Question

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

Answer 1

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 \[a\times b=0.\] 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 \[{{i}^{2}}=-1.\] Now, the equation given in the question is
\[z+{{z}^{3}}=0\]
We will take z common out from the left-hand side of the above equation. Thus, we will get,
\[z\left( 1+{{z}^{2}} \right)=0\]
Now, we know that if \[a\times b=0\] 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 \[{{z}^{2}}+1=0.\] On subtracting 1 from both the sides, we will get,
\[\Rightarrow {{z}^{2}}+1-1=0-1\]
\[\Rightarrow {{z}^{2}}=-1\]
\[\Rightarrow {{z}^{2}}={{\left( i \right)}^{2}}\]
\[\Rightarrow z=\pm i\]
Now, we know that z = x + iy. Thus, we will get,
\[\Rightarrow \left( x+iy \right)=\pm i\]
Now, the above equation will satisfy only when the real and imaginary parts are separately equal. This, x = 0 and \[y=\pm 1.\]
From both cases, we can conclude that x must be zero so that the equation \[z+{{z}^{3}}=0\] 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.