Grade: 12 Subject: Mathematics Unit: Precalculus Completion SAT: AdvancedMath ACT: Math

Unit Quiz: Precalculus Completion

Instructions

  • Time: 50 minutes
  • Questions 1-5: No calculator
  • Questions 6-12: Calculator allowed
  • Show all work for full credit

Section A: No Calculator

1. Evaluate: lim(x→3) (x² - 9)/(x - 3)

2. Find the 10th term of the arithmetic sequence: 5, 8, 11, 14, ...

3. Evaluate: lim(x→∞) (3x² + 2x)/(x² - 1)

4. Find the sum of the infinite geometric series: 12 + 4 + 4/3 + 4/9 + ...

5. For f(x) = (x² - 4)/(x - 2), identify any holes or vertical asymptotes.

Section B: Calculator Allowed

6. Find the sum of the first 20 terms of the arithmetic sequence where a₁ = 7 and d = 3.

7. A geometric sequence has a₁ = 5 and r = 1.5. Find the sum of the first 8 terms. Round to two decimal places.

8. Evaluate: lim(x→0) sin(3x)/x

9. Find all horizontal and vertical asymptotes of f(x) = (2x² - 8)/(x² - 4x + 3)

10. The 5th term of a geometric sequence is 48 and the 8th term is 384. Find the common ratio and the first term.

11. Use the definition of limit to determine if lim(x→2⁺) 1/(x-2) exists. If so, find its value.

12. An investment of $1000 earns 5% interest compounded annually. Write a sequence formula for the value after n years and find the value after 10 years.

Answer Key

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  1. 6 — Factor: (x+3)(x-3)/(x-3) = x+3; as x→3, this equals 6
  2. 32 — a₁₀ = 5 + (10-1)(3) = 5 + 27 = 32
  3. 3 — Divide by x²: (3 + 2/x)/(1 - 1/x²) → 3/1 = 3 as x→∞
  4. 18 — First term a = 12, r = 1/3; S = a/(1-r) = 12/(2/3) = 18
  5. Hole at x = 2 — f(x) = (x+2)(x-2)/(x-2) = x+2 with removable discontinuity at x=2; no vertical asymptotes
  6. 710 — S₂₀ = 20/2(2(7) + 19(3)) = 10(14 + 57) = 710
  7. ≈ 127.99 — S₈ = 5(1.5⁸ - 1)/(1.5 - 1) ≈ 5(25.63 - 1)/0.5 ≈ 127.99
  8. 3 — Using lim(x→0) sin(x)/x = 1: lim sin(3x)/x = lim 3·sin(3x)/(3x) = 3·1 = 3
  9. HA: y = 2; VA: x = 1 and x = 3 — Degrees equal so HA is ratio of leading coefficients; VA where denominator = 0
  10. r = 2, a₁ = 3 — 384/48 = r³ so r = 2; 48 = a₁·2⁴ so a₁ = 3
  11. +∞ (does not exist as a finite limit) — As x approaches 2 from the right, 1/(x-2) → +∞
  12. aₙ = 1000(1.05)ⁿ; a₁₀ ≈ $1628.89 — Geometric sequence with r = 1.05