FEBRUARY 7, 08 Invitational Sickles The abbreviation NOTA means None of These Answers and should be chosen if choices A, B, C and D are not correct Solve for over the Real Numbers: ln( ) ln() The trigonometric Arcfunctions, such as, have the traditional restricted ranges Arc tan( ) Diagrams may not be drawn to scale e e e e Arc cos 0 then which must be true of? 0 0 In, RS=0, sin R, and are both acute, and the area of is 0, then find the square of the length of side ST 7 7 T RST RST 80 0 R A triangle with side lengths, and 7 has what area? 9 9 A vector 9, has magnitude m and direction for 0 Use 0 as an approimation for Arc cos to give the approimated value of m( ) 9 0 A parabola has directri equation y and focus coordinates (, 0) Which is an equation for this parabola? y 0 y 8 8 0 y 8 80 0 8 y 80 0 7 A polar equation r 0 sin is graphed on the y -coordinate plane The maimum y -intercept of the graph is M and the minimum y intercept of the graph is m Give the value of M m 8
FEBRUARY 7, 08 Invitational Sickles 8 A pile of 9 potatoes is on a table Murdock starts peeling at potatoes per minute, and keeps up this rate for the entire peeling event After minutes Nathan starts helping Murdock at the rate of potatoes per minute After all the potatoes were peeled, how many did Murdock peel? 90 7 9 o 9 In, mt 0, TU=00, and UG=0 Use 09 as an approimation for, and/or 0 as an approimation for to determine the number of possible distinct triangles which eist with these measurements none infinite TUG cos 0 o sin 0 o The major ais of an elliptical arch is on the ground level (Note that the diagram is not drawn to scale) Two lights ( and in the diagram), each meters from the vertical ais, are placed so that the sum of the distances from the lights to any point (P) on the arch is always 0 m Tell the height of the arch m from the center of the arch (All measures are planar) 9 9 L L y NOT drawn to scale!!! P L L 0 When solved over the Real Numbers, the equation log ( ) log ( ) 0 has solution set S Find the sum of the elements of set S For i, i =? ( i ) ( i ) ( i ) ( i ) In circle R with radius, the radian measures Q R R of arcs QS and PQ S are 9 and, respectively 0 0 Find the area of the sector bounded by arc PS and radii RP and RS 0 9 P
FEBRUARY 7, 08 Invitational Sickles The function 7 shares eactly one zero with the function k 8 8 All of P( ) P( ) the zeros of both functions are rational If the coefficient k is an integer, then give the value of k a b f( ) has a graph with 9 eactly three asymptotes with equations p, p, and y p The -intercepts of f are, and p 0 Find the value of pa b 0 For 0, value of cos 7 9 9 cos Give the 9 7 9 7 A circle is tangent to both the - and the y-aes Its center is in Quadrant IV The circle passes through the point, with a radius greater than Give the radius of the circle (, ) 8 Which of the following is an equation of an asymptote of the hyperbola with equation 9y 8y 9 0? y y y y 0 9 The graph of y Psin( ) Q has a range of [ 0,00] If P and Q are both integers, give the greatest possible value of Q P 0 0 A parabola with equation f ( ) a b c has eactly one intercept and eactly one y intercept, both of which are shared by the function g ( ) That is, the graph of g has the same and y intercepts Give the value of ab c 0
FEBRUARY 7, 08 Invitational Sickles If n e epression for n ln nln then which is an in terms of n? n ln 9 n ln Which is equivalent to the epression 7 8? For 0, sin( ) sin ( ) Give the sum of the possible values of In RST, side length r is opposite R, side length t is opposite T and side length s is opposite S If r, s and t, then give tan( T ) Simplify, where defined: csc( ) cos sin( ) tan( ) sin( ) sec( ) csc( ) cos( ) If sin( ) cos( ) 0 then which of the following must be true? I tan is always positive II sin( ) is always negative III cos( ) is always negative III only II only I, II only I, III only 7 What is the product of the solutions for the equation log log when solved over the positive Reals?
8 What is the average (arithmetic mean) of the solutions of the equation sin( ) 0,? for FEBRUARY 7, 08 Invitational Sickles 9 sin cos tan = 0 Which epression is NOT equivalent to cos for all real values of for which sin( )cos( ) 0? sin sec cot( )sin( ) sin
FEBRUARY 7, 08 Invitational Sickles E 7 D A 9 C C D 8 B C 0 A D D 9 C D B 7 B A 0 A D D 8 A C A 7 A B 9 E B C 8 A A 0 A CE This is just asking the restricted range of the Arccosine function, and if is negative, then the Arccosine value will be in Quadrant II, Using Heron s Formula, S=(++7)/ Area = 9(9 )(9 )(9 7) = 9()()() = Magnitude = 9 = m Since the vector is toward QIV, with a reference of Arc cos, the direction 7 angle is = m( ) = 0 0 7 0 = ln( ) ln() ln e ln( ) ln( e) -=e =e+ If area is 0 then the X S height to RS is h 0 and 0 (0 h) R T gives the height to RS is Since sin(r) is ¼, then consider the right triangle RXT and RT= Now use the law of Cosines 00 (0)() cos R ( ST) 0 = 80 Halfway between directri y and focus (, 0) is verte (, ), with the parabola opening downward Equation then is y ( ), with the coefficient / coming from the distance from V to F as /(p) y ( ) y 8 y 8 8 0 7 The y-intercepts occur at at Mm and These radii are 8 and, but, the y-value is (0, -) So the ma is M=8 and the min is m= - = +=8 8 Murdock will peel potatoes in the first minutes So we will have 9 - = remaining Then together they will peel 8 per minute /8= min of peeling Murdock will have peeled +() potatoes = +9 = 7 potatoes 9 The height from U is 00sin0 which is approimately Since the 0 is less than 00 but greater than, there are two possible triangles 0 For alog ( ), a a 0 (a)( a) 0 a or a log ( ) gives log ( ) gives = The sum is i = cis cis using DeMoivre s Th = i The foci are at (,0 ), so c= Sum of distances from the foci T 00 U G
FEBRUARY 7, 08 Invitational Sickles to the curve = a = 0 So a=0 y and 00 b 00 b y So b=8 Now let = 00 9 y 00 9 y 8 = 0 Arc PS has measure 9 0 0 0 7 Q R 0 0 9 9 /0 which is S / of the circle Area is = The first polynomial 7 factors to ( )( )( ) Zeros are -/, and The second polynomial k 8 8 cannot have -/ as a factor, with leading coefficient We only consider and as the possible zero they have in common Using, k 8 8 0 gives k= - 8 8 Divide by (-) and get ( ) which has irrational zeros So k is not - Net use zero 8 k 7 8 0 gives k=8 k= This is the answer, but to verify, the polynomial 8 8 divided by (-) does product zeros,, which does have only one common zero with the first polynomial So k= Asymptotes of f are =, = - and y=a So a p gives p= and a=/ If the roots of f are and -, then P a b f( ) a b b 0 at = 9 and = - That gives b = -/ The sum is 0 and times p we still get 0 cos = 7 9 cos 7 If the circle is in QIV then it has center (a, -a) to be tangent to both aes The radius would be a ( a) ( y a) a Use (, ) to get ( ) ( ) a a a a a a a a a a 0 ( a )( a) 0 We are told the radius is greater than, so r= 8 9y 8y 9 0 ( 8 ) 9( y y) 9 ( 8 ) 9( y y ) 9 9 ( ) 9( y ) ( ) ( y) Asymptotes are 9 y ( ) Simplify to y ( ) y and y 9 If P is positive then the ma occurs at PQ 00 and the min at P Q 0 Add to get Q=90, Q= P= If P is negative then ma is at -P+Q=00 and P+Q= -0 Q=90 Q= and P= - Q P is -0 = - In second case +0= 0 A function parabola with eactly one -intercept must have verte on the -ais If the intercepts are the same as g ( ) then the verte is (, 0) and y k( ) The y-intercept is -/ and so k is negative k(0 )
FEBRUARY 7, 08 Invitational Sickles k= -/ ( ) y y ab c= n ln e ln n (ln ln ) n n n ln = ln ln 9 7 8 = = If sin 0 then sin( ) sin ( ) (sin )(sin ) 0 Since sin 0 we discount the second factor, and so sin gives and For sin 0 sin( ) sin ( ) (sin )(sin ) 0 Since sin 0 we discount the second factor and sin, so 7 and Sum is 9 () cost 0cos T cost R sint = csc( ) cos sin( ) tan( ) cos( )sin sin ( )cos( ) sin( ) = T so tant = S = csc sin=cos at check intervals The interval, makes so we sin( ) cos( ) >0 true In this interval, tan is always positive, so I is true For II,, and in this interval sin() is not always negative II is false However, in this interval, cos() is always negative III is true I, III true 7 log log log log log log (log ) (log ) log log Let a log a alog (log ) 0 ( a log )( a log ) 0 log log or log log gives = or The product is 7 8 sin( ),,, 7 and,,, The sum is, and divided by gives an average of 9 E sin cos tan = = 0 Choice A is always positive, so it is only equal to cos() when cosine is positive So sin = cos NO YES NO cos