List-I EM-Wave | List-II Wavelength Range |
---|---|
(A) Infra-red | (III) 1 mm to 700 nm |
(B) Ultraviolet | (II) 400 nm to 1 nm |
(C) X-rays | (IV) 1 nm to \(10^{-3}\) nm |
(D) Gamma rays | (I) \(<10^{-3}\) nm |
The correct matching between the EM waves and their wavelength ranges is as follows:
- (A) Infra-red corresponds to \( 1 \, \text{mm} \) to \( 700 \, \text{nm} \), which is (III).
- (B) Ultraviolet corresponds to \( 400 \, \text{nm} \) to \( 1 \, \text{nm} \), which is (II).
- (C) X-rays correspond to \( 1 \, \text{nm} \) to \( 10^{-3} \, \text{nm} \), which is (IV).
- (D) Gamma rays correspond to wavelengths less than \( 10^{-3} \, \text{nm} \), which is (I).
Thus, the correct matching is:
\[ \text{(A)-(III), (B)-(II), (C)-(IV), (D)-(I)} \]
Infrared radiation has the longest wavelength among the options, while gamma rays have the shortest wavelength, corresponding to their relative energy levels, with gamma rays being the most energetic and infrared the least.
The dimension of $ \sqrt{\frac{\mu_0}{\epsilon_0}} $ is equal to that of: (Where $ \mu_0 $ is the vacuum permeability and $ \epsilon_0 $ is the vacuum permittivity)
The unit of $ \sqrt{\frac{2I}{\epsilon_0 c}} $ is: (Where $ I $ is the intensity of an electromagnetic wave, and $ c $ is the speed of light)
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: