Question:
If $(3 + 4i)^{2025} = 5^{2023}(x + iy)$, then find $\sqrt{x^2 + y^2}$.
If $(3 + 4i)^{2025} = 5^{2023}(x + iy)$, then find $\sqrt{x^2 + y^2}$.
Show Hint
Use modulus properties of complex numbers and powers: $|z^n| = |z|^n$ to simplify such problems.
Updated On: Jun 4, 2025
- 5
- 25
- 125
- 625
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Given:
\[
(3 + 4i)^{2025} = 5^{2023} (x + iy).
\]
First, find the modulus of $3 + 4i$:
\[
|3 + 4i| = \sqrt{3^2 + 4^2} = 5.
\]
Therefore,
\[
|(3 + 4i)^{2025}| = |3 + 4i|^{2025} = 5^{2025}.
\]
From the equation:
\[
5^{2025} = 5^{2023} |x + iy| \implies |x + iy| = \frac{5^{2025}}{5^{2023}} = 5^2 = 25.
\]
Hence,
\[
\sqrt{x^2 + y^2} = 25.
\]
So, the correct answer is 25.
Was this answer helpful?
0
0
Top Questions on Complex numbers
- If \(1, \omega, \omega^2\) are the cube roots of unity, then \( \left(1\left(2+\frac{1}{\omega}\right)\left(2+\frac{1}{\omega^2}\right) + 2\left(3+\frac{1}{\omega}\right)\left(3+\frac{1}{\omega^2}\right) + 3\left(4+\frac{1}{\omega}\right)\left(4+\frac{1}{\omega^2}\right) + \dots + 10 \text{ terms}\right) = \)
- AP EAPCET - 2025
- Mathematics
- Complex numbers
- Evaluate the infinite series: \[ \left|\begin{array}{ccc} 2 & 1 & \frac{1}{3} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{3} & \frac{1}{2} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{4} & \frac{1}{9} \\ 3 & 1 & 1 \\ \end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{4} & \frac{1}{27} \\ 3 & 1 & 1 \\ \end{array}\right| + \cdots = ? \]
- AP EAPCET - 2025
- Mathematics
- Complex numbers
- \( (1+\sqrt{3}i)^6 - (\sqrt{3}+i)^6 = \)
- AP EAPCET - 2025
- Mathematics
- Complex numbers
- If \( \alpha, \beta \) are the roots of the equation \( x^2 + bx + c = 0 \) satisfying the conditions \( \alpha+\beta=5 \) and \( \alpha^3+\beta^3=60 \), then \( 3c+2 = \)
- AP EAPCET - 2025
- Mathematics
- Complex numbers
- For any two non-zero complex numbers \(z_1\) and \(z_2\), if \(|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2\), then
- AP EAPCET - 2025
- Mathematics
- Complex numbers
View More Questions
Questions Asked in AP EAPCET exam
- The sequence of reagents required to convert aniline to benzoic acid is:
- AP EAPCET - 2025
- Reaction mechanisms
- In a binomial distribution, if \(n=4\) and \( P(X=0) = \frac{16}{81} \), then \( P(X=4) = \)
- AP EAPCET - 2025
- Poisson distribution
- If $\sinh^{-1}(x) = \log 3$ and $\cosh^{-1}(y) = \log\left(\frac{3}{2}\right)$, then $\tanh^{-1}(x - y) =$
- AP EAPCET - 2025
- Hyperbolic Functions
- $L_1 = ax - 3y + 5 = 0$ and $L_2 = 4x - 6y + 8 = 0$ are two parallel lines. If $p, q$ are the intercepts made by $L_1 = 0$ and $m, n$ are the intercepts made by $L_2 = 0$ on the X and Y coordinate axes, respectively, then the equation of the line passing through the points $(p, q)$ and $(m, n)$ is
- AP EAPCET - 2025
- Coordinate Geometry
- The radius and the height of a right circular solid cone are measured as 7 feet each. If there is an error of 0.002 ft for every feet in measuring them, then the error in the total surface area of the cone (in sq. ft) is
- AP EAPCET - 2025
- Geometry
View More Questions
