Comparing the given equation with the standard equation for a standing wave:
\[ \frac{2\pi nt}{\lambda} = \omega t, \quad \frac{2\pi x}{\lambda} = kx \]
where \( \omega \) is the angular frequency and \( k \) is the wave number.
Analyzing the dimensions:
\[ \left[\frac{n}{\lambda}\right] = [\omega] = [T^{-1}] \]
For the other terms:
\[ [nt] = [\lambda] = [L], \quad [n] = [\lambda \omega] = [LT^{-1}], \quad [x] = [\lambda] = [L] \]
Conclusion:
Hence, the dimensions of \( n / \lambda \) are [T].
A sub-atomic particle of mass \( 10^{-30} \) kg is moving with a velocity of \( 2.21 \times 10^6 \) m/s. Under the matter wave consideration, the particle will behave closely like (h = \( 6.63 \times 10^{-34} \) J.s)
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32