alleryStart learning →Find the modulus of the complex number $z = 3 + 4i$.
- $12$
- $5$
- $7$
- $\sqrt{7}$
Show answer and solution
Correct answer: B. $5$
$|z| = \sqrt{x^2 + y^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
CG PET Complex Numbers and Quadratic Equations — practice questions
Explore 49 free Complex Numbers and Quadratic Equations questions in CG PET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
Practise Complex Numbers and Quadratic Equations with Quill →Complex Numbers and Quadratic Equations questions and exercises with answers
Showing questions 1–30 of 49.
Find the modulus of the complex number $z = 3 + 4i$.Read the answer and worked solution →What is the principal argument of the complex number $z = 1 + i$?Read the answer and worked solution →If $\omega$ is a non-real cube root of unity, find the value of $1 + \omega + \omega^2$.Read the answer and worked solution →Using De Moivre's theorem, evaluate $\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)^6$.Read the answer and worked solution →The locus of the point $z$ in the Argand plane satisfying $|z - 2| = |z + 2|$ is:Read the answer and worked solution →If $z = \cos\theta + i\sin\theta$, find an expression for $z^n + \dfrac{1}{z^n}$.Read the answer and worked solution →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 →If z = cos(π/7) + i sin(π/7), then z⁷ equals:Read the answer and worked solution →For complex z, |z + i| = |z - i| describes:Read the answer and worked solution →|z|² = z × z̄. If |z|² = 4 and Re(z) = 1, then Im(z):Read the answer and worked solution →
Worked practice questions
What is the principal argument of the complex number $z = 1 + i$?
- $\dfrac{\pi}{3}$
- $\dfrac{\pi}{4}$
- $\dfrac{\pi}{6}$
- $\dfrac{\pi}{2}$
Show answer and solution
Correct answer: B. $\dfrac{\pi}{4}$
$\tan\theta = \dfrac{y}{x} = 1$, and $z$ is in the first quadrant, so $\theta = \dfrac{\pi}{4}$.
If $\omega$ is a non-real cube root of unity, find the value of $1 + \omega + \omega^2$.
- $0$
- $3$
- $-1$
- $1$
Show answer and solution
Correct answer: A. $0$
The three cube roots of unity are $1, \omega, \omega^2$, and they are roots of $z^3 - 1 = (z-1)(z^2 + z + 1) = 0$.
Vieta's formulas on $z^2 + z + 1$ give $\omega + \omega^2 = -1$, so $1 + \omega + \omega^2 = 0$.
Equivalently, the sum of *all* $n$-th roots of unity is zero for $n \ge 2$.
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