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Math ACT Book

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Math ACT Book

Hard ACT Problems

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. This is a short test. There are several full-length ACT math tests packed with only extremely challenging problems, making this the best math ACT prep material available anywhere!

 

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. Suppose you can pick exactly 2 of 7 ice creams and 1 of 3 toppings. How many possible sundaes can you have?

  1. 13
  2. 14
  3. 42
  4. 63
  5. 147

 

2. What is the equation of the perpendicular bisector of the line segment between (1,2) and (3,10)?

  1. y=-\dfrac{x}{4}+\dfrac{13}{2}
  2. y=\dfrac{x}{4}-\dfrac{13}{2}
  3. y=4x-26
  4. y=-4x+26
  5. y=-4x+\dfrac{13}{2}


3. A cube has a volume of 125 cubic feet. What is its surface area in square feet?

  1. 100
  2. 125
  3. 150
  4. 175
  5. 200


4. What is the equation of x^2 + 3x + 7 = y reflected about the y-axis?

  1. x^2+3x-7
  2. x^2-3x+7
  3. x^2-3x-7
  4. -x^2+3x+7
  5. -x^2-3x-7


5. a:b=3:2, b:c=4:7, c:d=5:1. What is a:d?

  1. 3:1
  2. 6:1
  3. 3:7
  4. 3:5
  5. 30:7


6. What is the slope of the line \dfrac{2x}{3} + \dfrac{4y}{5} = \dfrac{11}{17}?

  1. \dfrac{5}{6}
  2. \dfrac{6}{5}
  3. -\dfrac{5}{6}
  4. -\dfrac{6}{5}
  5. 5


7. What is (3a^3b^5)^4?

  1. 81a^{12}b^{20}
  2. 7a^7b^9
  3. 12a^{12}b^{20}
  4. 81a^7b^9
  5. 12a^7b^9


8. (3a - 2b)^2 = ?

  1. 9a^2+12ab+4b^2
  2. 9a^2-12ab+4b^2
  3. 9a^2+4b^2
  4. 9a^2-4b^2
  5. 6a-4b


9. Simplify \dfrac{1-\cos^2{x}}{\tan{x}}

  1. \cot{x}
  2. \tan{x}
  3. \sin{x}
  4. \sin{x}\cos{x}
  5. \cos{x}


10. (2a +7b -3c) - 2(-3a +4b -c)

  1. -8a+b+c
  2. 8a+b+c
  3. 8a+b-c
  4. 8a-b-c
  5. -8a-b-c


11. What is \dfrac{4 + 3i}{5 + 2i}?

  1. \dfrac{26 + 7i}{29}
  2. \dfrac{4}{5}+\frac{3}{2}i
  3. \dfrac{4}{5}-\frac{3}{2}i
  4. \fdrac{4 + 6i}{5}
  5. \dfrac{4 - 6i}{5}


12. The 1^{st} term in a geometric sequence is 16 and the 2^{nd} term is 24. What is the 7^{th} term?

  1. 729
  2. \dfrac{729}{2}
  3. \dfrac{729}{4}
  4. \dfrac{729}{8}
  5. \dfrac{729}{16}


13. The 3^{rd} term in a geometric sequence is 9 and the 6^{th} term is \dfrac{125}{3}. What is the 8^{th} term?

  1. 625
  2. \dfrac{625}{3}
  3. \dfrac{3125}{27}
  4. \dfrac{625}{27}
  5. \dfrac{625}{81}


14. The 1^{st} term in an arithmetic series is 11 and the last is 172. The sum of the series is 2196. What is the 4^{th} term?

  1. 30
  2. 32
  3. 34
  4. 36
  5. 38


15. What is the area of the circle having (-3,7) and (2,11) as endpoints of the diameter?

  1. \dfrac{41 \pi}{4}
  2. \dfrac{17 \pi}{4}
  3. \dfrac{41 \pi}{2}
  4. \dfrac{17 \pi}{2}
  5. \dfrac{15 \pi}{4}


16. 2^{x^2 +3x -15} = 8. What is a positive solution?

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5


17. Which fraction is equal to \dfrac{1}{2^{40}} + \dfrac{1}{2^{42}}?

  1. \dfrac{1}{2^{41}}
  2. \dfrac{3}{2^{42}}
  3. \dfrac{3}{2^{40}}
  4. \dfrac{5}{2^{41}}
  5. \dfrac{5}{2^{42}}


18. f(x)=2x+1, g(x)=5x+k. For what value of k does f(g(x))=g(f(x))?

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5


19. f(x)=7x-4. What is f(f(x))?

  1. 49x+32
  2. -49x-32
  3. -32x+49
  4. 32x+49
  5. 49x-32


20. You roll a die and flip 2 coins. What is the probability you roll a 6 and both coins come up heads?

  1. \dfrac{1}{2}
  2. \dfrac{1}{4}
  3. \dfrac{1}{6}
  4. \dfrac{1}{12}
  5. \dfrac{1}{24}


21. \angle A of hexagon ABCDEF has measure 300^{\circ}. What is the average of the measures of the other 5 angles?

  1. 60^{\circ}
  2. 72^{\circ}
  3. 84^{\circ}
  4. 48^{\circ}
  5. 36^{\circ}


22. \log_8{x} = -\dfrac{7}{3}. What is x?

  1. \dfrac{1}{128}
  2. \dfrac{1}{64}
  3. \dfrac{1}{32}
  4. \dfrac{1}{16}
  5. \dfrac{1}{8}


23. Which of the following is equivalent to \log{ \dfrac{a^{11}b}{c^{4}d} }?

  1. 11\log{a}+\log{b}-4\log{c}-\log{d}
  2. 11\log{a}+\log{b}+\log{c}-4\log{d}
  3. 11\log{a}+\log{b}+\log{c}+4\log{d}
  4. 11\log{a}-\log{b}-\log{c}-4\log{d}
  5. 11\log{a}-\log{b}+\log{c}+4\log{d}


24. A cube with sides of length 6 is inscribed in a sphere. What is the surface area of the sphere (surface area of a sphere = 4 \pi r^2)?

  1. 102 \pi
  2. 104 \pi
  3. 106 \pi
  4. 108 \pi
  5. 110 \pi


Answer key: 1D, 2A, 3C, 4B, 5E, 6C, 7A, 8B, 9D, 10D, 11A, 12C, 13C, 14B, 15A, 16C, 17E, 18D, 19E, 20E, 21C, 22A, 23A, 24D

 
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