MATH 204 Final, Fall 2008

American University of Beirut Mathematics Department Time: 2 hrs Fall Semester 2008-2009 Math 204 Final Exam Name : _______________________ ID#: _________________________ • Instructors: Mrs. M. Itani Hatab ,Miss Lina Rahal & Miss Manal Salam • Circle your problem solving section: Section 5 : M @10:00 Section 6 : M @11:00 Section 7 : W @10:00 Section 8 : W @11:00 Section9 : Th @3:30 Section 10 : Th @2:00 Section 11 : Tu @12:00 Section 12 : Tu @11:00 • Answer table for Part I 1 9 17 2 10 18 3 11 19 4 12 20 5 13 21 6 14 22 7 15 23 8 16 24 # of correct answers : ----------------------- # of wrong answers : ------------------------- Grade of Part I Total / 60 1. 2. 3. 4. 5. Grade of Part II / 40 Final Grade 2 Part I (60%) : 24 multiple choice questions with 2.5 points for each correct answer. Circle the correct answer then, copy your answers as a, b, c or d on the table provided on page 1: 1. For a standard normal distribution P (-1.3 < z < 0.45 ) = a) 0.5768 b) 0.1451 c)0.2296 d) 0.7251 --------------------------------------------------------------------------------------------------------------------- 2. If P(B) = 0.4 , ( P A / B) ′ = 0.75 and P(AU B) = 0.7 , then P (A) = a) 0.4 b) 0.3 c) 0.5 d) 0.2 --------------------------------------------------------------------------------------------------------------------- 3. Given the matrices, B =                 − − 2 3 1 3 4 0 1 5 , and D =       − − − 6 4 3 1 3 5 n m n If 3B = D T then the constant m = a) 5 b) 3 c) -3 d) -5 --------------------------------------------------------------------------------------------------------------------- 4. =             =       − − − = − then x and A x If A , 1 2 2 1 2 3 , 3 4 1 2 1 a) 2 b) -1 c) 1 d) 2 1 --------------------------------------------------------------------------------------------------------------------- 5. If , ( ) 3) 3 ( 5) 3 (2 2 2 ∫ ∫ ∫ = + + + + + b a b a a b f x dx dx x x dx x x then f (x) can be a) x x 2 − 2 b) 8 6 3 2 x + x − c) x2 − 2 d) x2 + 2 --------------------------------------------------------------------------------------------------------------------- 6. If the matrix A=( ) aij 5×4 is defined as a    − − = if j i s odd i j if j is even j ij 2 1 2 then 24 33 a a + =…. a) − 2 b) 8 c) 6 d) 24 --------------------------------------------------------------------------------------------------------------------- 7. If ( )( )( ) 3 2 1 3 2 1 1 2 − + − − + = − − − + x C x B x A x x x x then …… a) C = 2 b) C = −5 c) C = 1 d) C = 5 --------------------------------------------------------------------------------------------------------------------- 3 8. If the area to the left of 0z on the standard normal curve is 0.0516 then the value of 0z is…. a) 1.63 b) − .1 63 c) − .1 96 d) 1.96 --------------------------------------------------------------------------------------------------------------------- 9. If f(x,y) = (ln ) 4 2 3 2 x e y yx y + − then 0,2 ) = xf ( a) 1 b) 0 c) 2 d) 5 --------------------------------------------------------------------------------------------------------------------- 10. There are 20 contestants in a beauty contest. In how many ways can the judges choose the three best contestants? a) 20! b)       = 3 20 20C3 c) 3 20 P d) 3 20 --------------------------------------------------------------------------------------------------------------------- 11. There are 20 contestants in a beauty contest. In how many ways can the judges choose the Beauty Queen, the first runner-up and the second runner-up? a) 20! b)       = 3 20 20C3 c) 3 20 P d) 3 20 --------------------------------------------------------------------------------------------------------------------- 12. A password consists of 2 distinct letters of the English alphabet followed by 2 digits such that the first digit is different from zero, then the number of passwords that can formed is….. a) ( ) ( ) 26 2 10 2 b) 25 26 9 10 × × × c)( ) ( ) 25 2 9 2 d) ( ) 10 2 26 25 × × --------------------------------------------------------------------------------------------------------------------- 13. X is a normally distributed variable with a mean equal to 40 and a standard deviation equal to 5. The probability that X is between 46.6 and 48 is a) 0.0542 b) 0.3462 c) 0.0386 d) 0.0575 --------------------------------------------------------------------------------------------------------------------- Consider the definite integral: I=∫ − 5 1 2) (25 dx x . 14. The approximation of the definite integral I using the trapezoidal rule with n=4 is: a) 42 b) 58 c) 38 d) 48 15. The approximation of the definite integral I using the Simpson’s rule with n=4 is: a) 3 136 b) 3 156 c) 3 126 d) 3 176 --------------------------------------------------------------------------------------------------------------------- 16. If e x x f 5 ) ( = − then ( ) ) = ( f 53 x a) 553 e 5x − b) 53e 5x − − c) 553 e 5x − − d) 5e 53x − − --------------------------------------------------------------------------------------------------------------------------- 4 17. All the critical points of the function 10 6 2 , ) ( 2 2 4 + − + − = y y x x f x y are………… a) (0, 3, 1) (1, 3; 0 ) (-1, 3, 0 ) b) (1, 3, 0) (-1, -3, 0 ) (0, 0, 10 ) c) (1, 3, 0) (-1, 3, 0 ) d) (0, 3, 1) (1, 3, 0 ) 18. The point (1, 3, 0 ) is …………… a) a saddle point b) a local minimum c) a local maximum d) not a critical point ------------------------------------------------------------------------------------------------------------------------------- The probability that a certain drug produces undesirable effects in all patients who use it is 0.2. 19. In a sample of six patients using the drug, the probability that exactly four patients have undesirable effects is ………. a) 0.01536 b) 0.4328 c) 0.2458 d) 0.3241 20. In a sample of six patients using the drug, the probability that exactly two patients don’t have undesirable effects is ………. a) 0.01536 b) 0.4328 c) 0.2458 d) 0.3241 21. In a sample of 30 patients using the drug, the average number of patients expected to have undesirable effects is ……. a) 4 b) 6 c) 8 d) 20 ------------------------------------------------------------------------------------------------------------------------------- X is a discrete random variable with the following probability distribution. 22. The mean of this distribution is ……….. a) 2.2 b) 0.33 c) 1.8 d) 2 23. The standard deviation of this distribution is …….. a) 0.85 b) 1.46 c) 0.75 d) 1 ------------------------------------------------------------------------------------------------------------------------------- 24. Given the matrix A =           − − 0 2 1 2 1 2 2 8 3 . The cofactor of the element a23 is ….. a) 3 − b) 3 c) 2 d) − 2 --------------------------------------------------------------------------------------------------------------------------- X 1 2 3 P(X) 0.2 0.4 0.4 5 Part II ( 40%) Answer each of the following questions, explain and show your work 1. Given the system of equations AX=B ,    = + + = − + + = + + 0 10 10 5 2 3 7 3 1 2 3 2 1 3 2 1 3 2 1 x x x x x x x x x a. Use the Gaussian method to find A−1 , where A is the matrix of coefficients. b. Use A−1 to solve the given system. ( 8 %) 6 2. Given the following figure: y g(x) = 4 A B C h(x) =x2 D x f(x) = x+2 Find the coordinates of the points of intersection A, B, and C and D Set the definite integral or the combination of definite integrals that give the area of the region bounded: a. the curve h(x) = x2, and the straight line f(x) = x+2 b. by the curve h(x) = x2, and the two straight lines f(x) = x+2 and g(x) = 4 . c. the curve h(x) = x2 ,the straight line f(x) = x+2 and the x-axis. (8%) 7 3. Evaluate each of the following integrals: a. ( ) dx x x ∫ + 5 ln 3 = b. x dx x∫ 3 ln = c. dx x e x ∫ 3 −2 = ( 12 % ) 8 d. ( ) ( ) dx x x x ∫ − − 3 2 2 2 = 4. Given the function       + + + = 3 3 2 5) (7 5) 5) ( ln (3 ( ) x x x f x find f ′(0) , the first order derivative of f (x) at x = 0. ( 4 % ) 9 5. Given the following table that summarizes some characteristics of 50 persons. Fill in the missing values and then answer the following a. Are the events of the set { B CY X} , , mutually exclusive, and/or collectively exhaustive? (Explain) b. Find: P (A) = P (C′ ) = P ( A′ ∪ X ) = P ( A′/X ′ ) = A B C Total X 8 5 Y 7 23 Total 16 50 ( 8 % )