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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 (1+i1i)3 in the form A + iB where A and B are real numbers.

Solution:

Given (1+i1i)3

Now 1+i1i

= (1+i)(1+i)(1i)(1+i)

= (1+i)2(12i2

= 1+2i+iˆ21(1)

= 1+2i12

= 2i2

= i

Therefore, (1+i1i)3 = i3= i2 ∙  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 z1 = 2 - 3i and z2 = -1 + 7i

Therefore, |z1| = 22+(3)2 = 4+9 = 13

And |z2| = (1)2+72 = 1+49 = 50 = 52

Therefore, the required modulus of the given complex quantity = |z1z1| = |z1||z1| = 13  ∙ 52 = 526

 

3. Find the modulus and principal amplitude of -4.

Solution:

Let z = -4 + 0i.

Then, modulus of z = |z| = (4)2+02 = 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 = (2)2+(23)2 = 4+12 = 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 = 232 = - √3 where, π2 < θ ≤ π.

Therefore, tan θ = - √3 = tan (π - π3) = tan 2π3

Therefore, θ = 2π3

Therefore, the required amplitude of -2 + 2√3i is 2π3.

 

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

(xx2+y2)+i(yx2+y2)

Therefore, using the above formula, we get

z1 = (442+(5)2)+i((5)42+(5)2)

= (416+25)+i(5)16+25)

= (441)+(541)i

Therefore, the multiplicative inverse of the complex number z = 4 - 5i is (441)+(541)i

 

6. Factorize: x2 + y2

Solution:

x2 - (-1) y2 = x2 - i2y2 = (x + iy)(x - iy)





11 and 12 Grade Math 

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