1. **Problem 27:** Light of wavelength $\lambda$ is incident on a slit of width $b$ forming a diffraction pattern with central maximum width $\theta$ radians, where $\theta = \frac{\lambda}{b}$. The wavelength changes to $\frac{2}{3}\lambda$ and slit width to $\frac{1}{6}b$. Find the new angle between the first diffraction minimum and the point of maximum intensity. 2. **Formula:** The angle for the first minimum in single-slit diffraction is given by $$\theta = \frac{\lambda}{b}$$ 3. **Calculate new angle:** $$\theta_{new} = \frac{\frac{2}{3}\lambda}{\frac{1}{6}b} = \frac{2}{3} \lambda \times \frac{6}{b} = 4 \times \frac{\lambda}{b} = 4\theta$$ 4. **Interpretation:** The new angle is $4\theta$, so the correct answer is D. $4\theta$. 5. **Problem 28:** Monochromatic light of wavelength $\lambda$ is incident normally on a diffraction grating with slit spacing $d = \frac{9}{2} \lambda$. Find the total number of maxima produced. 6. **Formula:** Maxima occur at angles $\theta$ satisfying $$n \lambda = d \sin \theta$$ where $n$ is the order of the maximum. 7. **Maximum order $n_{max}$:** Since $\sin \theta \leq 1$, maximum $n$ satisfies $$n_{max} \lambda \leq d \Rightarrow n_{max} \leq \frac{d}{\lambda} = \frac{9}{2} = 4.5$$ 8. **Number of maxima:** $n$ can be integer values from $0$ to $4$ (positive orders) and also negative orders $-1$ to $-4$, plus the central maximum at $n=0$. Total maxima = $2 \times 4 + 1 = 9$. 9. **Answer:** C. 9