Name: AK-Nummer: Ergänzungsprüfung January 29, 2016

Transcription

1 INSTRUCTIONS: The test has a total of 20 pages including this title page and 9 questions which are marked out of 10 points; ensure that you do not omit a page by mistake. Please write your name and AK-Nummer on every page. You may use scrap paper for rough work, but only the answers written on the test paper will be considered for marks. Therefore, show all of your work on the test paper itself; you may use the back of the paper if you need extra space, but be sure to indicate this clearly if you do. Write as neatly and clearly as possible. Calculators are not needed and are not allowed. Likewise, the use of cell phones or any other electronic devices is not allowed during the test. You may not consult your notes, textbooks, or any other pre-prepared written material during the test. You have 3 hours to complete all of the questions; you are free to leave as soon as you are finished. 1/20

2 1. a) (5 Marks) Solve the following system of equations for (x, y, z) by your preferred method. A. x + y + z =4 B. x + 3y + 3z =10 C. 2x + y z =3 2/20

3 b) (5 Marks) Does the following linear system have: a) no solutions; b) exactly one solution; c) infinitely many solutions? Support your answer with a graph. If the linear system has one solution, derive that solution. If it has infinitely many solution give a particular solution. A. y 3x = 2 B. 3y 9x =20 3/20

4 4/20

5 2. a) (5 Marks) Give the formula, in the form y = c + bx + ax 2, of the quadratic function which has zeros x = 3 and x = 1 and takes value y = 1 when x = 1. Sketch the graph of the function and 2 give the coordinates of the vertex. 5/20

6 6/20

7 b) (5 Marks) Solve the following cubic equation x 3 7x x 9 = 0 7/20

8 3. a) (5 Marks) Fill in the coordinates on the unit circle corresponding to the given angle. (0, 1) (, ) (, ) 3π 4 π 6 ( 1, 0) (1, 0) 4π 3 7π 4 (, ) (, ) (0, 1) 8/20

9 b) (5 Marks) Draw the graph of y = 3 cos ( 2 ( x π 2 )) so that one can see one entire period. State the period of the function, the location of all zeros, and the value and location of all extreme points. 9/20

10 4. a) (5 Marks) State whether or not the following limits exist as real numbers. If they exist as real numbers, find the limits. If they do not exist, state whether or not the limits diverge to ±. 2 ln(x) (i) lim x 10 log(10x) + e (ii) lim x 1 x 2 3x+2 x 2 +4x 5 10/20

11 b) (5 Marks) Use the squeeze theorem to show that the following limit is zero. lim x 0 x 2 csc ( 1 x ) 11/20

12 5. Find y. In part c) your answer may contain x and y terms. a) (3 Marks) y = tan(x) 1+cos(x) b) (3 Marks) y = 1 x /20

13 c) (4 Marks) xe y = y sin(x) 13/20

14 6. (10 Marks) Analyse the function y = x2 x 2. x + 3 Find: a) The location of maxima and minima (you do not need to compute the maximal or minimal values though, just the location i.e. the x-coordinate); b) The location of inflection points; c) The location of saddle points; d) The formulas of any asymptotes (vertical, horizontal, or slant). 14/20

15 15/20

16 7. Evaluate the given integrals. a) (3 Marks) 1 0 (1 + x) 4 dx b) (3 Marks) s2 s ds 16/20

17 c) (4 Marks) e 2x 1+e 4x dx 17/20

18 8. (10 Marks) Evaluate the integral x 2 2x 1 (x 1) 2 (x 2 + 1) dx. 18/20

19 19/20

20 9. a) (5 Marks) For a probability measure P one of the following is always true for two events A and B P(A B) P(A) + P(B) 1, P(A B) P(A) + P(B) 1. or Which inequality is always true? Why? b) (5 Marks) There are two boxes, Box O and Box E. Box O contains 1 black ball and 3 white balls and Box E contains 2 black balls and 4 white balls. Suppose we select a box with equal probability 1/2, say by flipping a coin, and that we then blindly draw a ball out of that box. What is the probability that the ball we draw is black? 20/20