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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 sum of first 100 odd numbers?
Sum of first n odd numbers=n². 100²=10000.
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2. What is the 8th term of 1,3,5,7,...?
a₈=1+(7)(2)=15.
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3. What is the value of log₂(√8)?
√8=2^(3/2). log₂=3/2.
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4. What type of conic is x²/9+y²/16=1?
Different denominators under x² and y², both positive: ellipse.
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5. What is the asymptote of y=2ˣ+3?
Exponential function 2ˣ has asymptote y=0. Adding 3 shifts to y=3.
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6. 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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7. What is the value of ∑(k=1 to 4) 2k?
2+4+6+8=20.
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8. What is the sum of the geometric series 1 + 2 + 4 + 8 + ... + 64?
S = a(rⁿ-1)/(r-1) = 1(2⁷-1)/(2-1) = 127.
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9. What is log₃(27)+log₃(3)?
3+1=4.
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10. Solve: 2ˣ = 16
16 = 2⁴. So x = 4.
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11. What is the value of ln(e⁵)?
ln(eˣ)=x. ln(e⁵)=5.
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12. Which of the following is an odd function?
f(-x)=(-x)³+(-x)=-x³-x=-(x³+x)=-f(x). Odd.
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13. What is the value of ∑(k=1 to 5) k²?
1+4+9+16+25=55.
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14. 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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15. How many real solutions does x² + 4 = 0 have?
x² = -4. No real solutions since square of real is non-negative.
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16. What is the vertex of y = -2(x+1)² + 7?
Vertex form y = a(x-h)²+k. h=-1, k=7. Vertex = (-1, 7).
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17. What is the 100th term of 5,8,11,14,...?
a₁₀₀=5+99(3)=5+297=302.
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18. What is the range of f(x)=2ˣ-5?
2ˣ>0 always. So 2ˣ-5>-5. Range: y>-5.
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19. What is log₃(81)?
3⁴ = 81, so log₃(81) = 4.
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20. What is (1+i)²?
(1+i)²=1+2i+i²=1+2i-1=2i.
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21. What is the product of (2+i) and its conjugate?
(2+i)(2-i)=4+1=5.
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22. A geometric sequence has a₁=5 and a₄=40. What is r?
5×r³=40. r³=8. r=2.
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23. 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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24. What is the equation of the parabola with focus (0,3) and directrix y=-3?
p=3. y=x²/(4p)=x²/12. Or x²=12y.
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25. What is the remainder when x⁴-1 is divided by (x+1)?
f(-1)=(-1)⁴-1=1-1=0.
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26. What is the horizontal asymptote of f(x) = 5x²/(x²+1)?
Equal degrees: horizontal asymptote = 5/1 = 5.
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27. Simplify: log(ab²)
log(ab²)=log(a)+log(b²)=log(a)+2log(b).
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28. What is the value of (1+i)⁴?
(1+i)²=2i. (2i)²=-4.
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29. What is the simplified form of (x³-8)/(x-2)?
x³-8 = (x-2)(x²+2x+4). Divide: x²+2x+4.
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30. What is the value of aₙ for n=5 in aₙ=(-1)ⁿ×2ⁿ?
(-1)⁵×2⁵=-1×32=-32.
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