The wavelength order for different types of electromagnetic waves from shortest to longest is as follows:
1. **Gamma rays** (λ1) have the shortest wavelength.
2. **X-rays** (λ2) have a slightly longer wavelength than gamma rays.
3. **Infrared waves** (λ3) have a longer wavelength than X-rays.
4. **Microwaves** (λ4) have the longest wavelength among these options.
Therefore, the ascending order of wavelength is:
λ1 < λ2 < λ3 < λ4.
Answer: λ1 < λ2 < λ3 < λ4
A bead of mass \( m \) slides without friction on the wall of a vertical circular hoop of radius \( R \) as shown in figure. The bead moves under the combined action of gravity and a massless spring \( k \) attached to the bottom of the hoop. The equilibrium length of the spring is \( R \). If the bead is released from the top of the hoop with (negligible) zero initial speed, the velocity of the bead, when the length of spring becomes \( R \), would be (spring constant is \( k \), \( g \) is acceleration due to gravity):
Let $ f: \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that $$ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) $$ for all $ x, y \in \mathbb{R} $. If $ f(0) = 1 $, then the value of $ 24f^{(4)}\left(\frac{5\pi}{3}\right) $ is: