Argument Of Complex Numbers
The principal argument of the complex number given by z = -1 + i times the square root of 3, expressed in radians within the standard interval, equals which value?
Select the correct option:
Solution
2π/3
The principal argument of a complex number is the angle its position vector makes with the positive real axis, measured in (-π, π], a definition essential to JEE Advanced polar work. For z = -1 + i√3, the real part is negative and the imaginary part is positive, placing z in the second quadrant. The reference angle satisfies tan(θ) = |√3 / -1| = √3, giving a base angle of π/3. In the second quadrant the principal argument is π - π/3 = 2π/3. Option π/3 ignores the quadrant and treats z as first-quadrant. Option -2π/3 corresponds to the third quadrant, wrong sign of imaginary part. Option 5π/6 uses an incorrect reference angle. Hence arg(z) = 2π/3. Plausibility check: the modulus is √(1 + 3) = 2, so z = 2(cos 2π/3 + i sin 2π/3) = 2(-1/2 + i√3/2) = -1 + i√3, exactly reproducing z and confirming the argument.
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About This Question
- Subject
- mathematics
- Chapter
- complex numbers and quadratic equations
- Topic
- argument of complex numbers
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
2π/3
The principal argument of a complex number is the angle its position vector makes with the positive real axis, measured in (-π, π], a definition essential to JEE Advanced polar work. For z = -1 + i√3, the real part is negative and the imaginary part is positive, placing z in the second quadrant. The reference angle satisfies tan(θ) = |√3 / -1| = √3, giving a base angle of π/3. In the second quadrant the principal argument is π - π/3 = 2π/3. Option π/3 ignores the quadrant and treats z as first-quadrant. Option -2π/3 corresponds to the third quadrant, wrong sign of imaginary part. Option 5π/6 uses an incorrect reference angle. Hence arg(z) = 2π/3. Plausibility check: the modulus is √(1 + 3) = 2, so z = 2(cos 2π/3 + i sin 2π/3) = 2(-1/2 + i√3/2) = -1 + i√3, exactly reproducing z and confirming the argument.
This easy difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of argument of complex numbers. It appeared in the 2025 exam.
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