( ) π B. C. 8 D. sin(45 ) cos 1. e D. e 02. Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, 2012

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1 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, 01 Answer choice E. "NOTA" denotes "None Of The Above" 1. Let x = sin(0 + 5 ) and y = cos( ). Evaluate x y D Given that f( x) = sin( x)cos( x), what is the period of f(x)? π π π D. π π. A point P is expressed in polar coordinates as follows: P =,. Which of the following is an equivalent expression for P in Cartesian coordinates? 1, 1, D. (, ) (, ). Let f( x) = x 1x+ 16. If the three roots of f( x) are a, b, and c, what is the maximum possible value of a b+ c? 1 8 D Find the inverse of ( ) 1 sin(5 ) cos 1 e. 0 e e 0 D. 1 e 0 6. Let n x yx ( ) = n! n= 0. Evaluate y (). π 8 D. e

2 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, Evaluate 1,, (,,5 ) x 5,1, D. 8. If ln( x) + ln( x ) = ln(1), x = D. 105 ± 9. Let a = log, b = log. If x = 5, x = a+ b a b 1 a b D. 1 + a b 10. Let c = cos(0 ) sin(0 ) and d = cos(80 ). Which of the following is an equivalent expression of c+ d? sin(60 ) cos(0 ) D. cos(60 ) 11. Given the system of equations: x+ y z = x y+ z = 6 x 5y z = 8 Which of the following is equivalent to x y? D Let vector A = 1,, and vector B =,0,1. Vectors A and B are parallel perpendicular skew D. anti-parallel

3 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, A triangle with side lengths 8, 5, and 91 is inscribed inside a circle of diameter d. What is the value of d? D Let vector C = 1,, and vector D = 1, 0,1. What is the tangent of the acute angle between C and D? 1 1 D. 15. x + x y y+ 7 = 1 can be expressed in the form Ax Bxy Cy Dx Ey F = 0, with ABCDEF,,,,, all sharing no common divisor other than 1. If What is the value of A + B + C D+ E+ F? D. 16. Given: yx ( ) = x 1x+ 1 x + x+. Find all the asymptotes of yx. ( ) x= 1, x= x= 1, x= x= 1, x=, y = x D. x= 1, x=, y = What is the value of ( i + i i )? i + i 7 i D. 7 + i x 18. Let x = arcsin(cos(arctan(sec( π )))), with x in radians. What is the value of π? D

4 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, Given conic A with equation evaluate C D π. E x y x 1y =, center B= ( CD, ), and area E, 1 1 D. 0. If f x 1 = x 1, what is/are the root(s) of f( x )? 1,1 1.5 D Let g be the solution of + = on the interval [0, π ]. Evaluate g π. cos( θ) sin ( θ) cos( θ) 1 D.. u =,,5 and v =,,. Evaluate v ( u + v ) + u ( u v ) D. 8. If a b a <, ab,,and a =, then b is contained on what interval? b ( 0, ) 1, 1, 1,0 0, D. ( ). Let gx ( ) = 5sin( x) + 1 cos( x). What is the amplitude of gx ( )? D If x and y are governed by: x= sin( t), y = 6 cos( t), what is the product of the lengths of the major and minor axes of the conic generated in the Cartesian Plane? D. 8

5 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, Early one morning, Mahesh, ever the diligent student, repelled from the roof of his house into his batmobile, then proceeded to school at a modest 0 mph. Upon arriving at school, he realized that he had forgotten his calculus homework! Fearing Mr. Bradford s furious anger over the neglected calculus, Mahesh returned home at a hurried 0 mph. Fortunately for Mahesh, Andrew was able to meet Mahesh exactly halfway home to give him his homework. (Andrew also corrected a few of Mahesh s egregious mistakes on the homework, to prevent Mahesh from still invoking Mr. Bradford s furious anger, but that s irrelevant.) Mahesh, throwing caution to the wind, then breaks the speed limit and returns to school at 10 mph. What is his average speed for the entire journey? (Hint: If Mahesh lives d miles from the school, his entire journey is only d.) 15mph 0mph 0mph D. 190 mph 7. Evaluate: ( 1 i ) D How many petals does the graph of r = sin( θ ) have? 0 D Let 5π arccos cos = x y π and arcsin sin = x + y. What is the value of x + 8 y? π 1 9 D Mr. Bradford, being the matchless mathematician that he is, decided to use his trusty compass to draw two concentric circles with radii n and y, such that y > n. He then uses his steady hands, pencil, and straightedge to construct a chord of circle with radius y such that the circle with radius n trisects the chord into segments of length 1. Since Mr. Bradford used his trusty compass and is unmatched in arithmetic skill, he readily determines that the sum of the radii is 6; that is, n+ y = 6. What is n y? D.

6 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, 01 SOLUTIONS 1. C x y = sin(75 )cos(75 ) = sin( 75 ) = sin(150 ) =. C 1 π f( x) = sin( x)cos( x) = sin( x), T = = π. D π π π π π, =, π + =, = cos, sin = (, ). B x 1x+ 16 = ( x+ )( x ), roots are -,, => ( ) + = 8 5. A 6. D 7. B 1 sin(5 ) cos ( 1) 0 =, e e n x x yx ( ) = = e, n! n= 0 y() = e e =, = e 1 ( e) i j k,,5 x 5,1,1 = 5 = ( 1 5 1) i ( 1 5 5) j+ ( 1 5 ) k = 8, 1, ( ) 8. A x ( x ) ( x x ) 1,,,,5 x 5,1,1 = 1,, 8, 1,16 = = 1 ln( ) + ln = ln = ln(1), x x = 1,by inspection, only valid solution occurs when x > 0, so, x > 0 because of the ln( x) term. 9. C x 5 x log 5 x(x ) = x x 1 = 0, and we take the positive solution since 1 log 5 log10 log 1 a = => = = = = log log b 10. C cos(0 ) + cos(80 ) = sin(50 ) + cos( ) = sin(50 ) + cos(50 ) cos(0 ) sin(50) sin(0) 1 1 = sin(50 ) + cos(50 ) sin(50 ) = cos(50 ) + sin(50 ) = cos(0 ) cos(50 ) + sin(0 ) + sin(50 ) = cos(0 50 ) = cos( 0 ) = cos(0 ) 11. A By Cramer s Rule, x = and y = , so x y = AB = + 0+ = 0, which implies that the two vectors are perpendicular 1. B 1. B The circle is circumscribed about a right triangle, so the diameter is the hypotenuse, d=91.

7 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, 01 CxD sinθ C D CxD 1. D tanθ = = = = = cosθ CD CD C D 15. C x x y y = => x + x= y y+ => x y + x+ y = A= 1, B= 0, C =, D=, E = 1, F = 7, and so that the 16. C, which implies A+ B+ C = = = 1 (NOTE: If rearranged D+ E+ F ( 7) x coefficient is positive, the ratio of the resulting coefficients remains the same.) x 1x+ 1 ( x+ )( x 1)( x ) yx ( ) = =, so clearly x=-1 and x=- are vertical asymptotes. x + x+ ( x+ 1)( x+ ) 6x + 18 Long division yields yx ( ) = x +, so that x + x+ y = x is an oblique asymptote of yx. ( ) ( i + i i ) = + 1 () i = 1+ 1 i = ( ) i = ( 7 ) i i 17. C ( ) 18. C π x 1 = = π π 8 π π arcsin(cos(arctan(sec( π )))) = arcsin(cos(arctan( 1))) = arcsin cos = arcsin = ( x+ ) ( y ) 19. A x + y + x 1y+ 6 = 0 => + = 1 => B= (, ), E = π = π C Dπ () π = = E π B f = f( x) = = +, x x x x => x= 0, x=, x=0 is an extraneous solution, and x = = C 9 6 f x x x x x x x ( ) = + = 0 => = ( + ) = 0 cos( θ) + sin ( θ) cos( θ) = cos θ sin ( θ) + sin ( θ) cos( θ) = cos θ cosθ = = + =, cosθ 1, cosθ + 1 = 0 => θ = π. cos θ cosθ (cosθ 1)(cosθ ) 0. D v ( u + v ) + u ( u v ) = v + u = 9+ 8 = 8

8 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, E a = -, so, <. b is strictly greater than 0, so < 16b, or b b 1 b >. Note, b 0, so b is 1 contained on the interval,0 ( 0, ).. C A = = 1 x y x y 5. D = sin(), t = cos() t => + = sin () t + cos() t = 1=> a=, b= 6, ( a)( b) = d d di d 6. C d dt d d = rt => t = i tt ti rj 0mph r => = = i = 0 => = t = d = t i 1 π π 1 1π 1π 1 7.D ( 1 i) = cos isin cos isin = = = 8. D A rose of form sin( nx ) has n petals for odd n, and n petals for even n. = 8 5π π π π 1π 5π 9. B arccos cos = = x y and arcsin sin = = x + y. => x= => y = x+ 8y 1π 10π => = = π π 0. B n = h + 6 and y = h + 18, subtracting yields y n = 88. y n = ( y+ n)( y n ) = 6( y n) = 88 => y n = 8. Solving the system made by y n= 8 and y+ n= 6 yields y = n = 1, so that n y = 08. and n 1 y h

9 Tampa Bay Tech Invitational PRE-CALCULUS Individual February 18, 01 TBT Invit /18/ 01 Pre-Calc Individual Answers 1) C ) C ) D ) B 5) A 6) D 7) B 8) A 9) C 10) C 11) A 1) B 1) B 1) D 15) C 16) C 17) C 18) C 19) A 0) B 1) C ) D ) E ) C 5) D 6) C 7) D 8) D 9) B 0) B