Questions are transcribed from NTA's official question paper. Diagrams are described in words; open the official PDF to see them. Every answer was solved twice independently and is shown only when both agree. Where they did not, we say so.
The equation of a plane progressive wave is given by $y = 5 \cos \pi \left(200t - \frac{x}{150}\right)$ where $x$ and $y$ are in cm and $t$ is in second. The velocity of the wave is _________ m/s.
(1) 120
(2) 150
(3) 200
(4) 300
Show answer and solution
Answer: Option (4)
Key idea: Wave velocity from wave equation
Write the given wave equation: $y = 5 \cos \pi \left(200t - \frac{x}{150}\right)$
Expand to standard form: $y = 5 \cos \left(200\pi t - \frac{\pi x}{150}\right)$
Compare with $y = A \cos(\omega t - kx)$ to get $\omega = 200\pi$ rad/s and $k = \frac{\pi}{150}$ rad/cm
A transverse wave on a string is described by $y = 3\sin(36t + 0.018x + \pi/4)$, where $x, y$ are in cm and $t$ in seconds. The least distance between the two successive crests in the wave is ____ cm. (Nearest integer)
($\pi = 3.14$)
Show answer and solution
Answer: 349
Key idea: Wavelength from wave equation using wave number
Identify the wave equation form: $y = A\sin(\omega t + kx + \phi)$
Compare with given equation $y = 3\sin(36t + 0.018x + \pi/4)$ to find wave number $k = 0.018$ rad/cm
The distance between two successive crests is the wavelength $\lambda$
Use the relation $k = \frac{2\pi}{\lambda}$, so $\lambda = \frac{2\pi}{k}$
