ALGEBRA II

2000 STATE MATH CONTEST FINALS

 

  1. Given the line, 2x + 4y = 3, what is the approximate distance between it and the origin?
    a) 0.75 b) 1.5 c) 0.45 d) 0.67 e) 0.5

  2. A circle has center C(-3, 1) and passes through the point P(7, 5). The radius of the circle is:
    a) 10 b) 14 c) d) e)

  3. If x4 – y4 = 45 and x2 – y2 = 5, then x2 + y2 equals:
    a) 15 b) 9 c) 10 d) 40 e) 50

  4. What is the numerical coefficient of in the expansion of ?
    a) 1 b) 65,780 c) 325 d) 14,950 e) 2,600

  5. The number of real solutions of the equation is:
    a) 0 b) 1 c) 2 d) 3 e) 4

  6. The capacity of a car’s radiator is nine liters. The mixture of antifreeze and water is 30% antifreeze. The temperature is predicted to drop rapidly requiring the mixture to be 65% antifreeze. How much of the mixture in the radiator must be drawn off and replaced with pure antifreeze?

    a) 3.5 liters

    		b) 4.5 liters c) 5.0 liters d) 6.0 liters e) none of these

  7. An airplane, flying with a tail wind, travels 1200 miles in 5 hours; the return trip, against the wind, takes 6 hours. Find the cruising speed of the plane and the speed of the wind (assume that both are constant).
    a) 220 mph, 20 mph
    b) 220 mph, 40 mph
    c) 230 mph, 20 mph
    d) 230 mph, 40 mph
    e) 240 mph, 20 mph

  8. If , then x is:
    a) b) c) d) e)

  9. The center (C) and vertices (V) of the ellipse are:
    a) C(-2,3); V(-2±3,3) b) C(2,3); V(2±3,3) c) C(-2,3); V(-2±3,-3)
    d) C(2,3); V(2±3,0) e) C(0,0); V(2±3,3)

  10. For what values of k does the equation have two imaginary roots?
    a) b) c) d) e)

  11. How many integers between 199 and 301 are divisible by 4 or 10?
    a) 26 b) 31 c) 35 d) 37 e) 39

  12. Define the binary operation @ for two real numbers x and y so that x @ y = x2 - y2. @ is:
    a) Commutative only
    b) Associative only
    c) Commutative and Associative
    d) Neither Commutative nor Associative
    e) Not enough information

  13. Five men and five women go into a movie theater and occupy a row of ten seats. If they choose their seats randomly, the probability that all of the men are sitting together and all of the women are sitting together is:
    a) 1/126 b) 2/191 c) 1/191 d) 5/126 e) 1/2

  14. Find all real number solutions of
    a) b) c) d) e)

  15. A fair die is rolled three times. The probability that you get a larger number each time is:
    a) 17/216 b) 9/216c) 7/17d) 9/26e) none of these

  16. Suppose and are both linear functions, with and . Find the sum of the slope and the y-intercept of .
    a) -2 b) -1c) 0d) 1e) none of these

  17. A quadratic equation y=ax2+bx+c is known to pass through the points (0, 5), (2, 11), and (-2, 15). Find the sum of a and b.
    a) -7 b) 1c) 2d) 3e) 4

  18. If for all , then

    a) b) c) d) e)

  19. Consider the non-decreasing sequence of positive integers 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,… in which the nth positive integers appears n times. The remainder when the 2000th term is divided by 5 is

    a) 0b) 1c) 2d) 3e) 4

  20. Find the area enclosed by the graph of the following relation:

    a) 30b) c) 15d) 60e) none of these

  21. Which value best approximates ?
    a) 1.499987 b) c) 3 d) infinitye) none of these

  22. Which recursive formula generated the sequence –2, –7, –9, –16, –25, –41?
    a)
    b)
    c)
    d)
    e)

  23. Solve:
    a) b) c) d) e)

  24. Simplify
    a) b) c) d) e) none of these

  25. Which of the following is an equation of a hyperbola with horizontal transverse axis and asymptotes ?
    a) b) c) d) e)

  26. Find the center and r, the radius, of .
    a) center (6,-3); r = 16 b) center (6,-3); r = 4
    c) center (-6,3); r = 16 d) center (-6,3); r = 4
    e) none of these

  27. Solve the equation , where . The sum of the solutions is:
    a) p b) approx. 0.85107 c) 2p d) p e) none of these

  28. You work at a T-shirt printing business. 4% of 1600 T-shirts shipped are printed improperly. If you randomly select 100 T-shirts (selecting a T-shirt and replacing it), what is the probability that at least one of them is printed improperly?
    a) 0.017 b) 0.917 c) 0.517d) 0.983e) 0.933

  29. The distance from Asheville to Raleigh is approximately 250 miles. Acme Bus Lines advertises that it can make the trip in 4.5 hours. But the driver drives 50 mph for the first 120 miles. Approximately how fast must he drive for the remainder of the trip to reach Raleigh in time?
    a) 62.3 mphb) 55 mphc) 60 mphd) 61.1 mphe) 61.9 mph

  30. The length of the minor axis of the ellipse having equation is which of these?
    a) 1b) 2c) 3d) 4e) 5

  31. The IQ scores of the population are normally distributed, having mean = 100 and standard deviation = 15. What percent of the population would you expect to have IQ scores greater than 85?

    a) 50%b) 68%c) 84%d) 95%e) 99%

  32. What is the number of solutions to the following system?
    a) 0b) 1c) 2d) 3e) 4

  33. Find the value of if .
    a) –2b) 0c) 1d) 2e) none of these

  34. State the inverse function of .
    a) b) c)
    d) e) none of these

  35. A hot-air balloon rises 80 feet in the first minute of flight. If in each succeeding minute the balloon rises only 90% as far as in the previous minute, what will be its maximum altitude if it is allowed to rise without limit?
    a) 88.8 feetb) 800 feetc) 888.8 feetd) 900 feete) 905 feet

  36. A triangle has angles that measure 48Ί, 79Ί, and 53Ί. Which set below is a possible set of approximate measures of the sides?
    a) {48.0, 63.4, 51.6} b) {48.0, 79.0, 53.0}
    c) {48.0, 27.7, 24.7} d) {48.0, 13.7, 43.2}
    e) {48.0, 80.0, 64.0}

  37. sin(x+30Ί) + cos(x+60Ί) = ?
    a) sin(x) + cos(x) b) sin(x) – cos(x)c) sin(x)
    d) cos(x) e) none of these

  38. The standard form of is:
    a) b) c) d) e)

  39. If , then , for , is:
    a) b) c)
    d) e)

  40. The 6th term of the recursively defined sequence with a1=11 and an=(an-1)/n is:
    a) b) c) d) e)