Moaser Test Yasmin - Complex Numbers PDF

## Unit 1: Complex Numbers ### Remember * The square of an imaginary number is a real number. * The sum and product of two complex numbers are complex numbers. * The conjugate of sum of two complex numbers is the sum of conjugates of the two complex numbers. * The conjugate of product of two compl...

## Unit 1: Complex Numbers ### Remember * The square of an imaginary number is a real number. * The sum and product of two complex numbers are complex numbers. * The conjugate of sum of two complex numbers is the sum of conjugates of the two complex numbers. * The conjugate of product of two complex numbers is the product of conjugates of the two complex numbers. ### Understand * Complex numbers can be represented in the form $ a+bi$ where a and b are real numbers, and $i$ is the imaginary unit. * The real part of $a+bi$ is $a$ and the imaginary part of $a+bi$ is $b$. * The conjugate of $a+bi$ is $a - bi$. * The absolute value of $a+bi$ is $\sqrt{a^2 + b^2}$. ### Exercise 1 **Multiple Choice Questions** 1. Which of the following is an imaginary number? * a) π * b) √15 * c) √(-5) * d) i² 2. i²⁴ = * a) -1 * b) i⁹ * c) -i * d) 1 3. The simplest form of the imaginary number i⁴⁵ is: * a) i * b) -1 * c) -i * d) 1 4. i⁻³⁰ = * a) 1 * b) -1 * c) -i * d) i 5. i¹⁹⁹ = * a) i * b) -i * c) i * d) -1 6. i²⁶ + i²⁸ = * a) i⁵⁴ * b) -i * c) zero * d) 2 7. i²¹ = * a) i²⁴ * b) 2 i * c) -2 i * d) -i 8. 5i⁷ + 4i¹⁸ = * a) 9i * b) -9i * c) i * d) -i 9. 1 + i + i² + i³ + i⁴ = * a) 4i + 1 * b) -1 * c) 1 * d) 5 10. If n ∈ Z, then i⁸ⁿ⁻³ = * a) i * b) -i * c) -1 * d) 1 11. If n ∈ Z, then i⁴ⁿ⁺⁴² = * a) 1 * b) -1 * c) -i * d) i 12. The additive inverse of the complex number (4 - 7i) is: * a) 4 + 7i * b) -4 + 7i * c) -4 - 7i * d) 4 - 7i 13. The conjugate of the number (3i + 4) is: * a) 3i + 4 * b) -3i - 4 * c) -3i + 4 * d) 3i - 4 14. The conjugate of the number (i - i²) is: * a) 1 - i * b) 1 + i * c) -i - 1 * d) i - 1 15. The conjugate of the number (-8) is: * a) 8i * b) -8i * c) -8 * d) 8 16. The conjugate of the number (2 + i)² is: * a) 2 + i * b) (2 + i)⁻¹ * c) 3 + 4i * d) 3 - 4i 17. √(-2 × 1 - 8) = * a) i * b) -2i * c) 4i * d) -4i 18. √(-15) × √(-12) = * a) 6√6 * b) -2i * c) 4i * d) -4i 19. √(-9) × √(i) = * a) 6√6 * b) -6√6 * c) -6√6i * d) -6√6i 20. (-4)(-6) = * a) -10i * b) 24i * c) -24i * d) -24 21. (-2i)³(-3i)² = * a) -72i * b) 72i * c) 72 * d) -72 22. (3 + 2i) + (2 - 5i) = * a) 5 + 2i * b) 5 - 3i * c) 3 - 5i * d) 5 + 3i 23. If X, y are real numbers and (2 + 5i) - (4 - 2i) = X + yi, then X + y = * a) 9 * b) -1 * c) 1 * d) 5 24. (12 - 5i√17) - (7 - i√81) = * a) 5 - 4i * b) -5 + 4i * c) 5 + 4i * d) -5 - 4i 25. 2 - (1 - 2i) + ( 4 - 5i) - (1 - 3i) = * a) 4i * b) -5i * c) 7i * d) 4 26. (4 - 3i)(4 + 3i) = * a) 25i * b) 14 * c) 14i * d) 25 27. If X, y are real numbers and (1 + i)(1 - i) = X + yi, then X + y = * a) 4 * b) 3 * c) 2 * d) 1 28. If X, y are real numbers and X + yi = 14√3 + 3i - 4, then X + y = * a) 3 * b) 5 * c) 3 + 2i * d) 5i 29. If X + yi = 1/i where X, y ∈ R, then X + y = * a) zero * b) 1 * c) -1 * d) 2 30. If 12 + 3ai = 4b - 27i, then a + b = * a) -9 * b) 1 * c) -6 * d) 6 31. If X, y are real numbers and 3X - 2yi = (5 - 2i)², then y - X = * a) 17 * b) -3 * c) 3 * d) 21 - 20i 32. The solution set of the equation : 9X² + 4 = 0 in the set of complex numbers is: * a) { } * b) {2/3 i, -2/3 i} * c) { i/3, -i/3} * d) { √2/3 i, -√2/3 i} 33. If X,y are real numbers and X - 2i√3 + yi = 3 - yi, then the conjugate of the number X+ yi is: * a) 3 - 2i * b) 3 + 2i * c) -3 - 2i * d) -3 + 2i 34. If x² - 2x + 2 = 0, then X = * a) 2 ± 2i * b) 2 ± i * c) 1 ± i * d) 1 ± 2i 35. The multiplicative inverse of the number i/2i +1 is: * a) -2 + i * b) -2 - i * c) 2 - i * d) 2 + i 36. If Z, is the conjugate of the number Z₂, then Z₁Z₂ + (Z₁ + Z₂) is: * a) a real number * b) an imaginary * c) complex, not real * d) undetermined. 37. All of the following are imaginary numbers except * a) √(-1) - 18 * b) i¹⁹ * c) (2 + 2i)⁴ * d) (1 + i)⁶ 38. All the following are not real numbers except * a) (1 + i)⁴ * b) √(-8) * c) i³ * d) √(-π) 39. 3 + 3i + 3i² + 3i³ = * a) zero * b) 3 * c) 12 * d) 12i ### Exercise 2 * An introduction to complex numbers **Multiple Choice Questions** 1. Which of the following is an imaginary number? * a) π * b) √15 * c) √(-5) * d) i² 2. i24 = * a) -1 * b) i⁹ * c) -i * d) 1 3. The simplest form of the imaginary number i45 is: * a) i * b) -1 * c) -i * d) 1 4. i⁻³⁰ = * a) 1 * b) -1 * c) -i * d) i 5. i¹⁹⁹ = * a) 1 * b) -i * c) i * d) -1 6. i²⁶ + i²⁸ = * a) i⁵⁴ * b) -i * c) zero * d) 2 7. i²¹ = * a) i²⁴ * b) 2 i * c) -2 i * d) -i 8. 5i⁷ + 4i¹⁸ = * a) 9i * b) -9i * c) i * d) -i 9. 1 + i + i² + i³ + i⁴ = * a) 4i + 1 * b) -1 * c) 1 * d) 5 10. If n ∈ Z, then i⁸ⁿ⁻³ = * a) i * b) -i * c) -1 * d) 1 11. If n ∈ Z, then i⁴ⁿ⁺⁴² = * a) 1 * b) -1 * c) -i * d) i