Mu Alpha Theta National Convention 2004 Theta Individual Test For all questions, answer E, “NOTA” means none of the above answers is correct.
m n m3 n3 1. Simplify: 1 . n m m2n mn2
m n m n n m a) -1 b) c) d) e) NOTA m n m n n m
2. On a special flight, there are 9 boys, 5 American children, 9 men, 14 Americans, 6 American males, and 7 foreign females. Find the total number of people on the plane. (For this problem, boys are under 18 and men are over 18).
a) 23 b) 27 c) 33 d) 36 e) NOTA
1 2 x 1 1 y 3. Solve for x: 5 2 x 3y y 2x 11 1 6 a) x b) x c) x d) x .324 e) NOTA 34 3 17
4. If a, b, c are 3 consecutive odd integers such that a < b < c, then find the value of a 2 2 b 2 c 2 .
a) 4 b) 6 c) 8 d) 10 e) NOTA
5. If f is a function of x only and g is a function of y only, determine f and g such that log f logg log(1 z), where z x xy y.
a) f x, g y b) f 1 x, g 1 y c) f 1 x, g 1 y d) f x 1, g y 1 e) NOTA
6. The U. S. Mint began producing quarters for the 50-state quarter program at the beginning of 1999. After making the quarters for a given state for 10 weeks, the Mint begins making quarters for another state. In what year will the U. S. Mint finish making the quarters honoring the 50th state?
a) 2007 b) 2008 c) 2009 d) 2010 e) NOTA
7. The mean of 20 positive integers, 18 of which are x and 2 of which are y, is 8, and x y . What is the minimum possible value of x – y?
a) –70 b) –80 c) –90 d) –100 e) NOTA
page 1 8. When Mrs. Hiller takes pictures, they turn out with probability 1/5. She wants to take enough pictures so that the probability of at least one turning out is at least 3/4. How few pictures can she take to accomplish this?
a) 4 b) 5 c) 6 d) 7 e) NOTA
9. The owner of an office building needs to purchase enough digits to label all the office doors from 100 thru 125 on the 1st floor and 200 thru 225 on the 2nd floor. The digits can only be purchased in a package that contains one of each digit 0 thru 9. How many packages must the owner buy?
a) 30 b) 40 c) 50 d) 60 e) NOTA
10. The binary number 10101001110 2 is equal to what number in base 8?
a) 2 4 4 2 8 b) 2 5 1 6 8 c) 2 7 1 1 8 d) 2 7 6 5 8 e) NOTA
7 11. Simplify as far as possible: 1 3 2 3 2
a) 2 b) 2 c) 4 d) 2 2 e) NOTA
2 x y , 12. If ( l o g 3 x ) ( l o g x 2 x ) ( l o g 2 x y ) l o g x x , and find y.
a) 3 b) log3 c) 9 d) log9 e) NOTA
5 5 13. If ( 2 3 x k ) P k and equals 45 when x = 1, what is P when x = 2? k 0 k 0
a) 5 b) 8 c) 40 d) 42 e) NOTA
14. The contrapositive of the statement: “If it is summer, then it is hot,” is: a) “If it is hot, then it is summer.” b) “If it is not summer, then it is not hot.” c) “If it is summer, then it is not hot.” d) “If it is not hot, then it is not summer.” e) NOTA
15. If the line x = k is tangent to x 2 2 x y 2 4 y 4 , then k =?
page 2 a) 1 or –2 b) 1 or 3 c) –1 or –3 d) 0 or 2 e) NOTA
16. Using the table shown, what are the truth values for r, s, t, and u respectively?
p q p q q p ( p q ) ( q p ) T T T (t) T T (r ) F T (u) F T (s) F F F F T T T
a) F,T,T,F b) F,F,T,F c) T,T,T,F d) T,T,F,F e) NOTA
( n 2 ) ! ( n 1 ) ( n 2 ) ! 17. n ! n ( n 1 ) ! ( n 1 )
n 2 n a) 1 b) 0 c) d) e) NOTA n n 1
18. If a b means a b , and if a b means b a , what is the value of [ ( 2 6 ) 3 ] 2 ?
a) 4 b) 8 c) 12 d) 16 e) NOTA
19. Into a box are put 10 smaller boxes. Each of these smaller boxes is either left empty or is filled with 10 still smaller boxes. Of all the boxes, exactly 6 have other boxes inserted into them. Of all the boxes, how many remain empty?
a) 50 b) 55 c) 60 d) 65 e) NOTA
20. A certain parabola passes thru the points (5, 1) and (13, -7) and has the y-axis as its directrix. What are the coordinates of all points at which the vertex of this parabola could be located?
1 1 a) ( , -2) & (1, 2) b) (4, 5) & (0, -2) c) ( , -2) & (0, -2) 2 2 1 d) (4, 5) & ( , -2) e) NOTA 2
21. After traveling 1.5 hours at a constant rate, a train suffers a breakdown. The repair takes .5 hour and then the train continues at 4/5 of its former rate. It arrives at its destination 3 hours late. Find the ratio of the length of the entire trip, in miles, to the starting rate, in mph.
page 3 1 1 2 2 2 3 2 3 a) b) c) d) e) NOTA 1 3 2 3
3 9 4 1 2 k 22. Solve for all real value(s) of x: 2 2 l o g 1 0 x k 0 a) 2 b) 10 c) 100 d) 200 e) NOTA
23. If i 1 , find the ordered pair of real numbers (a, b) for which a + 4i and b + 5i are the roots of x 2 ( 1 0 9 i ) x ( 4 4 6 i ) 0 .
a) (2, 3) b) (3, 2) c) (4, 6) d) (6, 4) e) NOTA
24. The sum of the ages of all the members of a family is now 79 years. The members of this family are a father, a mother, a daughter, and a son. The father is 4 years older than his wife, and the daughter is 3 years older than the son. Six years ago, the sum of the ages of the members of the family was 56 years. Find the present age, in years, of the father.
a) 31 b) 35 c) 36 d) 39 e)NOTA
p 25. Find the value of with integers p & q, both relatively prime, and q is positive, q p such that l o g 1 2 ( l o g 7 ) ( l o g 5 ) ( l o g 4 ) 3 3 7 5 q 1 3 a) 1 b) c) 2 d) e) NOTA 2 2
x 1 0 26. Using the graph of the function f ( x ) . Find the sum of the horizontal x 2 and vertical asymptotes.
a) 0 b) 1 c) 2 d) 4 e) NOTA
27. A rectangular page is designed to contain 48 sq. in. of print. The margins on each side of the page are 1.5 inches and the margins at the top and bottom are each 1-inch. What should the dimensions of the page be so that the minimum amount of paper is used (to the nearest tenth)?
a) 8.5 by 5.6 b) 11.5 by 7.6 c) 12.6 by 8.4 d) 13.5 by 9.6 e) NOTA
page 4 28. I) Cramer’s Rule cannot be used to solve a system of linear equations if the determinant of the coefficient matrix is zero. II) In a system of linear equations, if the determinant of the coefficient matrix is zero, then the system has no solution.
If statement I is true, write the digit 1, if it is false, write the digit 2. If statement II is true, write the digit 3, if it is false, write the digit 4. Write your answer as a 2-digit number, with the 1st digit you get from statement I and the 2nd digit you get from statement II.
a) 13 b) 24 c) 23 d) 14 e) NOTA
29. Using the following 5 statements, let a true statement = 1 and a false statement = 0, find the sum of the trues and falses of these statements. e x 1. l o g b 2 x 2 x 2. e x 1 3. l n ( x y ) l n x l n y b e 1 0 4. l n ( x y ) l n ( x y ) 5. l o g 1 0 1 l o g 1 0 x x
a) 1 b) 2 c) 3 d) 4 e) NOTA
30. Solve for all real x: e 2 x 6 4 e x 6 5 e 6 0 .
a) ln 5 b) ln 1 c) ln 1 & ln 5 d) null set e) NOTA
3 c3 3c2d 3cd2 d3 Tiebreaker 1 Simplify 2 2 2 2 c d c d
x3 Tiebreaker 2 Solve for x: xlog x 100
page 5 Mu Alpha Theta National Convention 2004 Theta Individual Answers
# Answer # Answer 1 C 18 D 2 C 19 B 3 A 20 D 4 C 21 C 5 B 22 C 6 B 23 D 7 A 24 B 8 D 25 A 9 E 26 A 10 B 27 B 11 C 28 D 12 C 29 C 13 E 30 A 14 D TB1 1 c d 15 D TB2 x=10, x=100 16 A 17 B
page 6 m n m3 n3 m2 mn n2 mn(m n) m n 1. C - 1 g n m m2n mn2 mn (m n)(m2 mn n2 ) m n
2. C—Using the given info, we can determine that there are 2 American boys and 7 foreign boys; 3 American girls and x foreign girls; 4 American men and 5 foreign men; and 5 American women and (7-x) foreign women. That gives a total of 14 Americans and 19 foreigners, which = 33. 1 2 1-y=2x+2 -2x-y=1 x 1 1 y 11 3 A Solving the system, x 5 2 34 2x-6y=5y-10x 12x-11y=0 x 3y y 2x 4. C—Using c a 4 a n d c a 2 b and squaring each gives: c 2 2 a c a 2 1 6 a n d c 2 2 a c a 2 4 b 2 . By addition, we get 2 ( a 2 c 2 ) 4 b 2 1 6 a 2 c 2 2 b 2 8 , a n d a 2 2 b 2 c 2 8 . 5. B-- l o g ( 1 z ) l o g ( 1 x x y y ) l o g ( ( 1 x ) ( 1 y ) ) l o g f l o g g l o g ( f g ) . So, f 1 x & g 1 y . 6. B—If the mint spends 10 weeks on a quarter for each state, it will take 10 x 50 = 500 weeks to finish all of them. Dividing by 52, it will be about 9.6 years, which still leaves us in 2008. 7. A—Since the mean is 8 and there are 20 values, 18x + 2y = 20(8) = 160. Since x and y are both +, the max for x is 8, but then y also = 8, but that is not allowed. So the max possible for x is now 7. In this case, y = 17 and x – y = -10. Using the min for x, which is 1, we get y = 71 and x – y = -70. 8. D—The only way Mrs. H won’t get 1 picture to turn out is if NO pictures turn out. The prob. of a x 4 1 pic. not turning out is 4/5, so we need to solve . Using logs or trial and error, we get x 6 . 2 , 5 4 so x = 7. 9. E—(52) The owner needs more 1’s and 2’s than any other digit. He will need 26 ones for the hundreds place of the numbers in the hundreds. He will need (10)(2) = 20 more 1’s for the tens place of the rooms from 10 to 19 on both floors. He will need (3)(2) = 6 more 1’s for the ones place on both floors. Together, that’s 52 ones. He will not need as many 2’s (numbers stop in the mid-20’s)….so he needs 52 pkgs.
10. B—The binary number is equivalent to 1358 in base 10. Converting that to base 8 gives 2516 8 . 7 7 ( 3 2 ) 11. C-- 1 3 2 1 3 2 1 3 2 3 2 4 3 2 9 2 l o g x l o g 2 x l o g y l o g x 2 l o g y 12. C—Using base change formula: 2 . So, l o g 3 l o g x l o g 2 x l o g x l o g 3 l o g y l o g 9 a n d y 9 . 5 5 5 5 5 13. E—(48) For x = 2, the eqn. becomes ( 8 k ) P k . This factors to 8 k P k . k 0 k 0 k 0 k 0 k 0 5 Simplifying, we get 8 P ( 6 ) ( 8 ) 4 8 k 0 14. D—If an expression can be represented by A B , then the contrapositive is B ' A ' . So, A = It is summer and B = It is hot, and the contrapositive is “if it is not hot, then it is not summer.” 15. D—This is a circle with center (1, -2) and r = 1. Since x = k is to the x-axis and tangent to the circle, the diff btwn the x-coord. of the center and k must be 1 (radius). So, x k 1 1 k 1 o r k 0 , 2 . 16. A—(r): Since a conditional is false only if a true p leads to a false q, then (r ) must be false. (s): If p is false, there is no way to say the conditional is false, so (s) is true. (t): A conjunction is true only if both statements are true. In this case q p has to be true. page 7 (u): Since p q is false and both p q and q p need to be true for the conjunction to be true, the conjunction is false. ( n 2 ) ! ( n 1 ) ( n 2 ) ! ( n 2 ) ! ( n 1 ) ! 17. B-- 0 . n ! n ( n 1 ) ! ( n 1 ) n ( n 1 ) ( n 2 ) ! n ( n 1 ) ( n 1 ) ! 18. D—Evaluate from the inside out, getting 2 6 6 4 a n d 6 4 3 4 , a n d 4 2 1 6 . 19. B—The largest box is not empty (it has 10 smaller boxes). Since exactly 6 boxes are not empty, 5 of the ten smaller boxes are empty. The other 5 smaller boxes each contain 10 empty boxes or 5 + (5)(10) = 55. 20. D—This parabola has the form ( y k ) 2 4 p ( x h ) . Since p = h, substituting in the given coords., 1 we get ( 1 k ) 2 4 h ( 5 h ) a n d ( 7 k ) 2 4 h ( 1 3 h ) . Solving these, we get ( h , k ) ( 4 , 5 ) , , 2 . 2 21. C—Let d be the distance traveled after the breakdown, and 5r = the original rate. Then, d d 1 d 1 . 5 ( 5 r ) 5 0 r 7 . 5 r 2 3 3 a n d d 5 0 r . The requested ratio is then . 5 r 4 r 2 5 r 5 r 2 22. C-- 2 4 1 2 2 ( 2 4 0 1 ) l o g x ( 1 2 2 2 2 3 . . . 2 3 9 ) l o g x ( 2 4 0 1 ) 2 l o g x a n d x 1 0 0 . 23. D—The sum of the roots (a + 4i) + (b + 5i) = 10 + 9i, so a + b = 10. The product of the roots (a + 4i)(b + 5i) = 4 + 46i, so ab – 20 = 4 and 5a + 4b = 46. Solving, we get (a, b) = (6, 4). 24. B—If all members of the family were alive 6 years ago, their age sum would have been 79 –24 = 55. This is NOT the case. The only possible way is that the son’s present age is 5. Thus, the daughter is 8, the mother 31 and the father is 35. l o g 7 l o g 5 l o g 4 p l o g 4 p p 25. A—Using the base change formula, l o g 1 2 l o g 4 . Also, 3 l o g 3 l o g 7 l o g 5 q l o g 3 q 3 q p p l o g 4 l o g 3 l o g 4 . Therefore, 1 and p = q = 1, since both are relatively prime. 3 3 3 q q 26. A—There are no vertical asymptotes (denom. 0) and the horiz. asymptotes are 1 , so the sum is 0. 4 8 27. B—Let A = ( x 3 ) ( y 2 ) ( x 3 ) ( 2 ) . The minimum y occurs when x 8.5. So, 8.5 + 3 x =11.5 and the corresponding value of y is 7.6. 28. D—Statement I is true, so the 1st digit should be 1 and statement II is false so the 2nd digit should be 4…giving us 14. 29. C—Statements 1, 2, and 5 are true and since each true statement = 1, then sum is 3. 30. A—Dividing out the e 6 gives us e 2 x 4 e x 5 0 . This is a simple quadratic equation, which factors and yields e x 5 a n d e x 1 . Throwing out the 2nd case ( e x can’t be negative), and solving the first case, gives us x = ln 5.
3 (c d)3 c d 1 TB 1 2 2 2 c2 d2 c d (c d )
TB 2 x3 x3 xlog x logxlog x log (logx)2 3log x log100 (logx)2 3log x 2 0 100 100 (logx-1)(logx-2)=0 x = 10, x = 100
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