We will learn step-by-step 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 z-plane 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 z-plane 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 non-zero 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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