University Of Houston Precalculus Contest 2016

Name: School:University of Houston High School Mathematics ContestPre-Calculus Exam – Spring 20161. arctan(1) arctan(2) arctan(3) A) ππB)2πC)4πD)3E) 2πF) none of these2. log 2 (log 4 (log 8 64)) A) 1B) 11C)21D)41E) 2F) none of these3. Simplify cot(arcsin(tan(arccos ( x )))) .A) xB)C)D)E)xx 122x 2 1x2 12x 2 11 x 2x1 x 2F) none of these

Name: School:4. Solve 3sin( 2θ ) 81cos(θ ) on the interval 2π θ 2π .A) θ {0, π }π 3π 3π πB) θ , , 0, , 22 2 2 3π π π 3π C) θ , , , 2 2 2 2ππ3π 3πD) θ , π , ,0, , π , 222 2π π π π 5π 3π 3π 5π, , , , , ,E) θ , 62 6 6 2 62 2F) none of theseπ 17π 5. List all x – intercepts for f (x) 3cos 2x on the interval 0 x . 3 6A)B)C)D)E)F)π 4π 7π,,3 3 3π 7π 13π 19π 25π 31π,,,,,12 12 12 12 12 12π 5π 9π 13π,,,6 6 66π 3π 5π 7π 9π 11π,,,,,4 4 4 4 4 4π 2π 4π 5π 7π,,,3 3 3 3 3none of these6. Suppose f (x) log(3 2x) 1 x and g(x) 2x 1 f , give the domain of (x) .x 3 g 3 3 A) , , 3 ( 3, ) 2 2 1 1 3 B) 1, , 2 2 2 1 1 3 C) , , 2 2 2 1 1 3 3 D) , , , 3 ( 3, ) 2 2 2 2 E) all real numbersF) none of theseUniversity of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 2 of 10

Name: School:7.limx 2 2 x 2 xA)B)C)D)E)F) 110does not exist2none of these8. Find the area of triangle XYZ if Y 60 , XY 5 and YZ 4.A) 12B) 10C) 10 3D) 5 3E) 5F) none of these9. CDE is a triangle with C 60o, D 45o, and DE 8 cm. Find DC.A)8 63B)8 23C)4 6 4 23D)8 6 8 234 63F) none of theseE)University of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 3 of 10

Name: School:10. limx (A)B)C)D)E)F))4x 2 3x 1 2x 141334320none of these11. In which of the following situations would f (g(x)) always be an even function?I.f (x) and g(x) are both oddII.A)B)C)D)E)F)f (x) even and g(x) oddIII.f (x) and g(x) are both evenIV.f (x) odd and g(x) evenI onlyII and III onlyII, III and IV onlyI, II, III and IVI and IV onlynone of these12. Give the range of g(x) 2 1 x 2 .A) 0, 2 B) [ 0,1]C) 0, 2)D) 1, 2 )E) 1, 2F) none of theseUniversity of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 4 of 10

Name: School:2 θ θ 13. Simplify cos sin 2 2 .A) 1 sin (θ )B) cos (θ ) sin (θ )C) cos (θ ) 1D) 11E)2F) none of these14. The wheel on a car has a diameter of 52 cm and rotates at a rate of 600 rpm (rotations perminute). What is the approximate linear speed (in cm per second) of a point on thewheel?A) 1633 cm/secB) 63 cm/secC) 816 cm/secD) 2543 cm/secE) 1540 cm/secF) none of these15. QRST is a quadrilateral with QR 12 cm, RS 2 43 cm, and ST 6 cm. Whendiagonal QS is drawn, RQS 60o and QST 45o. Find QT.A) 2 58 21 2B) 14 43C)232 42 2D) 84 3E) 14 3 84 2F) none of these22() x y 16. One foci on the hyperbola 1 is the point 2 3, 0 . Find a . a 2 A) 12B) 4C) 2 2D) 3E) 0F) none of theseUniversity of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 5 of 10

Name: School:17. Find the tenth term of 8, 4,2,.1A)64B) 121C)10241D)16E) 16F) none of thesecos(nπ )3nn 0 18. Find the sum: 433B)4A)433D) 4E) does not existF) none of theseC) 19. Suppose g is a function such that g(g(y)) y . Find g(g(g(g(!(g(y)!) ." # %g appears n times yA) g(y) yB) g(y)n oddn evenn evenn oddC) yD) g(y)E) cannot be determinedF) none of theseUniversity of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 6 of 10

Name: School:20. The expression n 4 2n 3 n 2 is equivalent tonA) 4k3k 1nB) 2k3k 1nC) 6k2k 1nD) 4k2k 1nE) 2k4k 1F) none of these21. Solve for x : e x 2 e x 2 e 2 xA) 0B) {1,ln(2)} 1 2 C) ln 2 1 2 D) 0,ln 2 (E) ln 1 2)F) none of these 22. Find the limit of the sequence: 2, 2 2 , 2 2 2 ,. . A) 1B) 2C)232E) does not existF) none of theseD)University of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 7 of 10

Name: School:23. Give the Cartesian coordinate form of r cos ( 2θ ) .A) x 2 y 2 2B)x 2 y2 x 2 y2C)(x2 y2 ) x 2 y23D) ( x 2 y 2 ) x 2 y 23E) x 2 y 2 2 x 2 y 2F) none of these24. Give the horizontal asymptote for f (x) arctan(x 3) 2 .A) y 0π2y 23 πy 24 πy 2none of theseB) y C)D)E)F)25. Solve for x : 4 x 2 x 3(log (1 log ( 2 log (1 log (1 )13 )13 )7) 17)A) log 2 1 13 1B)C)D)E)2244F) none of theseUniversity of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 8 of 10

Name: School:26. Which of the following polynomials hasA)B)C)D)E)F)3 2 as one of its roots?x 5x 6x 4 10x 2 1x 3 2x 2 63x 3 12x 2 1x 4 2x 3 6none of these4227. If f (x) x 2 x , then f (x 1) A) f (x)B) f (x)C) f (x 1)D) f ( x)E) f (x) 1F) none of these28. Simplify:A)B)C)D)E)F) π sec 2 tan 1 1 4 π2161 1π40none of these29. Give an equation in Cartesian coordinates for the curve parameterized byx 3 2sec(t), y tan(t) 1 .(x 3)2 (y 1)2 1A)4(x 3)2 (y 1)2 12C) (3 2x)2 (y 1)2 1B)D) (3 x)2 (y 1)2 4E) (3 x)2 (y 1)2 4F) none of theseUniversity of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 9 of 10

Name: School:30. A party of hikers walks 8 kilometers from camp on course 30! , then turns and walks 6kilometers on a course 135! . Find the magnitude of the net displacement from camp.(Note that course is measured clockwise from due north.)A) 100 24 2 24 6B) 10C)100 24 2 24 3D)100 12 2 12 3E)112F) 100 24 2 24 6G) none of theseEND OF EXAM J University of HoustonHigh School Mathematics ContestPre-Calculus Exam, Spring 2016Page 10 of 10

University of Houston Pre-Calculus Exam, Spring 2016 High School Mathematics Contest Page 2 of 10 4. Solve 3sin(2θ . University of Houston Precalculus Contest 2016 Author: Bekki George Subject: High School Mathematics Contest -