ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 5): 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. For the roots α > β of x² − 4x + 1 = 0, the value of α − β is:
Answer: D. 2√3
Explanation: (α − β)² = (α + β)² − 4αβ = 16 − 4 = 12, and α > β gives α − β = √12 = 2√3 (roots 2 ± √3).
Q2. The roots of x² − kx + 24 = 0 are in the ratio 2 : 3. For positive k, the value of k is:
Answer: B. 10
Explanation: Roots 2r and 3r give 6r² = 24, so r = 2 (positive k) and k = 5r = 10 (roots 4 and 6).
Q3. The positive value of p for which the roots of x² − px + 15 = 0 differ by 2 is:
Answer: B. 8
Explanation: (α − β)² = p² − 60 = 4, so p² = 64 and p = 8 (roots 3 and 5).
Q4. If α, β are the roots of 2x² − 5x + 1 = 0, then α/β + β/α equals:
Answer: B. 21/2
Explanation: α + β = 5/2, αβ = 1/2, so (α² + β²)/(αβ) = (25/4 − 1)/(1/2) = (21/4)(2) = 21/2.
Q5. If α, β are the roots of x² − 3x + 1 = 0, the quadratic equation with roots α² and β² is:
Answer: D. x² − 7x + 1 = 0
Explanation: α² + β² = 9 − 2 = 7 and α²β² = 1, so the equation is x² − 7x + 1 = 0.
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Q6. The sum of all values of k for which x² − (k + 2)x + 4 = 0 has equal roots is:
Q7. The roots of x² + mx + 9 = 0 are real if and only if:
Q8. The solution of √(x + 9) − √x = 1 is:
Q9. The sum of the real solutions of 4ˣ − 6·2ˣ + 8 = 0 is:
Q10. The sum of the squares of the roots of 2x² − 5x + 2 = 0 is:
Q11. The product of all real solutions of x² − 3|x| − 10 = 0 is:
Q12. The solution of (x − 1)/(x + 2) = (x + 3)/(x − 4) is:
Q13. If x³ + ax + b leaves remainder 0 on division by x − 2 and remainder 6 on division by x + 1, then a + b equals:
Q14. The value of k for which x + 2 is a factor of x³ + kx² − 4x − 12 is:
Q15. The remainder when x⁹⁹ is divided by x² − 1 is:
Q16. Using synthetic division by x − 3, the quotient of 2x³ − 3x² − 11x + 6 is:
Q17. If 1/((x − 1)(x − 2)(x − 3)) = A/(x − 1) + B/(x − 2) + C/(x − 3), then B equals:
Q18. If (2x + 3)/(x − 1)² = A/(x − 1) + B/(x − 1)², then B equals:
Q19. In the partial fractions of 1/(x²(x + 1)) = A/x + B/x² + C/(x + 1), the value of A is:
Q20. If 4x/((x + 1)(x² + 3)) = A/(x + 1) + (Bx + C)/(x² + 3), then C equals:
Q21. When x³/(x² − 4) is expressed in partial fractions, the coefficient of 1/(x − 2) is:
Q22. The sum of the non-real roots of x³ = 8 is:
Q23. If α and β are the roots of x² − 2x − 1 = 0, then α³ + β³ equals:
Q24. The roots of x² + bx + c = 0 are consecutive integers. Then b² − 4c equals:
Q25. The sum of the squares of all four roots of (x² − 3x)² − 2(x² − 3x) − 8 = 0 is:
Q26. If x + y = 5 and xy = 6, then x³ + y³ equals:
Q27. If x² + y² = 25 and x + y = 7 with x > y, the value of x − y is:
Q28. The product of all four roots of x⁴ − 13x² + 36 = 0 is:
Q29. The cubic x³ − 5x² + 8x − 4 = 0 has a repeated root 2. The remaining root is:
Q30. One root of 5x² − 3x − 2 = 0 is 1 because the coefficients sum to zero. The other root is: