Rational and Sequence Assessment Worksheet
Rational & Sequence Assessment
Standardized Mastery Evaluation • Version 1.0
Name:
Date:
Testing Protocol:
Solve each problem in the provided work area. Choose the best answer. All rational expressions must be in simplest form. For sequences, assume \(n \geq 1\).
Part I: Rational Expressions
1. The variables \(x\) and \(y\) vary inversely. If \(x = 7\) and \(y = 2\), which equation relates \(x\) and \(y\)?
A) \(y = \frac{14}{x}\)
B) \(y = \frac{3.5}{x}\)
C) \(y = 14x\)
D) \(y = x + 5\)
2. Given \(x\) and \(y\) vary inversely where \(x = -6\) and \(y = 5\), find \(y\) when \(x = -2\).
A) \(y = -30\)
B) \(y = 15\)
C) \(y = -10\)
D) \(y = \frac{5}{3}\)
3. Consider the function \(y = \frac{1}{x-3} + 4\). Identify the vertical and horizontal asymptotes.
A) \(x=3, y=4\)
B) \(x=-3, y=4\)
C) \(x=4, y=3\)
D) \(x=3, y=-4\)
4. In the function \(y = \frac{1}{x-h} + k\), if the graph is shifted 5 units left and 2 units down, what are the values of \(h\) and \(k\)?
A) \(h=5, k=2\)
B) \(h=-5, k=-2\)
C) \(h=-5, k=2\)
D) \(h=5, k=-2\)
5. Simplify the expression: \(\frac{3x^3}{6y^2} \cdot \frac{4y^5}{7x^4}\)
A) \(\frac{2y^3}{7x}\)
B) \(\frac{12y^3}{42x}\)
C) \(\frac{2y^7}{7x^7}\)
D) \(\frac{7y^3}{2x}\)
6. Simplify the expression: \(\frac{x^2}{x-2} \cdot \frac{x^2+3x-10}{x^2+2x}\)
A) \(x+5\)
B) \(\frac{x+5}{x}\)
C) \(x-5\)
D) \(\frac{x(x+5)}{x+2}\)
7. Perform the division: \(\frac{3h^3}{7j^5} \div \frac{18h}{21j^3}\)
A) \(\frac{h^2}{2j^2}\)
B) \(\frac{h^2j^2}{2}\)
C) \(\frac{2h^2}{j^2}\)
D) \(\frac{h^4}{2j^8}\)
8. Simplify: \(\frac{x^2+3x+2}{x^2-1} \cdot \frac{x^2-6x+5}{x^2+5x+6}\)
A) \(\frac{x-5}{x+3}\)
B) \(\frac{x+2}{x+3}\)
C) \(\frac{(x+1)(x-5)}{(x-1)(x+3)}\)
D) \(\frac{x-5}{x-3}\)
9. Subtract the rational expressions: \(\frac{2x+5}{x+3} - \frac{x-2}{x+3}\)
A) \(1\)
B) \(\frac{x+7}{x+3}\)
C) \(\frac{3x+3}{x+3}\)
D) \(\frac{x-7}{x+3}\)
10. Determine the Least Common Denominator (LCD) of \(\frac{1}{x+2}\) and \(\frac{3}{x^2-4}\).
A) \(x^2-4\)
B) \((x+2)(x^2-4)\)
C) \(x+2\)
D) \((x+2)^2(x-2)\)
11. Find the sum: \(\frac{1}{x+2} + \frac{3}{x^2-4}\)
A) \(\frac{x+1}{x^2-4}\)
B) \(\frac{4}{x^2+x-2}\)
C) \(\frac{x+5}{x^2-4}\)
D) \(\frac{x+2}{x^2-4}\)
12. Subtract: \(\frac{x^2+x-6}{x^2+9x+14} - \frac{8}{x+7}\)
A) \(\frac{x-11}{x+7}\)
B) \(\frac{x+3}{x+7}\)
C) \(\frac{x^2+x-14}{x^2+9x+14}\)
D) \(\frac{x-1}{x+7}\)
13. A company has a fixed cost of $300 and produces muffins for $0.12 each. What is the expression for the average cost \(\bar{C}\) to produce \(x\) muffins?
A) \(\bar{C} = \frac{300 + 0.12x}{x}\)
B) \(\bar{C} = 300x + 0.12\)
C) \(\bar{C} = \frac{0.12x}{300}\)
D) \(\bar{C} = \frac{x}{300 + 0.12x}\)
14. Using the formula from Question 13, how many muffins must be produced for the average cost to be $0.25?
A) \(2,308\)
B) \(1,200\)
C) \(2,500\)
D) \(1,538\)
15. Solve for \(x\): \(\frac{x}{x-2} = \frac{2}{x-2} + 2\)
A) \(x = 2\)
B) \(x = 6\)
C) No Solution
D) \(x = 4\)
16. Simplify the complex fraction: \(\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}\)
A) \(\frac{y+x}{y-x}\)
B) \(\frac{x+y}{x-y}\)
C) \(1\)
D) \(-1\)
17. Determine the domain of the function \(f(x) = \frac{x^2-9}{x^2-x-6}\).
A) All reals except \(3, -2\)
B) All reals except \(-3, 2\)
C) All reals except \(3\)
D) All real numbers
18. In the function \(f(x) = \frac{x-3}{x^2-9}\), which feature is found at \(x=3\)?
A) Vertical Asymptote
B) Removable Discontinuity (Hole)
C) Horizontal Asymptote
D) x-intercept
19. Person A paints a room in 4 hours. Person B paints it in 6 hours. How long does it take working together?
A) \(5.0\) hours
B) \(2.4\) hours
C) \(10.0\) hours
D) \(3.0\) hours
20. If \(z\) varies jointly with \(x\) and \(y\), and \(z=24\) when \(x=2, y=3\), find the constant of variation \(k\).
A) \(k = 4\)
B) \(k = 6\)
C) \(k = 2\)
D) \(k = 12\)
Part II: Sequences & Series
21. Find the next term in the arithmetic sequence: \(2, 9, 16, 23, \dots\)
A) \(30\)
B) \(31\)
C) \(29\)
D) \(32\)
22. What is the rule for the \(n\)th term of the sequence: \(2, 9, 16, 23, \dots\)?
A) \(a_n = 7n - 5\)
B) \(a_n = 7n + 2\)
C) \(a_n = 2n + 7\)
D) \(a_n = 7n + 5\)
23. Write a rule for the \(n\)th term of the sequence: \(-3, 6, -9, 12, \dots\)
A) \(a_n = 3n(-1)^n\)
B) \(a_n = -3n\)
C) \(a_n = 3n(-1)^{n-1}\)
D) \(a_n = (-3)^n\)
24. Evaluate the sum: \(\sum_{n=1}^{7} (2n - 3)\)
A) \(35\)
B) \(49\)
C) \(21\)
D) \(42\)
25. Evaluate the geometric sum: \(\sum_{k=1}^{6} 9(-2)^{k-1}\)
A) \(-189\)
B) \(189\)
C) \(-567\)
D) \(-171\)
26. Classify the sequence: \(-20, -16, -12, -8, \dots\)
A) Arithmetic
B) Geometric
C) Neither
D) Exponential
27. Find the rule for the \(n\)th term of the geometric sequence: \(-1, 4, -16, 64, \dots\)
A) \(a_n = -1(-4)^{n-1}\)
B) \(a_n = (-4)^n\)
C) \(a_n = -1(4)^{n-1}\)
D) \(a_n = -4(-1)^{n-1}\)
28. In an arithmetic sequence, \(a_9 = 25\) and \(d = 4\). Determine the general rule for \(a_n\).
A) \(a_n = 4n - 11\)
B) \(a_n = 4n + 9\)
C) \(a_n = 4n + 25\)
D) \(a_n = 4n - 7\)
29. Find the 10th term (\(a_{10}\)) for the sequence: \(\frac{1}{5}, \frac{2}{10}, \frac{3}{15}, \dots\)
A) \(\frac{1}{5}\)
B) \(\frac{10}{50}\)
C) Both A and B
D) \(\frac{1}{50}\)
30. Find the sum of the series: \(1 + 2 + 3 + \dots + 12\)
A) \(78\)
B) \(144\)
C) \(66\)
D) \(91\)
Mathematics Laboratory • Structural Analysis Dept
File Ref: ALG2_UNIT_78_ASSESS
Assessment Answer Key
Teacher Answer Key
Confidential
Rational & Sequence Assessment • Version 1.0
Part I: Rational Expressions
1. A Inverse Variation
2. B Variation Solving
3. A Graphing Shifts
4. B Constants h, k
5. A Simplify Monomials
6. A Rational Mult.
7. A Rational Div.
8. A Poly Mult.
9. B Subtraction
10. A LCD ID
11. A Rational Addition
12. A Rational Subt.
13. A Avg Cost Modeling
14. A Solving Averages
15. C Extraneous Solutions
16. A Complex Fractions
17. A Domain Restrictions
18. B Holes vs Asymptotes
19. B Work Rate
20. A Joint Variation
Part II: Sequences & Series
21. A Next Term
22. A nth Term Rule
23. A Alternating nth
24. A Sigma Notation
25. B Geometric Sum
26. A Classification
27. A Geometric nth
28. A Finding Rule
29. C Pattern Recognition
30. A Arithmetic Sum
Grading Scale
- 27-30 CORRECT: EXCELLENT (A)
- 24-26 CORRECT: PROFICIENT (B)
- 21-23 CORRECT: DEVELOPING (C)
- < 21 CORRECT: REVIEW NEEDED (F)
Explanatory Notes
Q15: Solving algebraically gives x = 2. However, substituting x = 2 into the original equation creates a zero denominator in the term x / (x - 2). This solution is extraneous, so there is No Solution.
Q19: The combined rate is 1/4 + 1/6 = 5/12 rooms per hour. To find the time for one room, take the reciprocal: 12/5 = 2.4 hours.
Q29: All terms follow the pattern n / (5n), which reduces to 1/5. Since 1/5 and 10/50 are mathematically equivalent, choice C is the correct selection.
Q30: Using the arithmetic series sum formula: \( S_{12} = \frac{12(1 + 12)}{2} = 78 \).
