AP EAMCET (EAPCET) Mathematic Multiple Choice Questions (MCQs)

AP EAMCET Mathematics - De Moivre's Theorem

1. 41. Which of the following represents De Moivre’s Theorem for a complex number \( z = r (\cos \theta + i \sin \theta) \)?
   • a) \( z^n = r^n (\cos n\theta + i \sin n\theta) \) - Answer
   • b) \( z^n = r^n (\cos n\theta - i \sin n\theta) \)
   • c) \( z^n = r (\cos n\theta + i \sin \theta) \)
   • d) \( z^n = r (\cos \theta - i \sin \theta) \)
2. 42. The n-th roots of unity are the solutions to which equation?
   • a) \( z^n = 1 \) - Answer
   • b) \( z^n = 0 \)
   • c) \( z^n = -1 \)
   • d) \( z^n = 2 \)
3. 43. For \( z = 1 + i \), what is the value of \( z^4 \) using De Moivre’s Theorem?
   • a) \( 16 \) - Answer
   • b) \( 4 \)
   • c) \( 1 + i \)
   • d) \( 8 \)
4. 44. If \( z = \cos \theta + i \sin \theta \), what is the value of \( z^n \) using De Moivre’s Theorem?
   • a) \( \cos n\theta + i \sin n\theta \) - Answer
   • b) \( \cos \theta - i \sin \theta \)
   • c) \( \cos \theta + i \sin \theta \)
   • d) \( \cos 2\theta + i \sin 2\theta \)
5. 45. What is the geometric interpretation of De Moivre’s Theorem?
   • a) The modulus of the complex number increases as \( n \) increases.
   • b) The argument of the complex number is multiplied by \( n \). - Answer
   • c) The real part increases exponentially.
   • d) Both the modulus and argument remain constant.
6. 46. Which of the following is an application of De Moivre’s Theorem?
   • a) Finding powers and roots of complex numbers. - Answer
   • b) Solving linear equations.
   • c) Solving quadratic equations.
   • d) Solving system of inequalities.
7. 47. If \( z = 2 (\cos 30^\circ + i \sin 30^\circ) \), what is \( z^3 \) using De Moivre’s Theorem?
   • a) \( 8 (\cos 90^\circ + i \sin 90^\circ) \) - Answer
   • b) \( 8 (\cos 30^\circ + i \sin 30^\circ) \)
   • c) \( 8 (\cos 60^\circ + i \sin 60^\circ) \)
   • d) \( 8 (\cos 120^\circ + i \sin 120^\circ) \)
8. 48. Which of the following is the correct formula for the n-th roots of unity?
   • a) \( \text{Root} = \cos \frac{2k\pi}{n} + i \sin \frac{2k\pi}{n} \), where \( k = 0, 1, 2, \dots, n-1 \) - Answer
   • b) \( \text{Root} = \cos \frac{k\pi}{n} + i \sin \frac{k\pi}{n} \), where \( k = 0, 1, 2, \dots, n-1 \)
   • c) \( \text{Root} = \cos \frac{2k}{n} + i \sin \frac{2k}{n} \), where \( k = 1, 2, \dots, n-1 \)
   • d) \( \text{Root} = \cos k\pi + i \sin k\pi \), where \( k = 0, 1, 2, \dots, n-1 \)
9. 49. What is true for the rational index in De Moivre’s Theorem?
   • a) The n-th root of a complex number can be found. - Answer
   • b) Only integer powers are valid.
   • c) It only applies to real numbers.
   • d) The formula \( z^n = r^n (\cos n\theta + i \sin n\theta) \) holds only for negative n.
10. 50. The n-th roots of unity lie on a:
    • a) Straight line.
    • b) Circle. - Answer
    • c) Parabola.
    • d) Hyperbola.

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