JEE MAIN - Mathematics (2017 (Offline) - No. 6)
Let $$\omega $$ be a complex number such that 2$$\omega $$ + 1 = z where z = $$\sqrt {-3} $$. If
$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$,
then k is equal to :
$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$,
then k is equal to :
z
-1
1
-z
Explanation
Given 2$$\omega $$ + 1 = z;
z = $$\sqrt 3 i$$
$$ \Rightarrow $$ $$\omega = {{\sqrt 3 i - 1} \over 2}$$
$$ \Rightarrow $$ As $$\omega $$ is complex cube root of unity.
$${\omega ^3} = 1$$
$$1 + \omega + {\omega ^2} = 0$$
$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$
$$ \Rightarrow $$ $$\left| {\matrix{ 1 & 1 & 1 \cr 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & \omega \cr } } \right| = 3k$$
Applying R1 $$ \to $$ R1 + R2 + R3
$$ \Rightarrow $$ $$\left| {\matrix{ 3 & 0 & 0 \cr 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & \omega \cr } } \right| = 3k$$
$$ \Rightarrow $$ $$3\left( {{\omega ^2} - {\omega ^4}} \right) = 3k$$
$$ \Rightarrow $$ $$\left( {{\omega ^2} - \omega } \right) = k$$
$$ \therefore $$ $$k = \left( {{{ - 1 - \sqrt 3 i} \over 2}} \right) - \left( {{{ - 1 + \sqrt 3 i} \over 2}} \right)$$
$$ \Rightarrow $$ k = $${ - \sqrt 3 i}$$ = -z
z = $$\sqrt 3 i$$
$$ \Rightarrow $$ $$\omega = {{\sqrt 3 i - 1} \over 2}$$
$$ \Rightarrow $$ As $$\omega $$ is complex cube root of unity.
$${\omega ^3} = 1$$
$$1 + \omega + {\omega ^2} = 0$$
$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$
$$ \Rightarrow $$ $$\left| {\matrix{ 1 & 1 & 1 \cr 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & \omega \cr } } \right| = 3k$$
Applying R1 $$ \to $$ R1 + R2 + R3
$$ \Rightarrow $$ $$\left| {\matrix{ 3 & 0 & 0 \cr 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & \omega \cr } } \right| = 3k$$
$$ \Rightarrow $$ $$3\left( {{\omega ^2} - {\omega ^4}} \right) = 3k$$
$$ \Rightarrow $$ $$\left( {{\omega ^2} - \omega } \right) = k$$
$$ \therefore $$ $$k = \left( {{{ - 1 - \sqrt 3 i} \over 2}} \right) - \left( {{{ - 1 + \sqrt 3 i} \over 2}} \right)$$
$$ \Rightarrow $$ k = $${ - \sqrt 3 i}$$ = -z
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