LUMS LCAT Mathematics Partial Fractions — Set 2

LUMS LCAT Mathematics Partial Fractions — Set 2

1. Question 1

   Q1. Resolve 1 / (x³ + x) into partial fractions.

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

   Answer: 1 / x - x / (x² + 1)

   Explanation: Factor x³ + x = x(x² + 1), then decompose into 1/x + Bx + C / (x² + 1).

2. Question 2

   Q2. The partial fraction decomposition of (x + 2) / (x² - x - 6) is

   • A) 1 / (x - 3) + 0 / (x + 2)
   • B) 4 / 5(x - 3) + 1 / 5(x + 2)
   • C) -1 / 5(x + 2) + 4 / 5(x - 3)
   • D) 1 / (x + 2) + 1 / (x - 3)

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

   Explanation: Factor denominator, then apply cover-up method to get A = 4/5 and B = 1/5.

3. Question 3

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

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

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

   Explanation: Use cover-up method or equate coefficients to find A = 2/3 and B = 1/3.

4. Question 4

   Q4. The partial fraction decomposition of 1 / (x - 2)²(x + 1) is

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

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

   Explanation: Apply partial fraction decomposition for repeated linear factors.

5. Question 5

   Q5. Resolve (x² + 3x + 2) / (x + 1)(x + 2)² into partial fractions.

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

   Answer: 1 / (x + 1) + 0 / (x + 2) + 0 / (x + 2)²

   Explanation: Simplify the expression before decomposing into partial fractions, (x + 1)(x + 2) / (x + 1)(x + 2)² = 1 / (x + 2).

6. Question 6

   Q6. The partial fraction decomposition of 1 / (x + 1)(x² + 1) is

   • A) 1 / 2(x + 1) + (-x + 1) / 2(x² + 1)
   • B) -1 / 2(x + 1) + (x + 1) / 2(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: Decompose 1 / (x + 1)(x² + 1) into A / (x + 1) + (Bx + C) / (x² + 1), then equate coefficients.

7. Question 7

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

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

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

   Explanation: Use cover-up method or equate coefficients to find A = -1 and B = 4.

8. Question 8

   Q8. The partial fraction decomposition of (x + 3) / (x² + 2x + 1) is

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

   Answer: 2 / (x + 1) + 1 / (x + 1)²

   Explanation: Factor denominator as (x + 1)², then decompose into A / (x + 1) + B / (x + 1)².

9. Question 9

   Q9. Resolve 1 / (x - 1)(x² + x + 1) into partial fractions.

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

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

   Explanation: Decompose 1 / (x - 1)(x² + x + 1) into A / (x - 1) + (Bx + C) / (x² + x + 1), equate coefficients.

10. Question 10

    Q10. The partial fraction decomposition of (2x + 1) / (x² - 1) is

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

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

    Explanation: Decompose (2x + 1) / (x - 1)(x + 1) into A / (x - 1) + B / (x + 1), then equate coefficients.

11. Question 11

    Q11. Resolve (x + 2) / (x + 1)(x² + 2x + 2) into partial fractions.

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

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

    Explanation: Decompose into A / (x + 1) + (Bx + C) / (x² + 2x + 2), equate coefficients to find A, B, C.

12. Question 12

    Q12. The partial fraction decomposition of 1 / (x + 2)(x + 3) is

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

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

    Explanation: Apply cover-up method or equate coefficients to find A = -1 and B = 1.

13. Question 13

    Q13. Resolve (x² + x + 1) / (x + 1)(x + 2) into partial fractions.

    • A) 1 + 1 / (x + 1) - 3 / (x + 2)
    • B) 1 + 1 / (x + 2) - 1 / (x + 1)
    • C) 1 - 1 / (x + 1) + 3 / (x + 2)
    • D) 1 + 3 / (x + 1) - 7 / (x + 2)

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

    Explanation: Divide to get 1 + 1/(x+1) - 3/(x+2) after partial fraction decomposition.

14. Question 14

    Q14. The partial fraction decomposition of (2x + 3) / (x + 1)(x + 2)(x + 3) is

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

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

    Explanation: Decompose into A / (x + 1) + B / (x + 2) + C / (x + 3), then equate coefficients.

15. Question 15

    Q15. The partial fraction decomposition of 1 / (x² + 5x + 6) is

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

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

    Explanation: Factor denominator, then decompose 1 / (x + 2)(x + 3) into A / (x + 2) + B / (x + 3).

16. Question 16

    Q16. Decompose 1 / (x - 1)(x + 2) into partial fractions.

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

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

    Explanation: By partial fraction decomposition, 1 / (x - 1)(x + 2) = A / (x - 1) + B / (x + 2). Solving for A and B gives A = 1/3, B = -1/3.

17. Question 17

    Q17. The partial fraction decomposition of (2x + 3) / (x² - 4) is

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

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

    Explanation: Decomposing into partial fractions, (2x + 3) / (x² - 4) = A / (x - 2) + B / (x + 2). We find A = 7/4, B = 1/4.

18. Question 18

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

    • 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 / 2(x - 1) - 1 / 2(x + 1)

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

    Explanation: First, divide the numerator by the denominator. Then, decompose the remainder into partial fractions to get the result.

19. Question 19

    Q19. Decompose 1 / (x + 1)(x + 2)(x + 3) into partial fractions.

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

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

    Explanation: Using partial fraction decomposition, we equate 1 / (x + 1)(x + 2)(x + 3) to A / (x + 1) + B / (x + 2) + C / (x + 3) and solve for A, B, C.

20. Question 20

    Q20. The partial fraction decomposition of (x + 2) / (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) + 1 / (x + 1)

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

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