ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Quadratic Equations and Partial Fractions (Set 7): 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. The equations x² + px + q = 0 and x² + qx + p = 0 (with p ≠ q) have a common root. Then p + q equals
Answer: C. −1
Explanation: Subtracting: (p − q)x + (q − p) = 0, so x = 1 (as p ≠ q). Substituting x = 1 into the first equation gives 1 + p + q = 0, so p + q = −1.
Q2. If α and β are the roots of x² − 5x + 3 = 0, the equation whose roots are 2α + 1 and 2β + 1 is
Answer: D. x² − 12x + 23 = 0
Explanation: Sum = 2(α + β) + 2 = 12. Product = 4αβ + 2(α + β) + 1 = 12 + 10 + 1 = 23. So the equation is x² − 12x + 23 = 0.
Q3. A ball thrown upward has height h = 40t − 5t² metres after t seconds. Its greatest height is
Answer: D. 80 m
Explanation: h = −5(t − 4)² + 80, so the maximum occurs at t = 4 s, where h = 80 m.
Q4. For which values of k is kx² − 4x + k > 0 for every real x?
Answer: C. k > 2
Explanation: We need k > 0 (parabola opens upward) and discriminant 16 − 4k² < 0, i.e. k² > 4. Together: k > 2. (k = 2 gives a double root where the expression equals 0, so k = 2 fails.)
Q5. The sum of the roots of x² − (k + 2)x + (2k − 1) = 0 equals the product of its roots. Then k equals
Answer: B. 3
Explanation: Sum = k + 2 and product = 2k − 1. Setting k + 2 = 2k − 1 gives k = 3.
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Q6. If x − 1/x = 2, then x⁴ + 1/x⁴ equals
Q7. The solution of √(2x − 1) + √(x − 1) = 5 is
Q8. The product of all real solutions of 9ˣ − 4·3ˣ⁺¹ + 27 = 0 is
Q9. The sum of all real roots of (x + 1)(x + 2)(x + 3)(x + 4) = 24 is
Q10. When x⁵ is divided by x² − 3x + 2, the remainder, evaluated at x = 3, equals
Q11. If x² − 3x + 2 is a factor of x⁴ + px² + q, then p + q equals
Q12. The remainder when 2x⁴ − 3x² + x − 6 is divided by 2x − 1 is
Q13. The remainder when x²⁰²⁶ + x + 1 is divided by x² + x + 1 is
Q14. If (x² + 1)/(x(x − 1)²) = A/x + B/(x − 1) + C/(x − 1)², then B equals
Q15. The value of Σ 1/(n(n + 1)(n + 2)) summed over n = 1, 2, 3, … (all positive integers) is
Q16. When the polynomial x⁵ + 1 is divided by x² + 1, the degree of the remainder is
Q17. The equation 2x² − 6x + m = 0 has two distinct positive roots exactly when
Q18. One root of a quadratic with rational coefficients is 1/(2 + √3). The quadratic (with leading coefficient 1) is
Q19. The roots of x² − (k − 2)x − (k + 1) = 0 are α and β. The sum of squares α² + β² is least when k equals
Q20. The cubic x³ − 7x² + 14x − 8 = 0 has three integer roots. The sum of their squares is
Q21. In (a − b)x² + (b − c)x + (c − a) = 0 with a ≠ b, one root is always
Q22. The sum of all real solutions of 3ˣ + 3²⁻ˣ = 10 is
Q23. The sum of all solutions of x − 5√x + 6 = 0 is
Q24. If α is a root of x² − 2x + 10 = 0, then |α| equals
Q25. If x² − y² = 15 and x + y = 5, then xy equals
Q26. The length of a rectangle exceeds its width by 5 cm and its area is 84 cm². Its perimeter is
Q27. How many real roots does the equation x⁴ + x² + 1 = 0 have?
Q28. When the improper fraction (x³ + 2x² − 1)/(x² + 1) is written as a polynomial plus a proper fraction, the proper fraction is
Q29. A quadratic f(x) = ax² + bx + c has f(0) = 3 and zeros at x = 1 and x = 3. Then f(2) equals
Q30. Let S and P be the sum and the product of the roots of x² − 5x + 6 = 0. The quadratic equation having S and P as its roots is