Question

equation .​Previous year IIT jee Question
Chapter :- complex number and quadratic​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}6i& - 3i&1\\4&3i& -1\\23&3& i\end{array}\right | \end{gathered} = x + iy$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \sf{ \:OP \: R_1 \: \to \: R_1 + R_2}}$

$\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}6i + 4& - 3i + 3i&1 - 1\\4&3i& -1\\23&3& i\end{array}\right | \end{gathered} = x + iy$

$\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}6i + 4&0&0\\4&3i& -1\\23&3& i\end{array}\right | \end{gathered} = x + iy$

On expanding along Row 1, we get

$\rm :\longmapsto\:(6i + 4)\begin{array}{|cc|}\sf 3i &\sf - 1 \\ \sf 3 &\sf i \\\end{array} = x + iy$

$\rm :\longmapsto\:(6i + 4)\bigg( {3i}^{2} - ( - 3)\bigg) = x + iy$

$\rm :\longmapsto\:(6i + 4)\bigg( - {3} + 3\bigg) = x + iy$

$\bf\implies \:x + iy = 0$

$\bf\implies \:x + iy = 0 + 0i$

$\bf\implies \:x = 0 \: \: and \: y = 0$

Hence, Option (d) is correct

Additional Information :-

$\boxed{ \sf{ \: |z| \: = \: |\overline{z}|}}$

$\boxed{ \sf{ \:z \: \overline{z} \: = \: { |z| }^{2}}}$

$\boxed{ \sf{ \: |z| \: = \: 1 \: \: \rm \implies\:\overline{z} \: = \: \frac{1}{z} }}$

$\boxed{ \sf{ \:\overline{z_1 \: + \: z_2} \: = \: \overline{z_1} \: + \: \overline{z_2}}}$

$\boxed{ \sf{ \:\overline{z_1 \: - \: z_2} \: = \: \overline{z_1} \: - \: \overline{z_2}}}$

Properties of determinants

1. The value of determinant remain unchanged if rows and columns are interchanged.

2. The value of determinant is 0, if two rows or columns are same.

3. The value of determinant is multiplied by (- 1), if successive row or column are interchanged.

4. The determinant value remains unchanged if respective rows or columns are added or subtracted.

5. The value of determinant is 0, if elements in any row or columns all are 0.