Understanding Bio-reactors:
Bio-reactors are specialized vessels used for carrying out biological reactions. They provide the ideal environment for microbial growth, fermentation, or other biochemical processes. These reactors have different components like an agitator system, oxygen delivery, temperature regulation, and sometimes foam control to optimize the reaction environment.
Explanation of the options:
Option 1: This is a correct statement. Bio-reactors indeed provide optimal growth conditions, such as temperature, pH, and oxygen, which are crucial for the desired biological product production.
Option 2: This is also a correct statement. Most commonly used bio-reactors are of the stirring type, known as stirred-tank bioreactors. These systems ensure proper mixing of culture media, nutrients, and cells for consistent growth and productivity.
Option 3: This is the incorrect statement. Bio-reactors are typically used for large-scale production, not for small-scale bacterial cultures. They are used in the production of bio-products, pharmaceutical proteins, and other industrial products in large quantities. Small-scale bacterial culture is generally performed in laboratory flasks or smaller vessels.
Option 4: This is a correct statement. Modern bio-reactors are equipped with an agitator system for proper mixing, an oxygen delivery system for cellular respiration, and a foam control system to prevent foam buildup that can disrupt the process.
Step 2: Conclusion:
Thus, the incorrect statement is Option 3 - "Bio-reactors are used to produce small scale bacterial cultures." Bio-reactors are more suited for large-scale production and are not typically used for small-scale bacterial cultures.
Answer: The incorrect statement is Option 3.
A sphere of radius R is cut from a larger solid sphere of radius 2R as shown in the figure. The ratio of the moment of inertia of the smaller sphere to that of the rest part of the sphere about the Y-axis is : 
AB is a part of an electrical circuit (see figure). The potential difference \(V_A - V_B\), at the instant when current \(i = 2\) A and is increasing at a rate of 1 amp/second is:
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