Complex Numbers and Quadratic Equations Class 11 Notes AP Inter 1st Year Maths Chapter 4

Students can go through AP Inter 1st Year Maths Notes 4th Lesson Complex Numbers and Quadratic Equations will help students in revising the entire concepts quickly.

Complex Numbers and Quadratic Equations Class 11 Notes AP Inter 1st Year Maths 4th Lesson

→ A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part, and b is called the imaginary part of the complex number.

→ Let z₁ = a + ib and z₂ = c + id. Then

• z₁ + z₂ = (a + c) + i(b + d)
• z₁z₂ = (ac – bd) + i(ad + bc)

→ For any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists the complex number \(\frac{a}{a^2+b^2}+i \frac{-b}{a^2+b^2}\), denoted by \(\frac {1}{z}\) or z^-1, called the multiplicative inverse of z such that (a + ib) = \(\left(\frac{a}{a^2+b^2}+i \frac{-b}{a^2+b^2}\right)\) = 1 + i0 = 1

→ For any integer k, i^4k = 1, i^4k+1 = i, i^4k+2 = -1, i^4k+3 = -i

→ The conjugate of the complex number z = a + ib, denoted by \(\overline{\mathrm{z}}\), is given by \(\overline{\mathrm{z}}\) = a – ib.

Example Problems

Question 1.
If 4x + i(3x – y) = 3 + i(-6). where x and y are real numbers, then find the values of x and y.
Solution:
We have 4x + i(3x – y) = 3 + i(-6) ………(1)
Equating the real and the imaginary parts of (1), we get
4x = 3, 3x – y = – 6, which, on solving simultaneously, give
x = \(\frac {3}{4}\) and y = \(\frac {33}{4}\)

Question 2.
Express the following in the form of a + bi:
(i) (-5i)(\(\frac {1}{8}\)i)
(ii) (-i)(2i)(\(\frac {1}{8}\)i)³
Solution:

Question 3.
Express (5 – 3i)³ in the form a + ib.
Solution:
We have, (5 – 3i)³ = 5³ – 3 × 5² × (3i) + 3 × 5(3i)² – (3i)³
= 125 – 225i – 135 + 27i
= -10 – 198i

Question 4.
Express (-√3 + √-2)(2√3 – i) in the form of a – ib.
Solution:
(-√3 + √-2)(2√3 – i) = (-√3 + √2i)(2√3 – i)
= -6 + √3i + 2√6i – √2i²
= (-6 + √2) + √3(1 + 2√2)i

Question 5.
Find the multiplicative inverse of 2 – 3i.
Let z = 2 – 3i
Then \(\overline{\mathrm{z}}\) = 2 + 3i and |z|² = 2² + (-3)² = 13
Therefore, the multiplicative inverse of 2 – 3i is given by
\(z^{-1}=\frac{\bar{z}}{|z|^2}=\frac{2+3 i}{13}=\frac{2}{13}+\frac{3}{13} i\)
The above work can also be reproduced in the following manner also

Question 6.
Express the following in the form a – ib
(i) \(\frac{5+\sqrt{2} i}{1-\sqrt{2} i}\)
(ii) i^-35
Solution:
(i) We have,

Question 7.
Find the conjugate of \(\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}\)
Solution:

Question 8.
If x + iy = \(\frac{a+i b}{a-i b}\), prove that x² + y² = 1.
Solution:
We have

Multiple Choice Questions

Question 1.
The value of i^-999 is
(1) 1
(2) -1
(3) i
(4) -i
Answer:
(3) i

Question 2.
The multiplicative inverse of 1 + i is
(1) \(\frac {1}{2}\)(1 – i)
(2) \(\frac {1}{2}\)(1 + i)
(3) 1 – i
(4) i
Answer:
(1) \(\frac {1}{2}\)(1 – i)

Question 3.
The modulus of 5 + 4i is
(1) 41
(2) -41
(3) √41
(4) √-41
Answer:
(3) √41

Question 4.
The value of √25 × √-9 is
(1) 15
(2) -15
(3) 15i
(4) None of these
Answer:
(2) -15

Question 5.
The conjugate of \(\frac{2 + i}{1-2 i}\) is
(1) \(\frac{4+3 i}{5}\)
(2) 4 – 3i
(3) \(\frac{4-3 i}{5}\)
(4) 1
Answer:
(3) \(\frac{4-3 i}{5}\)

Question 6.
The value of (z + 3) (\(\overline{\mathrm{z}}\) + 3) is equivalent to
(1) |z + 3|²
(2) |z – 3|
(3) z² + 3
(4) None of these
Answer:
(1) |z + 3|²

Question 7.
Let x, y ∈ R, then x + iy is a non-real complex number if
(1) x = 0
(2) y = 0
(3) x ≠ 0
(4) y ≠ 0
Answer:
(4) y ≠ 0

Question 8.
If a + ib = c + id then
(1) a² + c² = 0
(2) b² + c² = 0
(3) b² + d² = 0
(4) a² + b² = c² + d²
Answer:
(4) a² + b² = c² + d²

Question 9.
The sum of the series i + i² + i³ …. upto 1000 terms is
(1) 1
(2) 0
(3) -1
(4) -2
Answer:
(2) 0

Question 10.
The complex number Z, which satisfies the condition \(\left|\frac{\mathbf{i}+\mathbf{z}}{\mathbf{i}-\mathbf{z}}\right|\) = 1, lies on
(1) Circle x² + y² = 1
(2) The X-axis
(3) The Y-axis
(4) The line x + y = 1
Answer:
(2) The X-axis