alleryStart learning →Find the modulus of the complex number $z = 3 + 4i$.
- $12$
- $5$
- $7$
- $\sqrt{7}$
Show answer and solution
Correct answer: B. $5$
$|z| = \sqrt{x^2 + y^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
UPCET / UPSEE Complex Numbers and Quadratic Equations — practice questions
Explore 49 free Complex Numbers and Quadratic Equations questions in UPCET / UPSEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
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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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