COMSATS Entry Test Mathematics: Partial Fractions MCQs

Practice Partial Fractions MCQs for COMSATS Entry Test Mathematics — topic-wise sets with solved answers.

COMSATS Entry Test Mathematics: Partial Fractions MCQs — sample questions

  1. Question 1

    Q1. Resolve (x + 1) / (x² + 1) into partial fractions.

    • A) (x + 1) / (x² + 1)
    • B) 1 / (x + i) + 1 / (x - i)
    • C) (x + 1) / (x + i) + (x + 1) / (x - i)
    • D) (1/2) / (x + i) + (1/2) / (x - i)

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

    Explanation: Use formula for partial fraction decomposition with irreducible quadratic factors: (x + 1) / (x² + 1) = A(x - i) + B(x + i).

  2. Question 2

    Q2. The partial fraction decomposition of 1 / (x - 1)(x + 2) is

    • A) 1/3(x - 1) + 1/3(x + 2)
    • B) 1/3(x - 1) - 1/3(x + 2)
    • C) -1/3(x - 1) + 1/3(x + 2)
    • D) 1/(x - 1) + 1/(x + 2)

    Answer: 1/3(x - 1) - 1/3(x + 2)

    Explanation: Apply formula for partial fraction decomposition with distinct linear factors: 1 / (x - 1)(x + 2) = A/(x - 1) + B/(x + 2).

  3. Question 3

    Q3. Resolve 3x / (x - 1)(x + 2) into partial fractions.

    • A) 1/(x - 1) + 2/(x + 2)
    • B) 1/(x - 1) - 2/(x + 2)
    • C) 2/(x - 1) + 1/(x + 2)
    • D) 2/(x - 1) - 1/(x + 2)

    Answer: 1/(x - 1) + 2/(x + 2)

    Explanation: Use partial fraction decomposition: 3x / (x - 1)(x + 2) = A/(x - 1) + B/(x + 2), then equate coefficients.

  4. Question 4

    Q4. The value of A in the partial fraction decomposition of 1 / (x - 2)²(x + 1) = A/(x - 2) + B/(x - 2)² + C/(x + 1) is

    • A) -1/9
    • B) 1/9
    • C) -1/3
    • D) 1/3

    Answer: -1/9

    Explanation: Clear fractions, then equate coefficients to find A: 1 = A(x - 2)(x + 1) + B(x + 1) + C(x - 2)².

  5. Question 5

    Q5. Resolve (x² + 1) / (x + 1)(x² + 4) into partial fractions.

    • A) 2/5(x + 1) + (3x - 2)/5(x² + 4)
    • B) 2/5(x + 1) + (3x + 2)/5(x² + 4)
    • C) 1/(x + 1) + x/(x² + 4)
    • D) 1/(x + 1) + (x + 1)/(x² + 4)

    Answer: 2/5(x + 1) + (3x - 2)/5(x² + 4)

    Explanation: Use partial fraction decomposition with distinct linear and irreducible quadratic factors.

  6. Question 6

    Q6. The partial fraction decomposition of (x + 2) / (x² - 4) is

    • A) 1/(x - 2) + 1/(x + 2)
    • B) 1/2(x + 2) + 1/4(x - 2)
    • C) 1/4(x - 2) + 1/4(x + 2)
    • D) 1/4(x + 2) + 3/4(x - 2)

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

    Explanation: First factor denominator, then apply partial fraction decomposition: (x + 2) / (x - 2)(x + 2) = A/(x - 2) + B/(x + 2).

  7. Question 7

    Q7. The value of B in 1 / (x - 1)²(x + 2) = A/(x - 1) + B/(x - 1)² + C/(x + 2) is

    • A) 1/3
    • B) 1/9
    • C) 1
    • D) 1/2

    Answer: 1/3

    Explanation: Clear fractions and substitute x = 1 to find B: 1 = B(1 + 2).

  8. Question 8

    Q8. Resolve 2x / (x + 1)(x² + 1) into partial fractions.

    • A) 1/(x + 1) + (x - 1)/(x² + 1)
    • B) 1/(x + 1) + (1 - x)/(x² + 1)
    • C) -1/(x + 1) + (x + 1)/(x² + 1)
    • D) -1/(x + 1) + (x - 1)/(x² + 1)

    Answer: -1/(x + 1) + (x + 1)/(x² + 1)

    Explanation: Partial fractions: 2x/((x+1)(x²+1)) = -1/(x+1) + (x+1)/(x²+1) after solving for constants.

  9. Question 9

    Q9. The partial fraction decomposition of 1 / (x + 1)(x + 2)(x + 3) is

    • A) 1/2(x + 1) - 1/(x + 2) + 1/2(x + 3)
    • B) 1/(x + 1) - 1/(x + 2) + 1/(x + 3)
    • C) -1/2(x + 1) + 1/(x + 2) - 1/2(x + 3)
    • D) 1/(x + 1) + 1/(x + 2) + 1/(x + 3)

    Answer: 1/2(x + 1) - 1/(x + 2) + 1/2(x + 3)

    Explanation: Apply partial fraction decomposition with distinct linear factors: 1 / (x + 1)(x + 2)(x + 3) = A/(x + 1) + B/(x + 2) + C/(x + 3).

  10. Question 10

    Q10. Resolve (2x + 1) / (x - 1)(x + 2) into partial fractions.

    • A) 1/(x - 1) + 1/(x + 2)
    • B) 1/(x - 1) + 2/(x + 2)
    • C) 2/(x + 2) + 1/(x - 1)
    • D) 2/3(x - 1) + 1/3(x + 2)

    Answer: 1/(x - 1) + 1/(x + 2)

    Explanation: Let (2x+1)/((x-1)(x+2)) = A/(x-1)+B/(x+2). Solving gives A=1 and B=1.

  11. Question 11

    Q11. The partial fraction decomposition of x / (x² + 2x + 1) is

    • A) 1/(x + 1) - 1/(x + 1)²
    • B) 1/(x + 1) + 1/(x + 1)²
    • C) -1/(x + 1) + 1/(x + 1)²
    • D) 1/(x + 1) - 2/(x + 1)²

    Answer: 1/(x + 1) - 1/(x + 1)²

    Explanation: First factor denominator, then apply partial fraction decomposition: x / (x + 1)² = A/(x + 1) + B/(x + 1)².

  12. Question 12

    Q12. The value of C in 1 / (x - 1)(x + 1)² = A/(x - 1) + B/(x + 1) + C/(x + 1)² is

    • A) 1/4
    • B) -1/4
    • C) -1/2
    • D) 1

    Answer: -1/2

    Explanation: Substitute x = -1 in the cleared equation: 1 = C(-2), so C = -1/2.

  13. Question 13

    Q13. Resolve (x + 3) / (x + 1)(x² + 4) into partial fractions.

    • A) 1/5(x + 1) + (4 - x)/5(x² + 4)
    • B) 2/5(x + 1) + (x + 1)/5(x² + 4)
    • C) 1/(x + 1) + x/(x² + 4)
    • D) 2/(x + 1) + (x - 1)/(x² + 4)

    Answer: 1/5(x + 1) + (4 - x)/5(x² + 4)

    Explanation: Use partial fraction decomposition with distinct linear and irreducible quadratic factors.

  14. Question 14

    Q14. The partial fraction decomposition of 1 / (x² - 1) is

    • A) 1/2(x - 1) - 1/2(x + 1)
    • B) 1/(x - 1) + 1/(x + 1)
    • C) 1/(x - 1) - 1/(x + 1)
    • D) -1/(x - 1) + 1/(x + 1)

    Answer: 1/2(x - 1) - 1/2(x + 1)

    Explanation: First factor denominator, then apply partial fraction decomposition: 1 / (x - 1)(x + 1) = A/(x - 1) + B/(x + 1).

  15. Question 15

    Q15. The value of A in 1 / (x + 2)(x - 1)² = A/(x + 2) + B/(x - 1) + C/(x - 1)² is

    • A) 1/9
    • B) -1/9
    • C) 1/3
    • D) -1/3

    Answer: 1/9

    Explanation: Substitute x = -2: 1 = 9A, so A = 1/9.

  16. Question 16

    Q16. The partial fraction decomposition of x² / (x + 1)(x + 2) is

    • A) 1 + 1/(x + 1) + 4/(x + 2)
    • B) 1 - 1/(x + 1) + 4/(x + 2)
    • C) 1 + 4/(x + 1) - 9/(x + 2)
    • D) 1 + 4/(x + 2) - 1/(x + 1)

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

    Explanation: First divide, then apply partial fraction decomposition: x² / (x + 1)(x + 2) = 1 + A/(x + 1) + B/(x + 2).

  17. Question 17

    Q17. Resolve (x + 2) / (x² + 4x + 4) into partial fractions.

    • A) 1/(x + 2) + 0/(x + 2)²
    • B) 1/(x + 2) - 0/(x + 2)²
    • C) 0/(x + 2) + 1/(x + 2)²
    • D) -1/(x + 2) + 1/(x + 2)²

    Answer: 1/(x + 2) + 0/(x + 2)²

    Explanation: (x+2)/(x+2)² simplifies to 1/(x+2).

  18. Question 18

    Q18. The partial fraction decomposition of 1 / (x + 1)(x² + 2x + 1) is

    • A) 1/(x+1)³
    • B) 1/(x + 1) + 1/(x + 1)² - 1/(x + 1)³
    • C) 1/(x + 1)² - 1/(x + 1)³
    • D) 1/(x + 1) - 1/(x + 1)² + 1/(x + 1)³

    Answer: 1/(x+1)³

    Explanation: Denominator factors as (x+1)³, so the fraction equals 1/(x+1)³.

  19. Question 19

    Q19. Resolve (2x + 3) / (x² - 4) into partial fractions.

    • A) 7 / 4(x - 2) + 1 / 4(x + 2)
    • B) 7 / 4(x + 2) + 1 / 4(x - 2)
    • C) -7 / 4(x + 2) + 1 / 4(x - 2)
    • D) -7 / 4(x - 2) + 1 / 4(x + 2)

    Answer: 7 / 4(x - 2) + 1 / 4(x + 2)

    Explanation: Factor denominator as (x - 2)(x + 2), then apply partial fractions to get A = 7/4 and B = 1/4.

  20. Question 20

    Q20. The partial fraction decomposition of (x² + 1) / (x + 1)(x - 1) is

    • A) 1 + 1 / (x + 1) + 1 / (x - 1)
    • B) 1 + 1 / (x - 1) - 1 / (x + 1)
    • C) 1 - 1 / (x + 1) + 1 / (x - 1)
    • D) 1 - 1 / (x - 1) + 1 / (x + 1)

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

    Explanation: Divide numerator by denominator to simplify, then apply partial fractions.

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