$A$ slit of width $a$ is illuminated by red light of wavelength $6500 \, \mathring{A}$. The first minimum will fall at $\theta = 30^{\circ}$ if $a$ is equal to:

Which of the following statements are correct with reference to single slit diffraction pattern?
$(I)$ The central maxima is twice as wide as the secondary maxima.
$(II)$ The intensity of secondary maxima decreases as we move away from the central maxima.
$(III)$ The width of the central maxima is independent of the slit width.
$(IV)$ The intensity of the central maxima is the same as that of the secondary maxima.

Light is incident normally on a diffraction grating through which the first order diffraction is seen at $32^\circ$. The second order diffraction will be seen at

Ray optics fails when

In a single slit diffraction pattern:

Light of wavelength $\lambda = 5000 \ \mathring{A}$ falls normally on a narrow slit. $A$ screen is placed at a distance of $1 \ m$ from the slit and perpendicular to the direction of light. The first minima of the diffraction pattern is situated at $5 \ mm$ from the centre of the central maximum. The width of the slit is ....... $mm$.