ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Number Systems and Complex Numbers (Set 7): 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. Evaluate (1 + i)²⁰.
Answer: A. −1024
Explanation: (1 + i)² = 2i, so (1 + i)²⁰ = (2i)¹⁰ = 1024·i¹⁰ = −1024.
Q2. The locus of the point z satisfying |z − 2| = |z + 2i| is a straight line with slope
Answer: A. −1
Explanation: With z = x + iy: (x − 2)² + y² = x² + (y + 2)² gives −4x + 4 = 4y + 4, i.e. y = −x, slope −1.
Q3. One of the square roots of 8 − 6i is
Answer: C. 3 − i
Explanation: (3 − i)² = 9 − 6i + i² = 8 − 6i. The other root is −3 + i.
Q4. For real θ (not a multiple of 2π), the real part of 1/(1 − cos θ + i sin θ) is
Answer: C. ½
Explanation: Multiply by the conjugate: (1 − cos θ − i sin θ)/[(1 − cos θ)² + sin²θ] = (1 − cos θ − i sin θ)/(2 − 2cos θ). The real part is (1 − cos θ)/(2(1 − cos θ)) = ½.
Q5. The number of non-real roots of x⁶ = 1 is
Answer: A. 4
Explanation: x⁶ = 1 has six roots in total, the sixth roots of unity. Only x = 1 and x = −1 are real, so 4 are non-real.
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Q6. The recurring decimal 0.454545… (45 repeating) equals the rational number
Q7. Which statement about irrational numbers is true?
Q8. If a > b and c < d, which inequality must be true?
Q9. For z = 2 − 3i, the value of z·z̄ + z + z̄ is
Q10. If z₁ = 2 + i and z₂ = 1 − 2i, then z₁·z̄₂ equals
Q11. If (x + iy)(2 − 3i) = 4 + 7i with x, y real, then x + y equals
Q12. Which of the following is a square root of −i?
Q13. If (√3 + √2)/(√3 − √2) = a + b√6 with a, b rational, then a + b equals
Q14. The value of i⁻²⁵ is
Q15. For z = −1 + i, the principal argument of z/z̄ is
Q16. If |z| = 1, the greatest possible value of |z + 1/z| is
Q17. The value of Σ iᵏ·(−i)ᵏ for k = 0 to 7 (eight terms) is
Q18. The principal argument of z = −sin 40° + i cos 40° is
Q19. If z + 1/z = √3, then z²⁰²⁶ + 1/z²⁰²⁶ equals
Q20. Let a be a rational number and b an irrational number. Which of the following is necessarily irrational?
Q21. The multiplicative inverse of (√3 − i)/2 is
Q22. The modulus of (1 + i)⁸(3 − 4i)/(1 − 2i) is
Q23. How many complex numbers z with |z| = 2 satisfy z² + z̄² = 0?
Q24. In ordered-pair form with (a, b)(c, d) = (ac − bd, ad + bc), the multiplicative inverse of (3, 4) is
Q25. The value of (cos 40° − i sin 40°)³ is
Q26. The value of (√3 − i)⁵ is
Q27. The product of the three cube roots of −27 is
Q28. The area of the triangle whose vertices are the three cube roots of unity 1, ω and ω² is
Q29. If ω is a complex cube root of unity, then (a + bω + cω²)/(c + aω + bω²) equals
Q30. If ω is a non-real cube root of unity, then [(1 + ω²)/(1 + ω)]²⁰²⁶ equals