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Question: Let w be a complex number such that \[2w + 1 = z\] where \[z = - \sqrt 3 \] , if \[\left| {\begin{ar...

Let w be a complex number such that 2w+1=z2w + 1 = z where z=−3z = - \sqrt 3 , if \left| {\begin{array}{*{20}{c}} 1&1&1 \\\ 1&{ - {w^2} - 1}&{{w^2}} \\\ 1&{{w^2}}&{{w^7}} \end{array}} \right| = 3k , the k is equal to:
A. –z
B. Z
C. -1
D. 1

Explanation

Solution

w is a complex number i.e. it can be express in the form a+bia + bi, where a and b are real numbers, and I represents the imaginary unit satisfying the equation i2=−1.{i^2} = - 1.
Because no real number satisfies the equation, it is called an imaginary number. For example, 7+3i7 + 3i is a complex number whose real part is 7 and imaginary part is 3. It is called a complex number because it consists of both real and imaginary parts.

Complete step by step solution:
Given: 2w+1=z and z=−32w + 1 = z{\text{ }}and{\text{ }}z = \sqrt { - 3}
We have been given that
2w+1=z2w + 1 = z
and z=−3=3iz = \sqrt { - 3} = \sqrt {3i}

2w+1=3i w=−1+3i2 ⇒w2=−1−3i2=−w−1.(i)  2w + 1 = \sqrt {3i} \\\ w = \dfrac{{ - 1 + \sqrt {3i} }}{2} \\\ \Rightarrow {w^2} = \dfrac{{ - 1 - \sqrt {3i} }}{2} = - w - 1 .\left( i \right) \\\

Now \left| {\begin{array}{*{20}{c}} 1&1&1 \\\ 1&{ - {w^2} - 1}&{{w^2}} \\\ 1&{{w^2}}&{{w^7}} \end{array}} \right| = 3k, has been given to us and

1&1&1 \\\ 1&w;&{{w^2}} \\\ 1&{{w^2}}&w; \end{array}} \right| = 3k$$ Putting the value of $${w^2} = - w - 1$$we got w. In the above determinant.

1\left( {{w^2} - {w^4}} \right) - 1\left( {w - {w^2}} \right) + 1\left( {{w^2} - w} \right) = 3k \\
3{w^2} - 3w = 3k \\
3\left( {{w^2} - w} \right) = 3k \\
k = {w^2} - w \\

$$k = - 1 - 2w $$(from equation (i) we get)

= - 1\left( { - 1 + \sqrt {3i} } \right) \\
k = - \sqrt {3i} \\
k = z \\

Thus, the value of k is –z, thus the determinant becomes. $$\left| {\begin{array}{*{20}{c}} 1&1&1 \\\ 1&{ - {w^2} - 1}&{{w^2}} \\\ 1&{{w^2}}&{{w^7}} \end{array}} \right| = 3x\left( { - z} \right)$$ Thus option (1) is correct. **Note:** In this type of question students often makes mistake in solving the complex number, so not make such mistakes and remember the $$\sqrt { - 3} $$ is not $$ - \sqrt 3 $$ but it’s $$\sqrt {3i} $$ not doing so will drastically decreases your chances of getting the correct answer.