Method of Partial fractions MCQs With Answer

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Method of Partial fractions MCQs With Answer is essential for B. Pharm students who apply algebra to pharmacokinetics, drug formulation and analytical calculations. This concise introduction explains how partial fraction decomposition breaks complex rational expressions into simpler terms for integration, inverse Laplace transforms and solving compartment models. You will learn when to use polynomial division, handle repeated linear or irreducible quadratic factors, and apply the Heaviside cover-up or coefficient-comparison methods. These skills strengthen calculations in drug release kinetics and PK/PD modeling. The following MCQs reinforce theory and step-by-step problem solving for practical pharmacy applications. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the primary purpose of the method of partial fractions?

  • To multiply polynomials efficiently
  • To decompose a rational function into simpler fractions for integration or inverse transforms
  • To solve systems of linear equations numerically
  • To approximate irrational numbers

Correct Answer: To decompose a rational function into simpler fractions for integration or inverse transforms

Q2. When must you perform polynomial long division before partial fraction decomposition?

  • When the degree of the numerator is less than the degree of the denominator
  • When the degree of the numerator is equal to the degree of the denominator
  • When the degree of the numerator is greater than or equal to the degree of the denominator
  • Never; division is not needed for partial fractions

Correct Answer: When the degree of the numerator is greater than or equal to the degree of the denominator

Q3. For the denominator (x)(x+2)(x+3) with distinct linear factors, the partial fraction form is:

  • A/(x) + B/(x+2) + C/(x+3)
  • A/x^2 + B/(x+2)^2 + C/(x+3)^2
  • A/(x(x+2)) + B/(x+3)
  • (Ax^2+Bx+C)/(x(x+2)(x+3))

Correct Answer: A/(x) + B/(x+2) + C/(x+3)

Q4. How is a repeated linear factor (x+1)^2 represented in partial fractions?

  • A/(x+1) + B/(x+1)^2
  • A/(x+1)^2 only
  • A/(x+1) + B/(x+1)^3
  • A/(x+1) + Bx/(x+1)^2

Correct Answer: A/(x+1) + B/(x+1)^2

Q5. For an irreducible quadratic factor (x^2+1), the partial fraction term takes the form:

  • A/(x^2+1)
  • (Ax+B)/(x^2+1)
  • A/x + B/(x^2+1)
  • (Ax^2+Bx+C)/(x^2+1)

Correct Answer: (Ax+B)/(x^2+1)

Q6. Which method quickly finds coefficients for simple distinct linear factors using substitution?

  • Polynomial long division
  • Heaviside cover-up method
  • Partial multiplication method
  • Quadratic completion

Correct Answer: Heaviside cover-up method

Q7. If you have the improper fraction (x^3+1)/(x+1), what is the first step?

  • Apply partial fractions directly
  • Perform polynomial long division to get a polynomial plus a proper fraction
  • Factor the numerator only
  • Differentiate numerator and denominator

Correct Answer: Perform polynomial long division to get a polynomial plus a proper fraction

Q8. Which equality is used when equating coefficients in partial fraction decomposition?

  • Differentiate both sides and equate derivatives
  • Set numerators equal after bringing to common denominator and equate coefficients of like powers of x
  • Integrate both sides and compare integrals
  • Multiply numerators only and compare constants

Correct Answer: Set numerators equal after bringing to common denominator and equate coefficients of like powers of x

Q9. In pharmacokinetics, why are partial fractions useful for inverse Laplace transforms?

  • They convert time-domain signals to frequency-domain
  • They simplify Laplace-domain rational functions into terms with known inverse transforms for solving compartment models
  • They are not useful for Laplace transforms
  • They avoid the need to integrate at all

Correct Answer: They simplify Laplace-domain rational functions into terms with known inverse transforms for solving compartment models

Q10. Decompose (3x+5)/(x(x+1)) into partial fractions. Which is correct?

  • 5/x – 2/(x+1)
  • 3/x + 5/(x+1)
  • 2/x + 1/(x+1)
  • -5/x + 8/(x+1)

Correct Answer: 5/x – 2/(x+1)

Q11. Decompose (2x+1)/(x^2-1) = (2x+1)/((x-1)(x+1)). What are A and B in A/(x-1)+B/(x+1)?

  • A=3/2, B=1/2
  • A=1/2, B=3/2
  • A=1, B=1
  • A=0, B=2

Correct Answer: A=3/2, B=1/2

Q12. For (x^2+2x+3)/(x+1), what is the result of polynomial division first?

  • x+1 plus remainder 2
  • x+1 with no remainder
  • x+1 plus remainder 1
  • x+1 plus remainder 3

Correct Answer: x+1 plus remainder 2

Q13. Which of the following denominators indicates you must include linear and quadratic terms in partial fractions?

  • (x-2)(x+3)
  • (x^2+4)(x+1)
  • x(x+1)(x+2)
  • (x+1)^3

Correct Answer: (x^2+4)(x+1)

Q14. The partial fraction form for (x+4)/(x(x+2)^2) includes which terms?

  • A/x + B/(x+2) + C/(x+2)^2
  • A/x + B/(x+2)^2 only
  • A/(x+2) + B/(x+2)^2
  • A/x^2 + B/(x+2)

Correct Answer: A/x + B/(x+2) + C/(x+2)^2

Q15. Which technique finds coefficients by plugging convenient x-values after clearing denominators?

  • Differentiation method
  • Cover-up (substitution) method or direct substitution
  • Matrix inversion
  • Graphical method

Correct Answer: Cover-up (substitution) method or direct substitution

Q16. Decompose (4x+1)/(x^2- x) = (4x+1)/(x(x-1)). What is correct?

  • 1/x + 3/(x-1)
  • 5/x -1/(x-1)
  • 1/x + 4/(x-1)
  • 4/x + 1/(x-1)

Correct Answer: 5/x -1/(x-1)

Q17. For partial fractions involving (x^2+1) repeated twice, the form includes:

  • (Ax+B)/(x^2+1) + (Cx+D)/(x^2+1)^2
  • A/(x^2+1) + B/(x^2+1)^2
  • A/(x^2+1)^3
  • (Ax^2+B)/(x^2+1) + C/(x^2+1)^2

Correct Answer: (Ax+B)/(x^2+1) + (Cx+D)/(x^2+1)^2

Q18. Which statement is true about uniqueness of partial fraction decomposition?

  • Decomposition is unique for a given proper rational function once denominator factorization is fixed
  • Decomposition can have many different correct sets of coefficients
  • There is no decomposition for irrational denominators
  • Uniqueness depends on the integrator used

Correct Answer: Decomposition is unique for a given proper rational function once denominator factorization is fixed

Q19. To integrate rational functions in drug dissolution models, partial fractions help because they:

  • Transform integrals into sums of elementary logarithmic and arctangent forms
  • Eliminate the need to integrate
  • Require numeric integration methods only
  • Convert integrals into differential equations

Correct Answer: Transform integrals into sums of elementary logarithmic and arctangent forms

Q20. Decompose 1/(x^2-2x+1) = 1/(x-1)^2. The partial fraction form is:

  • A/(x-1) + B/(x-1)^2
  • A/(x-1)^2 only
  • A/(x-1) only
  • A/x + B/(x-1)

Correct Answer: A/(x-1) + B/(x-1)^2

Q21. For the fraction (Ax+B)/(x^2+4), which inverse Laplace transform pair is easiest to use after decomposition?

  • e^{-4t} and sin(4t)
  • cos(2t) and sin(2t) type transforms because x^2+4 corresponds to s^2+2^2
  • Transforms for polynomials only
  • No Laplace pair is available

Correct Answer: cos(2t) and sin(2t) type transforms because x^2+4 corresponds to s^2+2^2

Q22. If you equate coefficients after decomposing, you get linear equations. Which method is commonly used to solve them?

  • Gaussian elimination or simple substitution
  • Numerical integration
  • Graph plotting
  • Prime factorization

Correct Answer: Gaussian elimination or simple substitution

Q23. Decompose (x+2)/(x^2+3x+2) = (x+2)/((x+1)(x+2)). What is correct?

  • 1/(x+1)
  • 1/(x+2)
  • 1/(x+1) + 0/(x+2)
  • 1/(x+1) + 1/(x+2)

Correct Answer: 1/(x+1)

Q24. In the cover-up method for A/(x-a) term, you compute A as:

  • The limit as x→a of (x-a) times the original rational function
  • The derivative of the denominator at x=a
  • The integral from 0 to a of the function
  • The value of the numerator at x=0

Correct Answer: The limit as x→a of (x-a) times the original rational function

Q25. Which decomposition is correct for (6)/(x(x+2))?

  • 3/x + 3/(x+2)
  • 2/x + 4/(x+2)
  • 6/x – 6/(x+2)
  • 1/x + 5/(x+2)

Correct Answer: 3/x + 3/(x+2)

Q26. For partial fractions of (2x^2+3x+1)/(x(x+1)(x+2)), how many unknown constants are expected?

  • 2 constants
  • 3 constants (one per linear factor)
  • 4 constants including quadratic terms
  • 1 constant only

Correct Answer: 3 constants (one per linear factor)

Q27. Which check should you perform after finding partial fraction coefficients?

  • Differentiate the decomposition
  • Multiply through by the common denominator and verify equality of numerators
  • Integrate each term immediately
  • Compute numerical approximations only

Correct Answer: Multiply through by the common denominator and verify equality of numerators

Q28. Decompose (5)/(x^2-1). Which is correct?

  • 5/(x-1) – 5/(x+1)
  • 2.5/(x-1) + 2.5/(x+1)
  • 2.5/(x-1) – 2.5/(x+1)
  • 5/(x^2-1) cannot be decomposed

Correct Answer: 2.5/(x-1) – 2.5/(x+1)

Q29. Which partial fraction term leads to an arctangent when integrated?

  • A/x
  • (Ax+B)/(x^2+a^2) where B=0 and A=0 is not necessary
  • (Ax+B)/(x^2+a^2) where denominator is irreducible quadratic
  • A/(x-a)

Correct Answer: (Ax+B)/(x^2+a^2) where denominator is irreducible quadratic

Q30. The fraction 1/(x^2+2x+5) is best handled by completing the square giving denominator (x+1)^2+4. The partial fraction approach yields:

  • A/(x+1) + B/((x+1)^2+4)
  • (Ax+B)/((x+1)^2+4)
  • A/(x+1)^2 + B/(x+1)
  • Cannot be decomposed

Correct Answer: (Ax+B)/((x+1)^2+4)

Q31. For the rational function with denominator (x-2)^3, the partial fraction expansion includes:

  • A/(x-2) only
  • A/(x-2) + B/(x-2)^2 + C/(x-2)^3
  • A/(x-2)^3 only
  • A/(x-2) + B/(x-2)^3

Correct Answer: A/(x-2) + B/(x-2)^2 + C/(x-2)^3

Q32. Which is a correct decomposition of (x+1)/(x^2+1)?

  • 1/2 * (2x)/(x^2+1) + 1/2 * 1/(x^2+1)
  • 1/(x^2+1) + x/(x^2+1)
  • (x+1) cannot be decomposed
  • x/(x^2+1) only

Correct Answer: 1/2 * (2x)/(x^2+1) + 1/2 * 1/(x^2+1)

Q33. When decomposing for inverse Laplace transforms, constant numerators over s+a correspond to:

  • Exponential terms e^{-at}
  • Sine functions
  • Polynomial functions of t
  • Delta functions only

Correct Answer: Exponential terms e^{-at}

Q34. If a rational function has irreducible quadratic factors, coefficients found will often lead to which time-domain terms after inverse Laplace?

  • Pure polynomials in t
  • Exponentials times sine or cosine
  • Only exponentials
  • No time-domain representation exists

Correct Answer: Exponentials times sine or cosine

Q35. Decompose (2x)/(x^2-1). Which is correct?

  • 1/(x-1) + 1/(x+1)
  • 1/(x-1) – 1/(x+1)
  • 2/(x-1) + 0/(x+1)
  • Cannot be decomposed

Correct Answer: 1/(x-1) + 1/(x+1)

Q36. For the fraction (3x^2+1)/(x(x^2+1)), the partial fraction structure is:

  • A/x + (Bx+C)/(x^2+1)
  • A/x + B/(x^2+1)
  • (Ax+B)/(x) + C/(x^2+1)
  • Cannot be decomposed because numerator degree equals denominator

Correct Answer: A/x + (Bx+C)/(x^2+1)

Q37. In practice, partial fractions help solve compartment model ODEs because they:

  • Convert system matrices to scalars
  • Allow inverse Laplace of rational transfer functions to yield time-domain responses
  • Make numerical simulation unnecessary
  • Directly provide dosage schedules

Correct Answer: Allow inverse Laplace of rational transfer functions to yield time-domain responses

Q38. Decompose (7x+5)/(x^2+3x+2). Which is correct?

  • 2/(x+1) + 5/(x+2)
  • 3/(x+1) + 4/(x+2)
  • 1/(x+1) + 6/(x+2)
  • 7/(x+1) -2/(x+2)

Correct Answer: 3/(x+1) + 4/(x+2)

Q39. Which of the following is NOT a valid step in partial fraction decomposition?

  • Factor the denominator fully over real numbers when possible
  • Use polynomial division if improper
  • Assume arbitrary forms for numerators consistent with factor types
  • Introduce higher power denominators arbitrarily beyond factor multiplicity

Correct Answer: Introduce higher power denominators arbitrarily beyond factor multiplicity

Q40. Decompose (x)/(x^2-4) into partial fractions. Which is correct?

  • 1/4 * (1/(x-2) + 1/(x+2))
  • 1/2 * (1/(x-2) – 1/(x+2))
  • 1/(x-2) – 1/(x+2)
  • x/(x^2-4) cannot be decomposed

Correct Answer: 1/4 * (1/(x-2) + 1/(x+2))

Q41. In biochemical kinetics, partial fractions can simplify integrals of rational rate laws to:

  • Closed-form expressions showing concentration vs. time
  • Only numerical approximations
  • Graphs without formulas
  • Non-integrable forms

Correct Answer: Closed-form expressions showing concentration vs. time

Q42. For (x^2+1)/(x(x^2+1)), the partial fraction result is:

  • 1/x
  • 1/x + 1/(x^2+1)
  • x/(x^2+1)
  • Cannot be decomposed

Correct Answer: 1/x

Q43. Which method is preferred when denominator roots are difficult to compute symbolically?

  • Numeric partial fraction using computed roots and solving linear systems
  • Always use cover-up method
  • Ignore factorization and integrate directly
  • Use prime factorization of coefficients

Correct Answer: Numeric partial fraction using computed roots and solving linear systems

Q44. Decompose (x^2+3)/(x(x^2+3)). Which is correct?

  • 1/x + 1/(x^2+3)
  • 1/x only
  • x/(x^2+3)
  • Cannot be decomposed

Correct Answer: 1/x

Q45. For partial fractions, what type of factor in the denominator leads to logarithmic terms on integration?

  • Irreducible quadratic factors
  • Linear factors (x-a)
  • High-degree polynomial factors only
  • Factors with complex coefficients only

Correct Answer: Linear factors (x-a)

Q46. Decompose (8)/(x^2-4x+3) = 8/((x-1)(x-3)). Which is correct?

  • 4/(x-1) – 4/(x-3)
  • 2/(x-1) + 6/(x-3)
  • 4/(x-1) + 4/(x-3)
  • -4/(x-1) + 12/(x-3)

Correct Answer: 4/(x-1) – 4/(x-3)

Q47. Which factorization step is critical before attempting partial fractions?

  • Factor numerator completely
  • Factor denominator completely into linear and irreducible quadratic factors over the reals
  • Convert polynomials to decimals
  • Differentiate the denominator

Correct Answer: Factor denominator completely into linear and irreducible quadratic factors over the reals

Q48. Decompose (9x+3)/(x^2+5x+6) = (9x+3)/((x+2)(x+3)). Which is correct?

  • 3/(x+2) + 6/(x+3)
  • 6/(x+2) -3/(x+3)
  • 3/(x+2) + 0/(x+3)
  • 9/(x+2) -6/(x+3)

Correct Answer: 3/(x+2) + 6/(x+3)

Q49. When is partial fraction decomposition not applicable directly?

  • When numerator degree is less than denominator degree
  • When denominator cannot be factored over the reals and complex decomposition is undesired, you may need complex partial fractions or complete-square forms
  • When denominator has linear factors only
  • It is always directly applicable without restriction

Correct Answer: When denominator cannot be factored over the reals and complex decomposition is undesired, you may need complex partial fractions or complete-square forms

Q50. Which best practice improves accuracy in manual partial fraction problems for pharmacokinetic modeling?

  • Skip verification if coefficients look reasonable
  • Always check by recombining terms and comparing to the original expression and, for Laplace use, verify inverse transforms
  • Use only numeric approximations without algebraic checks
  • Assume symmetric coefficients by inspection

Correct Answer: Always check by recombining terms and comparing to the original expression and, for Laplace use, verify inverse transforms

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