Florida Regional Competition March 08. Given: sin ( ) sin α = for 0 < α <, and β = for < β <. ( ) A = sin( α ). B = cos α. C = ( ) cos β. D = sec cos ( β ) ( ) α. Find the product ABCD, in the form q p for a positive prime p. --------------------------------------------------------------------------------------------------------------------------. Let C denote the conic determined by x + y x 5= 0, and C denote the conic determined by ( ) x ( y ) =. A = the distance between foci of the graph of C. B = the distance between the centers of the graphs of C and C. C = the product of the slopes of the asymptotes of the graph of C. D = the length of the transverse axis of the graph of C. Find the product ABCD. ------------------------------------------------------------------------------------------------------------------------. ( ) f() x = 8 x and gx ( ) = log ( x). A = the solution, x, of f( x ) = when solved over the Real Numbers. B = the solution, x, of gx= ( ), when solved over the Real Numbers. ( ) C = ( ) f g. D = the value of x for which f ( x) =. Find the value of the product (0ABCD).
Florida Regional Competition March 08. Consider Δ PQR with QR=0, and Δ FGH with GH= and FH=0. 5 A = the length of PR when sin( P ) =. B = the value of sin( m P+ m H) when PQ=0. A+B = a + b c for prime c. Give the value of ac b. P Q Not drawn to scale. F 0 R 0 G H ------------------------------------------------------------------------------------------------------------------------- 5. f( x) = x + x x x+ with roots of f are x, x, x and x, for x x x x. gx ( ) = sin ( x) sin( x). A = x x. B = the sum of the values of x over, where gx= ( ) 0. C = the value of x over, where gx= ( ). Find the value of AB in fraction form. C ------------------------------------------------------------------------------------------------------------------------------. Consider Δ PQR with PQ=0, QR=. P A = the length of PR if B = cos( Q) if PR=. m Q o = 0. C = sin ( R) if cos( P) =, and C > 0. o D square units = the area of Δ PQR if m Q= 0. 0 Q R Find ABCD.
Florida Regional Competition March 08 7. Consider the parametric equations: x= t+ and y= t, which has a graph with a minimum point on the xy-coordinate plane of Pxy. (, ) Consider the parametric equations: x= t and y= t + t which has a graph on the xy-coordinate plane with x-intercept at point Q. The y-intercept of this graph is R. t 0. A = the x-coordinate of point P. B = the y-coordinate of point P. C = the x-coordinate of point Q. D = the y-coordinate of point R. Find the value of A+B+C+D. ------------------------------------------------------------------------------------------------------------------------------ 8. Consider the equation for relation R : 8x= y y. The axis of symmetry of the graph of R has equation y = A. The vertex of the graph of R has coordinates ( BC, ). The focus of the graph of R has coordinates ( DC, ). Find the value of ABCD. ----------------------------------------------------------------------------------------------------------------------------- 9. The polar coordinates,, written in rectangular form, is (A, B). The graph of the polar equation r = cos( θ) intersects the graph of r = C times. The graph of the polar equation distance of D between vertices. r = is a conic section with a 9cos θ sin θ Find (A + B)(C + D). ------------------------------------------------------------------------------------------------------------------------- x x5 x 0. Consider f( x) = and gx ( ) =. x 5 8 x x A: the equation of the vertical asymptote of the graph of f is x = A. B: the equation of the horizontal asymptote of the graph of hx ( ) = xgx g ( ) is y = B. C: if the removable discontinuity of the graph of f is at x= k then C = lim f ( x ). x k D = g( f( )). If g( f( )) does not exist, then let D = 00. Find the value of ABCD.
Florida Regional Competition March 08. i = and cisθ = cosθ + i sinθ. A = ( ) i. B: the three cube roots of i are rcis( θ ), rcis( θ ) and rcis( θ ) 0< θ < θ < θ <, and all values of r n positive. B = θ. C: the set of fourth roots of 8 for all values of probability that, for is { acis( β ), acis( β ), acis( β ) and ( ) acis β }, o β between 0 and n 0, and all values of a n positive. C is the B for n =,,, is between 00 o and 00 o. n Find the value of (ABC). --------------------------------------------------------------------------------------------------------------------------. A = the approximation of. by using the sum of the first three terms of the B = expansion of n= 9 n +. (Note: The terms are in order of decreasing powers of.) C and D: The equation of the line tangent to is y = Cx + D. f( x) x x = at the point (, f ( ) ) Give the value of AB(C + D). ----------------------------------------------------------------------------------------------------------------------
Florida Regional Competition March 08. In a particular Literature class of 0 people, 0 people were taking Algebra and Biology. people were taking Algebra and Chemistry. 8 people were taking Biology and Chemistry. people were taking Algebra, Biology and Chemistry. The sections marked x have an equal number of students. Alg. x x x Biol. x = the number of students in that Literature class who are taking exactly one member of the set {Algebra, Biology, Chemistry} Chem. A = the probability that a person from that Literature class is taking Literature and exactly two other classes from the set {Algebra, Biology, Chemistry}. B = the maximum possible number of people in that Literature class who are also taking Biology. C = the probability that a person from that Literature class is taking four total classes. D = the number of people in that Literature class who are not taking any classes from the set of classes {Algebra, Biology, Chemistry}, if x is the maximum possible value. A Give the value of B D C + +. ---------------------------------------------------------------------------------------------------------------------------- P. 0 R Q Δ PQR has PQ=, PR=0, and m R o =. A: Use sin 0.5 to get an approximation for the value of sin Q. A = that approximation to the tenth place. B: Use sin 0.5 to give the value of the height from point P in the triangle Δ PQR (to the line containing side QR ). B = the value of that height. Give the value of A+B. May not be drawn to scale.
Florida Regional Competition March 08 v v 5. p= ij and q = i+ 8j denote two vectors in standard position. A = the vector, written as ai + bj, which has the same direction as p v and has length 0. B = the dot product of vectors p v and q v. C = find the cosine of the angle between vectors p v and q v. Find the product of the vector determined by part A, and the scalar BC. Give your answer in fraction form, with format ai + bj. --------------------------------------------------------------------------------------------------------------------------
Florida Regional Competition March 08 ANSWERS Part A Part B Part C Part D Question Answer Note. ABCD Answer must be 9 fraction. 8 ABCD. 0(ABCD) 5. + ---- ---- see 79 problem 0 5. ---- AB Answer must be C 9 fraction. 5 5 ABCD 5 5 7. 7 A+B+C+D 7 8. 0.5.5 ABCD 9. + (A+B)(C+D) 0 0. 5 ABCD 5. 7 ---- (ABC). 9 AB(C+D). 0 A B D 5 0 C + +. 0.9 5. --- --- A+B. 5. 8i j 8 --- BC(A) 9 9 i j 5 5 5 fraction form, i,j form needed
Soultions Pre-Calculus Team Questions Florida Regional Competition March 08. A: sin( α ) = sinαcosα = =. 9 B = cos α = sinα =. C = cos( β ) = D = cos β sin β = = = = sec( β ) cos( α ). sec( β ) sec β = and cosα =. cos( α ) = = Final Answer is ABCD = ( ) 9 = ( x) y. C is rewritten from x + y x 5= 0 to ( x ) + y = or + = C is rewritten from ( ) ( x) ( y) x ( y ) = to =. A: foci are a distance of = from the center, or from each other. B: distance between (, 0) to (, ) is. B=. C: asymptotes of C have slope ± and the product is D: transverse axis is horizontal and has length = 8. Final Answer is ABCD = () (8) = x. A: ( ) ( x) 8 =. =. + x =. x=. / B: log ( x ) =. log ( x ) =. = x. x =. g = ( ) C: ( ) log ( ) = log = log = log / x =. f () = 8 =. log =. 0 ( ) f which is 8 ( ) = D: f ( x) = gives the same result as () Final Answer is 0(ABCD) = 0() ( ) = 5. 5. A: If sin( P ) = then tan(p)= 5. 0 = 5. PR= PR B: sin( m P+ m H) = sin Pcos H + cos Psin H. Since PQ=0, this is a 0-0-90 triangle with sinp=0 o. + + =. 5 5 0
Florida Regional Competition March 08 + + A+B = + = = a+ b c for prime c. 0 0 Final answer is ac = ( )/ = () = 79. b 0 0 5. A: Factor f( x) = x + x x x+ to ( x+ )( x+ )( x+ )( x+ ) and x x x x x, x, x and x are respectively, -, -, and. x x = - (-) =. B: sin ( x) sin( x) = 0. sin x(sin x ) = 0. sin x = 0 and sin x =. Over,, the solutions are 5,, and the sum is. C: gx= ( ). sin ( x) sin( x) = 0. (sin x )(sin x+ ) = 0. sin x = at, and the other factor gives an extraneous answer. C=. ( ) C. Final answer is AB = =. 9. A: Using the Law of Cosines, + 00 (0)() cos0 = ( PR). +0= ( PR ). PR = A =. B: Using the Law of Cosines, 00 + (0)() cosq =. (0)() cosq = 8. 8 cosq = =. 0 5 sin R sin P sin R C: =. 5 = since sinr=cosr. sinr = 0 0. D: Area = (0)()sin0 0 = = 5 5 Final Answer is ABCD = () ( 5 ) = 5 5. 7 7. For the first set of equations, x= t+ so t = x. With y= t, y= ( x). The minimum point of this graph is (, -) so A = and B = -. Note that this point coincides with t=0, and it the minimum if the time values are limited to non-negative or not. For the second set of equations, x= t and y= t + t and since t 0, the x values are from [-, ). t = ( x+ ). y= ( x+ ) + ( x+ ) = ( x+ ) (( x+ ) + ). There is only one value at which y=0 and so Q has coordinates (-, 0) and C= -. The y-intercept at R occurs when x=0, so t=. y= + = 7. D=7. The Final Answer is A+B+C+D = + - + - + 7 = 7. 8. The graph of R : 8x= y y is a parabola opening to the right. 8x+ = y y+
9. 0. Pre-Calculus Team Questions Florida Regional Competition March 08 8( x+ 0.5) = ( y ). The axis of symmetry is y = and since this is y = A, A=. The vertex is ( BC, ) so B= -0.5 and C=. The focus is ( DC, ) and the focus is units from the vertex (from 8/), to the right. So D= -0.5+=.5. Final Answer is ABCD = ( 0.5)()(.5) =. = and sin = sin cos cos sin = =. cos = cos cos + sin sin = + = +. So A = cos = +. B = sin =. The graph of r = cos( θ) is a three-leaved rose, with center at the pole. The graph of r = is a circle centered at the pole. There are intersection points. C=. x y r = gives 9r cos θ r sin θ =. 9x y =. = 9cos θ sin θ 9 The distance D between vertices is =. D=. Final Answer is (A + B)(C + D) = ( )( + ) = 0. x x5 ( x 5)( x+ ) f( x) = = x 5 ( x 5)( x+ 5) x ( x ) gx ( ) = = 8 x x ( x )( x+ ) A = -5 because the non-canceled factor is (x+5) in the denominator. x x B = lim =. Horizontal asymptote is y= -. x 8xx ( x 5)( x+ ) C: k = 5 as we have a canceled factor of (x-5), and lim = = x 5 ( x 5)( x+ 5) 0 5 D: f ( ) = 0. g( f( )) = g (0) =. Final Answer is (ABCD) = 5( ) = 5. A: Using DeMoivre s Theorem, ( ) / i = ( 00 o o cis ) cis800 cis0 = = =. / B: ( i) = cis = cis. The cube roots are spaced / apart, so the next 7 roots are + =, and + =. θ = 7 = B. / C: ( ) ( ) / 8 = 8cis80 o = cis 5 o for the principal root. Add 90 for each subsequent fourth root, to get root angles 5, 5, 5, and 5. Between 00 and 00 degrees
Florida Regional Competition March 08 are out of, for a probability of ½. 7 () (ABC) Final Answer is = =.. A: The first three terms of the expansion of + have a sum of 0 C (,0)() (/ ) + C (,)() (/ ) + C (,)() (/ ) = + + =. /9 B: The sum of this infinite geometric series is = = / 9 C, D: The slope of the tangent line at (x, y) will be x so at the point (,), the line has equation y = ( x+ ) or in form y = Cx + D, C= - and D= -9. Final Answer is AB(C + D) = ½( - + -9) = -.. A: The football-shaped intersections are shown to the right. Exactly two other classes would mean they are in the 8, and 0 sections, for a total of. x 8 x That is a probability of /0 = /5. Alg. B: Biology would be +x students. Since x+ 0, x and so we have a max of x= in biology. B = +=0 C: /0 = /0 0 x D: See part B, to get x=. That gives +()= 8 Chem. and the number of people outside of the other classes is 0-8 =. Final Answer is A B D C + + = /5 0 0 + + = + = + =. /0 5 0. A: =. sinq= 5. = 0.88... which is 0.9 to the tenth place. 0.5 sin Q ht B: sin = so the height from point P is approximated by 0.5(0)=5.. 0 Biol. Final Answer is 0.9 + 5. =. v 5. A: p= ij has length 5, so to get length 0, we multiply by : A = 8i j. v v B: p= ij and q = i+ 8j have dot product + - = -8. C: Using the dot product from part B, and the product of the magnitudes of each gives 8 cosθ = =. 50 5 Final Answer is 8 (8i j) = 5 9 9 i 5 5 j