Question

Let w be a complex number such that \[2w + 1 = z\] where \[z = - \sqrt 3 \] , 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

Answer 1

Hint: w is a complex number i.e. it can be express in the form \[a + bi\], where a and b are real numbers, and I represents the imaginary unit satisfying the equation \[{i^2} = - 1.\]
Because no real number satisfies the equation, it is called an imaginary number. For example, \[7 + 3i\] 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: \[2w + 1 = z{\text{ }}and{\text{ }}z = \sqrt { - 3} \]
We have been given that
\[2w + 1 = z\]
and \[z = \sqrt { - 3} = \sqrt {3i} \]
\[
  2w + 1 = \sqrt {3i} \\
  w = \dfrac{{ - 1 + \sqrt {3i} }}{2} \\
   \Rightarrow {w^2} = \dfrac{{ - 1 - \sqrt {3i} }}{2} = - w - 1 .\left( i \right) \\
 \]
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
\[ \Rightarrow \left| {\begin{array}{*{20}{c}}
  1&1&1 \\
  1&w&{{w^2}} \\
  1&{{w^2}}&w
\end{array}} \right| = 3k\]
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.