ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 1)

Take the ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 1): 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.

Q1. The roots of the equation x² − 5x + 6 = 0 are:
- A. −2 and −3
- B. 1 and 6
- C. −1 and 6
- D. 2 and 3 ✓
Answer: D. 2 and 3
Explanation: x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 or x = 3.
Q2. The sum of the roots of 2x² − 7x + 3 = 0 is:
- A. 3/2
- B. 7/2 ✓
- C. −7/2
- D. 7
Answer: B. 7/2
Explanation: Sum of roots = −b/a = −(−7)/2 = 7/2. (3/2 is the product.)
Q3. The roots of x² + 4x + 5 = 0 are:
- A. complex conjugates ✓
- B. real, distinct and rational
- C. real and equal
- D. real, distinct and irrational
Answer: A. complex conjugates
Explanation: Discriminant b² − 4ac = 16 − 20 = −4 < 0, so the roots are a pair of complex conjugates (−2 ± i).
Q4. For what values of k does x² − kx + 9 = 0 have equal roots?
- A. k = 6 only
- B. k = ±3
- C. k = ±9
- D. k = ±6 ✓
Answer: D. k = ±6
Explanation: Equal roots require b² − 4ac = 0: k² − 36 = 0 ⇒ k = ±6.
Q5. The quadratic equation whose roots are 2 + √3 and 2 − √3 is:
- A. x² − 4x + 1 = 0 ✓
- B. x² − 4x + 7 = 0
- C. x² − 4x − 1 = 0
- D. x² + 4x + 1 = 0
Answer: A. x² − 4x + 1 = 0
Explanation: Sum = 4 and product = 4 − 3 = 1, so the equation is x² − 4x + 1 = 0.

The other 25 questions

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Q6. If α and β are the roots of x² − 3x + 1 = 0, then α² + β² equals:
- A. 11
- B. 5
- C. 9
- D. 7
Q7. If α and β are the roots of x² + 5x + 2 = 0, then 1/α + 1/β equals:
- A. 2/5
- B. 5/2
- C. −5/2
- D. −2/5
Q8. If ω is a complex cube root of unity, the value of (1 + ω − ω²)(1 − ω + ω²) is:
- A. 1
- B. 4
- C. −4
- D. 0
Q9. The product of the three cube roots of unity is:
- A. −1
- B. 0
- C. 1
- D. 3
Q10. The four fourth roots of unity are:
- A. 1, −1, i, −i
- B. 1, 1, −1, −1
- C. 1, i, ω, ω²
- D. 1, −1, ω, ω²
Q11. The solution set of 2²ˣ − 5·2ˣ + 4 = 0 is:
- A. {1, 4}
- B. {1, 2}
- C. {0, 4}
- D. {0, 2}
Q12. The solution set of the equation √(x + 7) = x + 1 is:
- A. {2, −3}
- B. {−3}
- C. {2}
- D. {3, −2}
Q13. The remainder when x³ − 2x² + 3x − 4 is divided by x − 2 is:
- A. 2
- B. 0
- C. 6
- D. −26
Q14. If x + 1 is a factor of x³ + kx² − x + 2, then k equals:
- A. 2
- B. −4
- C. −2
- D. 0
Q15. Dividing x³ − 6x² + 11x − 6 by x − 1 (for example by synthetic division) gives the quotient:
- A. x² + 5x + 6
- B. x² − 5x + 6
- C. x² − 5x − 6
- D. x² − 7x + 18
Q16. If α and β are the roots of x² − 6x + 8 = 0, then α²β + αβ² equals:
- A. 14
- B. 36
- C. 64
- D. 48
Q17. If one root of x² + px + 12 = 0 is 3, then p equals:
- A. 4
- B. −4
- C. 7
- D. −7
Q18. The roots of ax² + bx + c = 0 (a ≠ 0) are reciprocals of each other when:
- A. c = 0
- B. c = a
- C. b = a
- D. b = 0
Q19. The roots of ax² + bx + c = 0 are equal in magnitude but opposite in sign when:
- A. c = 0
- B. a = c
- C. b = 0
- D. b = c
Q20. If x + y = 5 and xy = 6, then x² + y² equals:
- A. 37
- B. 19
- C. 25
- D. 13
Q21. If 1/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2), then A equals:
- A. 1
- B. −1/3
- C. 1/2
- D. 1/3
Q22. The partial fraction decomposition of (3x + 1)/((x + 1)(x − 1)) is:
- A. 1/(x + 1) + 2/(x − 1)
- B. −1/(x + 1) + 4/(x − 1)
- C. 1/(x + 1) − 2/(x − 1)
- D. 2/(x + 1) + 1/(x − 1)
Q23. The partial fraction form of the improper fraction (x² + 1)/(x² − 1) is:
- A. 1 + 1/(x − 1) − 1/(x + 1)
- B. 1/(x − 1) − 1/(x + 1)
- C. 1 − 1/(x − 1) + 1/(x + 1)
- D. 1 + 2/(x − 1) − 2/(x + 1)
Q24. The correct form of partial fractions for x/((x − 1)²(x + 2)) is:
- A. A/(x − 1) + B/(x − 1)² + C/(x + 2)
- B. A/(x − 1) + B/(x + 2)
- C. A/(x − 1)² + B/(x + 2)
- D. (Ax + B)/(x − 1)² + C/(x + 2)²
Q25. If 1/((x + 1)(x² + 1)) = A/(x + 1) + (Bx + C)/(x² + 1), then A equals:
- A. 1
- B. 1/2
- C. 1/4
- D. −1/2
Q26. Which of the following is a proper rational fraction?
- A. (4x² + 5)/(2x + 1)
- B. (x³ − 1)/(x² + x)
- C. (x² + 1)/(x² − 4)
- D. (2x + 1)/(x² + 3)
Q27. If α and β are the roots of x² − 4x + 3 = 0, the equation whose roots are α + 1 and β + 1 is:
- A. x² − 6x + 8 = 0
- B. x² − 5x + 4 = 0
- C. x² − 2x = 0
- D. x² − 6x + 3 = 0
Q28. The solution set of x⁴ − 5x² + 4 = 0 is:
- A. {±2, ±i}
- B. {±1, ±2}
- C. {±1, ±4}
- D. {1, 4}
Q29. The equation x² + 2x + k = 0 has real and distinct roots when:
- A. k < 1
- B. k = 1
- C. k > 1
- D. k < −1
Q30. If ω is a complex cube root of unity, then ω⁹⁹ + ω¹⁰⁰ + ω¹⁰¹ equals:
- A. 0
- B. −1
- C. 1
- D. 3