ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 4): 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 3x² − 7x + 2 = 0, the value of α³ + β³ is:
Answer: B. 217/27
Explanation: α + β = 7/3 and αβ = 2/3. Then α³ + β³ = (α + β)³ − 3αβ(α + β) = 343/27 − 3(2/3)(7/3) = 343/27 − 126/27 = 217/27.
Q2. The equation x² + (k − 3)x + k = 0 has equal roots for two values of k. Their product is:
Answer: D. 9
Explanation: Setting the discriminant to zero: (k − 3)² − 4k = k² − 10k + 9 = 0. The roots are k = 1 and 9 with product 9 (from the constant term).
Q3. Let α, β be the roots of x² − 4x + 2 = 0. The quadratic equation having α² and β² as its roots is:
Answer: A. x² − 12x + 4 = 0
Explanation: α + β = 4 and αβ = 2, so α² + β² = 16 − 4 = 12 and α²β² = 4. The new equation is x² − 12x + 4 = 0.
Q4. The roots of x² − 6x + k = 0 differ by 2. The value of k is:
Answer: B. 8
Explanation: If the roots are r and r + 2, then 2r + 2 = 6, so r = 2 and the roots are 2 and 4. Hence k = 2 × 4 = 8.
Q5. The roots of 3x² − 2√6 x + 2 = 0 are:
Answer: A. real and equal
Explanation: The discriminant is (2√6)² − 4(3)(2) = 24 − 24 = 0, so the roots are real and equal (each is √6/3, which is irrational).
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Q6. Given ω³ = 1 and ω ≠ 1, simplify 1/(1 + ω) + 1/(1 + ω²).
Q7. The sum of the cubes of all five fifth roots of unity is:
Q8. The solution of √(x + 7) − √(x − 1) = 2 is:
Q9. The sum of all real solutions of 2·2²ˣ − 9·2ˣ + 4 = 0 is:
Q10. If x³ − 3x² + kx − 6 is exactly divisible by x − 2, then k equals:
Q11. The remainder when x⁴ − 2x³ + 3 is divided by x + 1 is:
Q12. If x − 1 and x + 2 are both factors of 2x³ + ax² + bx − 6, then a − b equals:
Q13. Using synthetic division, the remainder when 2x³ − 5x² + 3x − 7 is divided by x − 2 is:
Q14. If x² + y² = 25 and x − y = 1, with x and y real, then xy equals:
Q15. The fraction (5x − 1)/((x + 1)(x − 2)) is resolved as A/(x + 1) + B/(x − 2). Find A + B.
Q16. Resolving (2x + 3)/(x − 2)² as A/(x − 2) + B/(x − 2)², find B.
Q17. Resolve 1/((x − 1)(x² + 1)) into A/(x − 1) + (Bx + C)/(x² + 1). Find A + B + C.
Q18. In the resolution of (x² + x + 1)/(x² − 1) into partial fractions, the coefficient of 1/(x − 1) is:
Q19. The correct form of the partial fractions of (x + 2)/(x²(x − 3)) is:
Q20. If one root of 2x² − 9x + k = 0 is twice the other, then k equals:
Q21. For the roots α, β of 5x² − 3x − 2 = 0, find 1/α + 1/β.
Q22. The roots α, β of x² − 5x + k = 0 satisfy α² + β² = 13. Find k.
Q23. For real k, the roots of x² − kx − 4 = 0 are:
Q24. If 2 + √3 is a root of x² + px + q = 0 with p and q rational, then p + q equals:
Q25. The only solution of √(3x + 1) = x − 1 is:
Q26. The product of all real roots of (x² + x)² − 8(x² + x) + 12 = 0 is:
Q27. Given ω³ = 1 and ω ≠ 1, simplify the quotient (1 + ω²)/(1 + ω).
Q28. The product of the two non-real roots of x³ = 27 is:
Q29. A polynomial f(x) leaves remainders 3 and 5 when divided by (x − 1) and (x − 2) respectively. Its remainder on division by (x − 1)(x − 2) is:
Q30. If α and β are the roots of 2x² − 5x + 3 = 0, the equation with roots 1/α and 1/β is: