Question

Find the real value of $a$ for which $3{\iota ^3} - 2a{\iota ^2} + \left( {1 - a} \right)\iota + 5$ is real.

Answer 1

To solve this question, we will use the concept of the real and imaginary part of complex numbers i.e., if it is given that a complex number is real, it means that its imaginary part is zero. So, in this question first of all we will use some predefined values that are: ${\iota ^3} = - \iota$ and ${\iota ^2} = - 1$ .we will substitute these values in the given expression and simplify it. After that we will separate the real and imaginary parts. And finally, we will equate imaginary parts equal to zero and hence we will get our required result.

Complete step by step answer:
We have given the expression as: $3{\iota ^3} - 2a{\iota ^2} + \left( {1 - a} \right)\iota + 5$.
And we are asked to find the value of $a$ such that the given expression is real.Firstly, let the given expression as equation $\left( 1 \right)$ i.e.,
$3{\iota ^3} - 2a{\iota ^2} + \left( {1 - a} \right)\iota + 5{\text{ }} - - - \left( 1 \right)$
Now we know that
${\iota ^3} = - \iota$ and ${\iota ^2} = - 1$
Therefore, on substituting these values in the equation $\left( 1 \right)$ we have
$3\left( { - \iota } \right) - 2a\left( { - 1} \right) + \left( {1 - a} \right)\iota + 5$
Now on multiplying the terms, we will get
$- 3\iota + 2a + \iota - a\iota + 5$
On subtracting the like terms, we get
$- 2\iota + 2a - a\iota + 5$

Now we know that the imaginary part is the coefficient of $\iota$ and the real part is which does not contain $\iota$. Therefore, on separating the real and imaginary part from the above equation we get
$\left( {2a + 5} \right) + \left( { - 2 - a} \right)\iota {\text{ }} - - - \left( 2 \right)$
Now, according to the question we have
$3{\iota ^3} - 2a{\iota ^2} + \left( {1 - a} \right)\iota + 5$ is real
$\therefore \operatorname{Im} \left[ {3{\iota ^3} - 2a{\iota ^2} + \left( {1 - a} \right)\iota + 5} \right] = 0$
From equation $\left( 2 \right)$ we have
$\Rightarrow \left( { - 2 - a} \right) = 0$
On adding $2$ both the sides, we get
$\Rightarrow - a = 2$
$\therefore a = - 2$

Hence, the real value of $a$ for which $3{\iota ^3} - 2a{\iota ^2} + \left( {1 - a} \right)\iota + 5$ is real is $- 2$.

Note: While solving these types of questions, one should know how to distinguish between real and imaginary parts. Always remember that is we have $\operatorname{Re} \left( z \right) = 0$ then the complex number is called purely imaginary while if we have $\operatorname{Im} \left( z \right) = 0$ then the complex number is called purely real. Also, one should the basic concepts or some basic values like $\iota = \sqrt { - 1} ,{\text{ }}{\iota ^2} = - 1,{\text{ }}{\iota ^3} = - \iota ,{\text{ }}{\iota ^4} = 1$ , after that it repeats itself.