WYSE Academic Challenge 2004 Sectional Mathematics Solution Set

Transcription

1 WYSE Academic Callenge 00 Sectional Matematics Solution Set. Answer: B. Since te equation can be written in te form x + y, we ave a major 5 semi-axis of lengt 5 and minor semi-axis of lengt. Tis means we ave a focal lengt of. Te eccentricity is terefore /5, or Answer: C. Te two data points are (0, 5750) and (0, 50). We get a slope of -,50 giving a line equation of y -50(x) , so y -50 (5) Answer: D. If vectors A and B bot point in te same direction and bot ave te same lengt, ten we can describe bot of tem as ai + bj + ck. Te lengt of te vectors could also be written as a + b + c, wic is equal to x. Te dot product of te two vectors would be a a + b b + c c, wic equals te lengt squared, or x.. Answer: D. Te sum of te numbers from to 6, so te probability of a or a is (+)// 5. Answer: A. If te equation could be written as Ax + Bxy + Cy + Dx + Ey + F 0, ten it is a conic section, and te discriminant B AC tells us wic conic section we ave. Te equation can be written as x 6xy + y + 0x + 0y 0, so it is a conic section. Te discriminant value is ( 6) 0. Since tis value equals zero, we ave a parabola. 6. Answer: C. (0.5) / x 0.5 ln(0.5) ln(0.5) x x ln(0.5) ln(0.5) x.788 ln(0.5) ln(0.5) 7. Answer: C. Te most direct way to find out if a matrix as an inverse is to determine weter or not its determinant is equal to zero. Te matrices ave te following determinants: i. x + y + z xyz, ii. 0, iii. 0, iv. xyz. Since only two of tem ave nonzero determinants, only two of te matrices ave inverses. [( x + ) 5( x + ) + ] ( x 5x + ) 8. Answer: D. x + x + 5x 5 + x + 5x x + 5 x Sectional Matematics Solution Set

2 9. Answer: D. Because we must consider weter or not we ave one or two o s in our arrangement wen we pick,, or letters, we end up wit te following total: C( 5,) P(,) + P(,) + C(,) P(,) + P(,) + C(,) P(,) + P(,) (te underlining sows different numbers of letters cosen). 0. Answer: D. A equation, unknown problem wit te two equations x + y 90 and.0(x) +.5(y).60 and x 66 and y. Answer: C. Te easiest way to solve tis problem is to divide te triangle up as sown in te drawing. Eac of te tree lines bisects bot of te angle and edge tat it touces. (Note tat only part of te lines ave been labeled, but all of te oters would be lengt a, b, or c) Since te area of a triangle is equal to one alf base times eigt, ten te a area is equal to c ( a + b). Because te triangle is equilateral, eac of te triangle s angles are 60 degrees, and because te extra lines bisect te angles, we ave six rigt triangles wit angles of 0 b and 60 degrees. c c Tis gives us te furter information tat b a and c b a. We can terefore rewrite te equation for te area as a a ( a + a) 5. Since te area of te circle is π a, knowing tat 5 a. 569 gives us an area of 0.59 for te circle.. Answer: A. A equation, unknown problem tat yields 6 units for # one, units for # two and units for # tree.. Answer: A. We would need to find te matrix tat matces up to A AB + BC. Te matrix tat matces tis description for answer A. Answer: B. sin(5/60) x / 7 miles and 7 * 580 * sin(5/60) Answer: D. Te cross product is equal to te determinant of te matrix i 0 j k. 6. Answer: B. Simply evaluate p y x at te points of intersection determined by te constraints, namely te points (, 0), (, 0), (/, /) and (, /) 00 Sectional Matematics Solution Set

3 7. Answer: A. 5x + A(x - ) + B(x + ) so A + B 5 and B A and A 8/, B 7/ 8. Answer: C. For most students, te easiest metod will probably involve first converting te points into Cartesian coordinates. We ten end up wit te points (, ) and (0, -). We can ten use te distance formula to find te overall distance of ( 0) + ( ( ) ). 65. Te oter option is to realize tat law of cosines can be π used were we ave a triangle wit lengts and wit an angle of radians (5 degrees). Our lengt is ten + cos Answer: D. To find distance between te two points, we simply use te distance formula, wic is (( t + ) t) + (5t (t )), wic simplifies down to 0 t + t + 5. Since we want tis value to be greater tan or equal to 7, we are basically solving for te inequality 0t + t Because te value inside te square root function will always be positive, we can safely square bot sides witout creating bad answers. After we do tis and move everyting to te left, we end up wit 0t + t 0. Tis factors into (5t + )( t ) 0. We will ten test around te roots. and to find were te function is positive, and end up wit answer D. 0. Answer: B. (66*67)/ (60*6)/ 90,06,880 77,56. Answer: E. Te easiest way to find te roots of tis equation would be using te quadratic ( 8) ± ( 8) 9 equation. Since we end up wit ±, we get te complex roots of + i and i.. Answer: D. x x x x 0 ( x )( x + ) 0 and x or x P( calculus and pysics).. Answer: B. P ( pysics given calculus) P( calculus).7. Answer: B. tan(6) /x and x /(tan(6)) 5. Answer: D. Te ard part of tis problem is to remember sum /(-C.R.) 6. Answer: B. 6 ( + x) + (8 +x) and x 7 7. Answer: E. An expected value problem. (.65)*(.0)+(.5)*(.06)+(.0)*(.5) Answer: B. 5*5 + 75*x 650, 75x 55 and x 00 Sectional Matematics Solution Set

4 9. Answer: D. First we need to get slopes of te lines, so we rewrite te two equations as y x + and m y x +. We ten use te expression m tanθ, + mm were m / and m /. Tis expression simplifies to ( / ) (/ ) 7 / 6 tan θ 7 / 6, so ϑ tan (/ ) * ( / ) 6 0. Answer: B. 000(.0) + 500(.0) + 00(.0) + x(.9) 9.50,.9x.50 and x 5. Answer: C. First we will find tat te area of te triangle is ½**8, or 8 square meters. We ten use Pytagorean teorem to find tat te tird side is equal to + 8, or. meters. We ten use te area formula again to set up te equation 8 BX.. Tis gives us te lengt of BX as Answer: E. Tk(# speakers)/(# workers), k(/), so k /. T (/)(/6).56. Answer: A. From te information given, we can tell tat te interior angles must be 60 and 0 degrees. If we call te lengt of te sorter sides x, ten te larger sides (wic we will treat as te bases) ave lengt of x, and te eigt of te parallelogram must be x. Since te area of a parallelogram is equal to base times te eigt, we can set up te equation x x 5, and tus x.95. Since te perimeter would be equal to 6x, we end up wit a perimeter of Answer: A. Can be tricky x x x x * x x x x x x x 5. Answer: B. Te distance between a point and a line is found using te expression Ax + By + C d were x, y 7, A /, B and C 6, (note tat we first we A + B ad to put te line into te form x y ). Tis distance expression ten simplifies ( / )() + ( )(7) / 9 down to d, / 9 / 9 ( / ) + ( ) + wic reduces down to Answer: D. Simply put, Force 78,000*sin(9), Answer: E. Since te external angle of a regular dodecaedron is simply 60/, or 0, we end up wit an internal angle measure of 80-0, or 50 degrees. 00 Sectional Matematics Solution Set

5 8. Answer: B. log x / x x Answer: B. Te first ting to note is tat any side of te square will be 8 units long, and te diagonal for te square will be 8 units long. We will call te radii of any of te circles x. We notice tat eiter diagonal passes troug tree circle centers. Because of tis, if we go along te diagonal of te square, we can create te equation 8 x + x + x + x + x + x 8. If we solve for x, we end up wit x. + Tis value simplifies to If we use tis to find te area of te circle, we end up wit 8.6 square units. / + / Answer: A.. 5 /.5 00 Sectional Matematics Solution Set