Class 11

According to the principle of superposition, when two or more waves overlap in space, due to each individual wave, at any point the displacement is the vector sum of the displacements. The principle of superposition also explains phenomena like interference where waves combine to amplify or cancel out each other. It can happen in both ways - constructively or destructively. In real life, the superposition can be seen in the noise cancellation technologies, sound beats and the colourful patterns created by overlapping water waves. The principle is the fundamental concept for understanding wave behaviour in acoustics, optics and even quantum physics.

Yes, NCERT Solutions are generally enough for Class 11 Maths exam preparation, especially for CBSE board exams. Shiskha has provided NCERT solutions are based on the latest CBSE syllabus, with step-by-step explanations that help build a strong conceptual understanding.

 In Chapter 4 Complex Numbers and Quadratic Equations of class 11 Maths, students are introduced to key concepts like real and imaginary numbers, the form a + ib, modulus, conjugate, and solving equations with complex roots. These solutions explain each problem methodically, making it easier to grasp abstract topics like the Argand plane or polar representation. Our NCERT Solutions include all the concepts required to ace the CBSE class 11 mathematics exams.

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