. (3) (e) There are 200 days in a school year. Given that Y denotes the number of days in the year that at least 700 students leave before 15:30, find (i) E(Y); (ii) P(Y > 150). (4) (Total 17 marks) 13. (a) Show that a Cartesian equation of the line, l1, containing points A(1, –1, 2) and B(3, 0, 3) x 1 y 1 z  2   has the form . 2 1 1 (2) (b) x 1 y  2 z  3   . Show that the lines l1 1 2 1 and l2 intersect, and find the coordinates of their point of intersection. An equation of a second line, l2, has the form (5) (c) Given that direction vectors of l1 and l2 are d1 and d2 respectively, determine d1 × d2. (3) (d) Show that a Cartesian equation of the plane, Π, that contains l1 and l2 is –x – y + 3z = 6. (3) (e) Find a vector equation of the line l3 which is perpendicular to the plane Π and passes through the point T(3, 1, –4). (2) (f) (i) Find the point of intersection of the line l3 and the plane Π. (ii) Find the coordinates of T′, the reflection of the point T in the plane Π. (iii) Hence find the magnitude of the vector TT  . (7) (Total 22 marks) IB Math HL - The Internal Assessment It’s time to start thinking again about our IAs. . . At the end of class last year you looked at the IBO’s guidelines, grading rubric, and examples. On this webpage: http://tinyurl.com/8p8q5ve The page is really for IB teachers, but it has lots of good information that can help you get started with this process. You want to start in the “General Guidance” section on the left. Read “Skills and Strategies Required by Students” and look over “Assessment Criteria.” You can read some or all of the other topics in the section as well, either now or later. Then move down to the “Assessed Student Work” section – these are sample projects, with grades and explanations, that you can read and compare. If you like, you can start with “Overview” and “Examples of Explorations.” You might want to look at the “Frequently Asked Questions” section – the first third to half will be useful for you. But clearly the nine examples themselves are the most interesting part of this webpage. For each example you can see the project itself, one with grading notes included, and a summary of the grading comments and the grade itself. Your assignment: Choose three of the examples from the website to read and study. For each of your three, write a paragraph explaining what you think will be most helpful to you about that example as you prepare to write your own Internal Assessment – for example, did you gain a better understanding of the expected structure of the work? do you now understand the differences between the grading categories? did your reading make you think of a topic you would like to use for your paper? Your work must be typed, and make sure each paragraph identifies the example discussed. Your three paragraphs will be due on the second day of class when we get back to school in September! We will be doing this later in September! Start thinking now!!!! Picking a topic for your Math HL Internal Assessment Remember! You’re going to need to write 6-12 pages about a specific mathematical topic. You’re going to need to      include correct math at or above a level used in our class include charts and/or graphs (almost definitely) show some personal engagement with your topic explain how you reflected on the results keep in mind the IB rubric to ensure you have included everything necessary So you should start by picking a topic. Here are some ideas: Think about general subject areas like sports, health, music, food, or history. Try narrowing in on an area that interests you, and think about how you could connect it to math. If you’ve ever asked or thought “When we would ever use “ and actually care about the answer (instead of asking it to be annoying) that might be a good thing to investigate. Here are some resources that might help you look for an idea: Books:     several books by Ian Stewart – about different mathematical topics Agnesi to Zeno by Sanderson Smith several books by Martin Gardner – puzzles and brainteasers The Joy of X: A Guided Tour of Math, from One to Infinity by Steven H. Strogatz Websites:    101QS.com – interesting pictures and questions Conceptispuzzles.com – includes Sudoku and many other kinds of puzzles The IB Website from your intro assignment: http://tinyurl.com/8p8q5ve Under the “Planning” section it has some suggestions for topics. You will be asked to pick your topic by the end of September, with a rough draft of your paper likely due in early January, and the final paper due in March. We will discuss more details in class!