LUMS LCAT Mathematics: Partial Fractions MCQs

LUMS LCAT 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.