If \( α,β,γ\) are the cube roots of \(-2\) ,then the value of \(xα+yβ+zγ/xβ+yγ+zα\) is (\(x,y,z\) are variables)
- \(\)\(e^{ iπ/3}\)
\(e^{2πi/3}\)
\(1\)
\(-1\)
\(e^{ 4iπ/3}\)
The Correct Option is
Solution and Explanation
\(\dfrac{ xα + yβ + zγ }{xβ + yγ + zα }\)
\(=\dfrac{x(−2)^{1/3} + y (−2^{1/3}ω ) + z ( −2^{1/3}ω 2 )} {x (−2^{1/3}ω) + y (−2^{1/3}ω^2 ) + z(−2)^{1/3} }\)
\(= \dfrac{(−2)^{1/3}(x + yω + zω^2 )ω}{ −2^{1/3}(xω+ yω^2 +z) ω }\)
\(= 1 ω = \dfrac{ω ^3}{ω} = ω^2\)
\(= e^{4πi/3}\)
So, the correct option is (E) : \(e^{4πi/3}\)
Top Questions on complex numbers
- If \(r = |z|,\ θ = arg(z)\) and \( z = 2 – 2\ tan (\frac {5\pi}{8})\) then find \((r, θ)\).
- JEE Main - 2024
- Mathematics
- complex numbers
- If \(z = x + iy, \quad xy \neq 0\) satisfies the equation \(z^2 + iz = 0\), then \(|z^2|\); equal to
- JEE Main - 2024
- Mathematics
- complex numbers
- Let \(z_1\) and \(z_2\) be two complex numbers such that \(z_1 + z_2 = 5\) and \(z_1^3+z_2^3 =20 +15i\), then the value of \(| z_1^4+z_2^4 |\) is equal to
- JEE Main - 2024
- Mathematics
- complex numbers
- Let f(x) = x4 + ax3 + bx2 + c be a polynomial with real coefficients such that f(1) = -9. Suppose that \(i\sqrt3\) is a root of the equation 4x3 + 3ax2 + 2bx = 0, where\(i=\sqrt{-1}\) . If a1, a2, a3 and a4 are all the roots of the equation f(x) = 0, then |a1|2 + |a2|2 + |a3|2 + |a4|2 is equal to ______.
- JEE Advanced - 2024
- Mathematics
- complex numbers
- If : \(|z + 1| = αz + ß(i + 1)\) and \(z = \frac 12 - 2i\), find \(α + ß\).
- JEE Main - 2024
- Mathematics
- complex numbers
Questions Asked in KEAM exam
The angle of minimum deviation for a prism of apex angle 60° and refractive index of \(\sqrt{2}\) is:
- KEAM - 2023
- Ray optics and optical instruments
- A thin particle moves from \((0,1)\) and gets reflected upon hitting the x-axis at \((√3,0)\). Then the slope of the reflected line is ?
- KEAM - 2023
- Slope of a line
- Suppose \(A=\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{bmatrix}\) is an adjoint of the matrix \(\begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}\). Then the value of \(\frac{a_!+b_2+c_3}{b_1a_2}\) is
- KEAM - 2023
- Matrices
- Let \(k\) be a real number such that \(\sin \dfrac{3π}{14} \cos \dfrac{3π}{14} = k \cos \dfrac{π}{14}\).Then the value of \(4k\) is
- KEAM - 2023
- Trigonometry
- An average frictional force of 80N is required to stop an object at a distance of 25m. If the initial speed of the object is 20m/s,the mass of the object is:
- KEAM - 2023
- work, energy and power
Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.
SUBSCRIBE TO OUR NEWS LETTER
