Expert: Richard J. Raridon - 4/29/2011

Question
If a varies directly as b, and a = 75 when b = 40, find a when b = 12.
  a = 6.4      a = 0.4
  a = 22.5      a = 2.25
2.   If y is directly proportional to the square root of x, and y = 25 when x = 3, find x when y = 100.
  
x = 3Ö   3
     x = 48
  x = 100      x = 12
3.   If w varies directly as 2x - 1, and w = 9 when x = 2, find x when w = 15.
  x = 2      x = 36
  x = 5      x = 3
4.   At the grocery mart, a 575g can of green beans cost $0.39. How much will an 810 g can of green beans cost?
  $0.52      $0.50
  $0.55      $0.53
5.   If z is inversely proportional to r, and z = 32 when r = 1.5, find r when z = 8.
  r = 40      r = 0.17
  r = 6      r = .4
6.   If y varies inversely as x and y = 24 when x = 3, what is y when x = 8?
  5      8
  3      9
7.   A number y varies inversely as the square root of x. If y = 3 when x = 64, what is x when y = 6?
  12      14
  16      18
Part 1-3
Type the answer to the problem in the text box below each item. Be sure legibly show all work in your notebook. Remember to include any applicable units. If there is no solution, type "no solution". If there is not enough information present to solve the problem, type "not enough information". (Each question is worth two points)
8.   Suppose that w varies directly as z2 and inversely as xy. When w = 10, x = 15, y = 2, and z = 5. Find z when w = 2, x = 8, and y = 27.
Part 2-1
Select the best answer from the choices provided. (Each question is worth one point)
9.   Divide:
9z    -    z2
z    -    3
  
-z    +    6    +       18    
z    -    3
     
z    +    6    +       18    
z    -    3
  z + 24      z - 24
10.   Use synthetic substitution to find P(c) for the given polynomial P(x) and the given number x:

P(x) = 1 - 7x - 4x2 + x3; c = 6
  P(6) = 115      P(6) = 114
  P(6) = 31      P(6) = 42
11.   Solve using the given root:

2z3 + z2 - 8z + 3 = 0; 1.5
  
-1 ± Ö   2
     
-1 ± 2 Ö   2
  -3 or 1      3 or -1
12.   Find a polynomial equation with integral coefficients that has the roots:

-2, 2, -3
  x3 + x2 - 4x - 12 = 0      x3 + 3x2 - 4x - 12 = 0
  x3 + x2 - 8x - 12 = 0      x3 - 3x2 + 4x - 12 = 0
13.   Find a polynomial equation with integral coefficients that has the roots:

-2, -i, i
  x3 + x + 2 = 0      x3 + 2x2 + 2 = 0
  x3 + 2x2 + x + 2 = 0      x3 - 2x2 - x - 2 = 0
14.   Find a polynomial equation with integral coefficients that has the roots:

1, -1.5, 2
  2x3 - 3x2 - 5x + 6 = 0      2x3 + 3x2 - 7x + 6 = 0
  2x3 - x2 - 5x + 6 = 0      2x3 + 3x2 - 5x - 6 = 0
15.   Solve for x:
x    +    1    =              2
-x    +    2       x
  x = ±4      x = 1, 4
  x = 1, -4      x = -1, 3
Part 2-2
Type the answer to the problem in the text box below each item. Be sure legibly show all work in your notebook. Remember to include any applicable units. If there is no solution, type "no solution". If there is not enough information present to solve the problem, type "not enough information". (Each question is worth two points)
16.   Divide:
2a3 + 8a2 - 3a + 8
a + 4

Part 3-1
Select the best answer from the choices provided. (Each question is worth one point)
17.   List the possible rational roots for the equation:

x4 + 3x2 - 8x + 10 = 0

  {±1, 5, 10}      {±1, ± 2, ± 5, ± 10}
  {1, 2, 5, 10}      No answer is correct.
18.   List the possible rational roots for the equation:

x4 + 2x + 2 = 0

  {-0.5}      No answer is correct.
  {±1, ± 2,}      {±1, ± 2, ± 0.5}
19.   Assume that y varies directly as x. If y = 3.2 when x = 0.2, find y when x = 1.6.
  y = 2.56      y = 0.01
  y = 0.1      y = 25.6
20.   Find all the roots of the equation.

x3 - 2x2 - x + 2 = 0
  No roots      0, -1
  -1, 1, 2      -1
21.   Use the Rational Roots Theorem to solve the equation for the rational roots

x3 - 3x2 + 4x - 12 = 0
  -3, 0      4, 12, 3
  3      3, ± 2i
22.   Use the Rational Roots Theorem to solve the equation for the rational roots.

4y5 + 8y4 - 29y3 - 42y2 + 45y + 54 = 0
  -3, 2, -1, ± 1.5      -3, 1.5, -1
  2, -1.5, -1, 3      8, -2, 3, 1.5
23.   Find all roots of the equation

3x2 - 11x - 4 = 0
  -3, 4      3, 4
  -1/3, 4      4
Part 3-2
Type the answer to the problem in the text box below each item. Be sure legibly show all work in your notebook. Remember to include any applicable units. If there is no solution, type "no solution". If there is not enough information present to solve the problem, type "not enough information". (Each question is worth two points)
24.   Solve the following equation:

2x3 - 5x2 - 11x - 4 = 0
Part 4-1
Select the best answer from the choices provided. (Each question is worth one point)
25.   Find the distance between the pair of points:

(-a,b) and (2a,4b)
  ab      
3Ö   a2 + b2
  3ab      a2 + b2
26.   Find the midpoint of the segment joining the points:

(0.5,-1) and (-1,1)
  (-2, 0)      (-0.25, 0)
  (-0.5, 0)      (0.25, 0)
27.   Find the coordinates of Q given that M is the midpoint of line segment PQ:

P(0, 0) and M(h, k)
  (2h, 2k)      (h, k)
  (-2k, -2h)      (2h, k)
28.   For the given vertices what type of triangle is formed:

Vertices:  A(2, -3), B(2, 3), C(5, 0)
  Isoceles right triangle      Isosceles
  Equilateral      Scalene right triangle
29.   Find an equation of the circle with the center and radius:

(-3, 1); r = 5
  (x - 3)2 + (y + 1)2 = 25      (x + 3)2 + (y - 1)2 = 25
  (x - 3)2 + (y + 1)2 = 5      (x + 3)2 + (y + 1)2 = 5
30.   Find an equation of the circle with the center and radius:

(-5,3); r = 0.167
  (x + 5)2 + (y - 3)2 = 0.028      (x + 5)2 + (y - 3)2 = 0.167
  (x - 5)2 + (y + 3)2 = 0.167      (x - 5)2 + (y + 3)2 = 0.028
31.   Find the center and radius of a circle with equation:

4x2 - 16x + 4y2 - 24y + 36 = 0

  C(2, 3); r = 2      C(2, 3); r = 3
  C(3, 2); r = 2      C(3, 2); r = 3
Part 5-1
Select the best answer from the choices provided. (Each question is worth one point)
32.   Find an equation of the ellipse having the given points as foci and the given sum of the focal radii

(-9, 0); (9, 0); 30
  
x2    +    y2    =    1
15    12
     
x2    +    y2    =    1
225    144
  
x2    -    y2    =    1
15    12
     
x2    -    y2    =    1
225    144
33.   Find an equation of the hyperbola described:

Asymptotes y = x and y = -x; foci (-4, 0) and (4, 0)
  
x2    -    y2    =    1
8    8
     
x2    +    y2    =    1
4    4
  
x2    -    y2    =    1
8    2
     
x2    +    y2    =    1
8    2
34.   Give the radius and the coordinates of the center of the circle:

(x - 1)2 + y2 = 144
  center: (0, 0), radius: 12      center: (0, 0), radius: 72
  center: (1, 0), radius: 12      center: (1, 0), radius: 72
35.   Solve by completing the square:

x2 - 4x = -3
  4, 2      1, 3
  2, 3      0, 1
36.   Use the information to write an equation of the parabola:

Focus: (-8, 0), Directrix: x = 8
  x2 = -8x      y2 = -8x
  y2 = x/-2      y2 = -32x
37.   Use the information to write an equation of the parabola:

Focus: (-8, -5), Vertex: (2, -5)
  (y + 5)2 = -40(x - 2)      (x + 5)2 = 4(y - 2)
  (y - 5)2 = -4(x - 2)      (x - 5)2 = 40(y - 2)
Part 5-2
Select the best answer from the choices provided. (Each question is worth two points)
38.   Choose the correct foci:

y2 = 5x2 + 25
  (0, ±5.48)      (0, 7.75)
  (±30, 0)      (0, ±30)
Part 6-1
Select the best answer from the choices provided. (Each question is worth one point)
39.   Solve:

c3 + c2 - 7c - 3 = 0, given root -3
  c = 3 or -1      c = -3, -0.41, 2.41
  c = 1      c = -1
40.   Solve:

y4 - 3y3 - 2y2 + 10y - 12 = 0, given root 1 + i
  {1 ± i, -2, 3}      {±1, ±2, ±3, 4}
  {±2, ±4}      {i ± 2, ±3}

Answer
write exponents as x^2 or y^3
write square roots as x^1/2
1. a=2.25
2. x = 4(3^1/2)
3. x=3
4. $0.55
5. r=6
6. y=9
7. x=16
8. z=6
9. 9z-z^2 = (3-z)(3+z)
10. P(6)=31
11. z = +/-(1/2)
12. just multiply (x-2)(x+2)(x+3) to get x^3+3x^2-4x-12
13. x^2+2x^2+x+2
14. 2x^3-3x^2-5x+6
15. written incorrectly
16. no solution
17. no answer
18. no answer
19. y=25.6
20. x = +/-1,2
21. x = +/-2i,3
22. -1,2
23. x = -1/3,4
24. x = -1/2,-1,4
25. 3(a^2+b^2)^1/2
26. -2.5,0
27. 2h,2k
28. isosceles
29. (x+3)^2+(y-1)^2 = 25
30. (x+5)^2+(y-3)^2 = 0.028
31. C(2,3), r=2
32. (x^2/225)+(y^2/144) = 1
33. (x^2/8)-(y^2/8) = 1
34. center(1,0),radius=12
35. x =1,3
36. y^2 = -8x
37. (y+5)^2 = -40(x-2)
38. 0,+/-5.48
39. c = -3,-0.41,2.41
40. 1+/-i,-2,3