( ) ( ) Pre-Calculus Team Florida Regional Competition March Pre-Calculus Team Florida Regional Competition March α = for 0 < α <, and

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1 Flrida Reginal Cmpetitin March 08 Given: sin ( ) sin π α = fr 0 < α <, and π β = fr < β < π ( ) A = sin( α ) C = cs( β ) D = sec( β ) cs( ) π B = cs α α Find the prduct ABCD, in the frm q p fr a psitive prime p Flrida Reginal Cmpetitin March 08 Given: sin ( ) sin π α = fr 0 < α <, and π β = fr < β < π ( ) A = sin( α ) C = cs( β ) D = sec( β ) cs( ) π B = cs α α Find the prduct ABCD, in the frm q p fr a psitive prime p

2 Flrida Reginal Cmpetitin March 08 Let C dente the cnic determined by + 5 = 0, and y 6 6( y ) = 6 C dente the cnic determined by ( ) A = the distance between fci f the graph f C B = the distance between the centers f the graphs f C and C C = the prduct f the slpes f the asympttes f the graph f C D = the length f the transverse ais f the graph f C Find the prduct ABCD Flrida Reginal Cmpetitin March 08 Let C dente the cnic determined by y + 5 = 0, and C dente the cnic determined by ( ) y 6 6( ) = 6 A = the distance between fci f the graph f C B = the distance between the centers f the graphs f C and C C = the prduct f the slpes f the asympttes f the graph f C D = the length f the transverse ais f the graph f C Find the prduct ABCD

3 Flrida Reginal Cmpetitin March 08 ( ) 8 = and g( ) = lg ( ) f ( ) A = the slutin,, f f ( ) = when slved ver the Real Numbers B = the slutin,, f g( ) =, when slved ver the Real Numbers ( ) f g C = ( ) D = the value f fr which f ( ) = Find the value f the prduct (0ABCD) Flrida Reginal Cmpetitin March 08 f ( ) ( ) = 8 and g( ) = lg ( ) A = the slutin,, f f ( ) = when slved ver the Real Numbers B = the slutin,, f g( ) =, when slved ver the Real Numbers ( ) f g C = ( ) D = the value f fr which f ( ) = Find the value f the prduct (0ABCD)

4 Flrida Reginal Cmpetitin March 08 Cnsider PQR with QR=0, and FGH with GH=6 and FH=0 5 A = the length f PR when sin( P ) = B = the value f sin( m P + m H ) when PQ=0 A+B = a + b c fr prime c Give the value f ac b P Nt drawn t scale F Q 0 R 0 G 6 H Flrida Reginal Cmpetitin March 08 Cnsider PQR with QR=0, and FGH with GH=6 and FH=0 5 A = the length f PR when sin( P ) = B = the value f sin( m P + m H ) when PQ=0 A+B = a + b c fr prime c Give the value f ac b P Nt drawn t scale F Q 0 R 0 G 6 H

5 Flrida Reginal Cmpetitin March 08 5 f ( ) = + + with rts f f are,, and, fr g( ) = sin ( ) sin( ) A = π π B = the sum f the values f ver, where g( ) = 0 π π C = the value f ver, where g( ) = Find the value f AB C in fractin frm Flrida Reginal Cmpetitin March 08 5 f ( ) = + + with rts f f are,, and, fr g( ) = sin ( ) sin( ) A = π π B = the sum f the values f ver, where g( ) = 0 π π C = the value f ver, where g( ) = Find the value f AB C in fractin frm

6 Flrida Reginal Cmpetitin March 08 6 Cnsider PQR with PQ=0, QR=6 A = the length f PR if B = cs( Q) if PR= C = sin ( R) if cs ( P) m Q = 0 =, and C > 0 D square units = the area f PQR if m Q = 0 Find ABCD Flrida Reginal Cmpetitin March 08 6 Cnsider PQR with PQ=0, QR=6 A = the length f PR if B = cs( Q) if PR= C = sin ( R) if cs ( P) m Q = 0 =, and C > 0 D square units = the area f PQR if m Q = 0 Find ABCD

7 Flrida Reginal Cmpetitin March 08 7 Cnsider the parametric equatins: = t + and y = t, which has a graph with a minimum pint n the y-crdinate plane f P(, y ) Cnsider the parametric equatins: = t and y = t + t which has a graph n the y-crdinate plane with -intercept at pint Q The y-intercept f this graph is R t 0 A = the -crdinate f pint P B = the y-crdinate f pint P C = the -crdinate f pint Q D = the y-crdinate f pint R Find the value f A+B+C+D Flrida Reginal Cmpetitin March 08 7 Cnsider the parametric equatins: = t + and y = t, which has a graph with a minimum pint n the y-crdinate plane f P(, y ) Cnsider the parametric equatins: = t and y = t + t which has a graph n the y-crdinate plane with -intercept at pint Q The y-intercept f this graph is R t 0 A = the -crdinate f pint P B = the y-crdinate f pint P C = the -crdinate f pint Q D = the y-crdinate f pint R Find the value f A+B+C+D

8 Flrida Reginal Cmpetitin March 08 8 Cnsider the equatin fr relatin R : 8 = y y The ais f symmetry f the graph f R has equatin y = A The verte f the graph f R has crdinates ( B, C ) The fcus f the graph f R has crdinates ( D, C ) Find the value f ABCD Flrida Reginal Cmpetitin March 08 8 Cnsider the equatin fr relatin R : 8 = y y The ais f symmetry f the graph f R has equatin y = A The verte f the graph f R has crdinates ( B, C ) The fcus f the graph f R has crdinates ( D, C ) Find the value f ABCD

9 Flrida Reginal Cmpetitin March 08 π 9 The plar crdinates,, written in rectangular frm, is (A, B) The graph f the plar equatin r = cs( θ ) intersects the graph f r = C times The graph f the plar equatin distance f D between vertices r 6 = is a cnic sectin with a 9cs θ sin θ Find (A + B)(C + D) Flrida Reginal Cmpetitin March 08 π 9 The plar crdinates,, written in rectangular frm, is (A, B) The graph f the plar equatin r = cs( θ ) intersects the graph f r = C times The graph f the plar equatin distance f D between vertices r 6 = is a cnic sectin with a 9cs θ sin θ Find (A + B)(C + D)

10 Flrida Reginal Cmpetitin March 08 0 Cnsider f ( ) = 5 5 and g( ) = 8 A: the equatin f the vertical asymptte f the graph f f is = A B: the equatin f the hrizntal asymptte f the graph f h( ) = i g( ) is y = B C: if the remvable discntinuity f the graph f f is at = k then C = lim f ( ) k D = g( f ( )) If g( f ( )) des nt eist, then let D = 00 Find the value f ABCD Flrida Reginal Cmpetitin March 08 0 Cnsider f ( ) = 5 5 and g( ) = 8 A: the equatin f the vertical asymptte f the graph f f is = A B: the equatin f the hrizntal asymptte f the graph f h( ) = i g( ) is y = B C: if the remvable discntinuity f the graph f f is at = k then C = lim f ( ) k D = g( f ( )) If g( f ( )) des nt eist, then let D = 00 Find the value f ABCD

11 Flrida Reginal Cmpetitin March 08 i = and cisθ = csθ + i sinθ A = ( i ) 6 B: the three cube rts f i are rcis ( θ ), r cis ( θ ) and r cis ( θ ) 0 < θ < θ < θ < π, and all values f r n psitive B = θ C: the set f furth rts f 8 fr all values f β n between 0 and, fr is { a cis ( β ), a cis ( β ), a cis ( β ) and ( ) a cis β }, 60, and all values f a n psitive C is the prbability that B n fr n =,,, is between 00 and 00 Find the value f (ABC) π Flrida Reginal Cmpetitin March 08 i = and cisθ = csθ + i sinθ A = ( i ) 6 B: the three cube rts f i 0 θ θ θ π are rcis ( θ ), r cis ( θ ) and r cis ( θ ) < < < <, and all values f n C: the set f furth rts f 8 fr all values f β n between 0 and r psitive B = θ, fr is { a cis ( β ), a cis ( β ), a cis ( β ) and ( ) a cis β }, 60, and all values f a n psitive C is the prbability that B n fr n =,,, is between 00 and 00 Find the value f (ABC) π

12 Flrida Reginal Cmpetitin March 08 A = the apprimatin f by using the sum f the first three terms f the B = epansin f n= 9 n + (Nte: The terms are in rder f decreasing pwers f ) C and D: The equatin f the line tangent t is y = C + D = at the pint (, f ( ) ) ( ) f Give the value f AB(C + D) Flrida Reginal Cmpetitin March 08 A = the apprimatin f by using the sum f the first three terms f the B = epansin f n= 9 n + (Nte: The terms are in rder f decreasing pwers f ) C and D: The equatin f the line tangent t is y = C + D = at the pint (, f ( ) ) ( ) f Give the value f AB(C + D)

13 Flrida Reginal Cmpetitin March 08 In a particular Literature class f 0 peple, 0 peple were taking Algebra and Bilgy peple were taking Algebra and Chemistry 8 peple were taking Bilgy and Chemistry peple were taking Algebra, Bilgy and Chemistry The sectins marked have an equal number f students = the number f students in that Literature class wh are taking eactly ne member f the set {Algebra, Bilgy, Chemistry} Alg Chem Bil A = the prbability that a persn frm that Literature class is taking Literature and eactly tw ther classes frm the set {Algebra, Bilgy, Chemistry} B = the maimum pssible number f peple in that Literature class wh are als taking Bilgy C = the prbability that a persn frm that Literature class is taking fur ttal classes D = the number f peple in that Literature class wh are nt taking any classes frm the set f classes {Algebra, Bilgy, Chemistry}, if is the maimum pssible value Give the value f A B D C + + Flrida Reginal Cmpetitin March 08 In a particular Literature class f 0 peple, 0 peple were taking Algebra and Bilgy peple were taking Algebra and Chemistry 8 peple were taking Bilgy and Chemistry peple were taking Algebra, Bilgy and Chemistry The sectins marked have an equal number f students Alg Bil = the number f students in that Literature class wh are taking eactly ne member f the set {Algebra, Bilgy, Chemistry} Chem A = the prbability that a persn frm that Literature class is taking Literature and eactly tw ther classes frm the set {Algebra, Bilgy, Chemistry} B = the maimum pssible number f peple in that Literature class wh are als taking Bilgy C = the prbability that a persn frm that Literature class is taking fur ttal classes D = the number f peple in that Literature class wh are nt taking any classes frm the set f classes {Algebra, Bilgy, Chemistry}, if is the maimum pssible value Give the value f A B D C + +

14 Flrida Reginal Cmpetitin March 08 P 0 6 R May nt be drawn t scale Q PQR has PQ=6, PR=0, and m R = A: Use sin 05 t get an apprimatin fr the value f sin Q A = that apprimatin t the tenth place B: Use sin 05 t give the value f the height frm pint P in the triangle PQR (t the line cntaining side QR ) B = the value f that height Give the value f A+B Flrida Reginal Cmpetitin March 08 P 0 6 R May nt be drawn t scale Q PQR has PQ=6, PR=0, and m R = A: Use sin 05 t get an apprimatin fr the value f sin Q A = that apprimatin t the tenth place B: Use sin 05 t give the value f the height frm pint P in the triangle PQR (t the line cntaining side QR ) B = the value f that height Give the value f A+B

15 Flrida Reginal Cmpetitin March 08 5 p = i j and q = 6i + 8 j dente tw vectrs in standard psitin A = the vectr, written as ai + bj, which has the same directin as p and has length 0 B = the dt prduct f vectrs p and q C = find the csine f the angle between vectrs p and q Find the prduct f the vectr determined by part A, and the scalar BC Give yur answer in fractin frm, with frmat ai + bj Flrida Reginal Cmpetitin March 08 5 p = i j and q = 6i + 8 j dente tw vectrs in standard psitin A = the vectr, written as ai + bj, which has the same directin as p and has length 0 B = the dt prduct f vectrs p and q C = find the csine f the angle between vectrs p and q Find the prduct f the vectr determined by part A, and the scalar BC Give yur answer in fractin frm, with frmat ai + bj