SmartPrep

ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Number Systems and Complex Numbers (Set 2)

Take the ECAT Mathematics — Number Systems and Complex Numbers (Set 2): 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. If z = (1 + 2i)(2 − i), then |z| equals:

    • A. √10
    • B. 5 ✓
    • C. 25
    • D. √5

    Answer: B. 5

    Explanation: |z₁z₂| = |z₁||z₂| = √5 · √5 = 5. (Indeed z = 4 + 3i and |4 + 3i| = 5.)

  2. Q2. If z = cos θ + i sin θ, then z + 1/z equals:

    • A. 0
    • B. 2 sin θ
    • C. 2 cos θ ✓
    • D. 2i sin θ

    Answer: C. 2 cos θ

    Explanation: 1/z = cos θ − i sin θ (by De Moivre with n = −1), so z + 1/z = 2 cos θ.

  3. Q3. If z = 3 − 2i, then z + z̄ equals:

    • A. 0
    • B. 6 ✓
    • C. 6 − 4i
    • D. −4i

    Answer: B. 6

    Explanation: z̄ = 3 + 2i, so z + z̄ = 6. In general z + z̄ = 2 Re(z).

  4. Q4. The value of (1 + i)/(1 − i) is:

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

    Answer: B. i

    Explanation: Multiply by the conjugate: (1 + i)²/((1 − i)(1 + i)) = 2i/2 = i.

  5. Q5. Using De Moivre's theorem, the value of (√3 + i)⁶ is:

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

    Answer: B. −64

    Explanation: √3 + i = 2(cos 30° + i sin 30°), so (√3 + i)⁶ = 2⁶(cos 180° + i sin 180°) = 64(−1) = −64.

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. The recurring decimal 0.1666… (only the 6 repeating) written as a fraction in lowest terms is:

    • A. 1/9
    • B. 3/20
    • C. 1/6
    • D. 1/60

  2. Q7. Which of the following numbers is rational?

    • A. √0.09
    • B. 1 + √2
    • C. π/2
    • D. √0.9

  3. Q8. For which real value of a is (2 + ai)/(1 − i) a real number?

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

  4. Q9. The multiplicative identity in the set of complex numbers, written as an ordered pair, is:

    • A. (1, 1)
    • B. (0, 1)
    • C. (0, 0)
    • D. (1, 0)

  5. Q10. If 2z + z̄ = 3 + i, then z equals:

    • A. 1 − i
    • B. 3 + i
    • C. 1 + i
    • D. 1 + i/3

  6. Q11. In the Argand plane, the set of all points z satisfying |z − 1| = |z + 1| is:

    • A. the line y = x
    • B. the imaginary (y-) axis
    • C. a circle of radius 1 centred at the origin
    • D. the real (x-) axis

  7. Q12. The principal argument of z = −2 − 2i is:

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

  8. Q13. The conjugate of 1/(3 + 4i) is:

    • A. 3 + 4i
    • B. (3 − 4i)/25
    • C. (3 + 4i)/25
    • D. (3 + 4i)/5

  9. Q14. The value of (1 + 2i)³ is:

    • A. −3 + 4i
    • B. 1 + 8i
    • C. −11 − 2i
    • D. −11 + 2i

  10. Q15. The square roots of −8i are:

    • A. ±(1 − i)
    • B. ±(2 + 2i)
    • C. ±2√2 i
    • D. ±(2 − 2i)

  11. Q16. The distance between the points representing z₁ = 1 + 2i and z₂ = 4 + 6i in the Argand plane is:

    • A. 25
    • B. √7
    • C. 5
    • D. 7

  12. Q17. For real numbers, the statement a + (b + c) = (a + b) + c is called the:

    • A. associative property of addition
    • B. commutative property of addition
    • C. additive identity property
    • D. distributive property

  13. Q18. The modulus of (1 + i)¹⁰ is:

    • A. 16
    • B. 10√2
    • C. 1024
    • D. 32

  14. Q19. The modulus of (2 + i)³/(1 + 2i)² is:

    • A. √5
    • B. 5
    • C. 5√5
    • D. 1

  15. Q20. The smallest positive integer n for which ((1 + i)/(1 − i))ⁿ = 1 is:

    • A. 8
    • B. 2
    • C. 1
    • D. 4

  16. Q21. The real part of 1/(2 + i) is:

    • A. 1/5
    • B. 2/5
    • C. 1/2
    • D. −1/5

  17. Q22. If z = x + iy (x, y real), then the real part of z² is:

    • A. x² + y²
    • B. x² − y²
    • C. 2xy
    • D. x² − 2xy + y²

  18. Q23. The number 22/7, often used for π, is:

    • A. a non-real number
    • B. a rational number
    • C. exactly equal to π
    • D. an irrational number

  19. Q24. How many complex numbers z satisfy z² + |z| = 0?

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

  20. Q25. The value of i⁻³ is:

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

  21. Q26. If (x + iy)(2 − i) = 5, where x and y are real, then xy equals:

    • A. −2
    • B. 5/2
    • C. 1
    • D. 2

  22. Q27. The polar form of −√3 + i is:

    • A. 2(cos 5π/6 + i sin 5π/6)
    • B. 2(cos 2π/3 + i sin 2π/3)
    • C. 2(cos π/6 + i sin π/6)
    • D. √2(cos 5π/6 + i sin 5π/6)

  23. Q28. Which real number has no multiplicative inverse?

    • A. −1
    • B. 1
    • C. 1/2
    • D. 0

  24. Q29. For every complex number z, which statement is always true?

    • A. |z + z̄| = 2|z|
    • B. |z̄| = |z|
    • C. z̄ = −z
    • D. |z²| = 2|z|

  25. Q30. The value of (1 − i)⁶ is:

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