Arkansas Council of Teachers of Mathematics Regional Exam. Pre-Calculus

Transcription

1 Arkansas Council of Teachers of Mathematics 0 Regional Exam Pre-Calculus For questions through 5, mark your answer choice on the answer sheet provided. After completing items through 5, answer each of the tie breaker items in sequential order (do # first, followed by #, and then # last). Be sure that your name is printed on each of the tiebreakers.. Which complex number has a modulus of A. 5i B. 5 i C. + 5i D. 5 + i. 9 and an argument of approximately 0.8?. Find the equation of the circle whose center is at (4, 5) and contains the point (8, ). A. x y B. x y C. x y D. x y E. x y Find all possible solutions, in radians, for: cosx + cos x = 0 A. n B. 5 n C. n D. All of these 4. Determine the value of the 0 th term that will appear in the given sequence {,,6,0,5, } A. 65 B. 5 C. 58 D. 45 pg.

2 x 5. On what domain does the function, f( x), have an inverse. x, A. B. C. 0,, D. F always has an inverse except when x i E. F never has an inverse y x Find the directrix of the parabola given by A. x = 0 B. y = 0 C. y = D. y = 4 E. x =. The number of trees (N) that can be found per acre is a function of the diameter (S) of the trees. The function that 0.0S describes this is approximated by N Find a function that describes the diameter of the trees (S) as a function of the number of trees per acre (N). log N 5 A. S.0 B. C. N 0 50 S.0 S N 00 D. S log N log 50 E. This function does not have an inverse. 8. Determine the coefficient in front of A. 455 B. 8 C.,640 D.,840 E.,60 x x for the polynomial given by 5 pg.

3 9. What is the range of the arcsin function, in radians? A. y B. y C. D. E. 0 y 0 y y 0. A rock is dropped from a tall tower. In the first second, the rock falls 6 feet. In the second second, the rock falls 48 feet. In the third second it falls 80 feet. In the fourth second, it falls feet. How long will it take for the rock to hit the ground if it is dropped from the top of a 50 foot building? A. 5 seconds B. 6.5 seconds C. 0 seconds D. 5. seconds E. 4.5 seconds. A new strain of the bird flu was found in an Arkansas town. The town has a population of 5000 people. The CDC (Center for Disease Control) will put the town under quarantine if more than 50% of the population develops the disease. After how many days will the town be under quarantine if the spread of the virus is found to be given by: 5000 n where n is the number of people with the disease and t is the number of days since the outbreak t e A. -. days B. 6.4 days C. 40 days D. 4 days. Two vectors act on each other to produce a resultant of.9n at 68.. One of the vectors is 0N at 40. What is the other vector? A. 8N, 60 B. 8N, 00 C..8N, 4. D. 49N, 68. pg.

4 . A house sits on a steep hill. The side of the house makes an angle of 0 with the ground going down the hill. A ladder is set up so that the base of the ladder is 5 feet from the wall of the house when measured along the ground. The ladder needs to go up to a height of feet on the wall. What angle will the ladder make with the ground? A. 4.5 B. 4.6 C. D. 8.9 E Find the sin arccos x A. 4 x B. x x 4 C. x 4 x D. x 4 x E. Not possible 5. A sheet of paper is moving at x ft/sec on a printing machine. If a sensor detects a problem, then the distance the sheet travels before the signal reaches the control center is given by R( x) x. Once the signal reaches the control center, the distance the sheet continues to travel before the machine can stop is given by S( x) x. Find a function, F(x), that 0 gives the total distance the sheet of paper travels once the problem is detected by the sensor. A. F( x) x x 0 B. F( x) x, x 0 0 C. F( x) x 0 D. F( x) x x 0 6. What is the sum of the coefficients when you expand x? A. 64 B. 49 C. 8 D. 8 E. 6 pg. 4

5 . Determine a formula that will compute the n th term for the given sequence:{,,6,0,5, } nn ( ) A. an B. an a n C. a a n n D. an ( n )( n ) n 8. A plane is heading due north at 500 km/hr. A wind of 48 km/hr is blowing from the southeast at 8 west of north. What is the resultant speed of the plane? A. 54 km/hr B. 548 km/hr C. 458 km/hr D. 49 km/hr E. 44 km/hr 9. Find the asymptotes of the hyperbola defined by the equation: 9x 4y 8x y 9 0 A. y x B. y x 6 and y C. y x D. y x and x 5 y x 0. Given that the sin s with s in quadrant III, find the tan s. A. 5 B. 5 5 C. D. E. 5. Find the equation in standard form of the conic section given by A. x y x6 y B x4 y C. 9 6 x6 y D x x y 6y 9. pg. 5

6 . Which of the following functions has a period of a and an amplitude of b? A. y bsin x a B. y sin x b a y bsin ax C. D. y asin x b y sin bx a E.. Determine a formula to compute the n th term for the recursive relation defined as a n a where. n A. n n an n B. an n n C. a n n n D. a n E. All of these a 4. Two trucks are pulling a stump out of the ground. Before the stump is pulled out, the one truck exerted 800 lbs of force and the other truck exerted 00 lbs of force. The angle between the chains connecting each truck to the stump is 8.5. What is the combined force acting on the stump? A lbs B. 84 lbs C. 88 lbs D. 00 lbs 5. Let f ( x) x 4 and let 8 A. 4 4 g( x) x. Evaluate f g (8) B. 4,064 C. D. E. 6 pg. 6

7 Tie Breaker Name: Suppose that a survey has been conducted to investigate the annual population movement between a city and its surrounding suburbs. It was found that 95% of the people who live in the city remained in the city the following year, while only 5% left for the suburbs. It was also found that 9% of those who live in the suburbs remain in the suburbs, but % left to live in the city. a) If these trends continue, then what is the population that civil engineers should expect for the city and the suburbs in years, if the initial population for the city is,000 and the suburbs is,000? b) If these trends continue, then what is the population that civil engineers should expect for the city and the suburbs in years, if the initial population of the city is 5,000 and the suburbs is,000? pg.

8 Tie Breaker Name: An ellipse has foci at (,) and (,). The major axis has a length of 8 units. Find the standard equation for this conic section. Graph the ellipse given in this problem. Find all x-intercepts and y-intercepts. pg. 8

9 Tie Breaker Name: Graph the function: f( x) x x x x 5x 5 Find all asymptotes and undefined points: Find all x-intercepts: Find all y-intercepts: Describe the end behavior of this graph: pg. 9

10 Answer Key:. B. D. D 4. E 5. B 6. C. D 8. C 9. E 0. D. D. B. E 4. C 5. A 6. D. A 8. A 9. D 0. B. C. A. D 4. C 5. E Tie Breaker : a) Year : The population of the city is given by: 000(.95)+000(.0) = 980 people and the population of the suburbs is given by 000 [000(.95)+000(.0)] = 00 people. In other words, the city would lose 5% (or 50 people) but they would gain % of the suburbs population (or 0 people). That would put the population of the city at 980 and the population of the suburbs at 00 people. Year : The population of the city is given by 000(.95+.0)(.95)+(.0)[(000)(.9)+(000)(.05)] = 96.6 or about 96 people. The population of the suburbs can be found by 6000 (000(.95+.0)(.95)+(.0)[(000)(.9)+(000)(.05)])= 08 people. In other words, the city would lose another 5% (or 49 people) but they would gain % of the suburbs population (or 0.6 people). That would mean that 96.6 (or 96) people would live in the city and 08 people would then live in the suburbs. b) Year : The population of the city is found by 5000(.95)+000(.0)=480 people. The population of the suburbs can be found by 6000 [5000(.95)+000(.0)]=0 people In other words, the city would lose 5% (or 50 people) but they would gain % of the suburbs population (or 0 people). That would put the population of the city at 480 and the population of the suburbs at 0 people. Year : The population of the city can be found by.95[5000(.95)+000(.0)]+.0[000(.9)+5000(.05)]=45.6 or about 458 people. That would give the population of the suburbs to be.9[000(.9)+5000(.05)]+.05[5000(.95)+000(.0)]= 4.4 or 4 people. In other words, the city would lose another 5% (or 9 people) but they would gain % of the suburbs population (or 6.6 people). That would put the population of the city at 4,5.6 or 4,58 people and the population of the suburbs would be,4 people. pg. 0

11 Tie Breaker : The foci are at (,) and (,) so the center is at, 4,. Since the center is at (4,) and one of the foci is at (,) that means that c=. Since the major axis is 8, we know that a=8 so a=4. Since a 6 b or b c, we get 9 b ( x 4) ( y ) That gives the standard form of the ellipse as 6 ( 4) ( y ) ( ) Finding the zeros: Let x = 0 and we get y 6 so y =. The y-intercept is at (0, ). ( x 4) ( y ) ( x 4) To find the x-intercept, let y=0 and we get 6 6 ( x 4) 6 4 can find that ( x 4) 6 48 x 4 and this gives us that 48 ( x 4) ( ) 48 ( x 4) ( y ) 0 x x 4 which is approximately.4 and 6.6 for the x- 6 6 coordinates. Therefore the x-intercepts are 4, 0 and 4, 0. y 0. From this we pg.

12 Tie Breaker : Graph: Find all asymptotes and undefined points: Find all x-intercepts: no x-intercepts y 0, y x and a hole in the graph at x= 4 Find all y-intercepts: (0,) Describe the end behavior of this graph: as x approaches infinity the graph goes to infinity. As x approaches negative infinity, the graph approaches negative infinity. pg.