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ACT Math - Advanced Math Practice Test
Test your Advanced Math skills with 30 random practice test questions.
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1. What is (2+3i)-(3-i)?
(2-3)+(3+1)i=-1+4i.
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2. What is the inverse of f(x)=10ˣ?
Inverse of 10ˣ is log₁₀(x) (common log).
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3. What is the 7th term of arithmetic sequence with first term 4 and common difference 5?
a₇ = 4 + (7-1)(5) = 4 + 30 = 34.
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4. If f(x) = eˣ, what is f'(x)?
The derivative of eˣ is eˣ — it is its own derivative.
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5. What is (a+bi)+(a-bi)?
Sum of conjugates=2a.
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6. Simplify: (5-2i)+(3+4i)
(5+3)+(−2+4)i=8+2i.
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7. 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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8. What is the horizontal asymptote of f(x) = 5x²/(x²+1)?
Equal degrees: horizontal asymptote = 5/1 = 5.
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9. What is the value of ln(e⁻³)?
ln(e^x)=x. ln(e⁻³)=-3.
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10. What is the 5th term of the sequence aₙ=2n²-1?
a₅=2(25)-1=49.
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11. What are the roots of x³ - 8 = 0 (real roots only)?
x³=8. Real cube root: x=2.
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12. What is the simplified form of (2+3i)/(1+i)?
Multiply by conjugate: (2+3i)(1-i)/((1+i)(1-i))=(2-2i+3i-3i²)/2=(2+i+3)/2=(5+i)/2=5/2+1/2i.
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13. What is the equation of the axis of symmetry for x=(y-3)²+2?
This is a horizontal parabola with vertex (2,3). Axis of symmetry: y=3.
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14. What is the sum of the series ∑(n=1 to ∞) (1/2)ⁿ?
S=a/(1-r)=(1/2)/(1-1/2)=1.
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15. What is the 8th term of 1,3,5,7,...?
a₈=1+(7)(2)=15.
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16. What is the value of ∑(k=0 to 4) 3^k?
1+3+9+27+81=121.
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17. What is the end behavior of f(x) = 2x⁴?
Positive leading coefficient + even degree: both ends rise.
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18. 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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19. Which of the following has a hole at x=3?
(x-3)(x+3)/(x-3)=x+3 has hole at x=3.
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20. What is the value of eˡⁿ⁽⁵⁾?
e^(ln x) = x. So e^(ln 5) = 5.
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21. What is log(10⁵)?
log₁₀(10⁵)=5.
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22. What is the sum of the first 6 terms of 1+3+9+27...?
S₆ = 1(3⁶-1)/(3-1) = (729-1)/2 = 728/2 = 364.
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23. What is the domain of f(x)=log(2x-6)?
2x-6>0→x>3.
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24. What type of function is f(x) = |x|?
f(x) = |x| is defined piecewise: x if x≥0, -x if x<0.
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25. If the 3rd term of a geometric sequence is 12 and r=2, what is the 1st term?
a₃ = a₁r² → 12 = a₁(4) → a₁ = 3.
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26. What is the simplified form of e^(ln(x)+ln(y))?
e^(ln(xy))=xy.
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27. What is the value of ⁵√(32)?
32=2⁵. ⁵√32=2.
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28. What is the sum of the first 20 terms of 1,2,3,...,20?
S=20×21/2=210.
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29. What is the formula for the nth term of geometric: a₁=2, r=3?
aₙ = a₁×r^(n-1) = 2×3^(n-1).
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30. What is log_a(1)?
logₐ(1) = 0 because a⁰ = 1.
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