MATHEMATICAL METHODS

Transcription

1 2018 Practice Eam 2B Letter STUDENT NUMBER MATHEMATICAL METHODS Written eamination 2 Section Reading time: 15 minutes Writing time: 2 hours QUESTION AND ANSWER BOOK Number of questions Structure of book Number of questions to be answered Number of marks A B Total 80 Students are permitted to bring into the eamination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set squares, aids for curve sketching, one bound reference, one approved technolog (calculator or software) and, if desired, one scientific calculator. Calculator memor DOES NOT need to be cleared. For approved computer-based CAS, full functionalit ma be used. Students are NOT permitted to bring into the eamination room: blank sheets of paper and/or correction fluid/tape. Materials supplied Question and answer booklet of 22 pages. Formula sheet. Working space is provided throughout the book. Instructions Write our student number in the space provided above on this page. Unless otherwise indicated, the diagrams in this book are not drawn to scale. All written responses must be in English. At the end of the eamination You ma keep the formula sheet. Students are NOT permitted to bring mobile phones and/or an other unauthorised electronic devices into the eamination room. TRIUMPH TUTORING 2017

2 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 2 SECTION A - Multiple-choice questions Instructions for Section A Answer all questions in pencil on the answer sheet provided for multiple-choice questions. Choose the response that is correct for the question. A correct answer scores 1; an incorrect answer scores 0. Marks will not be deducted for incorrect answers. No marks will be given if more than one answer is completed for an question. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 The quadratic function f : D! R, f() = ( 3) 2 +4has range ( 5, 4]. A possible domain D of the function is A. [3, 6) C. (3, 6] B. R D. ( 5, 4] E. ( 6, 3] Question 2 John is taking a maths eam. The multiple choice section on the eam has 22 questions which each have 5 potential answers. John is a ver poor maths student and guesses ever answer; that is, he has a 20% chance to answer an given question correctl. Given that he got at least 8 correct answers, the probabilit that he got eactl 11 correct answers is closest to A C B D E Question 3 The simultaneous linear equations 2 +(m 3) = 4 3 m (m 4) +3 =8 where m is a real number, have infinitel man solutions for A. m 2 R\{1, 0} C. m 2 R\{1} B. m 2 R\{0} D. m =6 E. m =1 SECTION A - continued

3 MATHMETH EXAM 2B - TRIUMPH TUTORING Question 4 The graph of a function f, with domain R, is shown below The graph which best represents f( 2)+1is A. B C. D E SECTION A - continued

4 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 4 Question 5 The gradient of a line perpendicular to the line which passes through (0, 3) and ( 9, 0) is A B C E. 3 2 Question 6 Consider the graph below. D The graph shown could have equation A. =2sin C. =2sin E. =2sin B. =2cos(3 ) D. =4cos Question 7 Let f : R +! R be a differentiable function. Then, for all 2 R +, the derivative of f(log e ()) with respect to is equal to A. f 0 (log e ()) 1 2 B. f 0 () 1 2 C. log e ()f 0 () 1 2 D. ef 0 (log e ()) 1 2 E. f 0 (log e ()) 1 2 SECTION A - continued

5 MATHMETH EXAM 2B - TRIUMPH TUTORING Question 8 The transformation T : R 2! R 2 that maps the curve with equation =cos() onto the curve with equation = 1 2 cos 3 is given b 2 apple apple 1 apple apple apple apple apple apple 0 A. T = B. T = apple apple apple apple apple apple apple apple C. T = D. T = apple apple apple apple 2 0 E. T = + 3 Question The upper limit of a 95% confidence interval of a particular surve is 47%. If 1,000 people participated in this surve, the sample proportion, ˆp, is closest to A B C D E Question 10 In a particular town, there are 20,000 citizens. From census data, it is known that 1 in 8 citizens are retired. There are 20 people in a cafe on a Frida morning. For this group of 20 people, ˆp is the random variable of the distribution of sample proportions of retired citizens. The value of Pr(ˆp >0.1) is closest to A B C D E Question 11 Consider the functions f :[ 1, 1)! R, f() = p +1 g :[ a, a]! R, g() =2cos() What is the maimum value of a for which f(g()) eists? 1 A. B C. E D SECTION A - continued

6 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 6 Question 12 Charlie loves to go horse riding. If he goes riding one week, the probabilit that he goes riding the net week is 0.9. If he doesn t go riding one week, the probabilit that he doesn t go riding the net is 0.2. Charlie didn t go riding this week. The probabilit that he goes riding eactl once in the net two weeks is A B C D E Question 13 Z 2 Z 2 If f() d =5and b 2 +5f() d =41, the value of b is 0 A. 1 C. 4 E. 8 0 B. 2 D. 6 Question 14 The derivative function g 0 () is shown below The integral of g 0 () is equal to g()+c. The equation g()+c =0has three solutions. The possible range of values of c is A. 2 <c<4 1 B. c< [ c> C. 3 <c<28 1 D. c< [ c> E. 3 <c< SECTION A - continued

7 MATHMETH EXAM 2B - TRIUMPH TUTORING Question 15 If f 0 () = , which one of the following graphs could represent the graph of = f()? A. B. C. D. E. Question 16 If the tangent to the graph of =log e (a), a 6= 0, at = b passes through the origin, b is equal to A. e 1 a 2 B C. e 1 a 2 D E. e 1 2 SECTION A - continued

8 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 8 Question 17 The inverse function of f : 3 A. f 1 : 2, 1! R, f 1 () = 3 B. f 1 : 2, 1 3 2, 1! R, f() = 1 1 (2 3) 2 2! R, f 1 () = C. f 1 : R\{0}! R, f 1 () = D. f 1 : R +! R, f 1 () = E. f 1 : R +! R, f 1 () = p 2 3 is Question 18 If f( +2)=( +4), then f() is equal to A C B D E. 2 4 Question 19 The height of an given tree in a particular forest area is given b X metres, where X is a normall distributed random variable with mean 8.4 metres and standard deviation k metres. A sample of 300 trees is researched and 11 of these trees are shorter than 7.3 metres tall. The value of k is A B C D E SECTION A - continued

9 MATHMETH EXAM 2B - TRIUMPH TUTORING Question 20 Part of the graph of f : R! R, f() =9 2 is shown below. A 1 A 2 k The area of the region A 1 is equal to the area of the region A 2. The value of k is A. 0 B. 18 C. 2 p 3 D. 6 E. 3 p 3 END OF SECTION A

10 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 10 SECTION B Answer all questions in the spaces provided. Instructions for Section B In all questions where a numerical answer is required, an eact value must be given unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 (7 marks) A tin can with a closed lid is made in the shape of a clinder as shown below, with the radius of the lid r cm and height h cm. r cm h cm The volume of the can is 192 cm 3. a. Find an epression for h in terms of r. b. Show that the total surface area of the can, A cm 2, is given b A =2 r r SECTION B - Question 1 - continued

11 MATHMETH EXAM 2B - TRIUMPH TUTORING c. Find the value of h for which the tin has the minimum possible total surface area. 3 marks SECTION B - continued

12 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 12 Question 2 (1) While Zoe and Jessica are riding a Ferris Wheel, the realise that as the wheel spins the height of their carriage above ground level, h metres, t minutes after the ride starts, can be modelled b the function h(t) =a b cos(ct). Once the ride starts, it takes the girls 2 minutes to reach the top. a. If the bottom of the wheel is 3 metres off the ground, and the diameter of the wheel is 16 metres, find the values of a, b and c. The wheel does not stop during a regular ride. 3 marks b. Sketch the graph of h against t for the interval 0 apple t apple 8 (a regular ride length) 3 marks on the aes below. Label all endpoints and ais intercepts. h t c. How man minutes have the girls been on the ride when the first reach a height of 8 metres? Give our answer correct to two decimal places. 1 mark SECTION B - Question 2 - continued

13 MATHMETH EXAM 2B - TRIUMPH TUTORING d. During a regular ride, for how long is Zoe and Jessica s carriage above a height of 14 metres? Give our answer correct to the nearest second. 3 marks e. How fast are Zoe and Jessica travelling while on the wheel? Give our answer in metres per second. SECTION B - continued

14 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 14 Question 3 (18 marks) Destin and Tom are a hiking couple. Earlier in the da, the enter Jurassic National Park at point B. The then ride their mountain bikes using sealed roads to the premium hiking location at point A, which is 20 km north of an intersection at point C. The sealed roads run along ACDB. There are off-road shortcut roads along AX, BY, and BZ. CD =70km, CX = YZ = ZD = km, AC =20km and BD =25km. A B 20 km 25 km C X Y Z D 70 km The trek is more difficult than the couple thought it would be, so the return to their bikes at point A later than the had planned. The know that the National Park closes its gates at a certain time, depending on the season. Sensors at the front gate at point B recognise when it gets too dark, and the gate automaticall closes so that no hikers can enter the park at night. The intensit of light in winter decreases over time according to the equation L :[0, 12]! R, L(t) =50(1+19e t ) where L is the intensit of light (lu) and t is the time in hours after 4:30pm. The gates shut when the light intensit reaches 80 lu. a. At what time will the gates shut and trap Destin and Tom in the park overnight, correct to the nearest minute? SECTION B - Question 3 - continued

15 MATHMETH EXAM 2B - TRIUMPH TUTORING To avoid being trapped in the park, the couple must leave point A as soon as possible. The jump on their mountain bikes and start pedalling, but the need to decide which wa to go. The can travel on sealed roads at 32 km/hour or the can travel through the bush via the off-road shortcuts at 25 km/hour. b. Show that Destin and Tom will remain trapped in the park if the onl take sealed roads. 1 mark In order to reach the gates in time, Destin and Tom take the off-road shortcut from A to X, ride along the sealed road from X to Z and then ride up another off-road shortcut from Z to B. c. Show that the time taken to reach the front gates, T hours, is given b T = 1 25 (p p )+ 1 (35 ) 16 d. Find the value of which allows the couple to reach the front gates at point B in the minimum time. Give our answer correct to the nearest metre. SECTION B - Question 3 - continued

16 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 16 e. Show that Destin and Tom can avoid being trapped. 1 mark f. Could the hikers still make it to the gate before it shuts if the took the route AXY B? Give the value for km that allows for this in the minimum time possible, and state this minimum time. 3 marks Destin and Tom decide that it would be best to hike in the summer net time because the would get more dalight. In summer, the equation that represents the intensit of light is slightl different to the equation in winter. However, the brightest and darkest points between 4:30pm and 4:30am are the same in both summer and winter. The light intensit in summer is modelled b the equation L = a + be ct, where L is the intensit of light in summer, t is the time after 4:30pm and a, b and c are real numbers. g. If the graph of the function modelling light intensit in summer passes through the point 2, , find the values of a, b and c. e marks SECTION B - Question 3 - continued

17 MATHMETH EXAM 2B - TRIUMPH TUTORING h. If the gates still shut when the light intensit reaches 80 lu, at what time will the gates be closed during summer, correct to the nearest minute? i. In summer, the park issues a warning through their PA sstem advising hikers to start heading back to the front gates. If this warning is issued at 8:45pm, what is the intensit of light in the park at this time, correct to two decimal places? SECTION B - continued

18 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 18 Question 4 (11 marks) A local hardware store is conducting a surve to find out how man households own a barbecue. a. The store discovers that an given household has a 40% chance to own a barbecue. i. If the store surves 15 people, what is the probabilit that none of them own a barbecue, correct to four decimal places? 1 mark ii. What is the probabilit that less than 8 of them don t own a barbecue? b. In a larger surve of 120 people, ˆp is the random variable of the distribution of sample proportions of households that own a barbecue. i. Find Pr ˆp apple 5, correct to four decimal places. 12 SECTION B - Question 4 - continued

19 MATHMETH EXAM 2B - TRIUMPH TUTORING ii. 45% of households in the larger surve own a barbecue. Find the 96% confidence interval for the estimate of the proportion of households in this town that own a barbecue, correct to four decimal places. 3 marks c. The store conducts a different surve to find out the proportion of households in the town that own a fire pit. After conducting the surve, the are a% certain that between 10% and 18% of households in the town own a fire pit. If 200 people are surveed, find the value of a, correct to one decimal place. 3 marks SECTION B - continued

20 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 20 Question 5 (1) The graph of f :(0, 1)! R, f() = log e () 2 is shown below. a. Find: i. the rule for the derivative function f 0 (). ii. the gradient of = f() at =2, correct to two decimal places. 1 mark SECTION B - Question 5 - continued

21 MATHMETH EXAM 2B - TRIUMPH TUTORING The tangent, = t(), at = k, passes through the origin. b. Find the value of k, correct to three decimal places. c. Find the rule for the function t(), giving an values correct to three decimal places. 1 mark Point P (p, f(p)) lie on the graph of f such that the distance from origin O to point P is at a minimum. d. Find the coordinates of point P, correct to three decimal places. 3 marks SECTION B - Question 5 - continued

22 2018 MATHMETH EXAM 2B - TRIUMPH TUTORING 22 The graph of g :(0, 1)! R, g() = log e () 2 +2is drawn on the same aes as shown below. A f() B g() Points A and B are the points of intersection between = f() and = g(). e. Find: i. the values of points A and B. ii. the area of the shaded region; the area bound b the graphs of = f() and = g(). 1 mark END OF QUESTION AND ANSWER BOOK

23 MATHEMATICAL METHODS Written eamination 2 FORMULA SHEET Instructions This formula sheet is provided for our reference. A questions and answer book is provided with this formula sheet. Students are NOT permitted to bring mobile phones and/or an other unauthorised electronic devices into the eamination room.

24 2018 MATHMETH EXAM FORMULA SHEET 2 Mensuration Formula Sheet area of a trapezium 1 2 (a + b)h volume of a pramid 1 3 Ah curved surface area of a clinder 2 rh volume of a sphere 4 3 r3 volume of a clinder r 2 h area of a triangle 1 bc sin(a) 2 volume of a cone 1 3 r2 h Calculus d Z n = n n 1 d d Z (a + b) n = an(a + b) n 1 d d Z e a = ae a d n d = 1 n +1 n+1 + c, n 6= 1 (a + b) n d = e a d = 1 a ea + c 1 a(n +1) (a + b)n+1 + c, n 6= 1 d d (log e ()) = 1 Z 1 d =log e ()+c, >0 d (sin(a)) = a cos(a) d Z sin(a) d = 1 a cos(a)+c d d (cos(a)) = a sin(a) Z cos(a) d = 1 a sin(a)+c d d (tan(a)) = a cos 2 (a) = a sec2 (a) product rule d (uv) =udv d d + v du d quotient rule d u = v du u dv d d d v v 2 chain rule d d = d du du d

25 MATHMETH EXAM FORMULA SHEET Probabilit Formula Sheet Pr(A) =1 Pr(A 0 ) Pr(A [ B) =Pr(A)+Pr(B) Pr(A \ B) Pr(A B) = Pr(A \ B) Pr(B) mean µ =E(X) variance var(x) = 2 =E((X µ) 2 )=E(X 2 ) µ 2 Probabilit distribution Mean Variance discrete Pr(X = ) =p() µ = p() 2 = ( µ) 2 p() continuous Z b Pr(a <X<b)= f() d µ = Z 1 f() d 2 = Z 1 ( µ) 2 f() d a 1 1 Sample proportions ˆP = X n mean E( ˆP )=p standard deviation sd( ˆP )= r p(1 p) n approimate confidence interval ˆp r r! p(1 p) p(1 p) z, ˆp + z n n END OF FORMULA SHEET