3rd ACT Math Test | Perfectmathsat

3rd ACT Questions Test

There is a new book, much of which is available for free here.

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

A. $15$

B. $-15$

C. $225$

D. $3 \sqrt {6}i$

E. $6 \sqrt {3}i$

$147$

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

A. $x=\dfrac{-1 \pm \sqrt{41}}{5}$

B. $x=\dfrac{-1 \pm \sqrt{5}}{41}$

C. $x=\dfrac{-8 \pm \sqrt{2}}{5}$

D. $x=\dfrac{-8 \pm \sqrt{5}}{2}$

E. $x=\dfrac{-2 \pm \sqrt{5}}{8}$

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

A. 2 $x$ + C > D or C + 2 $x$ > D

B. 2 $x$ + C > D or C – 2 $x$ > D

C. 2 $x$ – C > D or C + 2 $x$ > D

D. 2 $x$ – C > D or C – 2 $x$ > D

E. $x$ + C > D or C + $x$ > D

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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?

A. 12

B. 21

C. 23

D. 32

E. 34

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

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

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

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

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

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

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

A. $x=1$

B. $y=1$

C. $2x+5y=16$

D. $5x+2y=16$

E. $x+y=5$

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

A. $71\pi$

B. $73\pi$

C. $75\pi$

D. $77\pi$

E. $79\pi$

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

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

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

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

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

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

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

A. $x=-\dfrac{2}{3}$

B. $x=-\dfrac{3}{2}$

C. $x=-\dfrac{4}{9}$

D. $x=-\dfrac{9}{4}$

E. $x=-\dfrac{3}{4}$

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

A. $1.86 \times 10^3$

B. $1.86 \times 10^5$

C. $5.9 \times 10^3$

D. $5.9 \times 10^5$

E. $5.9 \times 10^{12}$

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

A. $\dfrac{a}{\sqrt{ 1- a^2}}$

B. $\dfrac{a}{\sqrt{ a^2-1}}$

C. $\dfrac{a}{\sqrt{ 1+ a^2}}$

D. $\dfrac{1-a^2}{\sqrt{ a}}$

E. $\dfrac{1+a^2}{\sqrt{a}}$

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

A. 34

B. 43

C. 45

D. 54

E. 90

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

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

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

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

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

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

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

A. –6, –5

B. –6, 5

C. 6, –5

D. 6, 5

E. –5, –6

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

A. $x^4+6x^3+16x^2+21x+1$

B. $x^4+12x^3+6x^2+18x+18$

C. $x^4+18x^3+6x^2+18x+12$

D. $x^4+6x^3+18x^2+18x+12$

E. $x^4+12x^3+18x^2+18x+6$

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

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

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

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

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

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

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

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

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

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

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

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

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

A. $36^\circ$

B. $72^\circ$

C. $108^\circ$

D. $144^\circ$

E. $162^\circ$

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

A. $-\dfrac{3}{5}$

B. $-\dfrac{5}{3}$

C. $-\dfrac{4}{7}$

D. $-\dfrac{7}{4}$

E. $-\dfrac{3}{7}$

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

A. $3C - 6 D$

B. $6C - 9D$

C. $9C - 12 D$

D. $12C - 15 D$

E. $15C - 18 D$

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

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

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

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

D. $\dfrac{y^{20}}{20 x^{16}}$

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

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

A. $4x^{10}$

B. $8x^{10}$

C. $16x^{20}$

D. $32x^{20}$

E. $64x^{25}$

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

A. $\sqrt{x}$

B. $\sqrt[3]{x}$

C. $\sqrt[4]{x}$

D. $\sqrt[5]{x}$

E. $\sqrt[6]{x}$

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

A. –2

B. –1

C. 1

D. 2

E. 3

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

A. 1

B. 3

C. 5

D. 7

E. 9

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

A. 1 and –1

B. 2 and –2

C. 3 and –3

D. 4 and –4

E. 5 and –5

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

A. $-\dfrac{1}{4}$

B. $-\dfrac{1}{8}$

C. $-\dfrac{1}{32}$

D. $-\dfrac{2}{5}$

E. $-\dfrac{5}{2}$

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

A. line

B. circle

C. ellipse

D. parabola

E. hyperbola

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

A. 0.258

B. 0.285

C. 0.528

D. 0.582

E. 0.825

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

3rd ACT Questions Test

There is a new book, much of which is available for free here.

The best and most challenging ACT math problems on the web in 2024.