ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Number Systems and Complex Numbers (Set 1)

Take the ECAT Mathematics — Number Systems and Complex Numbers (Set 1): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer…

What this test covers

• 30 ECAT Mathematics

Sample questions with answers

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

1. Q1. Which of the following numbers is irrational?

   • A. 0.333…
   • B. √12 ÷ √3
   • C. √5 ✓
   • D. √(16/9)

   Answer: C. √5

   Explanation: √5 cannot be written as p/q. √(16/9) = 4/3, 0.333… = 1/3 and √12 ÷ √3 = √4 = 2 are all rational.

2. Q2. The multiplicative inverse of 3 − 4i is:

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

   Answer: A. (3 + 4i)/25

   Explanation: 1/(3 − 4i) = (3 + 4i)/((3 − 4i)(3 + 4i)) = (3 + 4i)/(9 + 16) = (3 + 4i)/25.

3. Q3. The value of i²⁰²⁷ is:

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

   Answer: A. −i

   Explanation: 2027 = 4 × 506 + 3, so i²⁰²⁷ = i³ = −i.

4. Q4. The modulus of (3 + 4i)/(1 − i) is:

   • A. 5/√2 ✓
   • B. 5√2
   • C. 5/2
   • D. 7/√2

   Answer: A. 5/√2

   Explanation: |(3 + 4i)/(1 − i)| = |3 + 4i| / |1 − i| = 5/√2.

5. Q5. The additive inverse of the complex number −2 + 5i is:

   • A. −2 − 5i
   • B. 2 + 5i
   • C. 2 − 5i ✓
   • D. (−2 − 5i)/29

   Answer: C. 2 − 5i

   Explanation: The additive inverse w satisfies z + w = 0, so w = −z = 2 − 5i. (−2 − 5i is the conjugate, not the additive inverse.)

The other 25 questions

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6. Q6. The conjugate of (2 + i)² is:

   • A. 3 + 4i
   • B. 5 − 4i
   • C. 4 − 4i
   • D. 3 − 4i

7. Q7. The recurring decimal 0.454545… written as a fraction in lowest terms is:

   • A. 45/99
   • B. 5/11
   • C. 4/9
   • D. 9/20

8. Q8. For real numbers, the statement a(b + c) = ab + ac is called the:

   • A. multiplicative identity property
   • B. commutative property of addition
   • C. associative property of multiplication
   • D. distributive property of multiplication over addition

9. Q9. Which of the following sets is NOT closed under subtraction?

   • A. The set of rational numbers
   • B. The set of real numbers
   • C. The set of natural numbers
   • D. The set of integers

10. Q10. The imaginary part of (2 + 3i)/(1 + i) is:

    • A. 3/2
    • B. 5/2
    • C. −1/2
    • D. 1/2

11. Q11. The modulus of z = −5 + 12i is:

    • A. 17
    • B. 7
    • C. 13
    • D. √119

12. Q12. The principal argument of z = −1 + √3 i is:

    • A. −π/3
    • B. 2π/3
    • C. π/3
    • D. 5π/6

13. Q13. If z = x + iy, then z·z̄ is equal to:

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

14. Q14. If (x − 1) + (y + 2)i = 3 − 4i, where x and y are real, then x + y equals:

    • A. −2
    • B. 10
    • C. −10
    • D. 2

15. Q15. The square roots of −5 + 12i are:

    • A. ±(3 + 2i)
    • B. ±(3 − 2i)
    • C. ±(2 + 3i)
    • D. ±(2 − 3i)

16. Q16. The value of i + i² + i³ + … + i¹⁰⁰ is:

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

17. Q17. The value of (1 + i)⁸ is:

    • A. −16
    • B. 16
    • C. 16i
    • D. 8

18. Q18. In the Argand plane, the set of points z satisfying |z − 2| = 3 is:

    • A. a circle with centre (0, 2) and radius 3
    • B. a straight line x = 2
    • C. a circle with centre (2, 0) and radius 3
    • D. a circle with centre (−2, 0) and radius 3

19. Q19. For any complex numbers z₁ and z₂, which statement is always true?

    • A. |z₁ + z₂| = |z₁| + |z₂|
    • B. |z₁ − z₂| = |z₁| − |z₂|
    • C. arg(z₁z₂) = arg z₁ · arg z₂
    • D. |z₁z₂| = |z₁||z₂|

20. Q20. The polar form of 1 + i is:

    • A. √2(cos π/2 + i sin π/2)
    • B. 2(cos π/4 + i sin π/4)
    • C. √2(cos 3π/4 + i sin 3π/4)
    • D. √2(cos π/4 + i sin π/4)

21. Q21. Using De Moivre's theorem, (cos 15° + i sin 15°)⁶ equals:

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

22. Q22. Over the complex numbers, a² + b² can be factorised as:

    • A. (ia + b)(ia − b)
    • B. (a + ib)(a − ib)
    • C. (a + b)²
    • D. (a + b)(a − b)

23. Q23. The roots of the equation z² − 2z + 5 = 0 are:

    • A. 2 ± i
    • B. 1 ± 2i
    • C. 1 ± 4i
    • D. −1 ± 2i

24. Q24. The non-terminating decimal 0.101001000100001… (one more zero each time) represents:

    • A. a natural number
    • B. a terminating decimal
    • C. an irrational number
    • D. a rational number

25. Q25. The value of √(−4) × √(−9) is:

    • A. −6
    • B. 6
    • C. −6i
    • D. 6i

26. Q26. In ordered-pair form, the product (2, 3)·(1, −1) equals:

    • A. (5, 1)
    • B. (5, −1)
    • C. (2, −3)
    • D. (−1, 5)

27. Q27. For a complex number z, the imaginary part Im(z) can be expressed as:

    • A. (z + z̄)/2
    • B. (z + z̄)/(2i)
    • C. (z − z̄)/(2i)
    • D. (z − z̄)/2

28. Q28. If z = 2 − 3i, then z² − 4z equals:

    • A. −5 − 12i
    • B. 3
    • C. −13
    • D. −13 − 24i

29. Q29. If |z| = 1 and z ≠ 0, then 1/z is equal to:

    • A. −z̄
    • B. z̄
    • C. −z
    • D. z

30. Q30. The property 'for any two real numbers a and b, exactly one of a < b, a = b or a > b holds' is called the:

    • A. transitive property
    • B. additive property
    • C. trichotomy property
    • D. Archimedean property