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Chapter Analysis
Intermediate12 pages • EnglishQuick Summary
This chapter on Complex Numbers and Quadratic Equations introduces complex numbers, a form of numbers expressed as a + ib where a and b are real numbers. The chapter covers the operations with complex numbers, such as addition, subtraction, multiplication, and division, and discusses the properties of these operations. It also explores the representation of complex numbers on the Argand plane and the concept of conjugate and modulus of complex numbers. Furthermore, it addresses quadratic equations and how complex numbers can be used to find solutions to equations with no real roots.
Key Topics
- •Definition and properties of complex numbers
- •Operations with complex numbers
- •Argand plane representation
- •Modulus and conjugate of complex numbers
- •Quadratic equations with complex roots
- •Power and roots of complex numbers
Learning Objectives
- ✓Understand the definition and components of complex numbers
- ✓Perform arithmetic operations with complex numbers
- ✓Visualize complex numbers on the Argand plane
- ✓Calculate the modulus and conjugate of complex numbers
- ✓Solve quadratic equations using complex numbers
- ✓Explore powers and roots of complex numbers
Questions in Chapter
Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.
Page 83
Express the following expression in the form of a + ib: (3 + 5i)(3 + 2i) - (3 - 2i)(3 - 5i).
Page 83
Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.
Page 83
If z1 = 2 - i, z2 = 1 + i, find the value of (z1 + z2)/(1 - z1 z2).
Page 86
For any two complex numbers z1 and z2, prove that Re(z1 z2) = Re z1 Re z2 - Im z1 Im z2.
Page 86
If α and β are different complex numbers with β ≠ 1, then find 1/(β - 1) - 1/(α - β).
Page 87
Additional Practice Questions
What is the sum of the complex numbers 4 + 3i and 2 - 5i?
easyAnswer: The sum is obtained by adding the real parts and the imaginary parts separately: (4 + 2) + (3i - 5i) = 6 - 2i.
How do you represent the complex number 3 + 4i on the Argand plane?
mediumAnswer: The complex number 3 + 4i represents a point in the Argand plane with the x-coordinate 3 and the y-coordinate 4, plotted as P(3, 4).
If z = 2 + 3i, find the conjugate of z.
easyAnswer: The conjugate of z = 2 + 3i is 2 - 3i.
Calculate the modulus of the complex number -1 + i.
mediumAnswer: The modulus is given by the square root of the sum of squares of the real and imaginary parts: √((-1)^2 + 1^2) = √2.
Find the product of (2 + i) and (3 - 4i).
mediumAnswer: Using the distributive property: (2 + i)(3 - 4i) = 2*3 + 2*(-4i) + i*3 + i*(-4i) = 6 - 8i + 3i + 4 = 10 - 5i.