ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 2)

Take the ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 2): 30 multiple-choice questions (40 minutes) with instant scoring and a full…

What this test covers

• 30 ECAT Mathematics

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

1. Q1. If α and β are the roots of x² − 5x + 3 = 0, then α/β + β/α equals:

   • A. 5/3
   • B. 31/3
   • C. 25/3
   • D. 19/3 ✓

   Answer: D. 19/3

   Explanation: α/β + β/α = (α² + β²)/(αβ) = [(α + β)² − 2αβ]/(αβ) = (25 − 6)/3 = 19/3.

2. Q2. The partial fraction decomposition of the improper fraction x³/(x² − 1) is:

   • A. x + 1/(2(x − 1)) + 1/(2(x + 1)) ✓
   • B. 1 + 1/(2(x − 1)) − 1/(2(x + 1))
   • C. x − 1/(2(x − 1)) + 1/(2(x + 1))
   • D. x + 1/(x − 1) + 1/(x + 1)

   Answer: A. x + 1/(2(x − 1)) + 1/(2(x + 1))

   Explanation: Divide: x³/(x² − 1) = x + x/(x² − 1), and x/((x − 1)(x + 1)) = (1/2)/(x − 1) + (1/2)/(x + 1).

3. Q3. The solution set of x^(2/3) − 5x^(1/3) + 6 = 0 is:

   • A. {−8, −27}
   • B. {4, 9}
   • C. {8, 27} ✓
   • D. {2, 3}

   Answer: C. {8, 27}

   Explanation: Put y = x^(1/3): y² − 5y + 6 = 0 gives y = 2 or 3, so x = 2³ = 8 or 3³ = 27.

4. Q4. One root of the quadratic equation ax² + bx + c = 0 (a ≠ 0) is zero when:

   • A. a = c
   • B. b² = 4ac
   • C. c = 0 ✓
   • D. b = 0

   Answer: C. c = 0

   Explanation: Putting x = 0 gives c = 0; equivalently the product of roots c/a must be 0.

5. Q5. If x + y = 7 and x² + y² = 25, then x³ + y³ equals:

   • A. 175
   • B. 133
   • C. 343
   • D. 91 ✓

   Answer: D. 91

   Explanation: xy = [(x + y)² − (x² + y²)]/2 = (49 − 25)/2 = 12; x³ + y³ = (x + y)³ − 3xy(x + y) = 343 − 252 = 91.

The other 25 questions

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6. Q6. The solution set of the equation x + 1/x = 5/2 is:

   • A. {5, 1/5}
   • B. {2, 1/2}
   • C. {1, 3/2}
   • D. {−2, −1/2}

7. Q7. The equation 2x² − 4x + k = 0 has real roots if and only if:

   • A. k ≤ 8
   • B. k ≥ 2
   • C. k < 2
   • D. k ≤ 2

8. Q8. The roots of the equation 3x² − 5x − 2 = 0 are:

   • A. 2 and 1/3
   • B. 2 and −1/3
   • C. −2 and 1/3
   • D. 3 and −2/3

9. Q9. The solution of the equation √(x + 5) + √x = 5 is:

   • A. x = 16
   • B. x = 4
   • C. x = 9
   • D. x = 1

10. Q10. If α and β are the roots of x² − 7x + 10 = 0, then (α − β)² equals:

    • A. 89
    • B. 29
    • C. 3
    • D. 9

11. Q11. The quadratic equation whose roots are the squares of the roots of x² − 3x + 2 = 0 is:

    • A. x² − 9x + 4 = 0
    • B. x² − 5x + 2 = 0
    • C. x² − 5x + 4 = 0
    • D. x² + 5x + 4 = 0

12. Q12. For a quadratic equation with rational coefficients, if the discriminant is a positive perfect square, the roots are:

    • A. complex conjugates
    • B. real and equal
    • C. real, rational and unequal
    • D. real, irrational and unequal

13. Q13. If (x² + 2)/((x − 1)(x² + 1)) = A/(x − 1) + (Bx + C)/(x² + 1), then C equals:

    • A. −3/2
    • B. 1/2
    • C. 3/2
    • D. −1/2

14. Q14. The partial fraction decomposition of (5x − 1)/((x + 1)(x − 2)) is:

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

15. Q15. If α and β are the roots of x² − 2x + 4 = 0, then α⁶ + β⁶ equals:

    • A. 64
    • B. 128
    • C. −128
    • D. 0

16. Q16. If the roots of x² − px + q = 0 differ by 1, then:

    • A. p² + 4q = 1
    • B. q² − 4p = 1
    • C. p² − 4q = 1
    • D. p² − 4q = 0

17. Q17. If x − 2 is a factor of x³ − 3x² + kx − 2, then k equals:

    • A. 5
    • B. 2
    • C. 3
    • D. −3

18. Q18. The product of the roots of 5x² + 3x − 10 = 0 is:

    • A. 3/5
    • B. 2
    • C. −2
    • D. −3/5

19. Q19. The quadratic equation whose roots are 3 and −5 is:

    • A. x² + 2x + 15 = 0
    • B. x² + 2x − 15 = 0
    • C. x² − 2x − 15 = 0
    • D. x² − 8x + 15 = 0

20. Q20. The remainder when 2x³ + x² − 5x + 1 is divided by x + 2 is:

    • A. −5
    • B. 3
    • C. −1
    • D. 31

21. Q21. If ω is a complex cube root of unity, then (1 + ω)⁵ equals:

    • A. −ω²
    • B. ω
    • C. ω²
    • D. −ω

22. Q22. If ω is a complex cube root of unity, then (1 + ω − ω²)³ equals:

    • A. 8
    • B. 0
    • C. −8
    • D. −8ω

23. Q23. If one root of 2x² − 9x + c = 0 is double the other, then c equals:

    • A. 9
    • B. 18
    • C. 9/2
    • D. 6

24. Q24. Using partial fractions, the sum 1/(1·2) + 1/(2·3) + 1/(3·4) + … + 1/(9·10) equals:

    • A. 9/10
    • B. 8/9
    • C. 10/11
    • D. 1

25. Q25. A rational fraction P(x)/Q(x) is called improper when:

    • A. degree of P(x) < degree of Q(x)
    • B. Q(x) has repeated factors
    • C. degree of P(x) ≥ degree of Q(x)
    • D. P(x) is a constant

26. Q26. Dividing 2x³ − 3x² + 4 by x + 1 (for example by synthetic division) gives the quotient:

    • A. 2x² + 5x + 5
    • B. 2x² − x − 1
    • C. 2x² − 5x + 5
    • D. 2x² − 5x − 5

27. Q27. The product of the four fourth roots of unity is:

    • A. −1
    • B. i
    • C. 0
    • D. 1

28. Q28. The partial fraction decomposition of x/(x − 1)² is:

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

29. Q29. The discriminant of the quadratic equation 2x² − 3x + 4 = 0 is:

    • A. 41
    • B. −41
    • C. −23
    • D. 23

30. Q30. If α and β are the roots of x² − 4x + 2 = 0, then α³ + β³ equals:

    • A. 40
    • B. 52
    • C. 56
    • D. 64