## Partial Fractions MCQs ECAT

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**ECAT**#### How To Solve Partial Fraction Problems?

### Question:

Which Partial Fraction Decomposition is Correct for the Given Rational Expression?

Rational expression: (3x^{2} + 5x + 2) / (x^{3} + 6x^{2} + 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:**

x^{3} + 6x^{2} + 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:

(3x^{2} + 5x + 2) / (x^{3} + 6x^{2} + 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:

3x^{2} + 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: (3x^{2} + 5x + 2) / (x^{3} + 6x^{2} + 11x + 6) = (1/x) + (2/(x + 1)) + (1/(x + 2)) + 0/(x + 3)

**Finally,** after some simplification, we arrive at: (3x^{2} + 5x + 2) / (x^{3} + 6x^{2} + 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:

∫ (4x^{2} + 5x + 2) / (x^{3} + 6x^{2} + 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:

∫ (x^{2} + 7x + 10) / (x^{3} + 3x^{2} + 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:

∫ (5x^{3} + 4x^{2} – 3x + 2) / (x^{2} – x – 2) dx is equal to:

- A) 2x
^{2}+ 3x + 2ln|x + 1| – 3ln|x – 2| + C - B) 2x
^{2}+ 3x – 2ln|x + 1| + 3ln|x – 2| + C - C) 5x
^{2}+ 2x – 3ln|x + 1| + 2ln|x – 2| + C - D) 5x
^{2}+ 2x + 2ln|x + 1| – 3ln|x – 2| + C

Correct answer: B) 2x^{2} + 3x – 2ln|x + 1| + 3ln|x – 2| + C

### Question 4:

∫ (6x + 3) / (x^{2} + 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:

∫ (3x^{2} + 2x + 1) / (x^{3} – 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:

(3x^{2} + 5x – 2) / (x^{3} + 3x^{2} + 2x)

Options:

- A) (3x + 2) / (x
^{2}+ 2x) - B) (3x – 1) / (x
^{2}+ x) - C) (3x – 2) / (x
^{2}+ 2x) - D) (3x + 1) / (x
^{2}+ x)

Correct answer: A) (3x + 2) / (x^{2} + 2x)

### In Addition, Another Set of MCQs:

- What is the partial fraction decomposition of the rational function (3x
^{2}+ 5) / (x^{3}+ 2x^{2}+ 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) / (x^{2} + 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 (2x^{2} – 3x + 1) / (x^{3} – x^{2} + x – 1)?

- (A) 1/(x – 1) + 1/(x
^{2}+ 1) - (B) 1/(x – 1) + (x – 1)/(x
^{2}+ 1) - (C) 1/(x – 1) + (2x – 1)/(x
^{2}+ 1) - (D) (x – 1)/(x
^{2}+ 1) + (2x – 1)/(x^{2}+ 1)

Correct Answer:(C)

4. Evaluate the integral of the rational function ∫(3x^{2} + 4) / (x^{3} + 2x^{2} + 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) / (x^{2} + 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) / (x^{2} + 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 (3x^{2} – 2x + 5) / (x^{3} – 4x^{2} + 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 ∫(x^{2} + 4x + 3) / (x^{3} + 6x^{2} + 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 (4x^{3} – 5x^{2} + 2x – 1) / (x^{4} – x^{2} )?

- (A) 1/x + 2/x
^{2}+ 3/x^{3}+ 4/x^{4} - (B) 1/x + 2/x
^{2}+ 3/x^{3}+ 4x/x^{4} - (C) 1/x + 2/x
^{2}+ 3/x^{3}+ 4x^{2}/x^{4} - (D) 1/x + 2/x
^{2}+ 3/x^{3}+ 4x^{3}/x^{4}

Correct Answer: (A)

10. Lastly, evaluate the integral of the rational function ∫(2x^{3} + 3x^{2} + 5x + 1) / (x^{4} + 4x^{3} + 6x^{2} + 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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