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Question: If \[z+{{z}^{3}}=0\] then which of the following must be true on the complex plane? \[\left( \text...

If z+z3=0z+{{z}^{3}}=0 then which of the following must be true on the complex plane?
(a) Re(z)<0\left( \text{a} \right)\text{ Re}\left( z \right)<0
(b) Re(z)=0\left( \text{b} \right)\text{ Re}\left( z \right)=0
(c) Im⁡(z)=0\left( \text{c} \right)\text{ }\operatorname{Im}\left( z \right)=0
(d) z4=1\left( \text{d} \right)\text{ }{{z}^{4}}=1

Explanation

Solution

Hint : To solve the given question, we will first find out what complex numbers are and what their general form is. Then we will take z common out from the above equation. After doing this, we will get the equation of the form a×b=0.a\times b=0. So, we will take up two cases in which we will equate a and b to zero separately. After doing this, we will get some common values from both cases. This common value will be the answer to the question given above.

Complete step-by-step answer :
Before we start to solve the given question, we must know what complex numbers are. A complex number is a number that can be expressed in the form of p + iq, where p and q are real numbers and i represents the imaginary unit, satisfying the equation i2=−1.{{i}^{2}}=-1. Now, the equation given in the question is
z+z3=0z+{{z}^{3}}=0
We will take z common out from the left-hand side of the above equation. Thus, we will get,
z(1+z2)=0z\left( 1+{{z}^{2}} \right)=0
Now, we know that if a×b=0a\times b=0 then either a = 0 or b = 0 or both. Thus, we will form two cases on the basis of this.
Case I: Let z = 0. We know that z can be written as z = x + iy. Thus, we will get the following equation.
x + iy = 0
Now, the above equation will be zero only if both the real and imaginary parts are equal. Thus, x = 0 and y = 0.
Case II: Let z2+1=0.{{z}^{2}}+1=0. On subtracting 1 from both the sides, we will get,
⇒z2+1−1=0−1\Rightarrow {{z}^{2}}+1-1=0-1
⇒z2=−1\Rightarrow {{z}^{2}}=-1
⇒z2=(i)2\Rightarrow {{z}^{2}}={{\left( i \right)}^{2}}
⇒z=±i\Rightarrow z=\pm i
Now, we know that z = x + iy. Thus, we will get,
⇒(x+iy)=±i\Rightarrow \left( x+iy \right)=\pm i
Now, the above equation will satisfy only when the real and imaginary parts are separately equal. This, x = 0 and y=±1.y=\pm 1.
From both cases, we can conclude that x must be zero so that the equation z+z3=0z+{{z}^{3}}=0 exists. Now, x is the real part of z. Thus, Re (z) = 0.
Hence, option (b) is the right answer.

Note : While applying the formula z = x + iy, we have assumed that both x and y belong to the real number. This assumption is necessary because if x and y are some complex functions then we cannot say that x is the real part of a complex number.