ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 3): 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. One root of x² − 9x + k = 0 is twice the other. The value of k is:
Answer: B. 18
Explanation: Roots α and 2α: 3α = 9 ⇒ α = 3, so the roots are 3 and 6 and k = 3 × 6 = 18.
Q2. The roots of x² − 8x + k = 0 differ by 2. The value of k is:
Answer: B. 15
Explanation: Roots a and a + 2: 2a + 2 = 8 ⇒ a = 3. The roots are 3 and 5, so k = 15.
Q3. The equation (m + 1)x² + 2(m + 3)x + (m + 8) = 0 has equal roots for m equal to:
Answer: B. 1/3
Explanation: D = 0 ⇒ 4(m + 3)² = 4(m + 1)(m + 8) ⇒ m² + 6m + 9 = m² + 9m + 8 ⇒ m = 1/3.
Q4. If α and β are the roots of x² − 6x + 4 = 0, then α³ + β³ equals:
Answer: C. 144
Explanation: α + β = 6, αβ = 4. α³ + β³ = (α + β)³ − 3αβ(α + β) = 216 − 72 = 144.
Q5. If α and β are the roots of x² − 6x + 4 = 0, then |α − β| equals:
Answer: C. 2√5
Explanation: (α − β)² = (α + β)² − 4αβ = 36 − 16 = 20, so |α − β| = √20 = 2√5.
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Q6. If α and β are the roots of x² − 3x + 1 = 0, the equation whose roots are α² and β² is:
Q7. The equation whose roots are the reciprocals of the roots of 2x² − 5x + 3 = 0 is:
Q8. Let ω ≠ 1 satisfy ω³ = 1. The product (1 − ω)(1 − ω²) is:
Q9. Let ω ≠ 1 satisfy ω³ = 1. The product (1 − ω + ω²)(1 + ω − ω²) is:
Q10. Let ω ≠ 1 satisfy ω³ = 1. The sum 1/(1 + ω) + 1/(1 + ω²) is:
Q11. The sum of the squares of all the roots of x⁴ − 5x² + 4 = 0 is:
Q12. The solution of √(2x + 3) = x is:
Q13. The solution of √(x + 5) + √x = 5 is:
Q14. If 4ˣ − 6·2ˣ + 8 = 0, the sum of all the values of x is:
Q15. The remainder when x⁹⁹ + 2 is divided by x + 1 is:
Q16. If x − 1 is a factor of x³ − 2x² + kx − 5, then k equals:
Q17. If x − 1 and x − 2 are factors of x³ + ax² + bx − 6, then a equals:
Q18. The sum of the coefficients of the quotient when x³ − 6x² + 11x − 6 is divided by x − 1 is:
Q19. If x + y = 7 and xy = 12, then |x − y| equals:
Q20. In x² − 6x + q = 0 one root is double the other. Then q equals:
Q21. The roots of x² − 6x + 7 = 0 are:
Q22. A quadratic equation with rational coefficients has 2 + √3 as a root. It is:
Q23. The minimum value of 2x² − 8x + 3 for real x is:
Q24. Write (4x + 5)/((x + 1)(x + 2)) as A/(x + 1) + B/(x + 2). The ordered pair (A, B) is:
Q25. In the partial fractions of (x + 3)/(x − 1)², the numerator over the (x − 1)² term is:
Q26. In the partial fraction decomposition of x³/(x² − 1), the polynomial part is:
Q27. In (3x + 2)/((x + 1)(x² + 1)) = A/(x + 1) + (Bx + C)/(x² + 1), the value of A is:
Q28. In the partial fractions of 1/(x(x + 1)(x + 2)), the coefficient of 1/(x + 1) is:
Q29. The number of real solutions of |x|² − 5|x| + 6 = 0 is:
Q30. Let ω ≠ 1 satisfy ω³ = 1. Then (1 + 2ω + 3ω²)(1 + 2ω² + 3ω) is: