Modulus And Conjugate Identity
For any complex number z, the expression z multiplied by its conjugate z-bar is always equal to which real quantity by the fundamental modulus identity?
Select the correct option:
Solution
∣z∣2
The product of a complex number with its conjugate is a foundational identity: if z = x + iy then z-bar = x - iy, a relationship used everywhere in JEE Advanced. Multiplying, z·z-bar = (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 + y^2, since i^2 = -1. This quantity x^2 + y^2 is precisely the square of the modulus, |z|^2, and is always a non-negative real number. The identity converts conjugate products into real magnitudes, simplifying rationalization and modulus computations. Option 2 Re(z) equals z + z-bar, not the product. Option |z| omits the square. Option z^2 is generally complex and unequal to the real product. Hence z·z-bar = |z|^2. Plausibility check: for z = 3 + 4i, z·z-bar = 9 + 16 = 25 = 5^2 = |z|^2, numerically confirming that the conjugate product returns the squared modulus.
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About This Question
- Subject
- mathematics
- Chapter
- complex numbers and quadratic equations
- Topic
- modulus and conjugate identity
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
∣z∣2
The product of a complex number with its conjugate is a foundational identity: if z = x + iy then z-bar = x - iy, a relationship used everywhere in JEE Advanced. Multiplying, z·z-bar = (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 + y^2, since i^2 = -1. This quantity x^2 + y^2 is precisely the square of the modulus, |z|^2, and is always a non-negative real number. The identity converts conjugate products into real magnitudes, simplifying rationalization and modulus computations. Option 2 Re(z) equals z + z-bar, not the product. Option |z| omits the square. Option z^2 is generally complex and unequal to the real product. Hence z·z-bar = |z|^2. Plausibility check: for z = 3 + 4i, z·z-bar = 9 + 16 = 25 = 5^2 = |z|^2, numerically confirming that the conjugate product returns the squared modulus.
This easy difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of modulus and conjugate identity. It appeared in the 2025 exam.
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