# The number of non-zero integral solutions of the equation |1 - 2i|x = 5x is

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NDA (Held On: 22 April 2018) Maths Previous Year paper
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1. Zero (No solution)
2. One
3. Two
4. Three

Option 1 : Zero (No solution)
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Electric charges and coulomb's law (Basic)
41133
10 Questions 10 Marks 10 Mins

## Detailed Solution

Concepts:

Let z = x + iy be a complex number.

• Modulus of z = $$\left| \text{z} \right|=\text{ }\!\!~\!\!\text{ }\sqrt{{{\text{x}}^{2}}+{{\text{y}}^{2}}}=\text{ }\!\!~\!\!\text{ }\sqrt{\text{Re}{{\left( \text{z} \right)}^{2}}+\text{ }\!\!~\!\!\text{ Im }\!\!~\!\!\text{ }{{\left( \text{z} \right)}^{2}}}$$
• arg (z) = arg (x + iy) = $${\tan ^{ - 1}}\left( {\frac{y}{x}} \right)$$

Calculation:

We have to find value of ′x′ for which should be a non-zero integral.

Given: ∣1 − 2i∣x = 5x

Let first find value of ∣1 − 2i∣

$$\Rightarrow {\rm{\;}}\mid 1{\rm{\;}} - {\rm{\;}}2{\rm{i}}\mid {\rm{\;}} = {\rm{\;}}\sqrt {{{\left( 1 \right)}^2} + {{\left( { - 2} \right)}^2}} {\rm{\;}} = {\rm{\;}}\sqrt 5$$​     ...(∵ z = x + iy ⇒ $$\left| {\rm{z}} \right| = {\rm{\;}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}}$$)

Now, ∣1 − 2i∣x = 5x

⇒ (√5) x = 5x

⇒ 5x/2 = 5x

Base is same so power is also equal

Therefore x/2 = x

⇒ x = 2x

∴ x = 0

Hence no non-zero integral solutions.
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