JEE Main - Complex numbers as ordered pairs of reals Online Test (MCQ Quiz)

1. What is the fundamental representation of a complex number in the form of an ordered pair of real numbers?

A) (a, b) B) {a, b} C) [a, b] D) <a, b>

2. In the ordered pair (a, b) representing a complex number, 'a' represents which part of the complex number?

A) Imaginary part B) Real part C) Modulus D) Argument

3. In the ordered pair (a, b) representing a complex number, 'b' represents which part of the complex number?

A) Real part B) Modulus C) Imaginary part D) Argument

4. How is the complex number 'a + ib' represented as an ordered pair of reals?

A) (b, a) B) (a, b) C) (a, -b) D) (-a, b)

5. What is the ordered pair representation of the complex number 5 - 3i?

A) (5, -3) B) (-3, 5) C) (5, 3) D) (-3, -5)

6. The complex number 7i can be represented as an ordered pair of reals. What is this ordered pair?

A) (7, 0) B) (0, 7) C) (0, -7) D) (7, 7)

7. What is the ordered pair representation of the real number 9?

A) (9, 9) B) (0, 9) C) (9, 0) D) (-9, 0)

8. If two complex numbers (a, b) and (c, d) are equal, what conditions must hold true?

A) a = c and b = d B) a = d and b = c C) a = -c and b = -d D) a + b = c + d

9. Consider the complex numbers z1 = (3, 4) and z2 = (x, y). If z1 = z2, what are the values of x and y?

A) x = 4, y = 3 B) x = -3, y = -4 C) x = 3, y = 4 D) x = -4, y = -3

10. How is the sum of two complex numbers z1 = (a, b) and z2 = (c, d) represented as an ordered pair?

A) (a+c, b+d) B) (ac, bd) C) (a+b, c+d) D) (ad, bc)

11. Let z1 = (2, 3) and z2 = (5, 1). What is the ordered pair representation of z1 + z2?

A) (10, 3) B) (7, 4) C) (2+5, 3+1) D) (7, 3)

12. How is the difference of two complex numbers z1 = (a, b) and z2 = (c, d) represented as an ordered pair?

A) (a-c, b-d) B) (ac, bd) C) (a-b, c-d) D) (ad, bc)

13. Let z1 = (6, 2) and z2 = (3, 5). What is the ordered pair representation of z1 - z2?

A) (3, -3) B) (18, 10) C) (3, 2) D) (6-3, 2-5)

14. How is the product of two complex numbers z1 = (a, b) and z2 = (c, d) represented as an ordered pair?

A) (ac - bd, ad + bc) B) (ac + bd, ad - bc) C) (ac, bd) D) (a+c, b+d)

15. Let z1 = (1, 2) and z2 = (3, 4). What is the ordered pair representation of z1 * z2?

A) (3, 8) B) (1*3 - 2*4, 1*4 + 2*3) C) (1*3 + 2*4, 1*4 - 2*3) D) (7, 10)

16. Calculate the product of z1 = (2, -1) and z2 = (-3, 5) in ordered pair form.

A) (-6, 5) B) (2*(-3) - (-1)*5, 2*5 + (-1)*(-3)) C) (-11, 13) D) (-1, 4)

17. What is the ordered pair representation of the complex number 'i'?

A) (1, 0) B) (0, 1) C) (i, 0) D) (0, i)

18. What is the ordered pair representation of the complex number '-i'?

A) (0, -1) B) (0, 1) C) (-1, 0) D) (1, 0)

19. What is the ordered pair representation of the complex number '-1'?

A) (0, -1) B) (-1, 0) C) (1, 0) D) (0, 1)

20. Consider the complex number z = (a, b). What is its conjugate represented as an ordered pair?

A) (a, -b) B) (-a, b) C) (-a, -b) D) (b, a)

21. What is the conjugate of the complex number z = (3, -5) in ordered pair form?

A) (-3, 5) B) (3, 5) C) (-3, -5) D) (5, -3)

22. If z = (a, b), what is the ordered pair representation of z + conjugate(z)?

A) (2a, 0) B) (0, 2b) C) (a, b) D) (2a, 2b)

23. If z = (a, b), what is the ordered pair representation of z - conjugate(z)?

A) (0, 2b) B) (2a, 0) C) (a, b) D) (0, 2a)

24. If z = (a, b), what is the ordered pair representation of i * z?

A) (-b, a) B) (b, -a) C) (a, -b) D) (-a, b)

25. Let z = (2, 3). What is the ordered pair representation of i * z?

A) (-3, 2) B) (3, -2) C) (2, -3) D) (-2, 3)

26. What is the ordered pair representation of the modulus of a complex number z = (a, b)?

A) (sqrt(a^2 + b^2), 0) B) (0, sqrt(a^2 + b^2)) C) (a^2 + b^2, 0) D) (sqrt(a^2 - b^2), 0)

27. The modulus of the complex number z = (3, 4) is represented as an ordered pair. What is it?

A) (5, 0) B) (0, 5) C) (7, 0) D) (25, 0)

28. Which property is satisfied by the ordered pair representation of complex numbers for addition? (z1 + z2) + z3 = z1 + (z2 + z3)

A) Commutative Property B) Associative Property C) Distributive Property D) Identity Property

29. Which property is satisfied by the ordered pair representation of complex numbers for addition? z1 + z2 = z2 + z1

A) Associative Property B) Commutative Property C) Distributive Property D) Inverse Property

30. What is the additive identity in the ordered pair representation of complex numbers?

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

31. What is the ordered pair representation of the additive inverse of z = (a, b)?

A) (-a, -b) B) (a, -b) C) (-a, b) D) (b, a)

32. Which property is satisfied by the ordered pair representation of complex numbers for multiplication? (z1 * z2) * z3 = z1 * (z2 * z3)

A) Commutative Property B) Associative Property C) Distributive Property D) Identity Property

33. Which property is satisfied by the ordered pair representation of complex numbers for multiplication? z1 * z2 = z2 * z1

A) Associative Property B) Commutative Property C) Distributive Property D) Inverse Property

34. What is the multiplicative identity in the ordered pair representation of complex numbers?

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

35. Which property relates addition and multiplication for ordered pair representations of complex numbers? z1 * (z2 + z3) = z1 * z2 + z1 * z3

A) Commutative Property B) Associative Property C) Distributive Property D) Identity Property

36. What is the ordered pair representation of the multiplicative inverse of a non-zero complex number z = (a, b)?

A) (a/(a^2+b^2), -b/(a^2+b^2)) B) (a/(a^2-b^2), b/(a^2+b^2)) C) (b/(a^2+b^2), a/(a^2+b^2)) D) (a/(a^2+b^2), b/(a^2+b^2))

37. Find the multiplicative inverse of the complex number z = (2, 1) in ordered pair form.

A) (2/5, -1/5) B) (1/5, 2/5) C) (2/3, -1/3) D) (1/3, 2/3)

38. The set of all ordered pairs of real numbers with defined addition and multiplication operations forms which algebraic structure?

A) A Field B) A Ring C) A Group D) A Vector Space

39. Consider the complex number z = (a, b). What is the ordered pair representation of z * conjugate(z)?

A) (a^2 + b^2, 0) B) (a^2 - b^2, 0) C) (a^2, b^2) D) (0, a^2 + b^2)

40. Let z1 = (1, 0) and z2 = (0, 1). What is z1 * z2 in ordered pair form?

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

41. Let z1 = (0, 1) and z2 = (0, 1). What is z1 * z2 in ordered pair form?

A) (1, 0) B) (-1, 0) C) (0, 1) D) (0, -1)

42. The ordered pair representation of a complex number is a direct consequence of extending the number system from real numbers to:

A) Integers B) Rational numbers C) Complex numbers D) Imaginary numbers

43. In the ordered pair (a, b), the term 'b' is multiplied by which fundamental imaginary unit?

A) j B) k C) i D) l

44. What is the ordered pair representation of the complex number 2 + 0i?

A) (0, 2) B) (2, 2) C) (2, 0) D) (0, 0)

45. If z = (a, b) is a complex number, then the expression a + ib is equivalent to:

A) (b, a) B) (a, b) C) (a, -b) D) (b, -a)

46. The geometric interpretation of a complex number (a, b) is a point in which plane?

A) Cartesian Plane B) Argand Plane C) Coordinate Plane D) Complex Plane

47. In the Argand plane, the horizontal axis represents the:

A) Imaginary part B) Modulus C) Argument D) Real part

48. In the Argand plane, the vertical axis represents the:

A) Real part B) Modulus C) Argument D) Imaginary part

49. The operation of complex number multiplication in ordered pair form, (a, b) * (c, d) = (ac - bd, ad + bc), is derived from expanding which expression?

A) (a + ib) * (c + id) B) (a - ib) * (c + id) C) (a + ib) * (c - id) D) (a - ib) * (c - id)

50. Which of the following is NOT a property of complex number addition represented as ordered pairs?

A) Closure B) Commutativity C) Associativity D) Non-distributivity

Complex numbers as ordered pairs of reals - Online Test

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