FAST-NUCES Entry Test Mathematics Partial Fractions — Set 3

Partial Fractions MCQs set 3 for FAST-NUCES Entry Test Mathematics — 20 solved questions.

FAST-NUCES Entry Test Mathematics Partial Fractions — Set 3

  1. Question 1

    Q1. Decompose (x + 1) / (x - 1)(x + 2) into partial fractions.

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

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

    Explanation: Using partial fractions, (x + 1) / (x - 1)(x + 2) = A / (x - 1) + B / (x + 2). Solving for A, B gives A = 2/3, B = 1/3.

  2. Question 2

    Q2. Resolve 1 / (x + 1)²(x - 2) into partial fractions.

    • A) 1 / 9(x - 2) - 1 / 9(x + 1) + 1 / 3(x + 1)²
    • B) 1 / 9(x + 1) + 1 / 9(x - 2) + 1 / 3(x + 1)²
    • C) -1 / 9(x + 1) + 1 / 9(x - 2) + 1 / 3(x + 1)²
    • D) 1 / 9(x - 2) + 1 / 9(x + 1) - 1 / 3(x + 1)²

    Answer: -1 / 9(x + 1) + 1 / 9(x - 2) + 1 / 3(x + 1)²

    Explanation: Decomposing 1 / (x + 1)²(x - 2) into partial fractions, we equate it to A / (x + 1) + B / (x + 1)² + C / (x - 2) and solve for A, B, C.

  3. Question 3

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

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

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

    Explanation: Decomposing into partial fractions, (2x + 1) / (x + 1)(x - 2) = A / (x + 1) + B / (x - 2). Solving for A, B gives A = -1/3, B = 5/3.

  4. Question 4

    Q4. Resolve 1 / (x² - 4)(x + 1) into partial fractions.

    • A) -1 / 5(x + 1) + 1 / 5(x + 2) + 1 / 20(x - 2)
    • B) -1 / 5(x + 1) - 1 / 20(x + 2) + 1 / 20(x - 2)
    • C) 1 / 5(x + 1) + 1 / 20(x - 2) - 1 / 20(x + 2)
    • D) -3 / 20(x + 1) + 1 / 12(x - 2) - 1 / 12(x + 2)

    Answer: -3 / 20(x + 1) + 1 / 12(x - 2) - 1 / 12(x + 2)

    Explanation: Decomposing 1 / (x² - 4)(x + 1) into partial fractions, we equate it to A / (x + 1) + B / (x - 2) + C / (x + 2) and solve.

  5. Question 5

    Q5. The partial fraction decomposition of (x² + x + 1) / (x - 1)(x + 1) is

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

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

    Explanation: Divide (x² + x + 1) by (x² - 1), then decompose the remainder into partial fractions.

  6. Question 6

    Q6. Decompose 1 / (x + 1)(x² - x + 1) into partial fractions.

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

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

    Explanation: Using partial fractions for a linear and an irreducible quadratic factor, 1 / (x + 1)(x² - x + 1) = A / (x + 1) + (Bx + C) / (x² - x + 1).

  7. Question 7

    Q7. Resolve (2x + 3) / (x + 2)(x - 1) into partial fractions.

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

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

    Explanation: Decomposing into partial fractions, (2x + 3) / (x + 2)(x - 1) = A / (x + 2) + B / (x - 1). We find A = -1/3, B = 5/3.

  8. Question 8

    Q8. The partial fraction decomposition of (x + 3) / (x + 1)(x - 2) is

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

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

    Explanation: Using partial fraction decomposition, (x + 3) / (x + 1)(x - 2) = A / (x + 1) + B / (x - 2). We get A = -2/3, B = 5/3.

  9. Question 9

    Q9. Decompose (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) + 1 / (x + 2)²

    Explanation: First, simplify the denominator, then decompose into partial fractions: (x + 2) / (x + 2)² = 1 / (x + 2).

  10. Question 10

    Q10. The partial fraction decomposition of 1 / (x + 1)(x + 3) is

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

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

    Explanation: Decomposing 1 / (x + 1)(x + 3) into partial fractions, we get 1 / 2(x + 1) - 1 / 2(x + 3).

  11. Question 11

    Q11. Resolve (x + 1) / (x + 2)(x - 3) into partial fractions.

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

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

    Explanation: Decomposing (x + 1) / (x + 2)(x - 3) into partial fractions, we find A = -1/5, B = 4/5.

  12. Question 12

    Q12. Decompose (x² + 2) / (x + 1)(x² + 1) into partial fractions.

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

    Answer: 1 / 2(x + 1) + (1 - x) / 2(x² + 1)

    Explanation: First, divide (x² + 2) by (x + 1)(x² + 1), then decompose the remainder into partial fractions.

  13. Question 13

    Q13. Resolve (2x + 3) / (x + 1)(x + 2) into partial fractions.

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

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

    Explanation: Using partial fraction decomposition: (2x + 3) / (x + 1)(x + 2) = A / (x + 1) + B / (x + 2), then solving for A and B.

  14. Question 14

    Q14. The partial fraction decomposition of x / (x² - 4) is

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

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

    Explanation: Decomposing into partial fractions: x / (x² - 4) = A / (x - 2) + B / (x + 2), then A = 1/4, B = -1/4.

  15. Question 15

    Q15. Resolve (x² + 1) / (x + 1)(x² + 2x + 1) into partial fractions.

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

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

    Explanation: Using partial fraction decomposition: (x² + 1) / (x + 1)³ = A / (x + 1) + B / (x + 1)² + C / (x + 1)³.

  16. Question 16

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

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

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

    Explanation: Using partial fraction decomposition for repeated linear factors: 1 / x(x + 1)² = A / x + B / (x + 1) + C / (x + 1)².

  17. Question 17

    Q17. Resolve (x + 3) / (x - 1)(x + 2) into partial fractions.

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

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

    Explanation: Decomposing into partial fractions and solving for A and B: (x + 3) / (x - 1)(x + 2) = A / (x - 1) + B / (x + 2).

  18. Question 18

    Q18. Resolve 1 / (x + 1)(x² + 1) into partial fractions.

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

    Answer: 1 / 2(x + 1) + (x - 1) / 2(x² + 1)

    Explanation: Using partial fraction decomposition for irreducible quadratic factors: 1 / (x + 1)(x² + 1) = A / (x + 1) + (Bx + C) / (x² + 1).

  19. Question 19

    Q19. The decomposition of (x + 1) / x(x² + 4) is

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

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

    Explanation: Decomposing into partial fractions: (x + 1) / x(x² + 4) = A / x + (Bx + C) / (x² + 4), solving for A, B, and C.

  20. Question 20

    Q20. Resolve (x² + 2) / (x + 1)(x² + 3x + 2) into partial fractions.

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

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

    Explanation: Decomposing into partial fractions: (x² + 2) / (x + 1)(x + 1)(x + 2) = A / (x + 1) + B / (x + 1)² + C / (x + 2).