ADVANCED FUNCTIONS (MHF 4U) FINAL EXAMINATION

Transcription

1 Canadian International Matriculation Programme Sunway College (KL) Sdn. Bhd. ADVANCED FUNCTIONS (MHF 4U) FINAL EXAMINATION Date/Day : 8 June 2015, Monday Time : 8.30 am am Length : 2 hours Lecturers : Ms. Dzura Mr.Nithyananthan Mr. Grandy Ms. Pecarski Mr. Welch (Please circle your teacher s name) Student Name : Section/Period : Please read the following instructions carefully before you begin the examination: 1. This exam paper has 15 printed pages including this cover page and the formula sheet. 2. The examination is worth 30 percent of your overall mark for this course. 3. The examination consists of THREE parts as follows: Parts Content Number of Questions Marks A MULTIPLE CHOICE B SHORT ANSWERS 9 32 C PROBLEM SOLVING 5 (choose only 4) 24 OVERALL COMMUNICATION 10 TOTAL Please enter your multiple choice answers on the sheet provided on page Solutions for Parts B and C must be written in the space provided. 6. Answer 4 questions only for Part C. 7. Scientific or Graphing Calculators and paper dictionary are permitted. NO SHARING is allowed. 8. All answers must be written legibly in black/blue pen or pencil. For office use only: K/U T/I COM APPL TOTAL MHF 4U FINAL EXAMINATION June 8, 2015

2 Multiple Choice [34 marks] Identify the choice that best completes the statement or answers the question. 1. Solve 2 log x log 3 = 2 A. 10 C B. 2 D. 2. Find the exact value of tan 5π 3 A. 3 C. B D Which of the following functions is divisible by x 3? A. f(x) = x 3 10x x 24 C. h(x) = 3x 3 6x 2 + 3x + 9 B. g(x) = x 3 4x 2 13x + 24 D. j(x) = x 3 9x x For what interval(s) is the graph of the rational function negative? A. x 4 C. x 4, x 1 B. x 1 D. x, x 4 MHF 4U FINAL EXAMINATION June 8, 2015 Page 2

3 Multiple Choice [34 marks] cont d: Identify the choice that best completes the statement or answers the question. 5. What statement about f(x) = 1000(1.3) x A. Instantaneous rate of change is 0 for some value of x. B. Instantaneous rate of change is the same for all x-values. C. Instantaneous rate of change is negative for all x-values. D. Instantaneous rate of change is positive for all x-values. 6. State the domain and range of f(x) = 2 log 4(x 12) A. B. domain {x R x >3} range {y } C. domain {x R x > 12} range {y } D. domain {x R x < 3} range {y } domain {x R x < 12} range {y } 7. If f(x) = x2 k + 1 and f 1 (5) = 6, find k A. 9 C. 6 B. 5 D Determine the interval(s) on which x 2 + 2x 3 > 0 A. x < 3, x > 1 C. x < 3, 3 < x < 1, x > 1 B. 3 < x < 1 D. x > 1 9. Solve the equation x+2 = x+7 x 7 x+3 A. x = 11 C. x = 7, 3 x = 55 B. 6 D. No solution MHF 4U FINAL EXAMINATION June 8, 2015 Page 3

4 Multiple Choice [34 marks] cont d: Identify the choice that best completes the statement or answers the question. 10. Which of the following is true? A. f(x) = 1 has two vertical asymptotes. 2 x B. f(x) = 1 decreases over its entire domain. x C. f(x) = 1 is always negative. x D. f(x) = 1 never increases. 2x 11. Which function has the largest period? A. y = 6 sin(3x) + 20 C. y = tan (x) B. y = 2 sin(0.5x) 11 D. y = 7 cos(πx) The function s(d) = 93 log(d) + 65 relates the speed of the wind s in miles per hour near the centre of a tornado to the distance it travels in d miles. If the tornado has sustained winds of 400mph, estimate the distance it has travelled. A. 307 miles C miles B miles D. 69 miles 13. The point (6,6) is on the graph y = f(x). Find the corresponding coordinates of this point on the graph y = 4f ( 1 x + 9) 7 3 A. (-9,17) C. (11,11) B. (17,-9) D. (9,17) 14. Which is the solution to 5 < 3x + 17 < 71 A. C B. D MHF 4U FINAL EXAMINATION June 8, 2015 Page 4

5 Multiple Choice [34 marks] cont d: Identify the choice that best completes the statement or answers the question. 15. Which function has a y-intercept of 2 3? A. f(x) = 3x + 2 5x 1 B. f(x) = 3x + 2 5x 1 C. f(x) = 6x + 5 9x 7 D. f(x) = 5x + 6 7x Which of the following trigonometric ratios does not have the same value as the others? A. sin ( 3π 4 ) C. cos (7π 4 ) B. cos ( 5π 4 ) D. sin ( π 4 ) 17. Solve log(x + 5) = log (3 2x). A C B D MHF 4U FINAL EXAMINATION June 8, 2015 Page 5

6 MULTIPLE CHOICE ANSWER SHEET NAME : PERIOD: Answer all multiple choice questions on this sheet: MHF 4U FINAL EXAMINATION June 8, 2015 Page 6

7 Part B- Short Answers [32 marks] 1. Functions f and g are two linear functions with a slope of 3. Function f has an initial value of 6 and function g passes through the origin. What is (f g)(2). [T/I : 2 marks] 2. Write the equation of the function shown below if it is known that f(0) = 18. [APPL : 3 marks] y x Determine an equation in the form f(x) = ax+b at x = 4 3 and y = 2 3, and a y-intercept of 3 4. for a function that has asymptotes cx+d Sketch a graph of your function. [APPL : 4 marks] 6 y 4 2 x MHF 4U FINAL EXAMINATION June 8, 2015 Page 7

8 Short Answers [32 marks] cont d: 4. Solve the following trigonometric equations over the indicated intervals. Express your final answer(s) as exact values. [T/I : 4 marks] cos x = 2 2, 2π < x < 2π sin 2x = 1, 0 < x < 2π 2 5. Determine the point(s) of intersection for the graphs of y = 2(3 x ) and y = 6(2 x ). Round your answer to one decimal place. [T/I : 3 marks] 6. Determine the properties of the graph below. [K/U : 3 marks] Domain: Range: Interval of increase: Interval of decrease: End behaviour: Initial value: Zero(s): MHF 4U FINAL EXAMINATION June 8, 2015 Page 8

9 Short Answers [32 marks] cont d: 7. The volume of a rectangular box is (x x x + 140)cm 2. Suppose the box is (x + 5)cm long and (x + 7)cm wide. Determine the height of the box. [APPL : 2 marks] 8. Solve x < x+4 x+4 x [T/I : 4 marks] 9. Prove the following identity: cos x cos 2x+2 = 1+cos x 3 sin x sin 2x sin x [T/I : 3 marks] 10. Solve log(x + 3) + log(x 2) = log (3x + 2) [T/I : 4 marks] MHF 4U FINAL EXAMINATION June 8, 2015 Page 9

10 Problem Solving [24 marks] Answer 4 questions only. 1. Mr. Peter is docking a motorboat. He turns off the power at t = 0 and lets the boat coast toward the dock. The distance d, in metres, between the boat and the dock as a function of time t, in seconds, is given by d(t) = 7t+70, 0 t 10. t+4 [APPL : 6 marks] a) What is the average velocity of the boat during the time interval from when Mr. Peter turns the boat off to when it meets the dock? b) What is the velocity of the boat exactly when he turns the boat off? c) Sketch a graph of the function, distance versus time. 20 y x MHF 4U FINAL EXAMINATION June 8, 2015 Page 10

11 Problem Solving [24 marks] cont d: Answer 4 questions only. 2. A clothing company opened an outlet at the Pyramid in The revenue of this company which has increased over the first six years is recorded in the table below. Number of years from 2009 Revenue (million ringgit) a) Use technology to compute the regression model representing the Revenue of the company as a function of the number of years (since 2009). [Note: Choose only the regressions shown below and give your answers correct to 3 decimals] i) Cubic regression ii) Logarithmic regression [Note: Ln ( x ) regression means log e ( x )] b) i) Which of the two models will predict that the company will stop making revenue sometime in the future. ii) Use the graphing calculator to estimate the year when this could happen. c) Identify the better regression model and justify your answer mathematically. [APPL : 6 marks] MHF 4U FINAL EXAMINATION June 8, 2015 Page 11

12 Problem Solving [24 marks] cont d: Answer 4 questions only. 3. Jo is deciding between two investment options for his money. Option 1: Invest a sum in an account that earns 6.5% per annum compounded annually. Option 2: Invest a sum in an account that earns 6% per annum compounded weekly. [APPL : 6 marks] a) If he intends to invest RM1000, which investment strategy would yield greater returns after 5 years? b) How long would it take each account to double his investment? c) How much more should he invest in Option 2 in order to have the same amount as in Option 1 after 5 years? MHF 4U FINAL EXAMINATION June 8, 2015 Page 12

13 Problem Solving [24 marks] cont d: Answer 4 questions only. 4. The graph below is the result of transformations applied to y = x 3. y [APPL : 6 marks] (-2, 0) x 2 (-1, -4) (0, -8) a) State the steps (in the correct order) that would transform the parent function into this graph. b) Determine the equation of the transformed polynomial. c) Identify the original coordinates of the point (-2,0). MHF 4U FINAL EXAMINATION June 8, 2015 Page 13

14 Problem Solving [24 marks] cont d: Answer 4 questions only. 5. In the Bay of Fundy the water level d varies sinusoidally with time t. Records show that the water is 2m deep at low tide and six hours later at high tide is 18m. [APPL : 6 marks] a) If at midnight the water level is at 10m and decreasing, sketch a graph representing the water level at t hours after midnight. 0 t 12 b) Create an equation representing d(t) using a sine function: Create an equation representing d(t) using a cosine function: c) When the water level is at least 6m deep, boats can dock at the marina on the shore. For how many hours each day are the boats able to dock? ***** END OF QUESTIONS ***** MHF 4U FINAL EXAMINATION June 8, 2015 Page 14

15 Compound Angle (Addition and Subtraction) Formulae: Addition Formula for Cosine cos(a + b) = cos a cos b - sin a sin b Subtraction Formula for Cosine cos(a b) = cos a cos b + sin a sin b Addition Formula for Sine sin(a + b) = sin a cos b + cos a sin b Subtraction Formula for Sine sin(a - b) = sin a cos b - cos a sin b Addition Formula for Tangent tan a tan b tan( a b) 1 tan a tan b Subtraction Formula for Tangent tan a tan b tan( a b) 1 tan a tan b Double Angle Formulae: sin (2x) = 2 sin x cos x cos (2x) = cos 2 x - sin 2 x cos (2x) = 2 cos 2 x 1 2 tan x tan(2x) cos (2x) = 1 2 sin 2 x 2 1 tan x Pythagorean Identity: sin 2 x + cos 2 x = 1 sin 2 x = 1 cos 2 x cos 2 x = 1 sin 2 x Quotient Identities: sin x tan x cos x cos x cot x sin x Reciprocal Identities: 1 sec x cos x 1 cot x tan x csc x 1 sin x MHF 4U FINAL EXAMINATION June 8, 2015 Page 15