Reasonable Rates

Hard ACT Math Questions: 28 Challenging Practice Problems

There is a new ACT Book , much of which is available for free here

Looking for hard ACT Math questions to challenge your problem-solving skills? This practice set includes 28 difficult ACT Math problems covering algebra, geometry, functions, trigonometry, exponents, and other advanced topics.

These challenging problems are designed for students who already understand the fundamentals and want to strengthen their mathematical reasoning while preparing for a high ACT Math score.

How to Use This ACT Math Practice Set

Work through the questions at your own pace.
Show your calculations and identify the concepts required for each problem.
Check your answers after attempting each question.
Review the solution to understand the reasoning behind the correct answer.
Revisit the topics you find most difficult before moving on to another practice test.

These questions are intentionally challenging and are intended for advanced practice. They are not a timed test and should not be used to predict your ACT score.

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

$ 15$
$-15$
$ 225 $
$ 3\sqrt{6}i $
$ 6\sqrt{3}i $

Show correct answerHide correct answer

E

2. $5x^2+2x-8=0$. Solve for $x$.

$ x=\dfrac{-1 \pm \sqrt{41}}{5}$
$x=\dfrac{-1 \pm \sqrt{5}}{41}$
$x=\dfrac{-8 \pm \sqrt{2}}{5} $
$ x=\dfrac{-8 \pm \sqrt{5}}{2}$
$x=\dfrac{-2 \pm \sqrt{5}}{8} $

Show correct answerHide correct answer

A

3. Which of the following is equivalent to $|2x- C| > D$, where $C$ and $D$ are constants.

$ 2x+ C > D \text{ or } C + 2x> D$
$ 2x+ C > D \text{ or } C – 2x> D$
$ 2x– C > D \text{ or } C + 2x> D $
$ 2x– C > D \text{ or } C – 2x> D $
$ + C > D \text{ or } C +x> D$

Show correct answerHide correct answer

D

4. What is the minimum value of $x$ for which when $x$ is divided by $7$ the remainder is $4$ and when $x$ is divided by 5 the remainder is 2?

$12$
$21$
$23$
$ 32$
$34 $

Show correct answerHide correct answer

D

5. $2 ^{x^2 + 3x + 2} = 32$, what is $x$?

$ x=\dfrac{-2 \pm \sqrt{3}}{21}$
$ x=\dfrac{-2 \pm \sqrt{21}}{3} $
$ x=\dfrac{-21 \pm \sqrt{2}}{3} $
$ x=\dfrac{-3 \pm \sqrt{2}}{21} $
$ x=\dfrac{-3 \pm \sqrt{21}}{2} $

Show correct answerHide correct answer

E

6. What is the equation of the tangent line to the circle $(x-2)^2 + (y-5)^2 = 16$ at the point $(2,1)$?

$ x=1 $
$ y=1 $
$ 2x+5y=16 $
$ 5x+2y=16 $
$ x+y=5 $

Show correct answerHide correct answer

C

7. What is the area of the circle $(x -4)^2 + (y + 11) ^ 2 = 71$?

$ 71\pi $
$ 73\pi $
$ 75\pi $
$ 77\pi $
$ 79\pi $

Show correct answerHide correct answer

A

8. What is the equation of the factored form with integer coefficients $\dfrac{5}{2}, -\dfrac{3}{5}, 7i$, and $-7i$?

$ (2x+5)(3x+5)(x^2+49)=0$
$ (2x+5)(3x+5)(x^2-49)=0 $
$ (2x-5)(3x+5)(x^2+49)=0 $
$ (2x-5)(5x+3)(x^2+49)=0 $
$ (2x-5)(3x-5)(x^2-49)=0 $

Show correct answerHide correct answer

D

9. $8 ^{x + 1} = \dfrac{1}{32^{x+3}}$. Solve for $x$.

$ x=-\dfrac{2}{3} $
$ x=-\dfrac{3}{2} $
$ x=-\dfrac{4}{9} $
$ x=-\dfrac{9}{4} $
$ x=-\dfrac{3}{4} $

Show correct answerHide correct answer

D

10. At $186,000$ miles per second, how far in miles does a beam of light travel in one year in scientific notation?

$ 1.86 \times 10^3 $
$ 1.86 \times 10^5 $
$ 5.9 \times 10^3 $
$ 5.9 \times 10^5 $
$ 5.9 \times 10^{12} $

Show correct answerHide correct answer

E

11. If $\sin{x} = a, 0 < x < \dfrac{\pi}{2}$, what is $\tan{x}$?

$ \dfrac{a}{\sqrt{ 1- a^2}} $
$ \dfrac{a}{\sqrt{ a^2-1}} $
$ \dfrac{a}{\sqrt{ 1+ a^2}} $
$ \dfrac{a}{\sqrt{ 1+ a^2}} $
$ \dfrac{1+a^2}{\sqrt{a}} $

Show correct answerHide correct answer

A

12. If $\dfrac{(x + 5)!}{(x + 2)!} = 210$, then what does $\dfrac{(x+8)!}{(x+6)!}$ equal?

$ 34 $
$ 43 $
$ 45 $
$ 54 $
$ 90 $

Show correct answerHide correct answer

E

13. What is the distance between the points $(a , 2a + 5)$ and $(4a, a -1)$?

$ \sqrt {6a^2 + 10a + 36} $
$ \sqrt {6a^2 + 36a + 10} $
$ \sqrt {10a^2 + 6a + 36} $
$ \sqrt {10a^2 + 36a + 6} $
$ \sqrt {10a^2 + 12a + 36} $

Show correct answerHide correct answer

E

14. $\sqrt[5]{x^2 + x + 2} = 2$. Solve for $x$.

$ -6, \quad-5 $
$ -6,\quad5 $
$ 6,\quad-5 $
$ 6,\quad5 $
$ -5, \quad-6 $

Show correct answerHide correct answer

B

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

$ x^4+6x^3+16x^2+21x+1 $
$ x^4+12x^3+6x^2+18x+18 $
$ x^4+18x^3+6x^2+18x+12 $
$ x^4+6x^3+18x^2+18x+12 $
$ x^4+12x^3+18x^2+18x+6 $

Show correct answerHide correct answer

A

16. What is $\dfrac{3 \times 10^a}{8 \times 10 ^ b}$ in scientific notation?

$ 3.75 \times 10 ^ { a - b - 1} $
$ 3.75 \times 10 ^ { a + b - 1} $
$ 3.75 \times 10 ^ { a - b + 1} $
$ 3.75 \times 10 ^ { a + b +1} $
$ 3.75 \times 10 ^ { -a - b - 1} $

Show correct answerHide correct answer

A

17. What is $-\dfrac{37\pi}{5}$ radians in degrees between $0^{\circ}$ and $360^{\circ}$?

$ 36^{\circ} $
$ 72^{\circ} $
$ 108^{\circ} $
$ 144^{\circ} $
$ 162^{\circ} $

Show correct answerHide correct answer

C

18. For what value of $x$ is $y$ a minimum $y=3x^2+10x+11$?

$-\dfrac{3 }{ 5 } $
$ -\dfrac{ 5 }{ 3 } $
$ -\dfrac{ 4}{ 7 } $
$ -\dfrac{ 7 }{ 4 } $
$ -\dfrac{ 3}{7 } $

Show correct answerHide correct answer

B

19. $B=1000^{3C-4D}$. What is the $\log_{10} B$

$ 3C-6D $
$ 6C-9D $
$ 9C-12D $
$ 12C-15D $
$ 15C-18D $

Show correct answerHide correct answer

C

20. What is $(2x^3 y^{-5})^{-4}$?

$ \dfrac{y^{16}}{20 x^{12}} $
$ \dfrac{y^{20}}{16 x^{12}} $
$ \dfrac{y^{12}}{16 x^{20}} $
$ \dfrac{y^{20}}{20 x^{16}} $
$ \dfrac{y^{16}}{12 x^{20}} $

Show correct answerHide correct answer

B

21. $f(x) = -2x^5$. What is $f(f(x))$?

$ 4x^{10} $
$ 8x^{10} $
$ 16x^{20} $
$ 32x^{20} $
$ 64x^{25} $

Show correct answer

22. $f(x) = \sqrt{x}$. What is $f(f(x))$?

$ \sqrt{x} $
$ \sqrt[3]{x} $
$ \sqrt[4]{x} $
$ \sqrt[5]{x} $
$ \sqrt[6]{x} $

Show correct answerHide correct answer

C

23. $y=bx+5$ goes through the point $(3,1)$. What is $b$?

$ -2 $
$ -1 $
$ 1 $
$ 2 $
$ 3 $

Show correct answerHide correct answer

A

24. What is the units digit of $53^{175}$?

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

Show correct answerHide correct answer

D

25. What are all the solutions of $|x^2| + 3|x| - 10 = 0$?

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

Show correct answerHide correct answer

B

26. What is $\log_4 {\dfrac{1}{32}}$?

$ -\dfrac{ 1 }{ 4 } $
$ -\dfrac{ 1 }{ 8 } $
$ -\dfrac{ 1 }{ 32 } $
$ -\dfrac{ 2 }{ 5 } $
$ -\dfrac{ 5 }{2 } $

Show correct answerHide correct answer

E

27. $5x^2 - 5y^2 + 10x - 10y = 23$ is $a(n)$?

$ \text{ line } $
$ \text{ circle } $
$ \text{ ellipse } $
$ \text{ parabola } $
$ \text{ hyperbola } $

Show correct answerHide correct answer

E

28. $f(x) = e^{3x+2} + 4$. What is $f^{-1}(20)$?

$ 0.258 $
$ 0.285 $
$ 0.528 $
$ 0.582 $
$ 0.825 $

Show correct answerHide correct answer

A

Answer key: 1.E, 2.A, 3.D, 4.D, 5.E, 6.B, 7.A, 8.C, 9.D, 10.E, 11.A, 12.C, 13.C, 14.B, 15.D, 16.A, 17.C, 18.B, 19.C, 20.B, 21.E, 22.C, 23.A, 24.D, 25.B, 26.E, 27.E, 28.A

Learn About ACT Test Dates