ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Number Systems and Complex Numbers (Set 5): 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 imaginary part of (1 + 2i)³ is:
Answer: A. −2
Explanation: (1 + 2i)³ = 1 + 6i + 12i² + 8i³ = 1 + 6i − 12 − 8i = −11 − 2i, so the imaginary part is −2.
Q2. The smallest positive integer n for which ((√3 + i)/2)ⁿ = −1 is:
Answer: C. 6
Explanation: (√3 + i)/2 = cos 30° + i sin 30°, so the nth power is cos 30n° + i sin 30n°, which equals −1 first when 30n = 180, i.e. n = 6.
Q3. If z + 1/z = 1, then the value of z¹⁰⁰ + 1/z¹⁰⁰ is:
Answer: C. −1
Explanation: z² − z + 1 = 0 gives z³ = −1, so z⁶ = 1. Then z¹⁰⁰ = z⁴ = −z, 1/z¹⁰⁰ = −1/z, so the sum is −(z + 1/z) = −1.
Q4. The locus of points z in the Argand plane satisfying |z − 1| = |z + i| is:
Answer: C. the line y = −x
Explanation: (x − 1)² + y² = x² + (y + 1)² gives −2x = 2y, i.e. y = −x.
Q5. Taking the principal value, Arg[(1 − i)(√3 + i)] equals:
Answer: A. −π/12
Explanation: arg(1 − i) = −π/4 and arg(√3 + i) = π/6; arguments add under multiplication: −π/4 + π/6 = −π/12.
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Q6. If x and y are real and (x + iy)(1 + i) = 3 − i, then x − y equals:
Q7. If |z₁| = 3 and |z₂| = 4, then |z₁ + z₂|² + |z₁ − z₂|² equals:
Q8. If |z| = 2, the least possible value of |z + 6 + 8i| is:
Q9. For which real k is (k + 2i)/(1 + i) purely imaginary?
Q10. The product of all five roots of the equation z⁵ = 32 is:
Q11. The number of complex numbers z satisfying z⁴ = z̄ is:
Q12. The cubic x³ − 5x² + 17x − 13 = 0 has the root 2 + 3i. Its real root is:
Q13. If z² = 7 − 24i, then |z| equals:
Q14. The complex number with modulus 2 and argument 2π/3 is:
Q15. The argument of −1 − i, measured in the interval [0, 2π), is:
Q16. If |z| = 4, the triangle with vertices 0, z and iz has area:
Q17. The distance between the points representing 2 − 3i and −1 + i is:
Q18. Which statement about nonzero complex numbers z₁, z₂ is NOT always true when principal arguments are used?
Q19. If |z| = 5 and Re z = 3 with Im z negative, then z is:
Q20. The value of (1 + i√3)⁹ is:
Q21. The value of (√3 + i)⁻⁶ is:
Q22. The value of 1·i + 2·i² + 3·i³ + … + 40·i⁴⁰ is:
Q23. For every real x, the modulus of (1 + ix)/(1 − ix) is:
Q24. Rationalising, 1/(√3 − √2) − 1/(√3 + √2) equals:
Q25. The sum of the cubes of all six sixth roots of unity is:
Q26. The imaginary part of (1 + i)⁴/(1 − i)² is:
Q27. How many nonzero complex numbers satisfy z² = z̄?
Q28. Pairs are multiplied by (a, b)·(c, d) = (ac − bd, ad + bc). The multiplicative inverse of (3, 4) is:
Q29. The recurring decimal 0.12333… (the digit 3 repeating) equals:
Q30. If a < b < 0, which inequality must be true?