University of Houston High School Math Contest Pre-Calculus Test

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1 University of Houston High School Math Contest 08 f ( x ) is a quadratic function satisfying Pre-Calculus Test remainder when f ( x ) is divided by x A) B) 7 C) 9 D) E) 4 Let M be a non-zero digit When the sum of the 50 numbers M digit MM digits MMM digits MMMM M 50 digits f 7, f () 5 and f (4) 6 Find the is divided by 9, the remainder is Find the sum of all possible values of M A) 7 B) 0 C) D) 5 E) 7 Let f ( x) 8 x e, gx ( ) log (0x ), and Find the value of f g h A) ln B) ln C) ln 4x hx ( ) 6x UH Math Contest 08 Pre-Calculus Test Page of

2 D) ln E) ln x 4 Let f ( x ) be a real valued function defined as: f( x) x 4 Find the range of this function A) B) C) D) 5 5, 8 8 5, 4 4, 8 8, E), Let f( x) x x x x x x x, where x Evaluate the following: f f 0 A) 4 B) 5 C) 6 D) 7 E) 0 UH Math Contest 08 Pre-Calculus Test Page of

3 6 Let a be a real number with 0 a 0 The vertical asymptote of the function x sin x f( x) ln(8x ) is also a vertical asymptote of gx ( ) x cos( a x) Find the sum of all possible values of a 8 A) B) 5 C) D) 4 E) 8 7 Let y be the product of all real solutions of the equation Evaluate the following expression: ln( y ) log y e ln 4 ln x x A) 5/ B) 9/ C) /6 D) 0/ E) 7/6 8 Let S be the set of all real solutions of the inequality: x x 6 x Let T be the domain of the function gx ( ) arcsin tan 4 Find S T 4 A),4 4 B), C), 4 D), 4 E), UH Math Contest 08 Pre-Calculus Test Page of

4 9 Let A, B, C be sets defined as: A( x, x): x B( x, x): x C ( x, x4): x If ( pq, ) A B and (,) rsb C, find the value of A) / B) /4 C) /4 D) 4/5 E) /5 p r q s 0 Let n be an integer and f ( n ) be defined as the first non-zero digit of the number n from the right For example, f (04500) 5 and () f x at 00 x 000 A) B) 4 C) 6 D) 8 E) f Find the value of Let abcd,,, be integers satisfying: alog0 blog000 clog00 5 dlog0 08 Evaluate 4abc d A) 807 B) 406 C) 04 D) 0090 E) 08 UH Math Contest 08 Pre-Calculus Test Page 4 of

5 Let p be a real number Given that the distance between the foci of the ellipse x y p p A) 4 B) 6 C) 8 D) 0 E) 4 Let is, find the length of the major axis of this ellipse 0 cos(40 ) x cos(55 ) cos(5 ) 0 tan 75 y sin 5 cos 5 and Which of the following is equivalent to x y? A) B) C) 4cos(40 )cos(75 ) cos(40 )sec(75 ) 4cos(0 )sec(75 ) D) cos(40 )cos5 E) 8cos(0 )sec(75 ) 4 4 Let Y be defined as Y cos( kx) for any real number x Express the value of A) Y Y B) C) Y 4 D) Y Y 4 E) k0 4 cos ( kx) in terms of m k UH Math Contest 08 Pre-Calculus Test Page 5 of

6 5 Let x be a real number in 0, satisfying sin( x) x sin 5 A) 0 5 B) 0 C) D) 5 5 E) 5 4 sec( x) tan x cot x sec( x) Evaluate: 6 Find the number of solutions of the equation cos(5 x) cos( x) cos( x) A) 5 B) 8 C) 9 D) E) 0, over the interval 7 Let z be a complex number and z be its conjugate Given: z i write the expression z 8i 9 A) 9i B) 4 i 08 9 C) 5 5 i 4 D) 5 5 i E) i in the form of a bi 8i 4zz i, UH Math Contest 08 Pre-Calculus Test Page 6 of

7 8 A circle passing through the point radius of this circle A) 8 B) 0 C) D) 5 E) 0 0,0 is tangent to the x axis at x 0 Find the 9 Let ABCD be a square as given on the figure below Given: DE 4, AF, BC 6 and mfge x Find the value of A) -/8 B) -5/4 C) -/8 D) -/4 E) -5/8 cot x cot x 0 Given: 4sin x tan x cot x, find the value of sin x A) B) C) D) E) UH Math Contest 08 Pre-Calculus Test Page 7 of

8 Triangle ABC is an isosceles right triangle with right angle B Given: E is the midpoint of the side AC and BD AC 4 Find DE x (Note: The image is not drawn to scale) D x C E A B A) 0 B) 8 5 C) 5 D) 4 4 E) 5 Evaluate the following: A) 5 B) 5 9 C) 5 D) 69 5 E) 5 sin arcsin sec arccos 5 UH Math Contest 08 Pre-Calculus Test Page 8 of

9 Amy and Bob are standing at the seashore miles apart The coastline is a straight line between them Both can see the same ship in the water The angle between the coastline and the line between the ship and Amy is 45 degrees; the angle between the coastline and the line between the ship and Bob is 75 degrees What is the distance between the ship and Bob? A) 6 miles B) 6 miles C) 6 miles D) miles E) miles 4 Let t be a real number satisfying cost tan t Find the value of the expression 4 cos t sin t sin t A) B) C) - D) E) 0 5 A regular polygon with side length r is inscribed in a circle with radius r How many sides does this polygon have? A) 6 B) 8 C) 9 D) E) 8 UH Math Contest 08 Pre-Calculus Test Page 9 of

10 6 Let u and v be two different vectors in the coordinate plane whose sum is equal to the zero vector Which of the following is/are true? I uv v II If u v(, 4), then u (,) III The angle between u and v is 8 A) I only B) III only C) I and II D) I and III E) II and III 7 Evaluate: i 9 A) i B) -04 C) 5 D) -5 E) i 8 Let, 4 and m cos cos Which of the following is/are true? I m n p II p m n III qm 0 IV p n q A) II and III only B) I and IV only C) II and IV only D) III only E) II only, n sin cos, p cos sin, q cos sin 4 UH Math Contest 08 Pre-Calculus Test Page 0 of

11 9 Drones A and B leave the same point at the same time Drone A flies 0 miles with a 0 0 bearing of N75 E Plane B flies 0 miles with a bearing of S5 W Find the distance between the drones A and B A) 0 B) 0 7 C) 0 D) 6 5 E) The lines d and d intersect at the point O with an angle of 0 as shown in the figure below Points A, A,, A 0 and B, B,, B 0 are marked with the following pattern: A line segment perpendicular to d is drawn from the point A to the point B ; A line segment perpendicular to d is drawn from the point B to the point A ; A line segment perpendicular to d is drawn from the point A to the point B ; and so on, until all 40 points are marked (The figure above is not complete due to limited space) If AB, find the sum: AB AB AB A0B0 4 A) 40 4 B) 8 0 C) 40 4 D) 6 4 E) 40 UH Math Contest 08 Pre-Calculus Test Page of

12 The following questions are part of this test, but they are not multiple choice For the following questions, write your answer on the answer sheet as a number For example: 5, 0, 45, -7, 5 7, 4 5 7, 4, /4, / or 50/ are acceptable answers Radical expressions should be reduced; for example, should be written as Show your work on the empty space below each question and write your final answer on the answer sheet Your work may be used to break ties On the figure below, ABC and DBE are triangles with AD BD and BE EC Given 6 BDE (Note: The image is not drawn to scale) AB, sin BAD and sin BCE B, find the area of the triangle 5 A D E C ANSWER: UH Math Contest 08 Pre-Calculus Test Page of

13 The height of water in a bay various with time and can be modeled by the function f ( t) asin( t) bcos( t), where a and b are real numbers and t 0 represents time The difference between the height of the waves at low tide and high tide is 0 feet Find the largest possible value of a b ANSWER: 5 4 x x 5x 0x0 Let f( x) 5 x 8x x 8x4 If this function intersects its horizontal asymptote, then list the x coordinates of all points of intersection (if any) If it does not intersect the horizontal asymptote, state none ANSWER: THE END! Check that you wrote your answers on the answer sheet; only the answer sheet will be graded UH Math Contest 08 Pre-Calculus Test Page of

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