An open pipe of length $33 \ cm$ resonates to a frequency of $1000 \ Hz$. The mode of vibration is: (velocity of sound $= 330 \ m/s$)

$A$ parallel plate capacitor has circular plates of $10\, cm$ radius separated by an air-gap of $1\, mm$. It is charged by connecting the plates to a $100\, V$ battery. What is the change in energy stored in the capacitor when the plates are moved to a distance of $1\, cm$ while remaining connected to the battery?

The equation $\sqrt{3} \sin x + \cos x = 4$ has

The wavelength of ${K_\alpha }$ $X-$rays produced by an $X-$ray tube is $0.76 \ \mathring A$. The atomic number of the anode material of the tube is:

$A$ semiconductor $X$ is made by doping a germanium crystal with arsenic $(Z=33)$. $A$ second semiconductor $Y$ is made by doping germanium with indium $(Z=49)$. The two are joined end to end and connected to a battery as shown. Which of the following statements is correct?

The shortest distance between the line $y - x = 1$ and the curve $x = y^2$ is: