NCERT

An inverse of a function does just opposite work as the original function does in other words it essentially "undoes" what the original function does. If a function f maps an input x to an output y (written as f (x) = y),

 then its inverse function is denoted as  f? ¹ (y) and  f? ¹ (y) = x

However, Students must be aware that not all functions have inverses. For an inverse function, the function must be one-one (injective) and onto (surjective): So that every possible output value is covered.

Together, these two properties make the function bijective, and only bijective functions have well-defined inverses. If a function isn't one-one (like f (x) = x²), it can't have a true inverse unless its domain is restricted.

A relation deals with, how elements from any set  (A) are related to elements of another set (B). there are various form of relations, such as Reflexive, Symatric, Transitive and Equivalance. On the other hand, Functions are special case of Relations where every element in Set A is related to exactly one element in Set B. Functions works like mathematics formulas, for a given set of input the out put will always remain the same. Students will be abel to check the type of relations, and application in class 11 Math Relation and Function chapter. We have provided Class 11 Math Relations and Functions NCERT Solutions on shiskha 

The kinetic theory talks about the molecular explanation for the classical gas laws:

Boyle's Law (P? 1/V at constant T): Gas molecules will exert more pressure when the volume of the container reduces as it has less space to move and hence it collides with the walls of the container. However, when the temperature is constant, the speed of the molecule will not change.

Charles's Law (V? T at constant P): With the increase in temperature, the speed and average kinetic energy of the molecules increase. The gas expands to maintain constant pressure and increases the volume.

The kinetic theory links the physical behaviour of gases with the molecular motion and supports the ideal gas equation.

Class 11 Maths Chapter 2 Relations and Functions explores how elements from one set can be connected or related to elements from another.  Students get to learn many concepts, can be checked below;

• Concept of the Cartesian product of sets
• Concept of Relations, and how to represent those connections using different methods like arrow diagrams or set notation.
• Idea of a function, and different types of functions such as one-one, onto, bijective, constant, and identity functions.
• Real-valued functions, like linear, quadratic, modulus, and the greatest integer function.

Students can use NCERT Solutions for Chapter 2 Relations and Functions to practice these concepts, and getting mastery in it.

NCERT Chapter 1 Sets in Class 11 Maths contains a total of 6 exercises along with 1 miscellaneous exercise at the end. These exercises are designed to cover all key concepts such as types of sets, representation methods, Venn diagrams, and operations on sets like union, intersection, and complement. students can check details below

• Exercise 1.1 consists 9 questions related to basic understanding, and set builder forms.
• Exercise 1.2 consists 6 questions about type of sets, empty set, power set, finite and infinite sets.
• Exercise 1.3 consists 9 questions, related to operations on sets, and venn diagrams.
• Exercise 1.4 consists 12 questions related to properties of set operations such as Distributive laws.
• Exercise 1.5 consists 7 questions based on union and intersection of sets.
• Exercise 1.6 consists 6 questions based on Compements and De-Morgan's Law
• Miscellaneous Exercise consists 13 questions based on mixed conceptual questiopns.

Students can use NCERT Solutions for Chapter 1 Sets to score well in the CBSE Class 11 Maths annual exam. the questions is how to do so? Well here is the answer;

Students should start by thoroughly reading the chapter from the NCERT textbook to understand the concepts like types of sets, set operations, and Venn diagrams. Then, refer to the Class 11 Sets NCERT Solutions provided by Shiksha to cross-check your answers and understand the step-by-step method of solving problems. Sets NCERT Solutions include detailed explanations and alternate methods that clarify complex problems. Students should focus on mastering all the solved examples, as they reflect the CBSE exam style and often help in tackling similar questions in exams. Most important part is practice, Practice each exercise without referring to the solution first, and then compare your answers to learn from any mistakes. Students should must practice the miscellaneous exercise, as it contains application-based and mixed-level problems from the whols chapter that test conceptual depth of student.