The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Use Numeric Column Types For Numeric Values (query line: 21): Referencing a numeric value (e.g. 2010) as a string in a WHERE clause might result in poor performance. Possible impacts of storing numbers as varchars: more space will be used, you won't be able to perform arithmetic operations, the data won't be self-validated, aggregation functions like SUM won't work, the output may sort incorrectly and more. If the column is numeric, remove the quotes from the constant value, to make sure a numeric comparison is done.

Optimal indexes for this query:

CREATE INDEX results_idx_examyear_examtype ON "results" ("examYear","examType");

The optimized query:

SELECT
        COUNT(*) 
    FROM
        results 
    WHERE
        EXISTS (
            SELECT
                1 
            FROM
                jsonb_array_elements(subjects) AS j(data) 
            WHERE
                (
                    data# >> '{subject}'
                ) LIKE '%MATH%' 
                AND (
                    data# >> '{grade}'
                ) IN (
                    'A', 'B', 'C'
                )
        ) 
        AND results."examYear" = '2010' 
        AND results."examType" = 'CSEE'