Q81.Let z = a + ib, b ≠0 be complex numbers satisfying z2 = ¯z ⋅21−|z| . Then the least value of n ∈N , such that zn = (z + 1)n , is equal to _____ .
What This Question Tests
This question combines properties of modulus and argument of complex numbers with an equation involving powers. It requires careful manipulation of complex number expressions and the application of De Moivre's theorem or geometric interpretation.
Concepts Tested
Formulas Used
|z| = √(a²+b²)
z² = |z|²(cos 2θ + i sin 2θ)
conjugate(z) = a - ib
z^n = (z+1)^n implies |z|=|z+1| and arg(z)=arg(z+1) + 2kπ/n
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📋 Question Details
- Chapter
- Complex Numbers
- Topic
- Properties of complex numbers, De Moivre's Theorem
- Year
- 2022
- Shift
- 28 Jul Shift 2
- Q Number
- Q81
- Type
- Numerical
- NCERT Ref
- Class 11 Mathematics Ch 5: Complex Numbers and Quadratic Equations
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