ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Number Systems and Complex Numbers (Set 6): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. If ω is a non-real cube root of unity, the value of (1 + ω)(1 + ω²)(1 + ω⁴)(1 + ω⁸) is:
Answer: D. 1
Explanation: ω⁴ = ω and ω⁸ = ω², so the product is [(1 + ω)(1 + ω²)]² . Now (1 + ω)(1 + ω²) = 1 + ω + ω² + ω³ = 0 + 1 = 1, so the product is 1.
Q2. If ω is a complex cube root of unity, the value of (3 + 2ω)(3 + 2ω²) is:
Answer: D. 7
Explanation: (3 + 2ω)(3 + 2ω²) = 9 + 6(ω + ω²) + 4ω³ = 9 − 6 + 4 = 7.
Q3. If ω is a non-real cube root of unity, the sum ω + ω² + ω³ + … + ω¹⁰⁰ (100 terms) equals:
Answer: B. ω
Explanation: Every three consecutive powers sum to 0, so the first 99 terms add to 0. The remaining term is ω¹⁰⁰ = ω⁹⁹·ω = ω.
Q4. If ω is a complex cube root of unity, the modulus of 1 + 2ω + 3ω² is:
Answer: C. √3
Explanation: Using ω² = −1 − ω gives 1 + 2ω + 3(−1 − ω) = −2 − ω. With ω = −½ + (√3/2)i this is −3/2 − (√3/2)i, whose modulus is √(9/4 + 3/4) = √3.
Q5. The remainder when x¹⁰⁰ + 1 is divided by x² + x + 1 is:
Answer: C. x + 1
Explanation: The roots of x² + x + 1 are ω and ω². Since ω¹⁰⁰ = ω, the remainder r(x) = ax + b satisfies r(ω) = ω + 1, so r(x) = x + 1.
Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.
Q6. The real part of (1 + i)⁻⁴ is:
Q7. The imaginary part of [(1 + i)/(2 − i)]³ is:
Q8. How many points (a, b) with integer coordinates satisfy a² + b² = 25?
Q9. The recurring decimal 0.1363636… (the digits 36 repeat after the 1) equals the fraction:
Q10. The recurring decimal 2.3151515… (the digits 15 repeat after 2.3) equals:
Q11. The 100th digit after the decimal point in the decimal expansion of 1/7 = 0.142857142857… is:
Q12. √(7 + 4√3) simplifies to:
Q13. If z + 1/z = √2, then z⁸ + 1/z⁸ equals:
Q14. For which real value of a is |z₁ + z₂| = |z₁ − z₂| when z₁ = 2 + ai and z₂ = 1 + 3i?
Q15. The product of the four complex fourth roots of −16 is:
Q16. The locus of z (z ≠ −1) for which (z − 1)/(z + 1) is purely imaginary and non-zero is:
Q17. Which of the cube roots of i is purely imaginary?
Q18. The modulus of (1 + i)⁵/(1 − i)³ is:
Q19. The real part of (−1 + i√3)⁵ is:
Q20. The value of (1 + √2)³ + (1 − √2)³ is:
Q21. If |z₁| = 3, |z₂| = 4 and |z₁ + z₂| = 5, then |z₁ − z₂| equals:
Q22. With multiplication of ordered pairs defined by (a, b)(c, d) = (ac − bd, ad + bc), the product (2, 3)(4, −1) is:
Q23. For non-zero real numbers a < b, which condition guarantees 1/a > 1/b?
Q24. Writing (1 + i√3)/(1 − i) as r(cos θ + i sin θ) with −π < θ ≤ π, the angle θ equals:
Q25. For 0 < α < π, the argument of 1 − cos α + i sin α is:
Q26. How many complex numbers z satisfy |z| = 1 and z⁶ = z̄?
Q27. The smallest positive integer n for which (1 + i)ⁿ is a positive real number is:
Q28. Which of these fractions has a terminating decimal expansion?
Q29. The value of i · i² · i³ · … · i¹⁰ is:
Q30. If z = (1 + i)/√2, then z²⁰²⁶ equals: