30 random questions |30 minutes | Questions are randomized every attempt. Good luck!
ACT Math - Advanced Math Practice Test
Test your Advanced Math skills with 30 random practice test questions.
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1. What is the common ratio in 100, 50, 25, 12.5?
r=50/100=1/2.
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2. What is the product of (2+i) and its conjugate?
(2+i)(2-i)=4+1=5.
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3. What is log(a) + log(b) equal to?
Product rule: log(a) + log(b) = log(ab).
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4. Simplify: log₂(32) - log₂(4)
log₂(32/4) = log₂(8) = 3.
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5. What is log₂(8) + log₂(4)?
log₂(8)=3, log₂(4)=2. Sum=5. Or log₂(32)=5.
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6. What is the vertical asymptote of f(x) = 1/(x-4)?
Denominator = 0 when x = 4. Vertical asymptote at x = 4.
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7. Simplify: (2i)³
(2i)³ = 8i³ = 8(-i) = -8i.
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8. What is the range of f(x)=2ˣ-5?
2ˣ>0 always. So 2ˣ-5>-5. Range: y>-5.
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9. What is the number of roots (counting multiplicity) of a degree-5 polynomial?
By the Fundamental Theorem of Algebra, a degree-n polynomial has exactly n roots.
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10. What is f⁻¹(x) if f(x) = 3x + 6?
y = 3x+6 → x = 3y+6 → y = (x-6)/3 = x/3 - 2.
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11. For what values is f(x)=(x+1)/(x²-4) undefined?
x²-4=0 → x=±2.
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12. What is the value of a₆ in a geometric sequence where a₂=6 and a₄=54?
r²=54/6=9→r=3. a₂=6→a₁=2. a₆=2×3⁵=486.
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13. What is the product (1+i)(1-i)(1+2i)?
(1+i)(1-i)=2. 2(1+2i)=2+4i.
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14. If f is an even function and f(2)=5, what is f(-2)?
Even: f(-x)=f(x). f(-2)=f(2)=5.
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15. What is the simplified form of (1-i)/(1+i)?
Multiply by conjugate: (1-i)²/((1+i)(1-i))=(1-2i-1)/2=-2i/2=-i.
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16. If f(x)=3ˣ, what is f(-2)?
3⁻²=1/3²=1/9.
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17. What is the general term of 1, -1, 1, -1,...?
a₁=1=(-1)^(1+1)=1✓. a₂=-1=(-1)^(2+1)=-1✓.
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18. What is the value of ∑(k=1 to 5) k²?
1+4+9+16+25=55.
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19. What is the inverse of f(x) = (x-3)/2?
y = (x-3)/2. Swap: x = (y-3)/2. Solve: y = 2x+3.
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20. What is log₂(32×8)?
log₂(32)+log₂(8)=5+3=8.
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21. What is the conjugate of 5-3i?
Conjugate changes the sign of the imaginary part: 5+3i.
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22. What is the sum of first 10 terms of 1+3+9+27... (geometric)?
S₁₀ = 1(3¹⁰-1)/(3-1) = (59049-1)/2 = 29524.
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23. What is the value of log(√10)?
log(10^(1/2))=1/2.
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24. What is the value of P(8,0)?
P(n,0)=n!/(n-0)!=n!/n!=1.
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25. What is the value of (2-3i)²?
4-12i+9i²=4-12i-9=-5-12i.
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26. What is the inverse of f(x)=eˣ⁺¹?
y=e^(x+1). x=e^(y+1). y+1=ln(x). y=ln(x)-1.
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27. What is the value of (1+i)⁴?
(1+i)²=2i. (2i)²=-4.
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28. What is the sum of first 5 terms of 2, -6, 18, -54...?
r=-3. S₅=2((-3)⁵-1)/(-3-1)=2(-243-1)/(-4)=2(-244)/(-4)=122. Wait: S₅=2(1-(-3)⁵)/(1-(-3))=2(1+243)/4=2(244)/4=122. Index 1.
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29. What are the zeros of f(x) = x²(x-3)(x+1)?
Set f(x)=0: x²=0→x=0, x-3=0→x=3, x+1=0→x=-1.
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30. What is the relationship between the number of roots and the degree of a polynomial?
FTA: a degree-n polynomial has exactly n roots.
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