Partial Fractions MCQs ECAT
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How To Solve Partial Fraction Problems?
Question:
Which Partial Fraction Decomposition is Correct for the Given Rational Expression?
Rational expression: (3x2 + 5x + 2) / (x3 + 6x2 + 11x + 6)
Options:
A) (1/x) + (2/(x + 1)) + (1/(x + 3))
B) (1/(x + 2)) + (2/(x + 3)) + (1/x)
C) (1/(x + 1)) + (2/(x + 2)) + (1/x)
D) (1/(x + 3)) + (2/(x + 2)) + (1/x)
Correct answer: A) (1/x) + (2/(x + 1)) + (1/(x + 3))
Explanation:
In the world of algebra, understanding partial fraction decomposition is crucial. Let’s break down the process step by step:
First and foremost, to decompose a given rational expression, the initial step involves factorizing the denominator.
So, we begin by factorizing the expression:
x3 + 6x2 + 11x + 6 = (x + 1)(x + 2)(x + 3)
Next, the magic of partial fractions comes into play. We express the original rational expression as the sum of its partial fractions:
(3x2 + 5x + 2) / (x3 + 6x2 + 11x + 6) = A/x + B/(x + 1) + C/(x + 2) + D/(x + 3)
To solve for A, B, C, and D, we multiply both sides of the equation by the common denominator:
3x2 + 5x + 2 = A(x + 1)(x + 2)(x + 3) + Bx(x + 2)(x + 3) + Cx(x + 1)(x + 3) + Dx(x + 1)(x + 2)
Now, equating coefficients of like powers of x on both sides gives us a system of equations.
As we solve this system, the values of A, B, C, and D come to light.
Upon solving the system, we find that A = 1, B = 2, C = 1, and D = 0.
Therefore, the correct partial fraction decomposition takes the form: (3x2 + 5x + 2) / (x3 + 6x2 + 11x + 6) = (1/x) + (2/(x + 1)) + (1/(x + 2)) + 0/(x + 3)
Finally, after some simplification, we arrive at: (3x2 + 5x + 2) / (x3 + 6x2 + 11x + 6) = (1/x) + (2/(x + 1)) + (1/(x + 2))
In conclusion, mastering partial fraction decomposition is a valuable skill in algebra, allowing you to dissect complex rational expressions with ease.
Partial Fraction MCQs Set-1
Question 1:
∫ (4x2 + 5x + 2) / (x3 + 6x2 + 11x + 6) dx is equal to:
- A) ln|x| + 2ln|x + 1| + ln|x + 2| + C
- B) 1/x + 2/(x + 1) + 1/(x + 2) + C
- C) 1/x + 2/(x + 1) + 1/(x + 3) + C
- D) 1/x + 2/(x + 2) + 1/(x + 3) + C
Correct answer: C) 1/x + 2/(x + 1) + 1/(x + 3) + C
Question 2:
∫ (x2 + 7x + 10) / (x3 + 3x2 + 2x) dx is equal to:
- A) ln|x| + 2ln|x + 1| – 2ln|x + 2| + C
- B) 1/x + 7/(x + 1) + 10/(x + 2) + C
- C) 1/x + 7/(x + 1) – 2/(x + 2) + C
- D) 1/x + 7/(x + 2) + 10/(x + 3) + C
Correct answer: A) ln|x| + 2ln|x + 1| – 2ln|x + 2| + C
Question 3:
∫ (5x3 + 4x2 – 3x + 2) / (x2 – x – 2) dx is equal to:
- A) 2x2 + 3x + 2ln|x + 1| – 3ln|x – 2| + C
- B) 2x2 + 3x – 2ln|x + 1| + 3ln|x – 2| + C
- C) 5x2 + 2x – 3ln|x + 1| + 2ln|x – 2| + C
- D) 5x2 + 2x + 2ln|x + 1| – 3ln|x – 2| + C
Correct answer: B) 2x2 + 3x – 2ln|x + 1| + 3ln|x – 2| + C
Question 4:
∫ (6x + 3) / (x2 + 2x – 8) dx is equal to:
- A) 3ln|x + 4| + 2ln|x – 2| + C
- B) 6ln|x + 4| + 3ln|x – 2| + C
- C) 3ln|x + 4| – 2ln|x – 2| + C
- D) 6ln|x + 4| – 3ln|x – 2| + C
Correct answer: C) 3ln|x + 4| – 2ln|x – 2| + C
Question 5:
∫ (3x2 + 2x + 1) / (x3 – 3x + 2) dx is equal to:
- A) ln|x| + ln|x – 1| + ln|x – 2| + C
- B) 3ln|x| + 2ln|x – 1| + ln|x – 2| + C
- C) 3ln|x| + 2ln|x – 1| – ln|x – 2| + C
- D) 3ln|x| + 2ln|x – 1| + 1/(x – 2) + C
Correct answer: C) 3ln|x| + 2ln|x – 1| – ln|x – 2| + C
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Question 6:
Simplify the following rational expression using the method of partial fractions:
(3x2 + 5x – 2) / (x3 + 3x2 + 2x)
Options:
- A) (3x + 2) / (x2 + 2x)
- B) (3x – 1) / (x2 + x)
- C) (3x – 2) / (x2 + 2x)
- D) (3x + 1) / (x2 + x)
Correct answer: A) (3x + 2) / (x2 + 2x)
In Addition, Another Set of MCQs:
- What is the partial fraction decomposition of the rational function (3x2 + 5) / (x3 + 2x2 + 3x)?
- (A) 1/(x + 1) + 2/(x + 2) + 3/(x + 3)
- (B) 1/(x + 1) + 2x/(x + 2) + 3/(x + 3)
- (C) 1/(x + 1) + 2/(x + 2) + 3x/(x + 3)
- (D) 1/(x + 1) + 2x/(x + 2) + 3x/(x + 3)
Correct Answer: (C)
2. What is the integral of the rational function Correct Answer: ∫(4x + 7) / (x2 + 3x + 2) using partial fractions?
- (A) ln|x + 2| + 3ln|x + 1| + C
- (B) 2ln|x + 2| + 3ln|x + 1| + C
- (C) 2ln|x + 2| + 2ln|x + 1| + C
- (D) 3ln|x + 2| + 2ln|x + 1| + C
Correct Answer: (B)
3. What is the partial fraction decomposition of the rational function (2x2 – 3x + 1) / (x3 – x2 + x – 1)?
- (A) 1/(x – 1) + 1/(x2 + 1)
- (B) 1/(x – 1) + (x – 1)/(x2 + 1)
- (C) 1/(x – 1) + (2x – 1)/(x2 + 1)
- (D) (x – 1)/(x2 + 1) + (2x – 1)/(x2 + 1)
Correct Answer:(C)
4. Evaluate the integral of the rational function ∫(3x2 + 4) / (x3 + 2x2 + x) using partial fractions.
- (A) ln|x + 1| + 2ln|x| + C
- (B) 2ln|x + 1| + ln|x| + C
- (C) ln|x + 1| + ln|x| + C
- (D) 2ln|x + 1| + 2ln|x| + C
Correct Answer: (B)
5. What is the partial fraction decomposition of the rational function (5x + 3) / (x2 + 5x + 6)?
- (A) 2/(x + 3) + 1/[x + 2].
- (B) 2/{x + 2) +1/[x + 3]
- (C) 2/(x + 3) + 2/(x + 2},
- (D) 1/{x + 2}+1/(x + 3)
Correct Answer: (B)
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6. Also, Evaluate the integral of the rational function ∫(2x + 1) / (x2 + 4x + 4) using partial fractions.
- (A) ln|x + 4| + 2ln|x + 2| + C
- (B) ln|x + 4| + ln|x + 2| + C
- (C) 2ln|x + 4| + ln|x + 2| + C
- (D) ln|x + 4| + ln|x + 2| + 2C
Correct Answer: (A)
7. Also, What is the partial fraction decomposition of the rational function (3x2 – 2x + 5) / (x3 – 4x2 + 3x)?
- (A) 2/(x – 3) + 1/(x + 1)
- (B) 1/(x – 3) + 2/(x + 1)
- (C) 2/(x – 3) + 2/(x + 1)
- (D) 1/(x – 3) + 1/(x + 1)
Correct Answer: (B)
8. Furthermore, let’s evaluate the integral of the rational function ∫(x2 + 4x + 3) / (x3 + 6x2 + 12x) using partial fractions.
- (A) ln|x| + 2ln|x + 2| + C
- (B) ln|x| + ln|x + 2| + C
- (C) 2ln|x| + ln|x + 2| + C
- (D) ln|x| + ln|x + 2| + 2C
Correct Answer: (A)
9. What is the partial fraction decomposition of the rational function (4x3 – 5x2 + 2x – 1) / (x4 – x2 )?
- (A) 1/x + 2/x2 + 3/x3 + 4/x4
- (B) 1/x + 2/x2 + 3/x3 + 4x/x4
- (C) 1/x + 2/x2 + 3/x3 + 4x2 /x4
- (D) 1/x + 2/x2 + 3/x3 + 4x3 /x4
Correct Answer: (A)
10. Lastly, evaluate the integral of the rational function ∫(2x3 + 3x2 + 5x + 1) / (x4 + 4x3 + 6x2 + 4x) using partial fractions.
- (A) ln|x| + 2ln|x + 1| + C
- (B) ln|x| + ln|x + 1| + C
- (C) 2ln|x| + ln|x + 1| + C
- (D) ln|x| + ln|x + 1| + 2C
Correct Answer: (B)
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