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ACT Math Problems -- Test 2

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I didn’t see much out there of hard math ACT problems, so I created these. I have studied thousands of real problems. Some of these involve more than one concept from real problems. Others are hard real problems made harder. This is intended to help students looking for top scores prepare. It should not be used to determine your expected score.

These tests should not be taken timed and should not be used to determine scores. They are all extremely difficult but realistic problems designed to challenge students going for top scores.

1. The distance between (3,4) and (x,11) is 11. What is one possible value of x?

A. 10
B. -5
C. 3 + 2 \sqrt{6}
D. 3 + 6 \sqrt{2}
E. 6 + 3 \sqrt{2}

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D

2. The set of values satisfying |3x -5| < 4 is

A. \dfrac{1}{2}<x<2

B. \dfrac{1}{3}<x<3

C. \dfrac{1}{4}<x<4

D. \dfrac{1}{5}<x<5

E. \dfrac{1}{6}<x<6

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B

3. What is the product of matrices of A=\left[ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 5 & 6 & 7 & 8\\ \end{array} \right] and B=\left[ \begin{array}{cc} 2 & 3 \\ 1 & 5 \\ 2 & 4 \\ 1 & 2 \\ \end{array} \right]?

A. \left[ \begin{array}{cc} 14 & 33 \\ 38 & 89 \\ \end{array} \right]

B. \left[ \begin{array}{cc} 14 & 38 \\ 33 & 89 \\ \end{array} \right]

C. \left[ \begin{array}{cc} 33 & 89\\ 14 & 38 \\ \end{array} \right]

D. \left[ \begin{array}{cc} 89 & 14 \\ 33 & 38 \\ \end{array} \right]

E. \left[ \begin{array}{cc} 38 & 33 \\ 89 & 14 \\ \end{array} \right]

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A

4. What is a solution of 3x^2-2x+5=0 ?

A. \dfrac{5 - i\sqrt{17}}{3}

B. \dfrac{4 - i\sqrt{10}}{2}

C. \dfrac{3 - i\sqrt{11}}{3}

D. \dfrac{2 - i\sqrt{13}}{2}

E. \dfrac{1 - i\sqrt{14}}{3}

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E

5. The area of a circle is 324\pi. What is its circumference?

A. 18\pi
B. 24\pi
C. 30\pi
D. 36\pi
E. 42\pi

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D

6. \sin{x}=\dfrac{4}{7}, 180^{\circ}>x>90^{\circ}. What is \tan{x}?

A. -\dfrac{3\sqrt{22}}{22}

B. \dfrac{3\sqrt{22}}{22}

C. -\dfrac{4\sqrt{33}}{33}

D. \dfrac{4\sqrt{33}}{33}

E. -\dfrac{7}{4}

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C

7. What is the digit in the one’s place of 7^{2000}?

A. 1
B. 3
C. 5
D. 7
E. 9

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A

8. \dfrac{5x - 3y}{4x + y} = \dfrac{2}{3}. What is \dfrac{y}{x} equal to?

A. \dfrac{3}{5}

B. \dfrac{5}{3}

C. \dfrac{5}{7}

D. \dfrac{7}{5}

E. \dfrac{7}{11}

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E

9. 11 is 8% of what?

A. 3 
B. 19
C. 88
D. 88.5
E. 137.5

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E

10. An account increased by 10% per year for 20 years. What was the total percent increase in the account over the 20 years?

A. 200%
B. 300%
C. 427%
D. 573%
E. 673%

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D

11. If you are driving at 50 mph, what is your speed in feet per second to the nearest foot? (5280 feet equals one mile)

A. 73 ft/sec
B. 77 ft/sec
C. 83 ft/sec
D. 87 ft/sec
E. 93 ft/sec

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A

12. What is the period of y = \cot{20x}?

A. \pi
B. 20
C. \dfrac{\pi}{20}
D. 20\pi
E. \dfrac{20}{\pi}

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C

13. What is i^{383}? (i=\sqrt{-1})

A. -1
B. -i
C. 1
D. i
E. \dfrac{-1}{i}

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B

14. Which of the following is a factor of 8x^3+27?

A. 4x^2+6x+9

B. 4x^2-6x+9

C. 9x^2+6x+4

D. 9x^2 - 6x + 4

E. 9x^2 - 6x - 4

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B

15. 8% of 50 is 12% of what?

A. 133.33
B. 33.33
C. 143.33
D. 146.67
E. 153.33

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B

16. Which quadratic equation has the complex number 3 + 5i as a solution?

A. x^2-2x+18 = 0
B. x^2-3x+22 = 0
C. x^2-4x+26 = 0
D. x^2-5x+30 = 0
E. x^2-6x+34 = 0

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E

17. What are the foci of the ellipse \dfrac{(x + 5)^2}{25} + \dfrac{(y - 8)^2}{49} = 1?

A. (5,-8 \pm 2 \sqrt{2})
B. (8,8 \pm 2 \sqrt{2})
C. (-5,8 \pm 2 \sqrt{2})
D. (-5,8 \pm 2 \sqrt{6})
E. (-5,5 \pm 2 \sqrt{2})

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D

18. What is the x-intercept of the ellipse \dfrac{(x + 5)^2}{25} + \dfrac{(y - 8)^2}{49} = 1?

A. -5 \pm \dfrac{5 \sqrt{7}}{15}i
B. -5 \pm \dfrac{5 \sqrt{15}}{7}i
C. -15 \pm \dfrac{5 \sqrt{5}}{7}i
D. -15 \pm \dfrac{5 \sqrt{7}}{5}i
E. -15 \pm \dfrac{7 \sqrt{5}}{7}i

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B

19. What are the vertices of the ellipse \dfrac{(x + 5)^2}{25} + \dfrac{(y - 8)^2}{49} = 1?

A. (-5,1), (-5,15)
B. (1,-5), (15,-5)
C. (5,-1), (5,-15)
D. (-5,1), (5,15)
E. (-1,5), (-15,5)

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A

20. A rectangle is 4 centimeters by 11 centimeters. What is its area in square millimeters?

A. 1100 mm^2
B. 2200 mm^2
C. 3300 mm^2
D. 4400 mm^2
E. 5500 mm^2

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D

21. If \log_{2}{300} - \log_{2}{75} = \log_{10}{x}, what is x?

A. 100
B. 200
C. 300
D. 400
E. 500

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A

22. Which of the following equals (x + 2i)^4?

A. x^4 -8x^3 i+24x^2+32x i-16
B. x^4 +8x^3 i-24x^2+32x i-16
C. x^4 -8x^3 i-24x^2-32x i-16
D. x^4 +8x^3 i+24x^2+32x i+16
E. x^4 +8x^3 i-24x^2-32x i+16

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E

23. Solve for x, \ln {(x + 5)} = 3?

A. e^5 - 3
B. e^5 + 3
C. e^3 - 5
D. e^3 + 5
E. 5e^3

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C

24. Solve for x, e^{x - 2} = 8?

A. ln{2} -8
B. ln{2} +8
C. ln{8} - 2
D. ln{8} + 2
E. 2ln{8}

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D

25. You have a 70% chance of winning each of 2 games. What is the chance you lose both game?

A. 5%
B. 6%
C. 7%
D. 8%
E. 9%

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E

26. If today in Saturday, what day of the week will it be 1000 days for now?

A. Monday
B. Tuesday
C. Wednesday
D. Thursday
E. Friday

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E

27. What is \log_{3}{\sqrt {243}}?

A. \dfrac{2}{5}
B. \dfrac{5}{2}
C. -\dfrac{2}{5}
D. -\dfrac{5}{2}
E. 81

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B

28. What is \log_{2}{\dfrac{1}{128}}?

A. -9
B. -7
C. -5
D. -3
E. -1

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B

29. If x^{5b} = 2, then x^{20b}= what?

A. 16
B. 17
C. 18
D. 19
E. 20

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A

30. A line has an x intercept of -6 and a slope of \dfrac{2}{3}. What is its y intercept?

A. 1
B. 2
C. 3
D. 4
E. 5

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D

31. What is the period of y = 4 \sin {8x}?

A. \dfrac{\pi}{2}
B. \dfrac{\pi}{4}
C. \dfrac{\pi}{8}
D. \dfrac{\pi}{16}
E. \dfrac{\pi}{32}

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B

32. What is the distance between 3 - 11i and -2 - 4i in the complex plane?

A. \sqrt{44}
B. \sqrt{54}
C. \sqrt{74}
D. \sqrt{84}
E. \sqrt{94}

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C

33. What is the domain of y = \log {( x^2 + 3x + 2)}?

A. (-\infty, -2) \cup (-1, \infty)
B. (-\infty, -2) \cup (1, \infty)
C. (-\infty, -1) \cup (2, \infty)
D. (-\infty, -2) \cup (2, \infty)
E. (-\infty, -1) \cup (1, \infty)\sqrt{94}

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A

34. How many different committees of 3 men and 2 women can be formed from a group of 8 men and 5 women?

A. 5
B. 6
C. 56
D. 65
E. 560\sqrt{94}

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E

34. How many different committees of 3 men and 2 women can be formed from a group of 8 men and 5 women?

A. 5
B. 6
C. 56
D. 65
E. 560\sqrt{94}

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E

35. What is the matrix product \left [\begin{array}{c} 3 \\ 4 \\ 5 \end{array} \right ] \left [\begin{array}{ccc} 1 & 2 & 3 \end{array} \right ] ?

A. \left [\begin{array}{ccc} 3 & 6 & 9 \\ 4 & 8 & 12 \\ 5 & 10 & 15 \\ \end{array} \right ]
B. \left [\begin{array}{ccc} 3 & 4 & 5 \\ 6 & 8 & 10 \\ 9 & 12 & 15 \\ \end{array} \right ]
C. \left [\begin{array}{ccc} 3 & 8 & 15 \\ \end{array} \right ]
D. \left [\begin{array}{c} 26 \\ \end{array} \right ]
E. \left [\begin{array}{c} 3 \\ 8 \\ 15 \\ \end{array} \right ]\sqrt{94}

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A

36. What is the solution set of |3x + 5| > -2?

A. No solution
B. (-2,5)
C. (-2,3) \cup (3,5)
D. (-\infty,-2) \cup (-2,3) \cup (3,5) \cup (5,\infty)
E. All reals

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E

37. If you triple the sides of a box, the surface area is multiplied by what factor?

A. 3
B. 6
C. 9
D. 18
E. 27

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C

38. The product of two numbers is 20 and their sum is 10. What is greater of the two numbers?

A. 2 + \sqrt{2}
B. 3 + \sqrt{3}
C. 5 + \sqrt{5}
D. 6 + \sqrt{6}
E. 7 + \sqrt{7}

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C

39. What is the measure of each interior angle of a regular octagon in degrees?

A. 45^\circ
B. 60^\circ
C. 120^\circ
D. 135^\circ
E. 150^\circ7 + \sqrt{7}

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D

40. {(\sqrt {2})}^a = \left( \dfrac{1}{32} \right) ^ b. What is the ratio \fdrac{a}{b}?

A. -10
B. -5
C. 0
D. 5
E. 107 + \sqrt{7}

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A

41. What is 37 parts per million in scientific notation?

A. 3.7 \times 10^{-6}
B. 3.7 \times 10^{-5}
C. 3.7 \times 10^{-3}
D. 3.7 \times 10^{5}
E. 3.7 \times 10^{6}7 + \sqrt{7}

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B

42. What is the sum of the first 100 positive integers?

A. 1010
B. 2020
C. 3030
D. 4040
E. 50507 + \sqrt{7}

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E

43. f(x) = \sqrt[5]{2x+11}. What is f^{-1}(x)?

A. f^{-1} (x) = \dfrac{x^2-5}{11}
B. f^{-1} (x) = \dfrac{x^2-11}{5}
C. f^{-1} (x) = \dfrac{x^5-2}{11}
D. f^{-1} (x) = \dfrac{x^5-11}{2}
E. f^{-1} (x) = \dfrac{x^{11}-2}{5}

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D

44. If \cot{x} = 5 and \sin {x} < 0, what is \sec{x}?

A. \dfrac{\sqrt{5}}{26}
B. \dfrac{\sqrt{5}}{62}
C. \dfrac{\sqrt{25}}{6}
D. \dfrac{\sqrt{26}}{5}
E. \dfrac{\sqrt{62}}{5}

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D

45. y = 8t + 5, x = 5 - 4t. What is y in terms of x?

A. y=2+15x
B. y=2-15x
C. y=15-2x
D. y=15+2x
E. y=\dfrac{15}{2x}

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C

46. f(x)=x^2+3x+2. What is f(x+a)?

A. f(x+a) = f(x)+f(a)
B. f(x+a) = f(x)+f(a)-1
C. f(x+a) = f(x)+f(a)+ax
D. f(x+a) = f(x)+f(a)+ax-1
E. f(x+a) = f(x)+f(a)+2(ax-1)

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E

47. The ratio of the radii of two spheres is 3:4. What is the ratio of their volumes?

A. 3:4
B. 9:16
C. 27:64
D. 1:7
E. 1:12

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C

48. The ratio of the radii of two spheres is 3:4. What is the ratio of their surface areas?

A. 3:4
B. 9:16
C. 27:64
D. 1:7
E. 1:12

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B

49. How many two digit positive numbers have a units digit twice the tens digit?

A. 4
B. 6
C. 8
D. 10
E. 12

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A

50. x^2 + 11x + g =0 will have one solution for what value of g?

A. \dfrac{2}{11}
B. \dfrac{4}{121}
C. \dfrac{11}{2}
D. \dfrac{121}{4}
E. \dfrac{121}{2}

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D

51. x + y = 3a, x - y = 5b. What is x in terms of a and b?

A. \dfrac{3a-5b}{2}
B. \dfrac{3a+5b}{2}
C. \dfrac{5b-3a}{2}
D. \dfrac{2}{3a+5b}
E. \dfrac{2}{3a+5b}

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B

52. If 3^{x^2 - 4} = 1, what are the possible values of x?

A. -1 and 1
B. -2 and 2
C. -3 and 3
D. -4 and 4
E. -5 and 5

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B

53. Solve | 3x + 5 | = | -7|.

A. \dfrac{2}{3},-4
B. \dfrac{3}{5},-7
C. \dfrac{3}{7},-5
D. \dfrac{3}{4},-2
E. \dfrac{5}{7},-3

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A

54. The endpoints of the diameter of a circle are (-7, 5) and (2, 11) What is the equation of the circle?

A. (x-3)^2+(y-8)^2=25
B. (x+7)^2+(y-5)^2=25
C. (x-2)^2+(y-11)^2=25
D. (x+3)^2+(y-8)^2=25
E. (x-3)^2+(y+8)^2=25

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D

55. 4^{x + 3} = 32 ^{5-x}. Solve for x.

A. \dfrac{3}{4}
B. \dfrac{4}{3}
C. \dfrac{5}{32}
D. \dfrac{7}{19}
E. \dfrac{19}{7}

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E

56. a + b = 12, \dfrac{a}{b} = \dfrac{5}{3}. What is a?

A. \dfrac{2}{15}
B. \dfrac{15}{2}
C. \dfrac{5}{12}
D. \dfrac{12}{5}
E. \dfrac{2}{3}

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B

57. y = -11 \sin { \left(\dfrac{3\pi }{4}x + 5\right) }. What are amplitude and period of the function?

A. 11,\dfrac{8}{3}
B. 11,\dfrac{5}{3}
C. 11,\dfrac{5}{4}
D. 5,\dfrac{3}{11}
E. 5,\dfrac{8}{11}

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A

58. What is the domain of y = \dfrac{x + 8}{ \sqrt {3x + 4}}?

A. x>-\dfrac{3}{4}
B. x>\dfrac{3}{4}
C. x>-\dfrac{4}{3}
D. x>\dfrac{4}{3}
E. x>-8

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C

59. y = x^2 + 3x + 8 is shifted up 2 and right 5. What is the new equation?

A. x^2-7x+20
B. x^2-8x+10
C. x^2+7x-20
D. x^2+8x+10
E. x^2-7x+8

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A

60. Using complex arithmetic, what is \sqrt {-3} \times \sqrt {-75}?

A. -25
B. -15
C. 15
D. 25
E. 225

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B

Answer key: 1D, 2B, 3A, 4E, 5D, 6C, 7A, 8E, 9E, 10D, 11A, 12C, 13B, 14B, 15A, 16E, 17C, 18B, 19A, 20D, 21A, 22E, 23C, 24D, 25E, 26E, 27B, 28B, 29A, 30D, 31B, 32C, 33A, 34E, 35A, 36E, 37C, 38C, 39D, 40A, 41B, 42E, 43C, 44D, 45C, 46E, 47C, 48B, 49A, 50D, 51B, 52B, 53A, 54D, 55E, 56B, 57A, 58C, 59A, 60B

 
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