Reasonable Rates

ACT Math Practice - Test 2

ACT Math Problems (Advanced ACT practice test math)

If you are looking for challenging ACT practice test math, this advanced practice set is designed specifically for high-achieving students aiming for top scores. This resource also works perfectly as daily ACT math practice, whether you are solving an ACT math question of the day or completing a full ACT practice math test for deeper preparation.

This collection, “ACT Math Problems – Test 2,” includes some of the most difficult ACT math questions, rewritten and optimized for deeper training and stronger problem-solving skills. The set functions as both targeted ACT math practice questions and a realistic ACT math practice test, helping students build confidence through consistent ACT practice math questions.

These questions are ideal for students targeting 30+ on ACT Math and wanting practice that goes beyond standard prep books. Students can use this material as a full ACT math test, a structured math ACT practice test, or as focused drills within their daily ACT preparation routine.

Why This ACT Math Practice Is Unique

Includes hard ACT Math questions similar to real exam patterns.
Problems are adapted from thousands of official-style questions and rewritten for higher difficulty.
Ideal for advanced students preparing for competitive score ranges.
Designed to support consistent ACT math practice through realistic and challenging problem sets.

Covers all core topics: algebra, geometry, trigonometry, functions, number theory, matrices, coordinate geometry, ratios, rates, and more.

This ACT book is a helpful resource for students looking to build strong foundations before test day.

Now, let’s begin with the math questions below.

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

$ 10 $
$ -5 $
$ 3+2\sqrt{6} $
$ 3+6\sqrt{2} $
$ 6+3\sqrt{2} $

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D

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

$ \dfrac{1}{2}< x < 2 $
$ \dfrac{1}{3}< x < 3 $
$ \dfrac{1}{4} < x< 4 $
$ \dfrac{1}{5} < x< 5 $
$ \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 $B=\left[ \begin{array}{cc} 2 & 3 \\ 1 & 5 \\ 2 & 4 \\ 1 & 2 \\ \end{array} \right]$?

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

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A

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

$ \dfrac{5 - i\sqrt{17}}{3} $
$ \dfrac{4 - i\sqrt{10}}{2} $
$ \dfrac{3 - i\sqrt{11}}{3} $
$ \dfrac{2 - i\sqrt{13}}{2} $
$ \dfrac{1 - i\sqrt{14}}{3} $

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E

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

$ 18\pi $
$ 24\pi $
$ 30\pi $
$ 36\pi $
$ 42\pi $

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D

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

$ -\dfrac{3\sqrt{22}}{22} $
$ $
$ \dfrac{3\sqrt{22}}{22} $
$ -\dfrac{4\sqrt{33}}{33} $
$ -\dfrac{7}{4} $

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C

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

$ 1 $
$ 3 $
$ 5 $
$ 7 $
$ 9 $

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A

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

$ \dfrac{3}{5} $
$ \dfrac{5}{3} $
$ \dfrac{5}{7} $
$ \dfrac{7}{5} $
$ \dfrac{7}{11} $

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E

9. $11$ is $8\%$ of what?

$ 3 $
$ 19 $
$ 88 $
$ 88.5 $
$ 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?

$ 200\% $
$ 300\% $
$ 427\% $
$ 573\% $
$ 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)

$ 73 \text{ ft/sec} $
$ 77\text{ ft/sec} $
$ 83\text{ ft/sec} $
$ 87\text{ ft/sec} $
$ 93\text{ ft/sec} $

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A

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

$ \pi $
$ 20 $
$ \dfrac{\pi}{20} $
$ 20\pi $
$ \dfrac{20}{\pi} $

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C

13. What is $i^{383}$? $(i=\sqrt{-1})$

$ -1 $
$ -i $
$ 1 $
$ i $
$ \dfrac{-1}{i} $

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B

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

$ 4x^2+6x+9 $
$ 4x^2-6x+9 $
$ 9x^2+6x+4 $
$ 9x^2 - 6x + 4 $
$ 9x^2 - 6x - 4 $

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B

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

$ 133.33 $
$ 33.33 $
$ 143.33 $
$ 146.67 $
$ 153.33 $

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B

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

$ x^2-2x+18 = 0 $
$x^2-3x+22 = 0 $
$ x^2-4x+26 = 0 $
$ x^2-5x+30 = 0 $
$ x^2-6x+34 = 0 $

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E

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

$ (5,-8 \pm 2 \sqrt{2}) $
$ (8,8 \pm 2 \sqrt{2}) $
$ (-5,8 \pm 2 \sqrt{2}) $
$ (-5,8 \pm 2 \sqrt{6}) $
$ (-5,5 \pm 2 \sqrt{2}) $

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D

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

$ -5 \pm \dfrac{5 \sqrt{7}}{15}i $
$ -5 \pm \dfrac{5 \sqrt{15}}{7}i $
$ -15 \pm \dfrac{5 \sqrt{5}}{7}i $
$ -15 \pm \dfrac{5 \sqrt{7}}{5}i $
$ -15 \pm \dfrac{7 \sqrt{5}}{7}i $

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B

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

$ (-5,1), (-5,15) $
$ (1,-5), (15,-5) $
$ (5,-1), (5,-15) $
$ (-5,1), (5,15) $
$ (-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?

$ 1100 \text{ $mm^2$} $
$ 2200 \text{ $mm^2$} $
$ 3300 \text{ $mm^2$} $
$ 4400 \text{ $mm^2$} $
$ 5500 \text{ $mm^2$} $

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D

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

$ 100 $
$ 200 $
$ 300 $
$ 400 $
$ 500 $

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A

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

$ x^4 -8x^3 i+24x^2+32x i-16 $
$ x^4 +8x^3 i-24x^2+32x i-16 $
$ x^4 -8x^3 i-24x^2-32x i-16 $
$ x^4 +8x^3 i+24x^2+32x i+16 $
$ x^4 +8x^3 i-24x^2-32x i+16 $

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E

23. Solve for $x, \ln {(x + 5)} = 3$?

$ e^5 - 3 $
$ e^5 + 3 $
$ e^3-5 $
$ e^3+5 $
$ 5e^3 $

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C

24. Solve for $x, e^{x - 2} = 8$?

$ \ln{2} -8 $
$ \ln{2} +8 $
$ \ln{8} -2 $
$ \ln{8} +2 $
$ 2\ln{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?

$ 5\% $
$ 6\% $
$ 7\% $
$ 8\% $
$ 9\% $

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E

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

$ \text{Monday} $
$ \text{Tuesday} $
$ \text{Wednesday} $
$ \text{Thursday} $
$ \text{Friday} $

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E

27. What is $\log_{3}{\sqrt {243}}$?

$ \dfrac{ 2}{ 5} $
$ \dfrac{5 }{2 } $
$ -\dfrac{ 2}{ 5} $
$ -\dfrac{ 5}{ 2} $
$ 81 $

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B

28. What is $\log_{2}{\dfrac{1}{128}}$?

$ -9 $
$ -7 $
$ -5 $
$ -3 $
$ -1 $

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B

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

$ 16 $
$ 177 $
$ 18 $
$ 19 $
$ 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?

$ 1 $
$ 2 $
$ 3 $
$ 4 $
$ 5 $

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D

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

$ \dfrac{\pi}{2 } $
$ \dfrac{\pi}{ 4 } $
$ \dfrac{\pi}{8 } $
$ \dfrac{\pi}{ 16 } $
$ \dfrac{\pi}{ 32 } $

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B

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

$ \sqrt{ 44 } $
$ \sqrt{ 54 } $
$ \sqrt{ 74} $
$ \sqrt{ 84} $
$ \sqrt{94 } $

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C

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

$ (-\infty, -2) \cup (-1, \infty) $
$ (-\infty, -2) \cup (1, \infty) $
$ (-\infty, -1) \cup (2, \infty) $
$ (-\infty, -2) \cup (2, \infty) $
$ (-\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?

$ 5 $
$ 6 $
$ 56 $
$ 65 $
$ 560\sqrt{94} $

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E

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

$ \left [\begin{array}{ccc} 3 & 6 & 9 \\ 4 & 8 & 12 \\ 5 & 10 & 15 \\ \end{array} \right ] $
$ \left [\begin{array}{ccc} 3 & 4 & 5 \\ 6 & 8 & 10 \\ 9 & 12 & 15 \\ \end{array} \right ] $
$ \left [\begin{array}{ccc} 3 & 8 & 15 \\ \end{array} \right ] $
$ \left [\begin{array}{c} 26 \\ \end{array} \right ] $
$ \left [\begin{array}{c} 3 \\ 8 \\ 15 \\ \end{array} \right ] \sqrt{94} $

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A

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

$ \text{No solution} $
$ (-2,5) $
$ (-2,3) \cup (3,5) $
$ (-\infty,-2) \cup (-2,3) \cup (3,5) \cup (5,\infty) $
$ \text{All reals} $

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E

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

$ 3 $
$ 6 $
$ 9 $
$ 18 $
$ 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?

$ 2+\sqrt{2 } $
$ 3+\sqrt{3 } $
$ 5+\sqrt{5 } $
$ 6+\sqrt{6 } $
$ 7+\sqrt{ 7} $

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C

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

$ 45^{\circ} $
$ 60^{\circ} $
$ 120^{\circ} $
$ 135^{\circ} $
$ 150^{\circ}7+\sqrt{7} $

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D

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

$ -10 $
$ -5 $
$ 0 $
$ 5 $
$ 107+\sqrt{7} $

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A

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

$ 3.7 \times 10^{-6} $
$ 3.7 \times 10^{-5} $
$ 3.7 \times 10^{-3} $
$ 3.7 \times 10^{5} $
$ 3.7 \times 10^{6} $

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B

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

$ 1010 $
$ 2020 $
$ 3030 $
$ 4040 $
$ 50507 + \sqrt{7} $

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E

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

$ f^{-1} (x) = \dfrac{x^2-5}{11} $
$ f^{-1} (x) = \dfrac{x^2-11}{5} $
$ f^{-1} (x) = \dfrac{x^5-2}{11} $
$ f^{-1} (x) = \dfrac{x^5-11}{2} $
$ f^{-1} (x) = \dfrac{x^{11}-2}{5} $

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D

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

$ \dfrac{\sqrt{5}}{26} $
$ \dfrac{\sqrt{5}}{62} $
$ \dfrac{\sqrt{25}}{6} $
$ \dfrac{\sqrt{26}}{5} $
$ \dfrac{\sqrt{62}}{5} $

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D

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

$ y=2+15x $
$ y=2-15x $
$ y=15-2x $
$ y=15+2x $
$ y=\dfrac{15}{2x} $

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C

46. $f(x)=x^2+3x+2$. What is $f(x+a)$?

$ f(x+a) = f(x)+f(a) $
$ f(x+a) = f(x)+f(a)-1 $
$ f(x+a) = f(x)+f(a)+ax $
$ f(x+a) = f(x)+f(a)+ax-1 $
$ 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?

$ 3: 4 $
$ 9:16 $
$ 27:64 $
$ 1:7 $
$ 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?

$ 3: 4 $
$ 9:16 $
$ 27:64 $
$ 1:7 $
$ 1:12 $

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B

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

$ 4 $
$ 6 $
$ 8 $
$ 10 $
$ 12 $

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A

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

$ \dfrac{2}{11} $
$ \dfrac{4}{121} $
$ \dfrac{11}{2} $
$ \dfrac{121}{4} $
$ \dfrac{121}{2} $

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D

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

$ \dfrac{3a-5b}{2} $
$ \dfrac{3a+5b}{2} $
$ \dfrac{5b-3a}{2} $
$ \dfrac{2}{3a+5b} $
$ \dfrac{2}{3a+5b} $

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B

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

$ -1 \text{ and } 1 $
$ -2 \text{ and }2 $
$ -3 \text{ and }3 $
$ -4 \text{ and }4 $
$ -5 \text{ and }5 $

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B

53. Solve $| 3x + 5 | = | -7|$

$ \dfrac{2}{3},-4 $
$ \dfrac{3}{5},-7 $
$ \dfrac{3}{7},-5 $
$ \dfrac{3}{4},-2 $
$ \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?

$ (x-3)^2+(y-8)^2=25 $
$ (x+7)^2+(y-5)^2=25 $
$ (x-2)^2+(y-11)^2=25 $
$ (x+3)^2+(y-8)^2=25 $
$ (x-3)^2+(y+8)^2=25 $

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D

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

$ \dfrac{ 3 }{4} $
$ \dfrac{ 4 }{3} $
$ \dfrac{ 5 }{32} $
$ \dfrac{ 7}{19} $
$ \dfrac{19 }{7} $

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E

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

$ \dfrac{2}{15} $
$ \dfrac{15}{2} $
$ \dfrac{5}{12} $
$ \dfrac{12}{5} $
$ \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?

$ 11,\dfrac{8}{3} $
$ 11,\dfrac{5}{3} $
$ 11,\dfrac{5}{4} $
$ 5,\dfrac{3}{11} $
$ 5,\dfrac{8}{11} $

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A

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

$ x>-\dfrac{3}{4} $
$ x>\dfrac{3}{4} $
$ x>-\dfrac{4}{3} $
$ x>\dfrac{4}{3} $
$ 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?

$ x^2-7x+20 $
$ x^2-8x+10 $
$ x^2+7x-20 $
$ x^2+8x+10 $
$ x^2-7x+8 $

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A

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

$ -25 $
$ -15 $
$ 15 $
$ 25 $
$ 225 $

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B

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