Professors' Performance Analysis

Percentage growth in feedback:

• Prof. Arithmetic (2013-14): $(6.5 - 4)/4 = 62.5\%$
• Prof. Algebra (2015-16): $(8 - 5)/5 = 60\%$
• Prof. Calculus (2012-13): $(7 - 4)/4 = 75\%$
• Prof. Calculus (2014-15): $(9 - 5.5)/5.5 = 63.63\%$ Thus, Professor Calculus during 2012-2013 shows the maximum percentage growth in feedback. Hence, option C is correct.

Checking each professor for years where research % decreased and feedback score increased:

• Prof. Arithmetic: 2 instances (2013-14, 2015-16)
• Prof. Algebra: 3 instances (2011-12, 2014-15, 2015-16)
• Prof. Geometry: 2 instances (2010-11, 2013-14)
• Prof. Calculus: 2 instances (2010-11, 2014-15) Total instances = $2 + 3 + 2 + 2 = 9$. Hence, option C is correct.

Research efficiency = $\frac{\text{Publications}}{\text{Research Hours}}$.

• Prof. Arithmetic: $RE = \frac{8}{90 \times (0.7 + 0.5 + 0.6)} = \frac{8}{18 \times 9}$
• Prof. Algebra: $RE = \frac{8}{90 \times (0.3 + 0.4 + 0.25)} = \frac{8}{9.5 \times 9}$
• Prof. Geometry: $RE = \frac{8}{90 \times (0.75 + 0.25 + 2)} = \frac{8}{12 \times 9}$
• Prof. Calculus: $RE = \frac{8}{90 \times (0.55 + 0.55 + 0.4)} = \frac{8}{15 \times 9}$ Prof. Arithmetic has the highest denominator, hence the lowest research efficiency. Hence, option A is correct.