DRAFT. Appendix H. Grade 12 Prototype Examination. Pre-calculus 30. Course Code For more information, see the Table of Specifications.

Transcription

1 Grade 1 Prototype Examination Pre-calculus 30 Course Code 846 Barcode Number DRAFT Appendix H For more information, see the Table of Specifications. Month Day Date of Birth November 013

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5 Pre-calculus 30 TIME: Two and One-Half Hours Calculating devices MUST meet the requirements of the Calculator Use Policy. Before an examination begins, devices must be removed from their cases and placed on the students desks for inspection by a mathematics or science teacher. Students must clear all programmable calculators, both graphing and scientific, of all information that is stored in memory. Internet access and electronic signals are not allowed. Cases must be placed on the floor and left there for the duration of the examination. Do not spend too much time on any one question. Read each question carefully. The examination consists of 40 multiple-choice followed by 10 numerical response questions of equal value which will be machine scored. Record your answers on the Student Examination Form which is provided. Each multiple choice question has four suggested answers, one of which is better than the others. Select the best answer and record it on the Student Examination Form as shown in the example below: Student Examination Form: Multiple Choice This examination is being written in the subject A. Chemistry. B. Pre-calculus. C. Workplace and Apprenticeship Mathematics. D. Foundations of Mathematics. Numerical Response Record your answer in the numerical response section on the answer sheet. What is 10% of $ 000? (Round to the nearest dollar.) 1. A B C D i DRAFT - November 013

6 What is 10% of $48.50? (Round to the nearest dollar.) What is 10% of 4 15? (Round to the nearest whole number.) Use an ordinary HB pencil to mark your answers on the Student Examination Form. If you change your mind about an answer, be sure to erase the first mark completely. There should be only one answer marked for each question. Be sure there are no stray pencil marks on your answer sheet. If you need space for rough work, use the space in the examination booklet beside each question. Do not fold either the Student Examination Form or the examination booklet. Check that all information at the bottom of the Student Examination Form is correct and complete. Make any necessary changes, and fill in any missing information. Be sure to complete the Month and Day of Your Birth section. ii DRAFT - November 013

7 Pre-calculus 30 Quadratic Formula For ax bx c 0, x 4 b b ac a Arc Length a r Trigonometry and Trigonometric Identities sin tan cos 1 csc = sin cos cot = sin 1 sec cos 1 cot tan sin cos 1 1 tan sec 1 cot csc cos ( A + B) = cos Acos B sin Asin B cos ( A B) = cos Acos B+ sin Asin B sin ( A + B) = sin Acos B+ cos Asin B sin ( A B) = sin Acos B cos Asin B tan A tan B tan( AB) 1 tan A tan B tan A tan B tan( AB) 1 tan A tan B sin sin cos tan cos cos 1 tan 1 tan Permutations, Combinations, and Binomial Theorem n P r n! ( n r)! n C r n! ( n r)! r! n C r n r ( ab) C a b C a b C a b... C a b The n n 0 n 1 1 n 0 n n 0 n 1 n n n th r term of the expansion of ( ) n a b is: n a 1 r b n( r1) r1 iii DRAFT - November 013

8 Ministry of Education Pre-calculus Insert iv DRAFT - November 013

9 Table of Trigonometric Ratios sin cos tan csc sec cot sin cos tan csc sec cot v

10 GRADE 1 DEPARTMENTAL EXAMINATION PRE-CALCULUS 30, NOVEMBER 013 VALUE 100 (50 ) Answer the following 50 questions on the computer sheet entitled Student Examination Form. MULTIPLE CHOICE 1. If fx ( ) x 3x 4 and gx ( ) 5x, what is fx ( ) gx ( )? A. x x B. x x 6 C. x 8x D. x 8x 6. What is y logb x rewritten in exponential form? A. B. C. D. x b y b b x b y y x y x DRAFT - November 013

11 3. The graph of y f( x ) is shown below. Which of the following graphs represents the inverse of this function? A. B. C. D. - - DRAFT - November 013

12 4. Functions y f( x ) and y g( x) are shown graphed below. If hx ( ) f( x) gx ( ), what is the value of h ( )? A. 4 B. 6 C. 8 D DRAFT - November 013

13 5. The graph of an exponential function with a base of is shown below. Which equation best describes this function? A. B. C. D. x y ( ) (0.5) x y ( ) (0.5) y y x1 (3)( ) x1 (3)( ) 6. The function fx ( ) log ( x) is transformed into a new function gx ( ) log ( x ). How has the graph of fx ( ) been translated? A. units up B. units down C. units to the left D. units to the right DRAFT - November 013

14 7. How will the graph of f ( x) x be transformed if the function is changed to gx ( ) x3 4? A. The new graph will shift 3 units to the left and 4 units up. B. The new graph will shift 3 units to the right and 4 units up. C. The new graph will shift 3 units to the left and 4 units down. D. The new graph will shift 3 units to the right and 4 units down. 8. Which equation best represents the graph shown below? A. B. C. D. f( x) ( x )( x 1) f( x) ( x )( x 1) ( x )( x 1) f( x) ( x )( x 1) f( x) DRAFT - November 013

15 9. Which of the following statements is true for the graph of any fourth degree polynomial function? A. There will be a minimum of 1 x-intercept. B. There will be a maximum of 1 x-intercept. C. There will be a minimum of 4 x-intercepts. D. There will be a maximum of 4 x-intercepts. 10. A polynomial function is sketched below. What is the equation of this polynomial function? A. yx( x 4) B. yx( x 4) C. yx ( x 4) D. yx ( x 4) DRAFT - November 013

16 11. What is the value of x in the equation 1 7 x 3? A. B. C. D How many distinct x-intercepts does the graph of have? x f( x) 5x 4 x 16 A. 1 B. C. 3 D What is the equation of the inverse relation of fx ( ) 4x 10? A. B. y y x 10 x 10 C. D. 5 y x 5 y x DRAFT - November 013

17 14. Which sketch best represents the function 5x 13 y? x 3 A. B. C. D DRAFT - November 013

18 15. Which of the following shows a correct estimation and explanation to approximate the value of x in the equation log3 0 x? A. x.3 because x must be somewhere between and 3 and it will be closer to. B. x.3 because x must be somewhere between and 3 and it will be closer to 3. C. x.7 because x must be somewhere between and 3 and it will be closer to. D. x.7 because x must be somewhere between and 3 and it will be closer to What is the value of x in the equation x x3 7 5? A. 7.0 B. 7.5 C D x A function is defined by f( x). On the graph of f( x ), where x 5x 6 are the vertical asymptote and the point of discontinuity (hole)? A. The vertical asymptote is at x = ; the point of discontinuity (hole) is at (3, 6). B. The vertical asymptote is at x = 3; the point of discontinuity (hole) is at (,5). C. The vertical asymptote is at x = ; the point of discontinuity (hole) is at ( 3,0). D. The vertical asymptote is at x = 3; the point of discontinuity (hole) 1 is at, DRAFT - November 013

19 18. The sketch of the function y f( x) is shown below. What will be the graph of y f( x)? A. B. C. D DRAFT - November 013

20 19. What is the solution set for the equation x13 x 1? A. {3} B. {4} C. { 4, 3} D. { 3, 4} 0. What is the solution set for the equation log ( x1) log ( x)? A. {} B. { 3} C. { 3, } D. { 3, 0, } DRAFT - November 013

21 1. Which of the following represents an angle in standard position measuring 5 radians? 4 A. B. C. D DRAFT - November 013

22 . Angle is in the second quadrant with value of cos? 7 sin. What is the exact 9 A. B. C. D Angle A has a measure of 4 3 radians. What are the exact values of cos A and sin A? A. B. C. D. 3 1 cos A and sin A 3 1 cos A and sin A 1 3 cos A and sin A 1 3 cos A and sin A 4. Which of the following pairs of trigonometric ratios have the same value as sec 307? A. sec ( 53 ) and sec ( 17 ) B. sec ( 53 ) and sec 17 C. sec 53 and sec ( 17 ) D. sec 53 and sec DRAFT - November 013

23 5. What are possible solutions for sin A, where 0A 360? A. 45 and 135 B. 45 and 315 C. 135 and 5 D. 5 and What characteristic is the same for the graphs of y = sin x and y = tan x? A. amplitude B. asymptotes C. period length D. x-intercepts 7. Angle is in the fourth quadrant with value of sin? 1 tan. What is the exact 5 A. B. C. D DRAFT - November 013

24 8. The terminal arm of angle A in standard position passes through the point ( 3, 7). What is the value of cos A? A. B. C. D What is the exact value of tan 10tan 60? 1 tan 10 tan 60 A. 0 B. C D What is cos sin sec simplified? A. 1 B. sin C. sec D. 1 sec DRAFT - November 013

25 sin cos 31. What are the non-permissible values for, 1 sin where 0? A. 0, B. C. D. 3, 4 4 3, 3 3, 4 3. Which of the following is a solution for tan 3 tan 0? A. 10 B. 135 C. 5 D DRAFT - November 013

26 33. Which of the following represents the graph of y cos x? 4 A. B. C. D DRAFT - November 013

27 34. Which of the following is a trigonometric identity? A. cos cot (cos tan sin ) B. cot sec cos tan C. D. cos csc 1 sin 1 sin cos sin 35. What are the exact measures of the angles that satisfy where 0? 3 csc, 3 A. B. C. D. 11, 6 6, , , What are the non-permissible values for sin tan sec, where tan csc 0 360? A. 0, 90, 180, 70 B. 90, 180, 70 C. 90, 70 D. 0, DRAFT - November 013

28 37. To accessorize her outfit, Jane will choose 1 of 4 handbags, 1 of 5 hats, and 1 of 3 coats. How many different outfits can Jane create by changing these accessories? A. 3 B. 1 C. 60 D How many different passwords can be made from all the letters in the word CALCULUS? A. 50 B C D What is the solution set for r given 7 Cr 1? A. {} B. {3} C. {, 5} D. {3, 4} DRAFT - November 013

29 40. Which of the following represents the 3rd term in the expansion 7 of ( 3 x)? 7 A. () ( 3 x ) 5 B. C. D. 7 () ( 3 x ) () ( 3 x ) () ( 3 x ) DRAFT - November 013

30 NUMERICAL RESPONSE Record your answer in the Numerical Response section of the Student Examination Form. 41. Functions fx ( ) and g( x ) are defined by fx ( ) 3x 5 and gx ( ) ( x) 3. What is the value of gf ( (4))? 1 4. What is the value of a in the equation log5 a log5 6 3 log5 3 log5 81? What is the remainder when x x 8x 4 is divided by x 3? 44. Kelsey invested $1 000 at an annual interest rate of 6% compounded monthly. The accumulated value of her investment, A, is given by 0.06 A n where n is the number of months. What is the fewest number of months required for Kelsey s investment to be worth more than $1 500? 45. What is 3 radians in degrees? (Round to the nearest degree.) DRAFT - November 013

31 46. How many solutions are there for 4sin 1 0, where 0 360? 47. The location of a dolphin moving in rhythmic fashion (above and below the surface of the water) is recorded over a time span of 4.0 seconds. The results are shown on the graph below. What is the length of time it takes the dolphin to complete one cycle? 48. A bicycle tire has a diameter of 74 mm. A point is marked on the outer edge of the tire. After the tire has turned 60, what is the arc length the point has moved? (Round to the nearest millitre.) 49. How many different ways could 4 members be selected from a cheerleading squad with 1 members? 50. What is the coefficient of the term containing x in the expansion 7 of ( x 3)? - - DRAFT - November 013

32 (See Explanation of Answers) GRADE 1 DEPARTMENTAL EXAMINATION Pre-calculus 30 PROTOTYPE EXAM Answer Key 1. C. 11. C. 1. B. 31. C B. 1. A.. B. 3. C D. 13. A. 3. C. 33. B B. 14. C. 4. B. 34. A C. 15. D. 5. D. 35. D D. 16. D. 6. D. 36. A B. 17. A. 7. A. 37. C A. 18. C. 8. B. 38. B D. 19. A. 9. A. 39. C C. 0. A. 30. C. 40. D Explanation of Answers 1. C. If fx ( ) x 3x 4 and gx ( ) 5x, then fx ( ) gx ( ) ( x 3x4) (5x ) x 8x. B. The expression y log b x can be rewritten as y b x. - i - Answer Key DRAFT - November 013

33 3. D. The function y = f(x) is an exponential function. The inverse of an exponential function is a logarithmic function. The inverse of any function can be found by reflecting the original function in the line y = x. 4. B. h(x) = f(x) + g(x) h( ) = f( ) + g( ) h( ) = h( ) = 6 5. C. x The graph of y is shifted to the right one unit and then stretched vertically by a factor of 3 to get the given graph. y x ii - Answer Key DRAFT - November 013

34 6. D. Comparing fx ( ) log xand gx ( ) log ( x ) the graph of f(x) will be translated units horizontally and since it is x the translation is to the right. 7. B. The graph of gx ( ) x3 4 is the graph of f( x) x with a horizontal shift of 3 units to the right and a vertical shift of 4 units up. 8. A. The graph has a horizontal asymptote of x = 0, vertical asymptotes of x = 1 and x = and no zeros. Therefore f( x) ( x )( x 1) 9. D. The number of distinct x-intercepts is less than or equal the degree of the function. For a fourth degree polynomial function, the maximum number of x-intercepts is 4. - iii - Answer Key DRAFT - November 013

35 10. C. The graph has a zero at x = 0 with a multiplicity of and a zero at x = 4 with a multiplicity of 1. yax ( x 4) The function begins in quadrant 3 and ends in quadrant 1 therefore a must be positive. yx ( x 4) 11. C. x x x x x 7 x 6 1. A. x 5x 4 ( x 1)( x 4) ( x 1) The graph of f( x), x 4 will have only x 16 ( x 4)( x 4) ( x 4) one x-intercept at ( 1,0). The function will have one point of discontinuity 3 (hole) at 4,. 8 - iv - Answer Key DRAFT - November 013

36 13. A. f( x) 4x 10 y 4x 10 To find the equation of the inverse relation, switch the x s and y s in the equation: x 4 y 10 x 10 4 y x 10 y 4 x 10 y 4 y x C. 5x 13 y will have a y-intercept at x 3 a vertical asymptote at x = ,, 3 an x-intercept at 13,0, 5 and 15. D. To solve log3 0 x, rewrite the equation into exponential form use whole number exponents to approximate the solution. x (3 0) and lies in this range and is closer to 7 than it is to 9 - v - Answer Key DRAFT - November 013

37 16. D. Since the bases are not powers of one another (5 7), you need to take the log of both sides. x x 3 log 7 log 5 Use the power law of logs or x log 7 ( x 3) log 5 x log 7 ( x 3) log 5 x log 7 x log 5 3 log 5 x( ) ( x 3)( ) x log 7 x log 5 3 log x x x(log 7 log 5) 3 log x log5 x x log 7 log 5 x A. x 9 ( x 3)( x 3) ( x 3) The graph of f( x), x 3 will have a x 5x 6 ( x )( x 3) ( x ) vertical asymptote at x. The function will have one point of discontinuity (hole) at (3, 6). 18. C. To obtain y f( x), from y f( x), the graph will be reflected in the y-axis and translated units down. y f( x) y f( x) - vi - Answer Key DRAFT - November 013

38 19. A. x 13 x 1 x 13 ( x 1) x 13 x x 1 x x 1 0 ( x 4)( x 3) 0 x 4 x 3 Verification: False, x = 4 is extraneous True for x = 3 The solution is {3} 0. A. log ( x1) log ( x) log ( x 1) log ( x ) log ( x 1)( x ) x x x x 6 0 ( x 3)( x ) 0 x 3 0 or x 0 x 3 x Verification: log ( 3 1) log ( 3 ) log ( 4) log ( 1) Undefined, x 3 is extraneous log ( 1) log ( ) log (1) log (4) 0 The solution set is {} 1. B (180 ) 4 5 (180 45) - vii - Answer Key DRAFT - November 013

39 . B. 7 sin 9 y r x y r x r y x (9) (7) x x 3 x 3 x 4 Because the terminal arm is in quadrant II: x 4 ; y 7; r 9 x 4 cos r 9 3. C. A cos A and sin A 3 - viii - Answer Key DRAFT - November 013

40 4. B. sec( 53 ) 1.66 sec sec( 17 ) 1.66 sec is in quadrant 4, so sec 307 must be positive. The ratios that equal sec 307 are sec ( 53 ) and sec D. Since sin A is a negative value the two possible solutions are in Quadrant 3 and Quadrant 4. The reference angle for sin A is D. y sin x y tan x Amplitude 1 undefined Asymptotes None Yes (example ) Period x-intercepts 0,,... 0,,,... The x-intercepts are the only characteristic which is the same for the graphs of y sin x and y tan x. - ix - Answer Key DRAFT - November 013

41 7. A. 1 tan, and is in the fourth quadrant, then x 5 and y 1 5 x y r 5 ( 1) r 169 r 13 r r y sin r 1 sin B. If the point is ( 3,7), then x 3 and y 7. x y r ( 3) 7 r 58 r 58 r x cos A r cos A cos A A. tan 10tan 60 tan (10 60 ) tan (180 ) 0 1 tan10tan 60 - x - Answer Key DRAFT - November 013

42 30. C. cos sin sec sin cos cos cos sin cos cos cos cos sin cos 1 cos sec 31. C. sin cos The non-permissible values of occur when the denominator of is 1 sin equal to zero. 1sin 0 sin 1 sin 1 sin 1 sin C. tan 3 tan 0 (tan )(tan 1) 0 tan 63.4, 43.4 tan 1 45, 5 - xi - Answer Key DRAFT - November 013

43 33. B. The graph of y cos x will have 4 amplitude =, period length =, phase shift (shift left), and no vertical shift A. Using a graphing calculator the expression that is an identity cos cot (cos tan sin ) y 1 cos y cot (cos tan sin ) x y = 1 x y = 1 Verify numerically, Answer A: cos 45cot45 (cos 45tan 45sin 45 )? (1) (1)? True Trigonometric identity - xii - Answer Key DRAFT - November 013

44 Answer B: cot 45sec 45cos 45tan 45? (1) 1 False Not a trigonometric identity Answer C: cos 45 csc 45 1? sin 45? 1? 4 1 4? False Not a trigonometric identity - xiii - Answer Key DRAFT - November 013

45 Answer D:? 1 sin30 sin ? 3 1 1? 3 1 cos False Not a trigonometric identity 35. D. 3 3 csc is the reciprocal of sin. 3 The angles will be 4 5 and A. To determine non-permissible values, look at each trigonometric function and assess which terms have non-permissible values. First term, left side: sin tan sin : no non-permissible values 1 : will have non-permissible values where tan is undefined and also tan where tan 0. tan is undefined at 90 and 70 tan 0 when 0and180 - xiv - Answer Key DRAFT - November 013

46 Second term, left side: tan csc tan : will have non-permissible values where tan is undefined tan is undefined at 90 and 70 1 : csc will have non-permissible values where csc 0 happen) and also where csc θ is undefined csc θ is undefined at 0 and 180 (which will never Right side: sec : will have a non-permissible value where sec 0 (which will never happen) and also where sec is undefined sec is undefined at 90 and 70 All Non-Permissible Values = 0, 90, 180, C. Number of handbags number of hats number of coats = = B. n! 8! 5040 r! s! t!...!!! 39. C 7C 0 1 7C3 35 7C 1 7 7C4 35 C C will also equal xv - Answer Key DRAFT - November 013

47 40. D. The 3rd term in the expansion of ( 3 x) 7 is found using 7 () ( 3 x ) Numerical Response: 78 fx ( ) 3x 5 and gx ( ) ( x) 3 f(4) 3(4) gf g ( (4)) (7) (7 ) Numerical Response: 18 1 log5 6 3 log5 3 log log 6 log 3 log 81 log 6 log 7 log log5 9 log 18 5 a = Numerical Response: 35 3 x x 8x 4 x 3 3 (3) (3) 8(3) or xvi - Answer Key DRAFT - November 013

48 44. Numeric Response: 8 A ( ) n n (1.005) n 1.5 (1.005) n log(1.5) log(1.005) log(1.5) n log(1.005) log(1.5) n log(1.005) 81.3 n need at least 8 months 45. Numerical Response: Numerical Response: 4 4sin 1 0 4sin 10 1 sin 4 1 sin 1 sin will occur in quadrants 1 and. 1 sin will occur in quadrants 3 and 4. - xvii - Answer Key DRAFT - November 013

49 47. Numerical Response: 4 One complete wavelength (cycle) is completed in 4 seconds. 48. Numerical Response: 379 Solution 1: Convert the measure of the central angle to radians x x 3 a r a (36) 3 a a 379 mm Solution : Use proportion arc length central angle = circumference full rotation a 60 (36) 360 a a 379 mm 49. Numerical Response: 495 1C xviii - Answer Key DRAFT - November 013

50 50. Numerical Response: Given ( x 3) the term containing x will be given by C ( x) (3) 5 1 x x xix - Answer Key DRAFT - November 013