2002 Mu Alpha Theta National Tournament Mu Level Individual Test

Transcription

1 00 Mu Alpha Theta National Tournament Mu Level Individual Test ) How many six digit numbers (leading digit cannot be zero) are there such that any two adjacent digits have a difference of no more than one? Consider that 0 can be adjacent to B) 87 C) 0 D) 88 ) Evaluate ( x )( + x) dx. 0 B) C) 4 D) ) Find the derivative with respect to x of sin x tan x + cos x sec x. cos x B) tan x C) sec x D) sin x d 4) Find dx [ cos( x) ]. cos( x ) B) -sin( x) C) sin(x)cos(x) D) -4sin(x)cos(x) x 5) Evaluate x e + dx. x x+ e + C B) e x + ( x + ) + C C) e x x + C x x+ D) e e ( x ) + e + C 6) Compute the ratio of the area to the perimeter of a triangle with sides measuring 0,, and 4. 6 B) 4 C) D) 6 7) Two marbles are drawn from a vase which contains 4 white marbles, 6 blue marbles, and 8 green marbles. What is the probability that if I pick two marbles, they will be the same color? B) 5 C) 5 D)

2 8) Find the second derivative of e x sin x cos x at e B) e C) e D) e. 9) Evaluate sin(x) lim. x 0 x 0 B) C) D) 0) e i = B) - C) e D) -e ) Evaluate: t t + dt B) C) D) ) Find sin x cos x sin x cos xdx. sin x cos x sin x cos x + C B) + C D) sin 4 (cos( x)) +C C) xsin x + x cos x + C ) Find a complex number Z such that Z = arccos () ln( ) i B) ln( + ) i C) ln( + ) D) - ln( + ) i 4) Given that a certain cylinder has a volume V, what is the minimum surface area that the cylinder can have? V B) 4 V V C) r D) 5 4 5) Find the remainder when x + 4x x + is divided by x. x B) - C) x D) x

3 6) What is the minimum distance from (,) to a point on the line y = -x+8. B) C) D) 7) Find the middle entry of the 00 th row of Pascal s Triangle. 00! 5!49! 00! B) (50!) C) 0 D) 89 8) Find the sum of the values for the solutions for x, y, and z: x+y+6z=0 x+4y+z= x+y+z=6 B) 6 C) 8 D) -4 9) How many rectangles can be drawn on an 8x8 chess board which have odd area? 00 B) 00 C) 00 D) 400 k + 0) Evaluate. k! k = 0 6 ) Evaluate B) sin() C) 5 sin() D) 4 e 00 n. n + 99/00 B) ½ C) /40000 D) 4/50 ) Determine the convergence or divergence of n +. n Converges B) Diverges C) It cannot be determined whether it converges or diverges

4 ) To construct a Koch Snowflake, first you draw an Equilateral Triangle. Then you trisect each of the sides, and construct on the middle segment another equilateral triangle. If this is repeated infinitely, the resulting shape will be a Koch Snowflake (see picture). If a Koch Snowflake is created on an equilateral triangle with sides of length, what is it s area? 4 / B) 5 C) 6 5 D) 4) Find the volume of the solid obtained by revolving about the line y = - the region between the graph of the equation y = x+ and the line y = - on the interval [0,]. 0 9 ( ) B) C) D) 5) Evaluate B) 6 5 C) D) Evaluate: B) C) D) 7. A person rolls six 4-sided dice each of which has sides labeled A, B, C, and D. What is the probability that there is a tie for the letter which occurs the most? B) C) D) log4n 40 = logn45. Find n. 4 B) C) 5 D) 60 E) 75

5 9. Find cot Arcsin 5 - B) C) / D) -/ 0. The trigonometric form of i is : cos + isin B) ( cos + isin) C) sin + icos D) (cos + isin ). Which of the following cannot be expressed as the difference of the squares of two nonconsecutive integers? A. 4 B. 44 C. 45 D. 48 E. All of these. If p and q are distinct primes each of which is greater than, let d denote the number of positive integers n such that p^*q^5 divided by n is a positive integer. Which of the following is d? A. 84 B. 65 C. 48 D.0 E. NOTA. The sum of the solutions to 000x^-500x+777=0 is : A. / B. 0 C. -/ D. 000 E. NOTA 4. A debating team with 4 members is to be chosen from among 0 students. Find the number of distinct possible teams. ( Of course two teams are considered the same if they have the same members, even if the members were chosen in different orders). A. 680 B). 000 C. 000 D E What is the sum of all three solutions to x^-x^-4x-6=0. B) - C) D) - 6. What is the magnitude of the difference between 05 and B) 469 C) 06 D) If x and y satisfy the equation y=(00-x)(x+00) then the largest possible value of y is closest to 5000 B)0,000 C) 5,000 D) 50,000 E) 00,000

6 8. If the point (0,y) is on the line passing through (0,0) and (50,00) then y is: -50/ B) 0/ C) 650/ D) = 50,000 B) 00,000 C) 500,500 D),500,000 E) 5,000, Harry, Burly, and Joe are to receive a combined total of five pennies from Mr. Wilson s will. Assuming each person receives at least zero pennies and no more than five pennies, the number of possible ways in which the treasure can be divided is closest to which number? 0 B) 0 C) 0 D) 40 E) Ten balls, numbered through 0, are placed in a bag. Draw one ball at random from the bag, put it back, and then draw a second time. What is the probability that 4 divides the product of your two selections? ½ B) ¼ C) 9/0 D) /4 4. An equilateral triangle is inscribed in a circle which is then inscribed in a square. What is the ratio of the area of the triangle to the area of the square? 4 B) 6 C) 5 6 D) 0 4. How many 5 digit palindromes are there? (Palindromes are numbers which read the same forwards and backwards). 000 B) 00 C) 87 D) Suppose a sequence is defined as follows. a =, an+ = It can be shown that the sequence is decreasing and 0 a n for all n. Find the limit of the sequence. + 5 B) C) 5 D) a n 45. How many positive integers are there whose digits are all different? 6,59 B) 9,864,00 C) 986,409 D) 8,877,690

7 46. A ball is dropped from a height of 00 feet onto a hard level surface. Suppose that each time it bounces, it rebounds to half of its previous height. If the ball continues to bounce indefinitely, find the total distance that it travels. 00 ft B) 00 ft C) 50 feet D) 400 ft 47. For what value of the constant c will there be no inverse for the matrix 4 c 7 - B) - C) 0 D) E) 48. Let x x x 8 = +. Then 84 = 44 B) 444 C) 448 D) For what positive constant c will the function value of -? f( x) = x + cx+ have minimum B) 6 C) 6 D) E) If x= log5 and y = log7, then log = 7 x x x+ A ) B) C) D) x-y+ E) (x-y) y y y

8 . B. A. C. A. E (0). A 4. D 4. D 5. D 5. E 6. A 6. A 7. A 7. D 8. D 8. A 9. D 9. C 0. B 40. B. B 4. C. A 4. B. B 4. D 4. B 44. C 5. E (x+) 45. D 6. E (sqrt()/) 46. B 7. B 47. B 8. B 48. C 9. D 49. E 0. E (e^) 50. D. E (/4040). A. E 4. C 5. C 6. A 7. A 8. E 9. A 0. A 00 Mu Individual Test