Tag: Partial Fractions

Short Description:
These MCQs are based on the latest Curriculum of Mathematics for SSC-II (Class 10) provided by the Federal Board of Intermediate and Secondary Education (FBISE). This set of 45 multiple-choice questions covers the concept of partial fractions, decomposition of rational expressions, proper and improper fractions, and application-based problems. Ideal for exam revision and self-assessment before your Class 10 exams.

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MCQs – Partial Fractions

1. The partial fraction decomposition of $\frac{3x + 5}{(x+1)(x+2)}$(x+1)(x+2)3x+5​ is:
   A. $\frac{1}{x+1} + \frac{2}{x+2}$x+11​+x+22​
   B. $\frac{2}{x+1} + \frac{1}{x+2}$x+12​+x+21​
   C. $\frac{3}{x+1} + \frac{5}{x+2}$x+13​+x+25​
   D. $\frac{5}{x+1} + \frac{3}{x+2}$x+15​+x+23​
2. Decompose $\frac{7}{x(x+2)}$x(x+2)7​ into partial fractions:
   A. $\frac{3}{x} + \frac{4}{x+2}$x3​+x+24​
   B. $\frac{4}{x} + \frac{3}{x+2}$x4​+x+23​
   C. $\frac{2}{x} + \frac{5}{x+2}$x2​+x+25​
   D. $\frac{5}{x} + \frac{2}{x+2}$x5​+x+22​
3. $\frac{2x+3}{x^2+5x+6}$x2+5x+62x+3​ can be decomposed as:
   A. $\frac{1}{x+2} + \frac{1}{x+3}$x+21​+x+31​
   B. $\frac{2}{x+2} + \frac{1}{x+3}$x+22​+x+31​
   C. $\frac{1}{x+2} + \frac{2}{x+3}$x+21​+x+32​
   D. $\frac{2}{x+2} + \frac{3}{x+3}$x+22​+x+33​
4. The fraction $\frac{5x+1}{x^2-x-6}$x2−x−65x+1​ decomposes into:
   A. $\frac{2}{x-3} + \frac{3}{x+2}$x−32​+x+23​
   B. $\frac{1}{x-3} + \frac{4}{x+2}$x−31​+x+24​
   C. $\frac{1}{x-3} + \frac{2}{x+2}$x−31​+x+22​
   D. $\frac{3}{x-3} + \frac{2}{x+2}$x−33​+x+22​
5. Decompose $\frac{4x+7}{x^2+5x+6}$x2+5x+64x+7​ into partial fractions:
   A. $\frac{1}{x+2} + \frac{3}{x+3}$x+21​+x+33​
   B. $\frac{4}{x+2} + \frac{3}{x+3}$x+24​+x+33​
   C. $\frac{2}{x+2} + \frac{5}{x+3}$x+22​+x+35​
   D. $\frac{3}{x+2} + \frac{4}{x+3}$x+23​+x+34​
6. Which of the following is a proper fraction?
   A. $\frac{x+2}{x^2+3x+2}$x2+3x+2x+2​
   B. $\frac{x^3+1}{x^2+2}$x2+2×3+1​
   C. $\frac{x^2+3}{x+1}$x+1×2+3​
   D. $\frac{x^3+2}{x^2+3}$x2+3×3+2​
7. The decomposition of $\frac{3x+2}{x^2+3x+2}$x2+3x+23x+2​ is:
   A. $\frac{1}{x+1} + \frac{2}{x+2}$x+11​+x+22​
   B. $\frac{2}{x+1} + \frac{1}{x+2}$x+12​+x+21​
   C. $\frac{3}{x+1} + \frac{2}{x+2}$x+13​+x+22​
   D. $\frac{1}{x+1} + \frac{3}{x+2}$x+11​+x+23​
8. The proper fraction is defined as a fraction where:
   A. Degree of numerator < degree of denominator
   B. Degree of numerator > degree of denominator
   C. Degree of numerator = degree of denominator
   D. None of these
9. Decompose $\frac{6x+11}{x^2+5x+6}$x2+5x+66x+11​ into partial fractions:
   A. $\frac{5}{x+2} + \frac{1}{x+3}$x+25​+x+31​
   B. $\frac{2}{x+2} + \frac{4}{x+3}$x+22​+x+34​
   C. $\frac{5}{x+2} + \frac{1}{x+3}$x+25​+x+31​
   D. $\frac{1}{x+2} + \frac{5}{x+3}$x+21​+x+35​
10. $\frac{x+3}{x^2-4}$x2−4x+3​ decomposes into:
    A. $\frac{2}{x-2} + \frac{1}{x+2}$x−22​+x+21​
    B. $\frac{1}{x-2} + \frac{2}{x+2}$x−21​+x+22​
    C. $\frac{1}{x-2} + \frac{1}{x+2}$x−21​+x+21​
    D. $\frac{3}{x-2} + \frac{0}{x+2}$x−23​+x+20​
11. Decompose $\frac{2x+5}{x^2-5x+6}$x2−5x+62x+5​:
    A. $\frac{1}{x-2} + \frac{1}{x-3}$x−21​+x−31​
    B. $\frac{3}{x-2} + \frac{-1}{x-3}$x−23​+x−3−1​
    C. $\frac{2}{x-2} + \frac{1}{x-3}$x−22​+x−31​
    D. $\frac{1}{x-2} + \frac{2}{x-3}$x−21​+x−32​
12. Which of the following is an improper fraction?
    A. $\frac{x^3+2x+1}{x^2+1}$x2+1×3+2x+1​
    B. $\frac{x+2}{x^2+1}$x2+1x+2​
    C. $\frac{2x+1}{x^2+3}$x2+32x+1​
    D. $\frac{x}{x^2+2}$x2+2x​
13. The decomposition of $\frac{5x+9}{x^2+4x+3}$x2+4x+35x+9​ is:
    A. $\frac{1}{x+1} + \frac{4}{x+3}$x+11​+x+34​
    B. $\frac{2}{x+1} + \frac{3}{x+3}$x+12​+x+33​
    C. $\frac{1}{x+1} + \frac{5}{x+3}$x+11​+x+35​
    D. $\frac{3}{x+1} + \frac{2}{x+3}$x+13​+x+32​
14. The fraction $\frac{3x^2+2x+1}{x^3+3x^2+2x}$x3+3×2+2x3x2+2x+1​ is:
    A. Proper
    B. Improper
    C. Constant
    D. None of these
15. Decompose $\frac{x+5}{x^2+6x+8}$x2+6x+8x+5​:
    A. $\frac{1}{x+2} + \frac{4}{x+4}$x+21​+x+44​
    B. $\frac{3}{x+2} + \frac{2}{x+4}$x+23​+x+42​
    C. $\frac{2}{x+2} + \frac{3}{x+4}$x+22​+x+43​
    D. $\frac{1}{x+2} + \frac{5}{x+4}$x+21​+x+45​
16. The decomposition of $\frac{4x+7}{x^2+3x-10}$x2+3x−104x+7​ is:
    A. $\frac{1}{x-2} + \frac{3}{x+5}$x−21​+x+53​
    B. $\frac{2}{x-2} + \frac{1}{x+5}$x−22​+x+51​
    C. $\frac{3}{x-2} + \frac{1}{x+5}$x−23​+x+51​
    D. $\frac{1}{x-2} + \frac{2}{x+5}$x−21​+x+52​
17. If a fraction is improper, it can be written as:
    A. Polynomial + proper fraction
    B. Proper fraction only
    C. Constant only
    D. None of these
18. Decompose $\frac{6x+13}{x^2+5x+6}$x2+5x+66x+13​:
    A. $\frac{7}{x+2} + \frac{-1}{x+3}$x+27​+x+3−1​
    B. $\frac{5}{x+2} + \frac{3}{x+3}$x+25​+x+33​
    C. $\frac{3}{x+2} + \frac{5}{x+3}$x+23​+x+35​
    D. $\frac{1}{x+2} + \frac{6}{x+3}$x+21​+x+36​
19. $\frac{x^2+3x+2}{x+1}$x+1×2+3x+2​ is:
    A. Improper
    B. Proper
    C. Constant
    D. None of these
20. The decomposition of $\frac{2x+1}{x^2+3x+2}$x2+3x+22x+1​ is:
    A. $\frac{1}{x+1} + \frac{1}{x+2}$x+11​+x+21​
    B. $\frac{2}{x+1} + \frac{-1}{x+2}$x+12​+x+2−1​
    C. $\frac{-1}{x+1} + \frac{2}{x+2}$x+1−1​+x+22​
    D. $\frac{1}{x+1} + \frac{2}{x+2}$x+11​+x+22​
21. Decompose $\frac{3x+7}{x^2+5x+6}$x2+5x+63x+7​
    A. $\frac{1}{x+2} + \frac{2}{x+3}$x+21​+x+32​
    B. $\frac{2}{x+2} + \frac{1}{x+3}$x+22​+x+31​
    C. $\frac{3}{x+2} + \frac{1}{x+3}$x+23​+x+31​
    D. $\frac{1}{x+2} + \frac{3}{x+3}$x+21​+x+33​
22. Decompose $\frac{4x+9}{x^2+7x+12}$x2+7x+124x+9​
    A. $\frac{1}{x+3} + \frac{3}{x+4}$x+31​+x+43​
    B. $\frac{3}{x+3} + \frac{1}{x+4}$x+33​+x+41​
    C. $\frac{2}{x+3} + \frac{2}{x+4}$x+32​+x+42​
    D. $\frac{4}{x+3} + \frac{1}{x+4}$x+34​+x+41​
23. Proper fraction:
    A. $\frac{x+2}{x^2+5x+6}$x2+5x+6x+2​
    B. $\frac{x^3+1}{x^2+1}$x2+1×3+1​
    C. $\frac{x^2+5}{x+2}$x+2×2+5​
    D. $\frac{x^3+2}{x^2+3}$x2+3×3+2​
24. Improper fraction:
    A. $\frac{x^3+2x+1}{x^2+1}$x2+1×3+2x+1​
    B. $\frac{x+1}{x^2+2}$x2+2x+1​
    C. $\frac{2x+1}{x^2+3}$x2+32x+1​
    D. $\frac{x}{x^2+2}$x2+2x​
25. Partial fraction decomposition of $\frac{5x+11}{x^2+6x+5}$x2+6x+55x+11​
    A. $\frac{1}{x+1} + \frac{4}{x+5}$x+11​+x+54​
    B. $\frac{3}{x+1} + \frac{2}{x+5}$x+13​+x+52​
    C. $\frac{2}{x+1} + \frac{3}{x+5}$x+12​+x+53​
    D. $\frac{4}{x+1} + \frac{1}{x+5}$x+14​+x+51​
26. Decompose $\frac{6x+13}{x^2+5x+6}$x2+5x+66x+13​
    A. $\frac{5}{x+2} + \frac{3}{x+3}$x+25​+x+33​
    B. $\frac{3}{x+2} + \frac{5}{x+3}$x+23​+x+35​
    C. $\frac{6}{x+2} + \frac{1}{x+3}$x+26​+x+31​
    D. $\frac{2}{x+2} + \frac{4}{x+3}$x+22​+x+34​
27. Fraction $\frac{x^2+3x+2}{x+1}$x+1×2+3x+2​ is:
    A. Improper
    B. Proper
    C. Constant
    D. None
28. Decompose $\frac{2x+3}{x^2+3x+2}$x2+3x+22x+3​
    A. $\frac{1}{x+1} + \frac{1}{x+2}$x+11​+x+21​
    B. $\frac{2}{x+1} + \frac{1}{x+2}$x+12​+x+21​
    C. $\frac{1}{x+1} + \frac{2}{x+2}$x+11​+x+22​
    D. $\frac{3}{x+1} + \frac{1}{x+2}$x+13​+x+21​
29. Fraction $\frac{3x+5}{x^2+4x+3}$x2+4x+33x+5​ decomposes into:
    A. $\frac{1}{x+1} + \frac{2}{x+3}$x+11​+x+32​
    B. $\frac{2}{x+1} + \frac{1}{x+3}$x+12​+x+31​
    C. $\frac{3}{x+1} + \frac{2}{x+3}$x+13​+x+32​
    D. $\frac{1}{x+1} + \frac{3}{x+3}$x+11​+x+33​
30. Improper fraction example:
    A. $\frac{x^3+2}{x^2+1}$x2+1×3+2​
    B. $\frac{x+2}{x^2+1}$x2+1x+2​
    C. $\frac{2x+1}{x^2+3}$x2+32x+1​
    D. $\frac{x}{x^2+2}$x2+2x​
31. Proper fraction example:
    A. $\frac{x+1}{x^2+2}$x2+2x+1​
    B. $\frac{x^3+2}{x^2+1}$x2+1×3+2​
    C. $\frac{x^2+3}{x+1}$x+1×2+3​
    D. $\frac{x^3+1}{x^2+2}$x2+2×3+1​
32. Decompose $\frac{4x+5}{x^2+5x+6}$x2+5x+64x+5​
    A. $\frac{1}{x+2} + \frac{3}{x+3}$x+21​+x+33​
    B. $\frac{2}{x+2} + \frac{2}{x+3}$x+22​+x+32​
    C. $\frac{3}{x+2} + \frac{1}{x+3}$x+23​+x+31​
    D. $\frac{4}{x+2} + \frac{1}{x+3}$x+24​+x+31​
33. Decompose $\frac{5x+8}{x^2+6x+8}$x2+6x+85x+8​
    A. $\frac{1}{x+2} + \frac{3}{x+4}$x+21​+x+43​
    B. $\frac{2}{x+2} + \frac{3}{x+4}$x+22​+x+43​
    C. $\frac{3}{x+2} + \frac{2}{x+4}$x+23​+x+42​
    D. $\frac{4}{x+2} + \frac{1}{x+4}$x+24​+x+41​
34. Fraction $\frac{x^2+3x+1}{x+1}$x+1×2+3x+1​ is:
    A. Improper
    B. Proper
    C. Constant
    D. None
35. Decompose $\frac{2x+7}{x^2+5x+6}$x2+5x+62x+7​
    A. $\frac{1}{x+2} + \frac{1}{x+3}$x+21​+x+31​
    B. $\frac{2}{x+2} + \frac{1}{x+3}$x+22​+x+31​
    C. $\frac{1}{x+2} + \frac{2}{x+3}$x+21​+x+32​
    D. $\frac{3}{x+2} + \frac{1}{x+3}$x+23​+x+31​
36. Improper fraction example:
    A. $\frac{x^3+1}{x^2+2}$x2+2×3+1​
    B. $\frac{x+2}{x^2+1}$x2+1x+2​
    C. $\frac{2x+1}{x^2+3}$x2+32x+1​
    D. $\frac{x}{x^2+2}$x2+2x​
37. Decompose $\frac{3x+7}{x^2+4x+3}$x2+4x+33x+7​
    A. $\frac{1}{x+1} + \frac{2}{x+3}$x+11​+x+32​
    B. $\frac{2}{x+1} + \frac{1}{x+3}$x+12​+x+31​
    C. $\frac{3}{x+1} + \frac{2}{x+3}$x+13​+x+32​
    D. $\frac{1}{x+1} + \frac{3}{x+3}$x+11​+x+33​
38. Decompose $\frac{4x+11}{x^2+5x+6}$x2+5x+64x+11​
    A. $\frac{1}{x+2} + \frac{3}{x+3}$x+21​+x+33​
    B. $\frac{2}{x+2} + \frac{3}{x+3}$x+22​+x+33​
    C. $\frac{3}{x+2} + \frac{2}{x+3}$x+23​+x+32​
    D. $\frac{4}{x+2} + \frac{1}{x+3}$x+24​+x+31​
39. Fraction $\frac{x^2+5x+6}{x+2}$x+2×2+5x+6​ is:
    A. Improper
    B. Proper
    C. Constant
    D. None
40. Decompose $\frac{2x+9}{x^2+5x+6}$x2+5x+62x+9​
    A. $\frac{1}{x+2} + \frac{3}{x+3}$x+21​+x+33​
    B. $\frac{2}{x+2} + \frac{1}{x+3}$x+22​+x+31​
    C. $\frac{1}{x+2} + \frac{2}{x+3}$x+21​+x+32​
    D. $\frac{3}{x+2} + \frac{1}{x+3}$x+23​+x+31​
41. Decompose $\frac{5x+12}{x^2+6x+8}$x2+6x+85x+12​
    A. $\frac{2}{x+2} + \frac{3}{x+4}$x+22​+x+43​
    B. $\frac{3}{x+2} + \frac{2}{x+4}$x+23​+x+42​
    C. $\frac{1}{x+2} + \frac{4}{x+4}$x+21​+x+44​
    D. $\frac{4}{x+2} + \frac{1}{x+4}$x+24​+x+41​
42. Fraction $\frac{x^2+2x+1}{x+1}$x+1×2+2x+1​ is:
    A. Improper
    B. Proper
    C. Constant
    D. None
43. Decompose $\frac{2x+5}{x^2+3x+2}$x2+3x+22x+5​
    A. $\frac{1}{x+1} + \frac{1}{x+2}$x+11​+x+21​
    B. $\frac{2}{x+1} + \frac{1}{x+2}$x+12​+x+21​
    C. $\frac{1}{x+1} + \frac{2}{x+2}$x+11​+x+22​
    D. $\frac{3}{x+1} + \frac{1}{x+2}$x+13​+x+21​
44. Decompose $\frac{3x+10}{x^2+5x+6}$x2+5x+63x+10​
    A. $\frac{1}{x+2} + \frac{3}{x+3}$x+21​+x+33​
    B. $\frac{2}{x+2} + \frac{3}{x+3}$x+22​+x+33​
    C. $\frac{3}{x+2} + \frac{2}{x+3}$x+23​+x+32​
    D. $\frac{4}{x+2} + \frac{1}{x+3}$x+24​+x+31​
45. Fraction $\frac{x^2+3x+2}{x+1}$x+1×2+3x+2​ is:
    A. Improper
    B. Proper
    C. Constant
    D. None

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Answers – List Form

1. A
2. A
3. C
4. C
5. B
6. A
7. A
8. A
9. C
10. C
11. B
12. A
13. B
14. B
15. B
16. C
17. A
18. A
19. A
20. B
21. A
22. B
23. A
24. A
25. B
26. A
27. A
28. C
29. B
30. A
31. A
32. B
33. B
34. A
35. C
36. A
37. B
38. B
39. A
40. C
41. B
42. A
43. B
44. B
45. A