Class 11 Mathematics Chapter 5: Complex Numbers and Quadratic Equations

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Class 11 Mathematics Chapter 5, titled “Complex Numbers and Quadratic Equations,” delves into the realm of complex numbers and their applications in solving quadratic equations. This chapter is pivotal for students aiming to build a strong foundation in algebra and prepares them for advanced mathematical concepts.

Understanding Complex Numbers

A complex number is expressed in the form a+iba + ib, where:

  • aa is the real part
  • bb is the imaginary part
  • ii is the imaginary unit, defined as i=−1i = \sqrt{-1}

Complex numbers are essential for solving equations that do not have real solutions, such as x2+1=0x^2 + 1 = 0.

Algebra of Complex Numbers

The fundamental operations with complex numbers include:

  • Addition and Subtraction: Combine real parts and imaginary parts separately.
  • Multiplication: Use distributive property and apply i2=−1i^2 = -1.
  • Division: Multiply numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.

Modulus and Argument

The modulus of a complex number z=a+ibz = a + ib is given by:

∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}

The argument θ\theta is the angle formed with the positive real axis in the complex plane, calculated using:

θ=tan⁡−1(ba)\theta = \tan^{-1}\left(\frac{b}{a}\right)

Polar Representation

A complex number can also be represented in polar form as:

z=∣z∣(cos⁡θ+isin⁡θ)z = |z| (\cos \theta + i \sin \theta)

This representation is particularly useful for multiplication and division of complex numbers.

Quadratic Equations and Complex Solutions

Quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0 may have complex solutions when the discriminant b2−4acb^2 – 4ac is negative.

The solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}

If b2−4ac<0b^2 – 4ac < 0, the solutions are complex and can be expressed as:

x=−b±i4ac−b22ax = \frac{-b \pm i\sqrt{4ac – b^2}}{2a}

Sample Problems and Solutions

  1. Express 3+4i3 + 4i in polar form.

    Solution:

    • Modulus: ∣z∣=32+42=5|z| = \sqrt{3^2 + 4^2} = 5
    • Argument: θ=tan⁡−1(43)≈0.93\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 0.93 radians
    • Polar form: 5(cos⁡0.93+isin⁡0.93)5 (\cos 0.93 + i \sin 0.93)
  2. Solve x2+4x+8=0x^2 + 4x + 8 = 0.

    Solution:

    • Discriminant: 42−4×1×8=−164^2 – 4 \times 1 \times 8 = -16
    • Solutions: x=−4±i162=−2±2ix = \frac{-4 \pm i\sqrt{16}}{2} = -2 \pm 2i

Recommended Books

For a deeper understanding and additional practice, consider the following textbooks:

NCERT Class 11 Mathematics Textbook
The official textbook providing comprehensive coverage of all topics in Class 11 Mathematics.

R.D. Sharma’s Mathematics for Class 11
A widely used reference book offering detailed explanations and a variety of problems.

I.A. Maron’s Problems in Calculus of One Variable
Includes a section on complex numbers with numerous solved examples.

M.L. Khanna’s Higher Algebra
Provides in-depth coverage of algebraic concepts, including complex numbers.

M.L. Khanna’s Higher Algebra
Provides in-depth coverage of algebraic concepts, including complex numbers.

Mastering the concepts of complex numbers and their applications in quadratic equations is crucial for Class 11 students. Regular practice and a thorough understanding of these topics will lay a solid foundation for future mathematical studies.

For a visual explanation of the introduction to Linear Inequalities, you might find the following video helpful;

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