1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if

Transcription

1 . Which of the following defines a function f for which f ( ) = f( )? a. f ( ) = + 4 b. f ( ) = sin( ) f ( ) = cos( ) f ( ) = e f ( ) = log. ln(4 ) < 0 if and only if a. < b. < < < < > >. If f ( ) = ( )( k) and if the remainder is 5 when f ( ) is divided by ( + ), then k = a. 4 b If f( ) = and g( ) =, then the solution set of f ( g( )) = g( f ( )) is a. 4 b. { 4 } {, 4}, Pre Calculus Test UH Math Contest 0 Page

2 5. If a, b, c, d, and e are real numbers and a 0, then the polynomial equation 7 5 a b c d e = 0 has a. only one real root. b. at least one real root. an odd number of non-real roots. no real roots. no positive real roots. 6. If f ( ) = e, which of the following lines is an asymptote to the graph of f? a. y = 0 b. = 0 y = y = ln = y ( ) 7. If f ( ) = + A + B and if f () = 5 and f ( ) = 5, what is the value of A B? a. -0 b. 0 - It cannot be determined from the information given 8. Let f ( ) = sin(arctan ). What is the range of f? π π a., b. ( 0, ] [ 0, ] (,) [,] Pre Calculus Test UH Math Contest 0 Page

3 a 9. If loga ( ) a. b a =, then a = 0. The fundamental period of the function defined by a. π π b. 6 6π = f( ) cos π is. If f ( ) + f( ) = f( + ) for all real numbers and, which of the following could define f? a. f ( ) = + b. f ( ) = f( ) = f ( ) = e f ( ) =. The graph of y = + 4 is symmetric to which of the following? I. The -ais II. The y-ais III. The origin a. I only b. II only III only I and II only I, II and III Pre Calculus Test UH Math Contest 0 Page

4 . The figure below shows the graph of a cosine function for one perio Which of the following is an equation for the graph? a. y = cos π π b. y = cos ( + ) y = cos( + ) y = cos π ( + ) ( ) π y = cos + 4. Find the length of an arc intercepted by a central angle of measure π if the radius of the circle is in. a. 4π in. b. 6π in. 8π in. 4 π in. 8 π in. Pre Calculus Test UH Math Contest 0 Page 4

5 5. The terminal side of an angle θ, in standard position passes through the point (7, -4). What is cscθ? a. b The graph of the polar equation r = cosθ + is a a. cardioid b. rose limacon spiral circle 7. In the epression u u, let csc u =. Which of these is equivalent to the resulting epression? a. sec (sec + tan ) b. csc sec csc (csc + cot ) cos (sin ) sin Pre Calculus Test UH Math Contest 0 Page 5

6 π 8. Suppose 0 < t <, f ( t) = sintcost+ tantcsct and sin t =. Which of this is the value of f ( t)? a. b π 9. Suppose f( ) = sin +. For which of the following values of does the function reach its maimum value? a. 5 π b. 7 π 7 π 6 8 π More than one of the answers given in A,B,C and D generates a maimum value Pre Calculus Test UH Math Contest 0 Page 6

7 0. Evaluate sin(4 + 0 )*sin(4 947 ) a. 4 b. 4. Suppose A = and B + =. Compute ( A B) +. a. b Simplify: log + log a. 0 b. 4 log - Pre Calculus Test UH Math Contest 0 Page 7

8 . If ln =.69 and ln =.098, find ln 6. a b The graph of f ( ) = e passes through the key point (0, ). Which of these gives the transformation 4 of the key point for g ( ) = e +? a. (-4, -) b. (4, 0) (-4, ) (4, -) (-4, 0) 5. State the domain of the function: a. (,) (, ) b. (, e) ( e, ) (,0) ( 0, ) (, ) [ 0, ) e f( ) = e + 6. Suppose a triangle has vertices (0, 0), (0, 4) and (, ). For each verte, write the equation of the line that contains that verte and is perpendicular to the opposite side of the triangle and then find the point of intersection of the three lines. a., b. (, ),,,. Pre Calculus Test UH Math Contest 0 Page 8

9 7. Form the difference quotient for f( ) = and evaluate it at =. a. b. 8 5h 64 40h 5 40h h 64 5h 40h Evaluate sin (6 A) sin ( A) + cos (6 A) + cos ( A) a. + cos (6 A) b. + cos(6 A) + sin(6 A) 0 9. Evaluate π 4π sin sec 6 5π + tan 4 a. Undefined b. 0 Pre Calculus Test UH Math Contest 0 Page 9

10 0. Suppose f( ) = 4. The linear function that is perpendicular to f and passes through the origin intersects f at a. b. 8, 5 5 8, 5 5 8, 7 7 8, , 7 7. Epress AC as a function of θ. C 4 5 θ A a. 9secθ b. 4secθ + 5cscθ 5secθ + 4cscθ 9cscθ + 5cscθ Pre Calculus Test UH Math Contest 0 Page 0

11 . Find the area of the shaded region. The radius of the circle is, and the measure of angle X in the obtuse triangle is given in radians. a. π X + sin X π X + sin X b. ( ) + sin( π X ) + 4sin( π X ) ( π X + sin X ). Write the equation of the conic section with eccentricity which has foci at ( ± 7,0). a. b. 7y = 4 4 y = + 4y = 4y = 4 + y = Pre Calculus Test UH Math Contest 0 Page

12 Tie-breaker: Show all steps that you use to answer this question: A rectangular piece of paper that measures a inches by b inches (a > b) is red on one side and white on the other. The paper is on a table with the red side up. Suppose you fold the paper along the diagonal without lifting the paper off the tabl What is the area of the red triangle that you now see? Pre Calculus Test UH Math Contest 0 Page