Pre-Calculus Exam 2009 University of Houston Math Contest. Name: School: There is no penalty for guessing.

Size: px

Start display at page:

Download "Pre-Calculus Exam 2009 University of Houston Math Contest. Name: School: There is no penalty for guessing."

Alice Hawkins
5 years ago
Views:

1 Pre-Calculus Exam 009 University of Houston Math Contest Name: School: Please read the questions carefully and give a clear indication of your answer on each question. There is no penalty for guessing. Judges will use written comments and/or calculations to settle ties. Good luck.

2 University of Houston High School Mathematics Contest Precalculus Exam Spring 009 Name: School: Grade: Teacher: Directions: You have 60 minutes to complete this exam. Calculators are not permitted. Your exam will be graded using the answer grid below. For each of the 0 questions, put the letter answer (A, B, etc.) in the corresponding box on the answer grid. You may change your answer, but illegible answers will be marked as incorrect. Answer grid:

3 . Is there a Real number t so that the points (, ) and (t, ) are one unit apart? Yes No. Given that a function and its inverse are defined over the Real numbers, and that f() = 6, f(6) =, and f( ) = 0, what is f ( )? 6 D. Cannot tell from this information 0 E. (, ) C.. Which of the following is the graph of f(x) = ln( x)? C. D.

4 4. What s the inverse function to y = 0.5ln( 4 x)? C. 4x 8x y = e + 4 D. y= 4 e 4x y = 4 e E. None of these 6x y = 4 e 5. Given that x = log b(), y = log b(), z = log b(4), w = log b(5), and p = log b(6). Which of the following is equivalent to log()? y D. y -x p x x+ w E. p + z C. x (z p) 6. Which of the following graphs is expressed by the function y = ½ sin (5πx)? C. D. E. None of these

5 7. Let f(x) be an odd function that has the property: f(x) - f( x) = sin (x). π What is the value of f? C. 4 D. E. None of these 8. Given i=, evaluate 009 k i. k= 0 i i C. 0 D. + i E. 9. Simplify cosθ+ + cosθ cosθ cosθ cos θ D. csc θ E. cot θ C. sin θ + 0. Use the fact that 9 is a root of the given polynomial to solve the equation: x + 7x 9x 9= 0. Which of the following is a solution? D. + E. None of these C. 4

6 . Which of the following are in the solution space of this system of equations? + < x y y x 0 I. (0, 0. ) II. (, 0) III. (, ) IV. (0.5, 0) V. ( 0., 0. 4 ) None of them D. IV and V only All of them E. IV only C. I and V only. If f(x+ ) =, what is f(x + )? x x 4x + 4 C. 4x 4 D. 4x E. None of these. Which of the following does not simplify to cos x for every value for which the expression is defined? tan x cot x csc x sin x cosx tan x D. All of these simplify to cos x E. None of these simplify to cos x cosx C. csc x cot x 5

7 4. What is the domain for f(x) =? x 5 5 (, ) D. (, ) (, ) (, ) E. None of these C. {, 4} 5. Which of these functions is/are even? I. f(x) x 4x = + II. f(x) = cosx III. f(x) x 64 = IV. f(x) = x 7 All of them are even. D. IV only II, III, and IV only E. II and IV only C. None of them are even. 6. If x + y x+ y = 4, find all possible values for (x y). (, ) D. { 4, 4} [ 4, 4] E. {0} C. {, } F. No solution 6

8 x 7. Let f(x) = x. Let f(x) = f f(x). Let f (x) = f f f (x) and so on in this pattern of compositions. What is f 0 (x)? C. 0 x 0 0 x 0 x x 0x 0 0x 0 D. x E. None of these 8. Below is a graph of two functions. The parabola is f(x) and the line is g(x). Assuming that all of the expressions below are integers, which of them has a value of? f g (8) I. 5f(0) g(0) III. ( ) II. ( ) f g () IV. (f + g) () (f g)() -9 All of them. I and II only C. III and IV only D. II and IV only E. None of the statements yields as an answer. 7

9 π 9. Use trigonometric substitution for x = tan θ, 0 < θ< to rewrite the expression 4+ x as a trigonometric function of θ. D. (sin θ + cos θ ) tan θ E. csc θ + C. sec θ 0. The displacement from equilibrium of an oscillating weight suspended by a spring is given by y(t) = cos(4t). Find the displacement when t = 9 π. 4 D. 8 8 E. 8 C. 8. Given: f( θ ) = sinθis an odd function and f ( θ ) = cos θ is an even function. What can be said about p( θ ) = sin( θ) cos( θ )? p is even p is odd C. p is neither even nor odd D. p is symmetric about the x-axis 8

10 . The angles of elevation θ and α to an airplane are being continuously monitored by two observation points that are miles apart. See figure. Which of the following is the equation that gives the distance d between the plane and point B in terms of θ and α? airplane A θ miles B α Distance = 4 cos( θ α) Distance = sin α+ sin θ C. Distance = sin θ sin( α θ) D. Distance = sin α cos θ E. It cannot be determined from the given information.. Find a positive value k such that real number solution. x + kx = k has exactly one + 5 k = D. k = 5 k = + E. k = + 5 C. k = + 9

11 4. Given a 0 o 60 o 90 o right triangle with perimeter + cm and area.5 cm. What is the length of the shortest side? cm 5 cm C. cm D. cm E. cm 5. Find consecutive integers n and n + such that this expression lies between them: log(.00). 0 and D. and and E. None of these C. and 6. Let f be the function defined below. Is this function one-to-one? x x 0 f(x) = lnx 0< x< x + x Yes No 7. Which of the following statements are true? I. ln x e = x for all Real numbers. II. The range of the function III. sec(arcsin(x )) = x e + y = x e (x x ) is all Real numbers. IV. (n + )! = n + for n a natural number (n)! None of them are true. D. I and IV are true. I and II are true. E. None of them are true. C. III and IV are true. 0

12 8. Solve the following system: = x + y = 5 x y 4 x > 0 and y 5 D. ln ln 5ln, ln 5 ln ln ln 5ln ln, ln ln ln ln E. (, ) ln ln ln 5 ln, ln 5,, ln 5 ln + ln C. ( ) 9. Solve for x: x x e = e + ln 4 D. + ln 5 C. + 5 ln 5 ln E. No solution 0. Let A denote the area of the right triangle in the first quadrant that is formed by the y- axis and the lines y = mx and y = m (note that m > 0). Express the area of the triangle as a function of m. A = m D. A = m A = m + E. It cannot be done. C. A = m

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.