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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Number Systems and Complex Numbers (Set 4)

Take the ECAT Mathematics — Number Systems and Complex Numbers (Set 4): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. The value of (1 + i)¹² is:

    • A. −64i
    • B. 64i
    • C. -64 ✓
    • D. 64

    Answer: C. -64

    Explanation: (1 + i)² = 2i, so (1 + i)¹² = (2i)⁶ = 64·i⁶ = 64·(−1) = −64.

  2. Q2. The real part of 1/(3 + 4i) is:

    • A. 3/5
    • B. −4/25
    • C. 3/25 ✓
    • D. 4/25

    Answer: C. 3/25

    Explanation: 1/(3 + 4i) = (3 − 4i)/25, whose real part is 3/25.

  3. Q3. The locus of the point z in the Argand plane satisfying |z − 3| = 2|z| is:

    • A. a circle with centre (1, 0) and radius 2
    • B. a straight line parallel to the imaginary axis
    • C. a circle with centre (−1, 0) and radius 2 ✓
    • D. a circle with centre (−1, 0) and radius 4

    Answer: C. a circle with centre (−1, 0) and radius 2

    Explanation: Put z = x + iy: (x − 3)² + y² = 4x² + 4y², so 3x² + 6x + 3y² = 9, i.e. (x + 1)² + y² = 4.

  4. Q4. The value of i + i² + i³ + … + i¹⁰³ is:

    • A. −1 ✓
    • B. −i
    • C. i
    • D. 0

    Answer: A. −1

    Explanation: Every block of four consecutive powers sums to 0. The first 100 terms give 0, leaving i¹⁰¹ + i¹⁰² + i¹⁰³ = i − 1 − i = −1.

  5. Q5. Which of the following is a square root of −7 − 24i?

    • A. 3 + 4i
    • B. 4 − 3i
    • C. 3 − 4i ✓
    • D. 4 + 3i

    Answer: C. 3 − 4i

    Explanation: (3 − 4i)² = 9 − 24i + 16i² = −7 − 24i. The option 3 + 4i gives −7 + 24i, and 4 ∓ 3i give 7 ∓ 24i.

The other 25 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

  1. Q6. Given ω³ = 1 with ω ≠ 1, simplify (1 + ω²)⁴.

    • A. ω
    • B. −1
    • C. 1
    • D. ω²

  2. Q7. Given ω³ = 1 with ω ≠ 1, find the product (1 − ω)(1 − ω²)(1 − ω⁴)(1 − ω⁵).

    • A. −9
    • B. 9
    • C. 0
    • D. 3

  3. Q8. Find the principal argument of the quotient (1 + i√3)/(1 − i).

    • A. 5π/12
    • B. 7π/12
    • C. −π/12
    • D. π/12

  4. Q9. If z + 1/z = 1, then z¹⁰⁰ + 1/z¹⁰⁰ equals:

    • A. −1
    • B. 1
    • C. 2
    • D. −2

  5. Q10. The modulus of (6 + 8i)(5 + 12i)/(3 − 4i) is:

    • A. 130
    • B. 65
    • C. 26
    • D. 13

  6. Q11. If z = (√3 + i)/2, the smallest positive integer n for which zⁿ is a real number is:

    • A. 3
    • B. 6
    • C. 4
    • D. 12

  7. Q12. The number of complex numbers z satisfying z² = z̄ (the conjugate of z) is:

    • A. 4
    • B. 1
    • C. 2
    • D. 3

  8. Q13. Find Im[(1 + i)²/(2 − i)].

    • A. 4
    • B. 4/5
    • C. 2/5
    • D. −2/5

  9. Q14. If √(15 − 8i) = a + ib with a > 0, then a + b equals:

    • A. −3
    • B. 3
    • C. 5
    • D. −5

  10. Q15. Find the product of all six complex roots of the equation z⁶ = 1.

    • A. −1
    • B. 0
    • C. i
    • D. 1

  11. Q16. The recurring decimal 0.4333… (only the 3 repeating) equals:

    • A. 13/30
    • B. 13/33
    • C. 43/99
    • D. 43/90

  12. Q17. Select the rational number from the following list.

    • A. 1/(√5 − √3)
    • B. √(3 + 2√2) − √(3 − 2√2)
    • C. √(3 + 2√2) + √(3 − 2√2)
    • D. (√2 + √3)²

  13. Q18. The principal argument of (−1 + i)⁵ is:

    • A. −π/4
    • B. 3π/4
    • C. π/4
    • D. 15π/4

  14. Q19. If |z₁ + z₂| = |z₁ − z₂| for non-zero complex numbers z₁, z₂, then z₁/z₂ is:

    • A. purely imaginary
    • B. purely real and negative
    • C. of modulus 2
    • D. purely real and positive

  15. Q20. If |z| = 1 and z ≠ −1, the number (z − 1)/(z + 1) is:

    • A. purely real
    • B. equal to z
    • C. purely imaginary
    • D. equal to 1

  16. Q21. If |z − 3 − 4i| = 2, the greatest value of |z| is:

    • A. 3
    • B. 5
    • C. 2
    • D. 7

  17. Q22. The points representing the roots of z⁴ = 16 in the Argand plane form a polygon of area:

    • A. 4
    • B. 16√2
    • C. 8
    • D. 16

  18. Q23. The value of 1·i + 2i² + 3i³ + … + 8i⁸ is:

    • A. 4 + 4i
    • B. 8
    • C. −4 + 4i
    • D. 4 − 4i

  19. Q24. Which statement about irrational numbers is FALSE?

    • A. Between any two real numbers there is an irrational number
    • B. The product of two irrational numbers is always irrational
    • C. The sum of a rational and an irrational number is irrational
    • D. The decimal expansion of an irrational number neither terminates nor repeats

  20. Q25. If a < b and c < 0, which inequality is correct?

    • A. a + c > b + c
    • B. ac < bc
    • C. ac > bc
    • D. a/c < b/c

  21. Q26. If (x + iy)(2 − 3i) = 4 + i, then y equals:

    • A. −14/13
    • B. 14/13
    • C. 1/13
    • D. 5/13

  22. Q27. For 0 < θ < π, the modulus of 1 + cos θ + i sin θ is:

    • A. 2cos(θ/2)
    • B. 2cos θ
    • C. cos θ
    • D. 2sin(θ/2)

  23. Q28. Given ω³ = 1 with ω ≠ 1, evaluate (3 + 2ω + ω²)(3 + 2ω² + ω).

    • A. 9
    • B. 6
    • C. 0
    • D. 3

  24. Q29. If 2 − 3i is a root of x² + px + q = 0 with real p and q, then q equals:

    • A. −4
    • B. 4
    • C. −13
    • D. 13

  25. Q30. Write −√3 + i in the form r(cos θ + i sin θ). The result is:

    • A. 2(cos 5π/6 + i sin 5π/6)
    • B. 2(cos 7π/6 + i sin 7π/6)
    • C. 2(cos π/6 + i sin π/6)
    • D. √2(cos 5π/6 + i sin 5π/6)