Complex Numbers
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Questions and Answers

What is the ordered pair representation of the imaginary unit i?

  • (1, 1)
  • (0, 1) (correct)
  • (1, 0)
  • (0, 0)
  • Which of the following statements about complex numbers is true?

  • A complex number can be written as a + bi, where a and b are real numbers. (correct)
  • A complex number is always an imaginary number.
  • A complex number is always purely imaginary.
  • A complex number is always a real number.
  • What is the condition for a complex number to be imaginary?

  • Its imaginary part is zero.
  • Its imaginary part is non-zero. (correct)
  • Its real part is non-zero.
  • Its real part is zero.
  • What is the relationship between the set of real numbers (R) and the set of complex numbers (C)?

    <p>R is a subset of C.</p> Signup and view all the answers

    How can a complex number be represented?

    <p>As an ordered pair of real numbers.</p> Signup and view all the answers

    What is the value of |z| if iz² - z = 0?

    <p>0 or 1</p> Signup and view all the answers

    If z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, where x₁, x₂, y₁, y₂ ∈ R, and 0 < x₁ < x₂, y₁ = y₂, then which of the following is true?

    <p>|z₁| &lt; |z₃| &lt; |z₂|</p> Signup and view all the answers

    If z = a + ib, what is the argument of z?

    <p>tan⁻¹(b/a)</p> Signup and view all the answers

    What is the value of i²?

    <p>-1</p> Signup and view all the answers

    What is the representation of a complex number z = a + ib in the complex plane?

    <p>A point P in the complex plane, where the angle made by the line OP with the real axis is the argument of z</p> Signup and view all the answers

    What is the value of i^2 in complex numbers?

    <p>-1</p> Signup and view all the answers

    Which of the following is a correct representation of a complex number?

    <p>x + iy</p> Signup and view all the answers

    What is the value of -a when a is a positive real number?

    <p>ia</p> Signup and view all the answers

    What is the property of the imaginary unit i?

    <p>i is neither 0, nor greater than 0, nor less than 0</p> Signup and view all the answers

    What is the set of complex numbers denoted by?

    <p>C</p> Signup and view all the answers

    What is the property of addition of complex numbers that satisfies z1 + z2 = z2 + z1?

    <p>Commutative property</p> Signup and view all the answers

    Which of the following is a property of multiplication of complex numbers?

    <p>Commutative property</p> Signup and view all the answers

    What is the additive inverse of a complex number z = a + ib?

    <p>-z</p> Signup and view all the answers

    What is the identity element for multiplication of complex numbers?

    <p>1 + 0i</p> Signup and view all the answers

    What is the property of complex numbers that satisfies z1(z2 + z3) = z1z2 + z1z3?

    <p>Distributive property over addition</p> Signup and view all the answers

    If z₁ = a + ib and z₂ = c + id, what is the value of z₁ + z₂?

    <p>(a + c) + i(b + d)</p> Signup and view all the answers

    What is the value of z₁ - z₂ if z₁ = 2 + 3i and z₂ = 1 + 2i?

    <p>1 + i</p> Signup and view all the answers

    What can be said about the imaginary part of a complex number if it is a purely imaginary number?

    <p>It is always non-zero</p> Signup and view all the answers

    If z is a complex number, what is the relationship between z and its conjugate?

    <p>z is always unequal to its conjugate</p> Signup and view all the answers

    What is the result of multiplying a complex number by its conjugate?

    <p>A real number</p> Signup and view all the answers

    If z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, where x₁, x₂, y₁, y₂ ∈ R and x₁ < x₂, y₁ > y₂, then which of the following is true?

    <p>arg(z₁) &lt; arg(z₂)</p> Signup and view all the answers

    If z = a + ib, where a, b ∈ R, then which of the following is a property of z?

    <p>z² = a² - b²</p> Signup and view all the answers

    If z = x + iy, where x, y ∈ R, then the principal value of arg(z) lies in which of the following intervals?

    <p>(-π, π]</p> Signup and view all the answers

    If z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, where x₁, x₂, y₁, y₂ ∈ R, then which of the following is true?

    <p>z₁ + z₂ = (x₁ + x₂) + i(y₁ + y₂)</p> Signup and view all the answers

    If z = x + iy, where x, y ∈ R, then which of the following is the polar form of z?

    <p>z = r(cos(θ) + i sin(θ))</p> Signup and view all the answers

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