Partial fraction – Introduction MCQs With Answer

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Partial fraction – Introduction MCQs With Answer

Partial fraction decomposition is a core tool in pharmaceutical mathematics, helping B.Pharm students simplify rational expressions for integration, inverse Laplace transforms and pharmacokinetic modeling. This introduction covers methods for decomposing into simple fractions, handling repeated linear factors, and treating irreducible quadratic factors often encountered in drug-release and compartmental models. Key skills include polynomial long division, the Heaviside cover-up method, and equating coefficients to find constants. Mastering these techniques improves problem-solving in pharmaceutical calculations and analytic modeling. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the main purpose of partial fraction decomposition?

  • To factor polynomials into irreducible factors
  • To express a rational function as a sum of simpler fractions
  • To find numerical roots of polynomials
  • To compute determinants of matrices

Correct Answer: To express a rational function as a sum of simpler fractions

Q2. Before applying partial fractions, which condition must the rational function satisfy?

  • Numerator degree must be greater than denominator degree
  • Numerator and denominator must be prime
  • Numerator degree must be less than denominator degree
  • Denominator must be linear

Correct Answer: Numerator degree must be less than denominator degree

Q3. If the numerator degree is equal to or greater than the denominator degree, what is the first step?

  • Apply Heaviside cover-up directly
  • Perform polynomial long division
  • Factor denominator completely
  • Differentiate numerator and denominator

Correct Answer: Perform polynomial long division

Q4. How is a simple linear factor (x – a) represented in partial fractions?

  • A/(x – a)^2
  • A/(x – a)
  • (Ax + B)/(x – a)
  • A(x – a)

Correct Answer: A/(x – a)

Q5. For a repeated linear factor (x – a)^2, what is the correct partial fraction form?

  • A/(x – a)^2 + B/(x – a)
  • A/(x – a) + B/(x – a)^2
  • A(x – a) + B(x – a)^2
  • A/(x – a) + B/(x – a)^3

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

Q6. How should an irreducible quadratic factor (ax^2 + bx + c) be represented?

  • A/(ax^2 + bx + c)
  • (Ax + B)/(ax^2 + bx + c)
  • (Ax^2 + Bx + C)/(ax^2 + bx + c)
  • A(x^2 + bx + c)

Correct Answer: (Ax + B)/(ax^2 + bx + c)

Q7. Which method is most efficient for finding constants when denominator has distinct linear factors?

  • Numerical integration
  • Heaviside cover-up method
  • Matrix inversion
  • Polynomial long division only

Correct Answer: Heaviside cover-up method

Q8. When equating coefficients, what is compared on both sides of the identity?

  • Values at x = 0 only
  • Coefficients of like powers of x
  • Integral values over an interval
  • Maximum and minimum roots

Correct Answer: Coefficients of like powers of x

Q9. Partial fractions help solve integrals of rational functions by converting them into:

  • Simple logarithmic and arctan forms
  • Exponential functions exclusively
  • Trigonometric series only
  • Higher-degree polynomials

Correct Answer: Simple logarithmic and arctan forms

Q10. In pharmacokinetics, partial fractions commonly assist when using:

  • Nonlinear mixed effects models
  • Inverse Laplace transforms for compartmental analysis
  • Direct spectral analysis
  • Clinical trial randomization

Correct Answer: Inverse Laplace transforms for compartmental analysis

Q11. Which technique is used when denominator contains an irreducible quadratic?

  • Cover-up method for that quadratic
  • Assume linear numerator (Ax + B) over the quadratic
  • Ignore the quadratic and factor remaining terms
  • Use repeated application of long division only

Correct Answer: Assume linear numerator (Ax + B) over the quadratic

Q12. If you decompose (3x + 2)/(x^2 – x – 2), after factoring denominator, which factors appear?

  • (x + 2)(x – 1)
  • (x – 2)(x + 1)
  • (x – 1)(x – 2)
  • (x + 1)(x + 2)

Correct Answer: (x – 2)(x + 1)

Q13. The cover-up method directly gives constant A for A/(x – a) by:

  • Equating coefficients at high powers of x
  • Multiplying both sides by (x – a) and substituting x = a
  • Integrating both sides from 0 to a
  • Taking derivative with respect to x and evaluating at x = a

Correct Answer: Multiplying both sides by (x – a) and substituting x = a

Q14. Which of the following is NOT a valid step in partial fraction decomposition?

  • Factor the denominator completely over reals
  • Use long division if numerator degree >= denominator degree
  • Assume numerator of same degree as denominator for each term
  • Solve for constants by matching coefficients or substitution

Correct Answer: Assume numerator of same degree as denominator for each term

Q15. For the rational function (x + 5)/(x^2 + 4x + 3), partial fractions result in which denominators?

  • x and x + 3
  • (x + 1) and (x + 3)
  • (x + 2) and (x + 2)
  • (x – 1) and (x + 3)

Correct Answer: (x + 1) and (x + 3)

Q16. When a factor is (x^2 + 1), the partial fraction numerator form should be:

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

Correct Answer: Ax + B

Q17. After decomposing, which property ensures the equality holds for all x except poles?

  • Identity of rational functions
  • Orthogonality of polynomials
  • Uniqueness of integration constants
  • Symmetry of coefficients

Correct Answer: Identity of rational functions

Q18. Partial fractions are particularly useful for inverse Laplace transforms because:

  • They convert functions into tabulated transform pairs
  • They reduce the order of differential equations
  • They remove all singularities
  • They directly integrate initial conditions

Correct Answer: They convert functions into tabulated transform pairs

Q19. If denominator has complex conjugate quadratic factors, the partial fraction numerators should be:

  • Constants only
  • Linear expressions for each quadratic
  • Quadratic expressions for each factor
  • Exponential expressions

Correct Answer: Linear expressions for each quadratic

Q20. Which equation represents the decomposition format for (2x+3)/(x(x+1))?

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

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

Q21. In solving for constants by equating coefficients, how many equations are needed?

  • One equation regardless of number of constants
  • At least as many independent equations as unknown constants
  • Twice the number of unknown constants
  • No equations, constants are arbitrary

Correct Answer: At least as many independent equations as unknown constants

Q22. The partial fraction of 1/(x^2 – 2x + 1) where denominator is (x – 1)^2 yields which form?

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

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

Q23. For integration, which partial fraction term integrates to a logarithm?

  • Ax + B over quadratic
  • A/(x – a)
  • A/(x – a)^2
  • Ax over quadratic

Correct Answer: A/(x – a)

Q24. Which term leads to an arctan result upon integration?

  • A/(x – a)
  • (Ax + B)/(x^2 + p^2)
  • A/(x – a)^2
  • Ax over linear

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

Q25. The Heaviside cover-up method cannot be directly used when:

  • Factors are distinct linear factors
  • Denominator has repeated linear factors
  • Denominator has real roots
  • Numerator degree is zero

Correct Answer: Denominator has repeated linear factors

Q26. Which is a correct partial fraction decomposition for (5x + 1)/(x^2 – 1)?

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

Correct Answer: A/(x – 1) + B/(x + 1)

Q27. When decomposing (x^2 + x + 1)/(x^3 – x), after factoring denominator x(x – 1)(x + 1), how many constants are needed?

  • One constant
  • Three constants
  • Two constants
  • Four constants

Correct Answer: Three constants

Q28. If you obtain a quadratic numerator over a quadratic denominator after long division, what is the next step?

  • Stop; decomposition is unnecessary
  • Use partial fractions assuming linear numerator over quadratic
  • Differentiate numerator and denominator
  • Attempt to factor numerator only

Correct Answer: Use partial fractions assuming linear numerator over quadratic

Q29. Which of these rational functions requires polynomial division before decomposition?

  • (2x + 1)/(x^2 + 3x + 2)
  • (x^3 + x)/(x^2 + 1)
  • 1/(x^2 + 1)
  • x/(x – 1)

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

Q30. In a two-compartment pharmacokinetic model, partial fractions help to:

  • Estimate sample size for trials
  • Express concentration as sum of exponentials after inverse Laplace
  • Convert concentrations to pH values
  • Eliminate clearance rate constants

Correct Answer: Express concentration as sum of exponentials after inverse Laplace

Q31. Which approach is best when denominator resists real factorization?

  • Assume complex constants and proceed
  • Treat irreducible quadratics with linear numerators
  • Use numeric integration instead of decomposition
  • Ignore those factors

Correct Answer: Treat irreducible quadratics with linear numerators

Q32. After decomposing, how can you verify your partial fraction result?

  • Differentiate both sides
  • Combine fractions and simplify to original rational function
  • Compare only leading coefficients
  • Integrate both sides over arbitrary limits

Correct Answer: Combine fractions and simplify to original rational function

Q33. The decomposition of 1/(x^2 – a^2) is equivalent to which expression?

  • 1/(x – a) + 1/(x + a)
  • A/(x – a) + B/(x + a) with A = 1/(2a), B = -1/(2a)
  • A/(x – a)^2 + B/(x + a)^2
  • (x)/(x^2 – a^2)

Correct Answer: A/(x – a) + B/(x + a) with A = 1/(2a), B = -1/(2a)

Q34. For the expression (x + 2)/(x^2 + 4), the partial fraction numerator should be:

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

Correct Answer: Ax + B

Q35. What is the main advantage of breaking a complex transfer function into partial fractions in drug modeling?

  • It increases the order of the system
  • It yields simpler time-domain terms interpretable biologically
  • It hides parameter values
  • It replaces parameters with logs

Correct Answer: It yields simpler time-domain terms interpretable biologically

Q36. When denominator has a repeated quadratic factor (x^2 + 1)^2, the partial fraction terms include:

  • (Ax + B)/(x^2 + 1) + (Cx + D)/(x^2 + 1)^2
  • A/(x^2 + 1) only
  • A/(x^2 + 1)^3
  • Ax + B over linear factors

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

Q37. Which of the following is a correct use of partial fractions in laboratory calculations?

  • Estimating pKa directly from spectra
  • Solving integrals that appear in clearance calculations
  • Designing capsules shapes
  • Measuring viscosity experimentally

Correct Answer: Solving integrals that appear in clearance calculations

Q38. For decomposition, why is it important to fully factor the denominator over the reals?

  • Because complex factors are not allowed
  • To assign correct form of numerators for each factor
  • It is not important; you can guess constants
  • Only to make the expression longer

Correct Answer: To assign correct form of numerators for each factor

Q39. When using cover-up for function A/(x – a)(x – b), how is A found?

  • Evaluate original function at x = a after multiplying by (x – a)
  • Integrate original function from a to b
  • Differentiate and evaluate at x = a
  • Set x to zero

Correct Answer: Evaluate original function at x = a after multiplying by (x – a)

Q40. If a partial fraction yields term (Ax + B)/(x^2 + 1), its integral includes:

  • Only logarithmic terms
  • A logarithmic term and an arctan term
  • Only polynomial terms
  • Only exponential terms

Correct Answer: A logarithmic term and an arctan term

Q41. Which decomposition would you use for (4x^2)/(x^3 – x)?

  • Constants over x, x – 1 and x + 1
  • Linear numerators over each linear factor
  • Quadratic over cubic factors
  • No decomposition required

Correct Answer: Constants over x, x – 1 and x + 1

Q42. In practice, partial fraction constants are often determined by:

  • Solving a system of linear equations obtained by coefficient comparison
  • Random selection
  • Graphical methods only
  • Using trigonometric substitution only

Correct Answer: Solving a system of linear equations obtained by coefficient comparison

Q43. If original rational function simplifies exactly to a polynomial after division, what remains for partial fractions?

  • No fractional part remains; decomposition is just polynomial
  • Infinite fractions remain
  • Need to add more factors artificially
  • Partial fractions cannot be used

Correct Answer: No fractional part remains; decomposition is just polynomial

Q44. The decomposition of (2x)/(x^2 – 4) after factoring is:

  • A/(x – 2) + B/(x + 2)
  • (Ax + B)/(x^2 – 4)
  • A/(x – 2)^2 + B/(x + 2)^2
  • x/(x^2 – 4)

Correct Answer: A/(x – 2) + B/(x + 2)

Q45. Which method can reduce algebra when solving for multiple constants?

  • Using complex analysis exclusively
  • Selecting convenient x-values and substitution
  • Ignoring half of the equations
  • Assuming constants equal to zero

Correct Answer: Selecting convenient x-values and substitution

Q46. For the rational function with denominator (x – 1)(x^2 + 1), how many unknown constants are in the decomposition?

  • Two constants
  • Three constants
  • Four constants
  • One constant

Correct Answer: Three constants

Q47. Which statement is true about uniqueness of partial fraction decomposition?

  • Decomposition is unique once denominator factorization and numerator forms are fixed
  • Multiple different decompositions can represent the same function arbitrarily
  • Uniqueness depends on choice of integration limits
  • Uniqueness fails when numerator has even degree

Correct Answer: Decomposition is unique once denominator factorization and numerator forms are fixed

Q48. In computations, rounding errors in constants may affect:

  • Only symbolic results, not numerical ones
  • Numerical predictions such as concentration-time profiles
  • No results since partial fractions are exact
  • Only graphical plots but not values

Correct Answer: Numerical predictions such as concentration-time profiles

Q49. Which is a correct partial fraction decomposition procedure summary?

  • Factor denominator, divide if needed, set form, solve constants, verify
  • Differentiate numerator, integrate denominator, guess constants
  • Only factor numerator, then integrate
  • Use numerical methods without algebraic steps

Correct Answer: Factor denominator, divide if needed, set form, solve constants, verify

Q50. For applied problems in pharmacy, mastering partial fractions helps most in:

  • Qualitative description of molecular structures
  • Analytical solution of linear systems in drug kinetics and signal transforms
  • Designing tablet coatings by color matching
  • Counting discrete trial participants

Correct Answer: Analytical solution of linear systems in drug kinetics and signal transforms

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