ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Number Systems and Complex Numbers (Set 3): 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. The recurring decimal 0.272727… expressed as a fraction in lowest terms is:
Answer: B. 3/11
Explanation: Let x = 0.2727…; 100x − x = 27, so x = 27/99 = 3/11.
Q2. The value of i⁴⁷ is:
Answer: A. −i
Explanation: 47 = 4 × 11 + 3, so i⁴⁷ = i³ = −i.
Q3. The value of i + i² + i³ + … + i²⁰²⁶ is:
Answer: C. −1 + i
Explanation: The first 2024 terms form 506 blocks of four, each summing to 0. Remaining terms: i²⁰²⁵ + i²⁰²⁶ = i + (−1) = −1 + i.
Q4. The value of (1 − i)¹⁰ is:
Answer: A. −32i
Explanation: (1 − i)² = −2i, so (1 − i)¹⁰ = (−2i)⁵ = −32i⁵ = −32i.
Q5. The value of [(1 + i)/(1 − i)]²¹ is:
Answer: D. i
Explanation: (1 + i)/(1 − i) = (1 + i)²/2 = i. Then i²¹ = i^(20+1) = i.
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Q6. The modulus of (3 + 4i)/(5 − 12i) is:
Q7. The principal argument of −1 − i (with −π < θ ≤ π) is:
Q8. If (x + iy)(2 − 3i) = 7 + 4i with x, y real, then x equals:
Q9. The square roots of 7 − 24i are:
Q10. If ω is a complex cube root of unity, the value of (1 + ω)³ is:
Q11. The real part of (1 + 2i)³ is:
Q12. The distance in the Argand plane between the points 3 + 4i and −1 + i is:
Q13. The locus of a point z in the Argand plane satisfying |z − 1| = |z + 1| is:
Q14. The equation |z − 2i| = 3 represents a circle with:
Q15. If z = cos(π/9) + i sin(π/9), then z³ + 1/z³ equals:
Q16. The product of all the sixth roots of unity is:
Q17. The product (√2 + √3)(√2 − √3) is:
Q18. Which pair of irrational numbers has a rational sum?
Q19. Which of the following decimals represents an irrational number?
Q20. The set ℕ of natural numbers is NOT closed under:
Q21. For a complex number z = 3 + 4i, the product z·z̄ equals:
Q22. If |z| = 1 and z = (3 + 4i)/5, then 1/z equals:
Q23. The number z = 1 − √3 i has modulus r and principal argument θ. Then rθ equals:
Q24. The value of (2 + i)² − (2 − i)² is:
Q25. For which real value of k is (k + 2i)/(3 − 4i) purely imaginary?
Q26. If z₁ = 2 + i and z₂ = 1 − 3i, then |z₁z₂| equals:
Q27. (cos 20° + i sin 20°)⁶ equals:
Q28. If ω is a complex cube root of unity, then ω¹⁰⁰ + 1/ω¹⁰⁰ equals:
Q29. Which of the following is correct for √(−4) × √(−9)?
Q30. How many of the three cube roots of −8 are real?