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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 the value of ⁵√(32)?
32=2⁵. ⁵√32=2.
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2. What is the value of C(10,10)?
C(n,n)=1.
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3. Which function grows fastest as x→∞?
Factorial grows faster than exponential which grows faster than polynomial.
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4. 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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5. What is the partial sum formula for geometric series?
Geometric sum: Sₙ=a₁(rⁿ-1)/(r-1) for r≠1.
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6. What is the product of roots of x³-6x²+11x-6=0?
Product of all roots=(-1)³×(-6)/1=6 (Vieta's).
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7. Simplify: log(100) + log(10)
log(100)=2, log(10)=1. Sum = 3.
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8. What is the value of |3+4i|?
|3+4i|=√(9+16)=5.
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9. What is the sum of the first n terms of arithmetic: a₁=1, d=2, n=10?
Sₙ = n/2(2a₁+(n-1)d) = 10/2(2+18) = 5(20) = 100.
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10. Simplify: log(ab²)
log(ab²)=log(a)+log(b²)=log(a)+2log(b).
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11. What is the range of f(x)=2ˣ-5?
2ˣ>0 always. So 2ˣ-5>-5. Range: y>-5.
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12. What is the simplified form of e^(ln(x)+ln(y))?
e^(ln(xy))=xy.
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13. What is the 100th term of 5,8,11,14,...?
a₁₀₀=5+99(3)=5+297=302.
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14. What is the value of (2-3i)²?
4-12i+9i²=4-12i-9=-5-12i.
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15. What is the sum of first 10 terms of 3+6+12+24...?
S₁₀=3(2¹⁰-1)/(2-1)=3(1023)=3069.
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16. What is the general formula for sum of first n terms of arithmetic series?
Sₙ=n/2(a₁+aₙ).
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17. What is the conjugate of 5-3i?
Conjugate changes the sign of the imaginary part: 5+3i.
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18. What is ln(e³)?
ln(eˣ) = x. ln(e³) = 3.
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19. What is the value of ∑(k=1 to 5) k²?
1+4+9+16+25=55.
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20. What is the value of i^(4n+3) for any integer n?
i^(4n+3)=(i⁴)ⁿ×i³=1×(-i)=-i.
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21. What is the sum of the geometric series 2+2/3+2/9+... (infinite)?
S=2/(1-1/3)=2/(2/3)=3.
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22. What is the common ratio of 5, 15, 45, 135?
r = 15/5 = 3.
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23. What is the value of (3+4i)+(5-2i)?
(3+5)+(4-2)i=8+2i.
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24. What is the equation of the asymptotes of xy=4?
Hyperbola xy=4 has asymptotes x=0 and y=0.
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25. What is the value of (−i)⁴?
(-i)⁴=((-1)(i))⁴=(-1)⁴(i)⁴=1×1=1.
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26. What is the 3rd term of geometric sequence with a=1000, r=0.1?
a₃=1000×(0.1)²=10.
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27. What is the value of P(5,3)?
P(5,3)=5!/(5-3)!=5×4×3=60.
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28. If f(x)=x² and g(x)=x+3, what is (f/g)(x)?
(f/g)(x)=x²/(x+3).
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29. What is the value of 4ˡᵒᵍ₄⁽³⁾?
a^(logₐ(x))=x. 4^(log₄(3))=3.
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30. What is the sum of the arithmetic series 10+12+14+...+100?
n=46. S=46/2(110)=2530.
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