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Which of the following statements are true ?
(1) The amplitude of the product of complex numbers is equal to the product of their amplitudes.
(2) For any polynomial f(x) =0 with real co-efficients, imaginary roots occurs in conjugate paris.
(3) Order relation exists in complex numbers whereas it does not exist in real numbers.
(4) The value of ω used as a cube root 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 p...
(1) The statement is true. The amplitude of a complex number z is the modulus or absolute value of z, denoted as |z|. For two complex numbers z1 and z2, the product z1z2 has an amplitude of |z1z2| = |z1||z2|, which follows from the properties of modulus and the definition of complex multiplication.

(2) The statement is true and is known as the complex conjugate root theorem. If a polynomial f(x) with real coefficients has a complex root z = a + bi, where a and b are real numbers, then its conjugate z* = a - bi is also a root of f(x). This is because the coefficients of f(x) are all real, so if z is a root, then its complex conjugate z* must also be a root to satisfy the equation f(x) = 0.

(3) The statement is false. Both real and complex numbers have a natural order relation based on their magnitudes, denoted as < or="" />. For example, 2 < 3="" and="" 2i="" />< 3i="" in="" the="" complex="" plane.="" however,="" the="" order="" relation="" in="" complex="" numbers="" is="" not="" a="" total="" order,="" meaning="" that="" some="" pairs="" of="" complex="" numbers="" cannot="" be="" compared="" with="" each="" other.="" for="" instance,="" there="" is="" no="" way="" to="" say="" whether="" 2="" +="" 3i="" is="" greater="" or="" less="" than="" 3="" +="" 2i="" without="" additional="" />

(4) The statement is incomplete. It seems to be referring to the limit of a complex function, but the rest of the expression is missing. Without additional information, we cannot determine whether the statement is true or false.
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Which of the following statements are true ?(1) The amplitude of the product of complex numbers is equal to the product of their amplitudes.(2) For any polynomial f(x) =0 with real co-efficients, imaginary roots occurs in conjugate paris.(3) Order relation exists in complex numbers whereas it does not exist in real numbers.(4) The value of ω used as a cube root of unity and as a fourth root of unity are different.a)(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 equal to the product of their amplitudes.(2) For any polynomial f(x) =0 with real co-efficients, imaginary roots occurs in conjugate paris.(3) Order relation exists in complex numbers whereas it does not exist in real numbers.(4) The value of ω used as a cube root of unity and as a fourth root of unity are different.a)(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 equal to the product of their amplitudes.(2) For any polynomial f(x) =0 with real co-efficients, imaginary roots occurs in conjugate paris.(3) Order relation exists in complex numbers whereas it does not exist in real numbers.(4) The value of ω used as a cube root of unity and as a fourth root of unity are different.a)(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 equal to the product of their amplitudes.(2) For any polynomial f(x) =0 with real co-efficients, imaginary roots occurs in conjugate paris.(3) Order relation exists in complex numbers whereas it does not exist in real numbers.(4) The value of ω used as a cube root of unity and as a fourth root of unity are different.a)(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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