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Partial Fractions MCQs ECAT

Partial Fractions MCQs ECAT

Partial Fractions MCQs ECAT

Looking for effective ECAT test preparation? Explore our free online ECAT Partial Fractions MCQs and discover our comprehensive online ECAT classes.

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:

  1. 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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Straight Line MCQs ECAT

Mathematics, MCQs Mathematics ECAT

Straight Line MCQs ECAT

MCQs Mathematics ECAT, get ready for your ECAT test preparation with Alsaudia Virtual Academy well experienced  Math tutors.

MCQs Mathematics ECAT – Set 1

  1. What is the equation of a straight line in slope-intercept form?

a) y = mx + c

b) x = my + c

c) y = cx + m

d) x = cy + m

Answer: a) y = mx + c

  1. Which of the following is the slope of a horizontal line?

a) 0

b) 1

c) -1

d) Undefined

Answer: a) 0

  1. Which of the following represents a vertical line?

a) y = 2x + 3

b) y = -3

c) x = 4

d) y = -x

Answer: c) x = 4

  1. What is the slope of a line parallel to the y-axis?

a) 0

b) 1

c) -1

d) Undefined

Answer: d) Undefined

  1. If two lines are perpendicular, what is the relationship between their slopes?

a) Their slopes are equal.

b) Their slopes are reciprocals of each other.

c) Their slopes are negative reciprocals of each other.

d) Their slopes have no relationship.

Answer: c) Their slopes are negative reciprocals of each other.

Best Preparation Guidance

  1. Which of the following is the point-slope form of a straight line equation?

a) y = mx + c

b) y – y₁ = m(x – x₁)

c) (y – y₁)² = m(x – x₁)

d) (y – y₁)³ = m(x – x₁)

Answer: b) y – y₁ = m(x – x₁)

  1. What is the x-intercept of a line represented by the equation y = 3x + 2?

a) (2, 0)

b) (0, 2)

c) (-2, 0)

d) (0, -2)

Answer: c) (-2, 0)

  1. Which of the following equations represents a line with a negative slope?

a) y = 2x + 5

b) y = -3x + 1

c) y = x² – 2x + 3

d) y = 4

Answer: b) y = -3x + 1

  1. What is the y-intercept of a line represented by the equation 2x – 3y = 6?

a) (2, 0)

b) (0, 2)

c) (0, -2)

d) (0, -2)

Answer: b) (0, -2)

  1. Which of the following statements is true about parallel lines?

a) They have the same y-intercept.

b) They have the same slope.

c) They intersect at a right angle.

d) They have opposite slopes.

Answer: b) They have the same slope.

Also, MCQs Mathematics ECAT – Set-2

  1. Which of the following equations represents a straight line?

a) y = x^2

b) y = 2x + 3

c) y = √x

d) y = sin(x)

Answer: b) y = 2x + 3

2. What is the slope of a vertical line? a) 0

b) 1

c) Undefined

d) Infinity

Answer: c) Undefined

3. Which of the following statements is true for two parallel lines?

a) They have the same slope.

b) They intersect at one point.

c) They have different slopes.

d) They have a negative slope.

Answer: a) They have the same slope.

4. What is the equation of a horizontal line passing through the point (0, 5)?

a) y = 0

b) y = 5

c) x = 5

d) x = 0

Answer: b) y = 5

5. If the equation of a line is y = -2x + 4, what is the slope of the line?

a) -4

b) -2

c) 2

d) 4

Answer: b) -2

Excel your Test Preparation

6. What is the equation of a line parallel to the line 3x – 4y = 7 and passing through the point (2, 5)?

a) 4x – 3y = 7

b) 4x + 3y = 7

c) 3x – 4y = 5

d) 3x + 4y = 7

Answer: b) 4x + 3y = 7

7. What is the distance between the points (1, 2) and (4, 6) on a straight line?

a) √17

b) √22

c) 5

d) 4

Answer: c) 5

8. If two lines are perpendicular to each other, what is the product of their slopes?

a) 0

b) 1

c) Undefined

d) Infinity

Answer: a) 0

9. What is the y-intercept of the line represented by the equation 2x + 3y = 9?

a) 2

b) 3 c) 9 d) -3

Answer: b) 3

10. If the slope of a line is 0, what can be said about the line?

a) It is horizontal.

b) It is vertical.

c) It is inclined.

d) It is parallel.

Answer: a) It is horizontal.

Another, MCQs Mathematics ECAT, Set 3

  1. What is the slope-intercept form of the equation of a straight line?

a) y = mx + b

b) y = ax^2 + bx + c

c) y = mx – b

d) y = a^x

Answer:  a) y = mx + b

2. Which of the following is the slope of a vertical line?

a) 0 b) 1

c) Undefined

d) Infinity

Answer:  c) Undefined

3. Two lines are parallel if and only if their slopes are:

a) Equal

b) Perpendicular

c) Undefined

d) Negative

Answer: a) Equal

4. What is the standard form of the equation of a straight line?

a) y = mx + b

b) y = ax^2 + bx + c

c) Ax + By = C

d) y = a^x

Answer: c) Ax + By = C

5. The equation of a line passing through the point (3, 4) with a slope of 2 is:

a) y = 2x + 1

b) y = 2x – 2

c) y = 4x + 3

d) y = -2x + 10

Answer: b) y = 2x – 2

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6. Which of the following equations represents a horizontal line?

a) y = -3x + 2

b) y = 5

c) y = x^2 + 3

d) y = 2x – 1

Answer: b) y = 5

7. The equation of a line parallel to the x-axis is:

a) y = 0

b) x = 0

c) y = x

d) x = y

Answer: a) y = 0

8. What is the slope of a line perpendicular to a line with slope m?

a) -m

b) m

c) 1/m

d) -1/m

Answer: d) -1/m

9. The distance between two parallel lines is measured along a line perpendicular to them. True or False?

a) True

b) False

Answer: a) True

10. If two lines are perpendicular to each other, the product of their slopes is:

a) -1

b) 0

c) 1

d) Undefined

Answer: a) -1

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