Class 11

Chapter 4 of class 11 Maths covers concepts such as complex numbers, imaginary unit i, algebra of complex numbers, polar representation, quadratic equations, and the nature of roots. Students can find details below;

• Definition of Complex Numbers: Introduction to the general form a + ib, where i is the imaginary unit (i² = –1).

• Algebra of Complex Numbers: Operations such as addition, subtraction, multiplication, and division of complex numbers.

• Complex Plane (Argand Plane): Graphical representation of complex numbers  using the x-axis (real part) and y-axis (imaginary part).

• Modulus and Conjugate of a Complex Number

  • Modulus = |z| =? (a² + b²)

  • Conjugate = a – ib

• Polar Representation of Complex Numbers in the form r (cos? + I sin? ) using modulus and argument (angle? ).

• Quadratic Equations with Real Coefficients

• Nature of Roots Based on Discriminant (? = b² – 4ac)

  • ? > 0: Real and distinct roots

  • ? = 0: Real and equal roots

  • ? < 0: Complex conjugate roots

• Solving Quadratic Equations in the Complex Number System

Yes, NCERT Solutions for Class 11 Trigonometric Functions are highly useful for JEE and other competitive exams.  NCERT Textbook help students develop a clear understanding of fundamental concepts such as trigonometric identities, angle transformations, and general solutions of equations, which form the backbone of many advanced problems in exams like JEE Main and Advanced.

The step-by-step explanations in NCERT Solutions provided by Shiksha helps students in logical problem-solving techniques. Students can build the conceptual strength and formula fluency required to tackle higher-level problems found in books like Cengage or previous year question papers by practicing NCERT questions thoroughly.

According to the kinetic theory of gases class 11, the Maxwell-Boltzmann distribution law describes the gas molecules' distribution of speeds (or velocities). All molecules of gas do not move at the same speed, some move fast, some are slow and most move at an average speed. This law provides a mathematical function that helps in finding at a given temperature, how many molecules have a particular speed. The Maxwell-Boltzmann distribution law helps in defining three important types of speeds:

• Average speed: Arithmetic mean of all speeds.
• Most probable speed: Speed at which the maximum number of molecules travel.
• Root mean square (RMS) speed: Square root of the average of squares of all speeds. 

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.