In a certain code language, 'CUTE' is written as 'FQWA' and 'BLUE' is written as 'EHXA'. How will 'VOTE' be written in that same code language?

Let's analyze the pattern by looking at the alphabetical positional shifts:
For 'CUTE' $\to$ 'FQWA':

* $\text{C (3)}\stackrel{+3}{\to }\text{F (6)}$
* $\text{U (21)}\stackrel{-4}{\to }\text{Q (17)}$
* $\text{T (20)}\stackrel{+3}{\to }\text{W (23)}$
* $\text{E (5)}\stackrel{-4}{\to }\text{A (1)}$
The pattern is $+3,-4,+3,-4$.

Let's verify with 'BLUE' $\to$ 'EHXA':

* $\text{B (2)}\stackrel{+3}{\to }\text{E (5)}$
* $\text{L (12)}\stackrel{-4}{\to }\text{H (8)}$
* $\text{U (21)}\stackrel{+3}{\to }\text{X (24)}$
* $\text{E (5)}\stackrel{-4}{\to }\text{A (1)}$

Applying the same pattern to 'VOTE':

* $\text{V (22)}\stackrel{+3}{\to }\text{Y (25)}$
* $\text{O (15)}\stackrel{-4}{\to }\text{K (11)}$
* $\text{T (20)}\stackrel{+3}{\to }\text{W (23)}$
* $\text{E (5)}\stackrel{-4}{\to }\text{A (1)}$

Thus, 'VOTE' is coded as 'YKWA'.