ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 6): 30 multiple-choice questions (40 minutes) with instant scoring and a full…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. If α and β are the roots of x² − 4x + 1 = 0, then α⁴ + β⁴ equals:
Answer: D. 194
Explanation: α + β = 4 and αβ = 1, so α² + β² = 16 − 2 = 14, and α⁴ + β⁴ = 14² − 2(αβ)² = 196 − 2 = 194.
Q2. If α and β are the roots of x² − 5x + 3 = 0, the quadratic equation whose roots are α/β and β/α is:
Answer: D. 3x² − 19x + 3 = 0
Explanation: Sum = (α² + β²)/(αβ) = (25 − 6)/3 = 19/3 and product = 1. The equation is x² − (19/3)x + 1 = 0, i.e. 3x² − 19x + 3 = 0.
Q3. For what value of k does x² − 6x + k = 0 have one root double the other?
Answer: C. 8
Explanation: Let the roots be r and 2r. Their sum 3r = 6 gives r = 2, so the roots are 2 and 4 and k = 2 × 4 = 8.
Q4. The equation x² + 2(k + 2)x + 9k = 0 has equal roots for two values of k. Their sum is:
Answer: D. 5
Explanation: Equal roots need 4(k + 2)² − 36k = 0, i.e. k² − 5k + 4 = 0, so k = 1 or 4 and the sum is 5.
Q5. The sum of all real solutions of √(3x + 1) − √(x − 1) = 2 is:
Answer: B. 6
Explanation: Squaring: 3x + 1 = 4 + 4√(x − 1) + x − 1 gives x − 1 = 2√(x − 1), so √(x − 1) = 0 or 2, i.e. x = 1 or 5. Both satisfy the original equation (√4 − 0 = 2 and √16 − √4 = 2), so the sum is 6.
Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.
Q6. The sum of the squares of the real solutions of 5ˣ + 5²⁻ˣ = 26 is:
Q7. The sum of the squares of all four roots of (x² + x)² − 8(x² + x) + 12 = 0 is:
Q8. In the resolution (x² + x + 1)/((x − 1)(x − 2)(x − 3)) = A/(x − 1) + B/(x − 2) + C/(x − 3), the value of B is:
Q9. If (2x² + 3)/((x − 1)²(x + 1)) = A/(x − 1) + B/(x − 1)² + C/(x + 1), the value of B is:
Q10. If 1/(x²(x + 1)) = A/x + B/x² + C/(x + 1), then A equals:
Q11. When (x² + 1)/(x² − x) is expressed in partial fractions, the coefficient of 1/(x − 1) is:
Q12. A polynomial p(x) leaves remainder 3 when divided by (x − 1) and remainder −3 when divided by (x + 2). The remainder when p(x) is divided by (x − 1)(x + 2) is:
Q13. If x + 2 is a factor of x³ + kx² − 4x − 12, then the other two roots of the cubic are:
Q14. If a, b, c are real, the equation (x − a)(x − b) + (x − b)(x − c) + (x − c)(x − a) = 0 has:
Q15. The equations x² − 5x + 6 = 0 and x² − 7x + k = 0 have a common root. The sum of all possible values of k is:
Q16. If α and β are the roots of 2x² − 3x − 4 = 0, then α²β + αβ² equals:
Q17. If α and β are the roots of x² − x − 1 = 0, then α⁶ + β⁶ equals:
Q18. The minimum value of the quadratic expression 2x² − 12x + 25 for real x is:
Q19. The number of integers x satisfying x² − 6x + 5 ≤ 0 is:
Q20. The product of all real roots of x² − 5|x| + 6 = 0 is:
Q21. The number of real ordered pairs (x, y) satisfying x² + y² = 25 and xy = 12 is:
Q22. The expression x² + mx + 9 is positive for every real x for how many integer values of m?
Q23. The value of 1/(1·3) + 1/(2·4) + 1/(3·5) + … + 1/(20·22) is:
Q24. If the roots of x² − px + q = 0 differ by 1, then p² − 4q equals:
Q25. If α and β are the roots of x² − 10x + 4 = 0, then (1/√α + 1/√β)² equals:
Q26. The sum of the cubes of the roots of x² − 2x − 4 = 0 is:
Q27. The quadratic with roots 3 + √2 and 3 − √2 and leading coefficient 1 is:
Q28. If x² − 3x + 1 = 0, then x² + 1/x² equals:
Q29. The quadratic equation ax² + bx + c = 0 (a ≠ 0) has roots that are reciprocals of each other if and only if:
Q30. The solution set of the equation x = √(x + 6) is: