AP ECET MATHEMATICS Multiple Choice Questions (MCQs)
AP ECET Mathematics - Partial Fractions
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What is the first step in resolving a given rational function into partial fractions?
- a) Factor the denominator
- b) Add the fractions
- c) Multiply the numerator
- d) Simplify the numerator
Answer: a) Factor the denominator
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For a rational function with a denominator of the form (x² + 1)(x + 1), the partial fraction decomposition would require how many terms?
- a) 1
- b) 2
- c) 3
- d) 4
Answer: b) 2
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What type of partial fraction decomposition is used when the denominator has repeated linear factors?
- a) Simple fraction decomposition
- b) Mixed fraction decomposition
- c) Generalized partial fractions
- d) Multiple partial fractions
Answer: c) Generalized partial fractions
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Which of the following is the correct partial fraction decomposition for the rational function 3x + 5/(x + 1)(x - 2)?
- a) A/x + 1 + B/x - 2
- b) A/x - 1 + B/x + 2
- c) A/x + 1 + B/x + 2
- d) A/x - 2 + B/x + 1
Answer: a) A/x + 1 + B/x - 2
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When resolving partial fractions, the form of the partial fractions will depend on the type of which factor?
- a) Numerator
- b) Denominator
- c) Coefficients
- d) Exponent
Answer: b) Denominator
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In the partial fraction decomposition of 1/(x - 2)(x + 3), what are the values of A and B if we equate the rational function to A/x - 2 + B/x + 3?
- a) A = 1, B = -1
- b) A = 2, B = 3
- c) A = 3, B = -2
- d) A = -3, B = 2
Answer: c) A = 3, B = -2
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What does the term "proper fraction" mean in the context of partial fractions?
- a) The numerator's degree is less than the denominator's degree
- b) The denominator is a prime number
- c) The numerator's degree is higher than the denominator's degree
- d) The fraction contains only whole numbers
Answer: a) The numerator's degree is less than the denominator's degree
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Which of the following is the correct decomposition for 4/x(x + 2)?
- a) A/x + B/x + 2
- b) A/x + B/x - 2
- c) A/x + 2 + B/x
- d) A/x + 1 + B/x + 2
Answer: a) A/x + B/x + 2
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What is the partial fraction decomposition for 2x + 3/(x + 1)(x + 2)?
- a)
- b)
- c)
- d)
Answer: b)
- a)
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Which of the following is true about the partial fraction decomposition of a rational function with repeated factors in the denominator?
- a) Each factor contributes a separate fraction
- b) Only linear factors contribute fractions
- c) Powers of each factor contribute fractions with higher degrees
- d) No partial fraction decomposition is possible
Answer: c) Powers of each factor contribute fractions with higher degrees
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When a rational function has a denominator of the form (x - 1)2(x + 3), how do we resolve it into partial fractions?
- a) The fraction will have terms of the form A/x - 1 +
- b) The fraction will have terms of the form
- c) The fraction will have terms of the form
- d) The fraction will have terms of the form
Answer: a) The fraction will have terms of the form
- a) The fraction will have terms of the form A/x - 1 +
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For a rational function of the form x^2 + 3x + 2/x^3 + x^2 - 2x, what is the first step in the partial fraction decomposition process?
- a) Factor the numerator
- b) Factor the denominator
- c) Multiply the numerator and denominator
- d) Simplify the coefficients
Answer: b) Factor the denominator
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Which of the following is the partial fraction decomposition for 1/(x - 3)(x + 4)?
- a)
- b)
- c)
- d)
Answer: a)
- a)
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