Question
If $$\alpha $$ is non-real and $$\alpha = \root 5 \of 1 $$ then the value of $${2^{\left| {1 + \alpha + {\alpha ^2} + {\alpha ^{ - 2}} - {\alpha ^{ - 1}}} \right|}}$$ is equal to
A.
4
B.
2
C.
1
D.
None of these
Answer :
2
Solution :
$$\eqalign{
& {\alpha ^5} = 1 \cr
& \therefore \,\,{\text{index}} = \left| {1 + \alpha + {\alpha ^2} + {\alpha ^3} - {\alpha ^4}} \right| \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left| {1 + \alpha + {\alpha ^2} + {\alpha ^3} + {\alpha ^4} - 2{\alpha ^4}} \right| \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left| {\frac{{1 - {\alpha ^5}}}{{1 - \alpha }} - 2{\alpha ^4}} \right| \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left| { - 2{\alpha ^4}} \right| \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2{\left| \alpha \right|^4} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2 \times 1 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2. \cr} $$