The process of precipitating colloidal sol by an electrolyte is called peptization.
Colloidal solution freezes at higher temperature than the true solution at the same concentration.
Surfactants form micelle above critical micelle concentration (CMC). CMC depends on temperature.
Micelles are macromolecular colloids.
The statement is False:
Peptization refers to the process of converting a precipitate into a colloidal sol, not the other way around. Peptization is the dispersion of fine particles into a colloidal form by adding a suitable dispersing agent (often a small amount of electrolyte). This is different from the process of coagulation, where a colloidal sol is converted into a precipitate. Therefore, the given statement is incorrect.
The statement is True:
Colloidal solutions do exhibit colligative properties, but with a slight difference from true solutions. For example, a colloidal solution may show a higher freezing point depression or lower vapor pressure than a true solution, as the dispersed particles in the colloidal system can affect the overall behavior of the solvent. However, the statement claims that the freezing point is higher in colloidal solutions, which is inaccurate in terms of freezing point depression, but may be a misunderstanding of the phenomenon.
Note: The correct assertion should be that colloidal solutions show some effects of colligative properties, but their freezing point depression is often less than that of true solutions at the same concentration due to the large size of the dispersed particles.
The statement is True:
Surfactants indeed form micelles above the critical micelle concentration (CMC). The CMC is the concentration at which surfactant molecules aggregate to form micelles, and it depends on various factors, including temperature. As the temperature increases, the CMC can change, generally decreasing as temperature increases because the surfactant molecules become more energetically favorable to form micelles at lower concentrations.
Therefore, the statement about micelles and their dependence on temperature is correct.
The statement is False:
Micelles are not macromolecular colloids. Macromolecular colloids refer to large molecules such as proteins or polymers that are naturally colloidal in nature. On the other hand, micelles are associated colloids, formed by the aggregation of surfactant molecules (amphiphilic molecules) in a solvent, where the hydrophobic tails aggregate inward and the hydrophilic heads face outward. This aggregation occurs only above the critical micelle concentration (CMC). Therefore, the statement about micelles being macromolecular colloids is false.
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity): 
Solutions are homogeneous mixtures of two or more substances, where the solute is uniformly dispersed in the solvent. Solutions can be classified into several types based on their composition and properties.
Understanding the different types of solutions is important for understanding their properties, behavior, and applications in various fields, such as chemistry, biology, and engineering.