Technological principles of whey powder manufacturing:

• Statement (A) is correct: pre-crystallization of lactose ($85\text{--}90\%$) into $\alpha$-lactose monohydrate prevents sticking and hygroscopicity, allowing concentrates up to $58\text{--}60\%\text{ TS}$ to be spray dried successfully.
• Statement (B) is correct: two-stage drying (spray chamber followed by vibrating fluid bed) permits drying at lower temperatures, preserving native whey protein functionality.
• Statement (C) is incorrect: two-stage drying reduces thermal energy consumption by $15\text{--}20\%$ because fluid-bed evaporative moisture removal is thermally far more efficient than chamber drying.
• Statement (D) is incorrect: operating processing cost is decreased, not increased, due to fuel and energy savings.

Biosynthesis of Vitamin B12 from whey using Propionibacterium:

• Statement (A) is correct: cheese whey supplemented with cobalt and dimethylbenzimidazole is an efficient substrate for vitamin B12 biosynthesis by Propionibacterium freudenreichii subsp. shermanii.
• Statement (B) is incorrect: Propionibacteria are anaerobic to aerotolerant/microaerophilic microorganisms, not aerobic growers.
• Statement (C) is correct: accumulating propionic and acetic acids drop culture pH and arrest growth, requiring fed-batch neutralization (e.g., with ammonium hydroxide) to maintain cell growth.
• Statement (D) is correct: corrinoid precursor synthesis and final assembly of the cobalamin nucleotide loop occur maximally during late exponential and stationary phases.

Calculation of butter over-run:

• Quantity of cream churned = $800\text{ kg}$.
• Fat percentage in cream = $45\%$.
  Total fat churned ($F$):
  $$F = 800\text{ kg} \times \frac{45}{100} = 360\text{ kg}$$
• Quantity of butter obtained ($B$) = $400\text{ kg}$.
• Over-run is defined as the percentage increase in butter weight over the fat churned:
  $$\%\text{ Over-run} = \frac{B - F}{F} \times 100 = \frac{400 - 360}{360} \times 100 = \frac{40}{360} \times 100 = 11.11\% \approx 11\%$$