We will learn stepbystep how to solve different types of problems on complex numbers using the formulas.
1. Express \((\frac{1 + i}{1  i})^{3}\) in the form A + iB where A and B are real numbers.
Solution:
Given \((\frac{1 + i}{1  i})^{3}\)
Now \(\frac{1 + i}{1  i}\)
= \(\frac{(1 + i)(1 + i)}{(1  i)(1 + i)}\)
= \(\frac{(1 + i)^{2}}{(1^{2}  i^{2}}\)
= \(\frac{1 + 2i + iˆ{2}}{1  (1)}\)
= \(\frac{1 + 2i  1}{2}\)
= \(\frac{2i}{2}\)
= i
Therefore, \((\frac{1 + i}{1  i})^{3}\) = i\(^{3}\)= i\(^{2}\) ∙ i =  i = 0 + i (1), which is the required form A + iB where A = 0 and B = 1.
2. Find the modulus of the complex quantity (2  3i)(1 + 7i).
Solution:
The given complex quantity is (2  3i)(1 + 7i)
Let z\(_{1}\) = 2  3i and z\(_{2}\) = 1 + 7i
Therefore, z\(_{1}\) = \(\sqrt{2^{2} + (3)^{2}}\) = \(\sqrt{4 + 9}\) = \(\sqrt{13}\)
And z\(_{2}\) = \(\sqrt{(1)^{2} + 7^{2}}\) = \(\sqrt{1 + 49}\) = \(\sqrt{50}\) = 5\(\sqrt{2}\)
Therefore, the required modulus of the given complex quantity = z\(_{1}\)z\(_{1}\) = z\(_{1}\)z\(_{1}\) = \(\sqrt{13}\) ∙ 5\(\sqrt{2}\) = 5\(\sqrt{26}\)
3. Find the modulus and principal amplitude of 4.
Solution:
Let z = 4 + 0i.
Then, modulus of z = z = \(\sqrt{(4)^{2} + 0^{2}}\) = \(\sqrt{16}\) = 4.
Clearly, the point in the zplane the point z =  4 + 0i = (4, 0) lies on the negative side of real axis.
Therefore, the principle amplitude of z is π.
4. Find the amplitude and modulus of the complex number 2 + 2√3i.
Solution:
The given complex number is 2 + 2√3i.
The modulus of 2 + 2√3i = \(\sqrt{(2)^{2} + (2√3)^{2}}\) = \(\sqrt{4 + 12}\) = \(\sqrt{16}\) = 4.
Therefore, the modulus of 2 + 2√3i = 4
Clearly, in the zplane the point z = 2 + 2√3i = (2, 2√3) lies in the second quadrant. Hence, if amp z = θ then,
tan θ = \(\frac{2√3}{2}\) =  √3 where, \(\frac{π}{2}\) < θ ≤ π.
Therefore, tan θ =  √3 = tan (π  \(\frac{π}{3}\)) = tan \(\frac{2π}{3}\)
Therefore, θ = \(\frac{2π}{3}\)
Therefore, the required amplitude of 2 + 2√3i is \(\frac{2π}{3}\).
5. Find the multiplicative inverse of the complex number z = 4  5i.
Solution:
The given complex number is z = 4  5i.
We know that every nonzero complex number z = x +iy possesses multiplicative inverse given by
\((\frac{x}{x^{2} + y^{2}}) + i (\frac{y}{x^{2} + y^{2}})\)
Therefore, using the above formula, we get
z\(^{1}\) = \((\frac{4}{4^{2} + (5)^{2}}) + i (\frac{(5)}{4^{2} + (5)^{2}})\)
= \((\frac{4}{16 + 25}) + i (\frac{5)}{16 + 25})\)
= \((\frac{4}{41}) + (\frac{5}{41})\)i
Therefore, the multiplicative inverse of the complex number z = 4  5i is \((\frac{4}{41}) + (\frac{5}{41})\)i
6. Factorize: x\(^{2}\) + y\(^{2}\)
Solution:
x\(^{2}\)  (1) y\(^{2}\) = x\(^{2}\)  i\(^{2}\)y\(^{2}\) = (x + iy)(x  iy)
11 and 12 Grade Math
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