alleryThe imaginary unit i is defined as:
- √(-1)
- √1
- 1/i
- -1
Show answer and solution
Correct answer: A. √(-1)
i = √(-1) by definition.
**So i² = -1. This is the foundation of complex numbers, allowing extension beyond real numbers.**
AP Intermediate 1st Year Complex Numbers and Quadratic Equations — practice questions
Explore 46 free Complex Numbers and Quadratic Equations questions in AP Intermediate 1st Year Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
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Showing questions 1–30 of 46.
The imaginary unit i is defined as:Read the answer and worked solution →For z = 3 + 4i, find |z|:Read the answer and worked solution →The conjugate of z = a + ib is:Read the answer and worked solution →i² + i³ equals:Read the answer and worked solution →In polar form z = r(cos θ + i sin θ), r is called the ___ and θ is the ___:Read the answer and worked solution →For z = a + ib, Re(z) and Im(z) are:Read the answer and worked solution →Compute (2 + i)(3 - 2i):Read the answer and worked solution →Find (1 + i)²:Read the answer and worked solution →Solve z² + 1 = 0:Read the answer and worked solution →Modulus of z1 z2 where z1 and z2 are complex:Read the answer and worked solution →Argument of z1 z2:Read the answer and worked solution →For z = 1 + i, find arg(z):Read the answer and worked solution →Find z̄ × z (where z = a + ib):Read the answer and worked solution →Convert i to polar form:Read the answer and worked solution →DeMoivre's theorem: (cos θ + i sin θ)ⁿ equals:Read the answer and worked solution →Compute (1 + i)⁸:Read the answer and worked solution →Cube roots of unity are 1, ω, ω². Find 1 + ω + ω²:Read the answer and worked solution →Find n-th roots of unity. For z = 1: how many distinct nth roots in complex plane?Read the answer and worked solution →Find roots of z² - 4z + 5 = 0:Read the answer and worked solution →If z and z̄ are roots of a quadratic with real coefficients, then sum z + z̄ equals:Read the answer and worked solution →Locus of z satisfying |z - 1| = |z - i|:Read the answer and worked solution →The value of i⁴ⁿ⁺³ (n a positive integer) isRead the answer and worked solution →The modulus of the complex number 3 + 4i isRead the answer and worked solution →The conjugate of (2 − 3i) isRead the answer and worked solution →If z = 1 + i, then z² equalsRead the answer and worked solution →The argument (principal value) of the complex number i isRead the answer and worked solution →The multiplicative inverse of (3 + 4i) isRead the answer and worked solution →For complex numbers, which relation is NOT defined?Read the answer and worked solution →The roots of x² + 1 = 0 areRead the answer and worked solution →If z = a + ib is purely imaginary, thenRead the answer and worked solution →
Worked practice questions
For z = 3 + 4i, find |z|:
- 5
- 7
- 3
- 4
Show answer and solution
Correct answer: A. 5
|z| = √(a² + b²) = √(9 + 16) = √25 = 5. The modulus is the distance from origin in the Argand plane.
The conjugate of z = a + ib is:
- a - ib
- -a + ib
- -a - ib
- a + ib
Show answer and solution
Correct answer: A. a - ib
Conjugate z̄ = a - ib (flip the sign of imaginary part).
**Geometrically:** reflection across the real axis in the Argand plane.
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