Problems on Complex Numbers

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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