Question:

The value of $\left| \frac{1+ i \sqrt{3}}{\left( 1 + \frac{1}{i+1}\right)^2} \right|$ is

Updated On: May 17, 2024
  • $20$
  • $9$
  • $\frac{5}{4}$
  • $\frac{4}{5}$
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The Correct Option is D

Solution and Explanation

$\left|\frac{1+i \sqrt{3}}{\left(1+\frac{1}{i+1}\right)^{2}}\right|= \left|\frac{(1+i \sqrt{3})(i+1)^{2}}{(i+2)^{2}}\right|\left(\because i^{2}=-1\right)$
$=\left|\frac{(1+i \sqrt{3})\left(i^{2}+1+2 i\right)}{\left(i^{2}+4+4 i\right)}\right|$
$=\left|\frac{(1+i \sqrt{3}) 2 i}{(4 i+3)}\right|$
$=\left|\frac{-2 \sqrt{3}+2 i}{3+4 i}\right|$
$=2\left|\frac{(-\sqrt{3}+i)(3-4 i)}{(3+4 i)(3-4 i)}\right|$
$=2\left|\frac{-3 \sqrt{3}+3 i+4 \sqrt{3} i+4}{25}\right|$
$=\frac{2}{25}|(4-3 \sqrt{3})+(3+4 \sqrt{3}) i|$
$=\frac{2}{25} \sqrt{(4-3 \sqrt{3})^{2}+(3+4 \sqrt{3})^{2}}$
$=\frac{2}{25} \sqrt{16+27-24 \sqrt{3}+9+48+24 \sqrt{3}}$
$=\frac{2}{25} \sqrt{100}=\frac{2 \times 10}{25}=\frac{4}{5}$
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Concepts Used:

Algebra of Complex Numbers

Algebra of complex numbers

1. Addition of two complex numbers:

Consider z1 and z2 are two complex numbers. 

For example, z1 = 3+4i and z2 = 4+3i

Here a=3, b=4, c=4, d=3

∴z1+ z2 = (a+c)+(b+d)i 

⇒z1 + z2 = (3+4)+(4+3)i 

⇒z1 + z2 = 7+7i 

Properties of addition of complex numbers

  • Closure law: While adding two complex numbers the resulting number is also a complex number.
  • Commutative law: For the complex numbers z1 and z2 , the commutation can be z1+ z2 = z2+z1
  • Associative law: While considering three complex numbers, (z1+ z2) + z?3 = z1 + (z2 + z3)
  • Additive identity: An Additive identity is nothing but zero complex numbers that go as 0+i0. For every complex number z, z+0 = z.
  • Additive inverse: Every complex number has an additive inverse denoted as -z.

2. Difference between two complex numbers

It is similar to the addition of complex numbers, such that, z1 - z2 = z1 + ( -z2) 

For example: (5+3i) - (2+1i) = (5-2) + (-2-1i) = 3 - 3i

3. Multiplication of complex numbers

Considering the same value of z1 and z2 , the product of the complex numbers are

z1 * z2 = (ac-bd) + (ad+bc) i

For example: (5+6i) (2+3i) = (5×2) + (6×3)i = 10+18i

 

 

Properties of Multiplication of complex numbers

Note: The properties of multiplication of complex numbers are similar to the properties we discussed in addition to complex numbers.

  • Closure law: When two complex numbers are multiplied the result is also a complex number.
  • Commutative law: z1* z2 = z2 * z1

Associative law: Considering three complex numbers, (z1 z2) z3 = z1 (z2 z3)

  • Multiplicative identity: 1+0i is always denoted as 1. This is multiplicative identity. This means that z.1 = z for every complex number z.
  • Distributive law: Considering three complex numbers, z1 (z2 + z3) =z1 z2 + z1 z3 and (z1+ z2) z3 = z1 z2 + z2 z3.

Read More: Complex Numbers and Quadratic Equations

4. Division of complex numbers

If z1 / z2 of a complex number is asked, simplify it as z1 (1/z2 )

For example: z1 = 4+2i and z2 = 2 - i

z1 / z2 =(4+2i)×1/(2 - i) = (4+i2)(2/(2²+(-1)² ) + i (-1)/(2²+(-1)² )) 

=(4+i2) ((2+i)/5) = 1/5 [8+4i + 2(-1)+1] = 1/5 [8-2+1+41] = 1/5 [7+4i]