Sure! Here are some important questions from the chapter on Complex Numbers and Quadratic Equations for Class 11, along with their answers. Note that these questions reflect fundamental concepts and standard results that could be emphasized in study materials like the NCERT textbook:
Important Questions and Answers
Question 1: Define a complex number. Give an example.
Answer:
A complex number is defined as a number of the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, \( i \) is the imaginary unit, and \( i^2 = -1 \). Here, \( a \) is called the real part, and \( b \) is called the imaginary part of the complex number.
Example: \( z = 3 + 4i \) (where \( a = 3 \) and \( b = 4 \)).
Question 2: What is the modulus of a complex number? Calculate the modulus of \( z = 3 - 4i \).
Answer:
The modulus of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \).
For \( z = 3 - 4i \):
\[ |z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \]
Question 3: Find the conjugate of the complex number \( z = 5 + 12i \).
Answer:
The conjugate of a complex number \( z = a + bi \) is given by \( \overline{z} = a - bi \).
For \( z = 5 + 12i \):
\[ \overline{z} = 5 - 12i. \]
Question 4: Show that the set of complex numbers is closed under addition and multiplication.
Answer:
Let \( z_1 = a + bi \) and \( z_2 = c + di \) be two complex numbers.
-
Addition:
\[ z_1 + z_2 = (a + c) + (b + d)i \]
Since \( a+c \) and \( b+d \) are real numbers, \( z_1 + z_2 \) is also a complex number. -
Multiplication:
\[ z_1 \times z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2 \]
\[ = (ac - bd) + (ad + bc)i \]
Here, \( ac-bd \) and \( ad+bc \) are real numbers, hence \( z_1 \times z_2 \) is also a complex number.
Thus, the set of complex numbers is closed under both addition and multiplication.
Question 5: If \( z_1 = 2 + 3i \) and \( z_2 = 4 - 5i \), find \( z_1 + z_2 \) and \( z_1 \times z_2 \).
Answer:
-
Addition:
\[ z_1 + z_2 = (2 + 3i) + (4 - 5i) = (2 + 4) + (3 - 5)i = 6 - 2i. \] -
Multiplication:
\[ z_1 \times z_2 = (2 + 3i)(4 - 5i) = 8 - 10i + 12i - 15i^2 = 8 + 2i + 15 = 23 + 2i. \]
Question 6: Solve the quadratic equation \( x^2 + 2x + 5 = 0 \) using the quadratic formula.
Answer:
The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
For \( a = 1, b = 2, c = 5 \):
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16. \]
\[ x = \frac{-2 \pm \sqrt{-16}}{2 \cdot 1} = \frac{-2 \pm 4i}{2} = -1 \pm 2i. \]
The solutions are \( x = -1 + 2i \) and \( x = -1 - 2i \).
Question 7: Find the roots of the equation \( x^2 - 4x + 5 = 0 \).
Answer:
Using the quadratic formula:
\[ a = 1, b = -4, c = 5. \]
\[ D = b^2 - 4ac = (-4)^2 - 4 \times 1 \times 5 = 16 - 20 = -4. \]
Since \( D < 0 \), the roots are complex.
\[ x = \frac{4 \pm \sqrt{-4}}{2 \cdot 1} = \frac{4 \pm 2i}{2} = 2 \pm i. \]
The roots are \( x = 2 + i \) and \( x = 2 - i \).
These questions should provide a strong foundation and practice for students studying complex numbers and quadratic equations in Class 11.