\(\frac{\gamma}{\beta}\)
To solve this problem, we need to understand the nature of the numbers involved. The cube roots of a number are solutions to the equation obtained by setting that number equal to \( x^3 \). Hence, we begin by recognizing that the cube roots of \(-8\) satisfy \( x^3 = -8 \).
Let's express -8 in polar form: \(-8 = 8 \text{cis} 180^\circ\) Taking the cube root, we have: \(\alpha = 2 \text{cis} (60^\circ), \quad \beta = 2 \text{cis} (180^\circ), \quad \gamma = 2 \text{cis} (300^\circ)\)
These roots form an equilateral triangle in the complex plane and satisfy: \(\alpha^3 = \beta^3 = \gamma^3 = -8 \quad \text{and} \quad \alpha \beta \gamma = -8\)
Given the expression: \(\frac{\alpha^2 p^2 + \beta^2 q^2 + \gamma^2 r^2}{\beta^2 p^2 + \gamma^2 q^2 + \alpha^2 r^2}\) It is often helpful to evaluate components like symmetry or factors in the roots to simplify.
Notice that: \(\alpha^2 = 4 \text{cis}(120^\circ), \quad \beta^2 = 4 \text{cis}(240^\circ), \quad \gamma^2 = 4 \text{cis}(0^\circ)\)
Therefore, the given expression: \(\frac{4 \text{cis}(120^\circ) p^2 + 4 \text{cis}(240^\circ) q^2 + 4 \text{cis}(0^\circ) r^2}{4 \text{cis}(240^\circ) p^2 + 4 \text{cis}(0^\circ) q^2 + 4 \text{cis}(120^\circ) r^2}\) simplifies (since \(\alpha^2, \beta^2, \gamma^2\) are symmetric) to: \(\frac{\gamma}{\beta}\)
Thus, the correct answer is: \(\frac{\gamma}{\beta}\).
Similar Questions
If
\(\left( \frac{1-i}{1+i} \right)^{2m} \left( \frac{1+i}{1-i} \right)^{2n} = 1\)
where \(i = \sqrt{-1}\), then what is the smallest positive value of \((m – n)\)?
- If \(z\) is any complex number and \(iz^3 + z^2 - z + i = 0\), where \(i = \sqrt{-1}\), then what is the value of \((|z|+1)^2\)?
- Let \(z_1\) and \(z_2\) be two complex numbers such that \(\left|\frac{z_1+z_2}{z_1-z_2}\right| = 1\), then what is \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\) equal to ?
- If \(\omega \ne 1\) is a cube root of unity, then what are the solutions of \((z-100)^3 + 1000 = 0\) ?
- What is \((1+i)^4 + (1-i)^4\) equal to, where \(i = \sqrt{-1}\)?
- If \(\omega \neq 1\) is a cube root of unity, then what is \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) equal to?
- What is the value of the sum
\(\sum_{n=1}^{20}(i^{n-1} + i^n + i^{n+1})\)
where \(i = \sqrt{-1}\)? - If \(x, y\) and \(z\) are the cube roots of unity, then what is the value of \(xy + yz + zx\)?
If A2 + B2 + C2 = 0, then what is the value of the following?
\(\Delta = \begin{vmatrix} 1 & \cos C & \cos B \\\ \cos C & 1 & \cos A \\\ \cos B & \cos A & 1 \end{vmatrix}\)If ω is a non-real cube root of unity, then what is a root of the following equation?
Important Questions from Complex Numbers
If A + iB = tan (x + iy), then the value of tan 2x is?
If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -
If x = cos θ + i sin θ, then the value of \({x^n} + \frac{1}{{{x^n}}}\) is:
If \(\left| {\begin{array}{*{20}{c}} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {20}&3&i \end{array}} \right| = x + iy\), then the values of x and y are:
If iz3 + z2 - z + i = 0, then the value of |z| is:
