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Which of the following statements are true?
1. The amplitude of the product of complex numbers is equals to the product of their amplitudes
2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs
3. Order relation exists in complex number whereas it does not exist in real numbers
4.The values of ω used as a cube roots of unity and as a fourth root of unity are different
1. The amplitude of the product of complex numbers is equals to the product of their amplitudes
2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs
3. Order relation exists in complex number whereas it does not exist in real numbers
4.The values of ω used as a cube roots of unity and as a fourth root of unity are different
- a)(1) and (2) only
- b)(2) and (4) only
- c)(3) and (4) only
- d)(1),(2) and (4) only
Correct answer is option 'B'. Can you explain this answer?
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Which of the following statements are true? 1. The amplitude of the pr...
True Statements and Explanation:
1. The amplitude of the product of complex numbers is equals to the product of their amplitudes.
- This statement is true.
- The amplitude of a complex number is its modulus or absolute value, which is the distance of the complex number from the origin in the complex plane.
- When two complex numbers are multiplied, their moduli are multiplied and their arguments are added.
- Hence, the amplitude of the product of two complex numbers is the product of their moduli.
2. For any polynomial f(x) = 0 with real coefficients, imaginary roots occur in conjugate pairs.
- This statement is true.
- If a polynomial has real coefficients, then non-real roots occur in conjugate pairs.
- This means that if a polynomial has an imaginary root a + bi, then its conjugate a - bi is also a root.
- This is because the coefficients of the polynomial are real and the complex roots must satisfy the polynomial equation.
3. Order relation exists in complex number whereas it does not exist in real numbers.
- This statement is false.
- Order relation refers to the ability to compare two numbers and determine which one is greater or less than the other.
- In real numbers, the order relation is well-defined and we can compare any two real numbers.
- However, in complex numbers, there is no natural way to define an order relation.
- We cannot say that one complex number is greater than or less than another complex number.
4. The values of ω used as a cube roots of unity and as a fourth root of unity are different.
- This statement is true.
- The cube roots of unity are 1, ω, and ω^2, where ω = e^(2πi/3).
- The fourth roots of unity are 1, i, -1, and -i, where i = e^(πi/2).
- Hence, the value of ω used as a cube root of unity is different from the value of ω used as a fourth root of unity.
Therefore, the correct answer is option B, which states that statements 2 and 4 are true.
1. The amplitude of the product of complex numbers is equals to the product of their amplitudes.
- This statement is true.
- The amplitude of a complex number is its modulus or absolute value, which is the distance of the complex number from the origin in the complex plane.
- When two complex numbers are multiplied, their moduli are multiplied and their arguments are added.
- Hence, the amplitude of the product of two complex numbers is the product of their moduli.
2. For any polynomial f(x) = 0 with real coefficients, imaginary roots occur in conjugate pairs.
- This statement is true.
- If a polynomial has real coefficients, then non-real roots occur in conjugate pairs.
- This means that if a polynomial has an imaginary root a + bi, then its conjugate a - bi is also a root.
- This is because the coefficients of the polynomial are real and the complex roots must satisfy the polynomial equation.
3. Order relation exists in complex number whereas it does not exist in real numbers.
- This statement is false.
- Order relation refers to the ability to compare two numbers and determine which one is greater or less than the other.
- In real numbers, the order relation is well-defined and we can compare any two real numbers.
- However, in complex numbers, there is no natural way to define an order relation.
- We cannot say that one complex number is greater than or less than another complex number.
4. The values of ω used as a cube roots of unity and as a fourth root of unity are different.
- This statement is true.
- The cube roots of unity are 1, ω, and ω^2, where ω = e^(2πi/3).
- The fourth roots of unity are 1, i, -1, and -i, where i = e^(πi/2).
- Hence, the value of ω used as a cube root of unity is different from the value of ω used as a fourth root of unity.
Therefore, the correct answer is option B, which states that statements 2 and 4 are true.
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Which of the following statements are true? 1. The amplitude of the product of complex numbers is equals to the product of their amplitudes 2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs 3. Order relation exists in complex number whereas it does not exist in real numbers 4.The values of ω used as a cube roots of unity and as a fourth root of unity are differenta)(1) and (2) onlyb)(2) and (4) onlyc)(3) and (4) onlyd)(1),(2) and (4) onlyCorrect answer is option 'B'. Can you explain this answer?
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Which of the following statements are true? 1. The amplitude of the product of complex numbers is equals to the product of their amplitudes 2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs 3. Order relation exists in complex number whereas it does not exist in real numbers 4.The values of ω used as a cube roots of unity and as a fourth root of unity are differenta)(1) and (2) onlyb)(2) and (4) onlyc)(3) and (4) onlyd)(1),(2) and (4) onlyCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following statements are true? 1. The amplitude of the product of complex numbers is equals to the product of their amplitudes 2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs 3. Order relation exists in complex number whereas it does not exist in real numbers 4.The values of ω used as a cube roots of unity and as a fourth root of unity are differenta)(1) and (2) onlyb)(2) and (4) onlyc)(3) and (4) onlyd)(1),(2) and (4) onlyCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements are true? 1. The amplitude of the product of complex numbers is equals to the product of their amplitudes 2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs 3. Order relation exists in complex number whereas it does not exist in real numbers 4.The values of ω used as a cube roots of unity and as a fourth root of unity are differenta)(1) and (2) onlyb)(2) and (4) onlyc)(3) and (4) onlyd)(1),(2) and (4) onlyCorrect answer is option 'B'. Can you explain this answer?.
Which of the following statements are true? 1. The amplitude of the product of complex numbers is equals to the product of their amplitudes 2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs 3. Order relation exists in complex number whereas it does not exist in real numbers 4.The values of ω used as a cube roots of unity and as a fourth root of unity are differenta)(1) and (2) onlyb)(2) and (4) onlyc)(3) and (4) onlyd)(1),(2) and (4) onlyCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following statements are true? 1. The amplitude of the product of complex numbers is equals to the product of their amplitudes 2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs 3. Order relation exists in complex number whereas it does not exist in real numbers 4.The values of ω used as a cube roots of unity and as a fourth root of unity are differenta)(1) and (2) onlyb)(2) and (4) onlyc)(3) and (4) onlyd)(1),(2) and (4) onlyCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements are true? 1. The amplitude of the product of complex numbers is equals to the product of their amplitudes 2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs 3. Order relation exists in complex number whereas it does not exist in real numbers 4.The values of ω used as a cube roots of unity and as a fourth root of unity are differenta)(1) and (2) onlyb)(2) and (4) onlyc)(3) and (4) onlyd)(1),(2) and (4) onlyCorrect answer is option 'B'. Can you explain this answer?.
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ample number of questions to practice Which of the following statements are true? 1. The amplitude of the product of complex numbers is equals to the product of their amplitudes 2.For any polynomial f (x)=0 with real coefficients ,imaginary roots occur in conjugate pairs 3. Order relation exists in complex number whereas it does not exist in real numbers 4.The values of ω used as a cube roots of unity and as a fourth root of unity are differenta)(1) and (2) onlyb)(2) and (4) onlyc)(3) and (4) onlyd)(1),(2) and (4) onlyCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice JEE tests.
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