It might be used. It depends on many factors, mostly your data(and statistics about your data), and your queries.
TL/DR; you need to test this on your own data and your own queries. The index might be used.
You should try it out on the data that you have or expect to have. It is very easy to create some test-data on which you can test your queries and try different indexes. You might also need to reconsider the order of the columns in the index, is col_1 really the best column to be first in the index?
Below is a very specific scenario from which we can only conclude that the index can be used, sometimes, in similar scenarios as yours.
Consider this scenario below; the table contains 1M rows and have only a single nonclustered index on (a, b, c). Note that the values in column D is very large.
The first 4 queries below used the index, only the fifth query did not.
Why?
Sql Server will need to figure out how to complete the query while reading the least amount of data. Sometimes it is easier for SQL Server to read the index instead of the table even when the query-filter does not completely match the index.
In Query 1 and 2 the query will actually do a Seek on the index which is quite good. It knows that column A is a good candidate to perform the Seek on.
In query 3 and 4 it needs to scan the entirety of the index to find the matching rows. It still used the index.
In query 5 SQL Server realizes that it is easier to scan the actual table instead of the index.
IF OBJECT_ID('tempdb..#peter') IS NOT NULL DROP TABLE #peter;
CREATE TABLE #peter(a INT, b INT, c VARCHAR(100), d VARCHAR(MAX));
WITH baserows AS (
SELECT * FROM master..spt_values WHERE type = 'P'
),
numbered AS (
SELECT TOP 1000000
a.*, rn = ROW_NUMBER() OVER(ORDER BY (SELECT null))
FROM baserows a, baserows b, baserows c
)
INSERT #peter
( a, b, c, d )
SELECT
rn % 100, rn % 10, CHAR(65 + (rn % 60)), REPLICATE(CHAR(65 + (rn % 60)), rn)
FROM numbered
CREATE INDEX ix_peter ON #peter(a, b, c);
-- First query does Seek on the index + RID Lookup.
SELECT * FROM #peter WHERE a = 55 AND c = 'P'
-- Second Query does Seek on the index.
SELECT a, b, c FROM #peter WHERE a = 55 AND c = 'P'
-- Third query does Scan on the index because the index is smaller to scan than the full table.
SELECT a, b, c FROM #peter WHERE c = 'P'
-- Fourth query does a scan on the index
SELECT a, b, c FROM #peter WHERE b = 22
-- Fifth query scans the table and not the index
SELECT MAX(d) FROM #peter
Tested on SQL Server 2014.